/
Author: Hartman P.
Tags: mathematics differential equations exact sciences problems in mathematics
Year: 1964
Text
Ordinary
Differential Equations
PHILIP HARTMAN
The Johns Hopkins University
John Wiley & Sons, Inc., New York • London • Sydney
Copyright © 1964 by John Wiley & Sons, Inc.
All Rights Reserved
This book or any part thereof
must not be reproduced in any form
without the written permission of the publisher.
Library of Congress Catalog Card Number: 64-23837
Printed in the United States of America
To the memory
of my parents
To the patience
of Sylvia, Judith, arid Marilyn
Preface
This book is based on lecture notes of courses on ordinary differential
equations which I have given from time to time for advanced under-
undergraduates and graduate students in mathematics, physics, and engineering.
It assumes a knowledge of matrix theory and, if not a thorough knowledge
of, at least a certain maturity in the handling of functions of real variables.
I was never tempted to scatter asterisks liberally throughout this book
and claim that it could serve as a sophomore-junior-senior textbook, for
I believe that a course of this type should give way to basic courses in
analysis, algebra, and topology;
This book contains more material than I ever covered in one year but
not all of the topics which I treated in the many courses. The contents of
these courses always included the subject matter basic to the theory of
differential equations and its many applications to other disciplines (as,
for example, differential geometry). A "basic course" is covered in
Chapter I; §§ 1-3 of Chapter II; §§ 1-6 and 8 of Chapter III; Chapter IV
except for the "Application" in § 3 and part (ix) in § 8; §§ 1-4 of Chapter
V; §§1-7 of Chapter VII; §§ 1-^ of Chapter VIII; §§ 1-12 of Chapter
X; §§1-4 of Chapter XI; and §§ 1-4 of Chapter XII.
Many topics are developed in depth beyond that found in standard
textbooks. The subject matter in a chapter is arranged so that more
difficult, less basic, material is usually put at the end of the chapter (and/
or in an appendix). In general, the content of any chapter depends only
on the material in that chapter and the portion of the "basic course"
preceding it. For example, after completing the basic course, an instructor
can discuss Chapter IX, or the remainder of the contents of Chapter XII,
or Chapter XIV, etc. There are two exceptions: Chapter VI, Part I, as
written, depends on Chapter V, §§5-12; Part III of Chapter XII is not
essential but is a good introduction to Chapter III.
Exercises have been roughly graded into three types according to
difficulty. Many of the exercises are of a routine nature to give the
student an opportunity to review or test his understanding of the tech-
techniques just explained. For more difficult exercises, there are hints in the
back of the book (in some cases, these hints simplify available proofs).
Finally, references are given for the most difficult exercises; these serve
Vll
viii Preface
to show extensions and further developments, and to introduce the
student to the literature.
The theory of differential equations depends heavily on the "integration
of differential inequalities" and this has been emphasized by collecting
some of the main results on this topic in Chapter III and § 4 of Chapter
IV. Much of the material treated in this book was selected to illustrate
important techniques as well as results: the reduction of problems on
differential equations to problems on "maps" (cf. Chapter VII, Appendix,
and Chapter IX); the use of simple topological arguments (cf. Chapters
VIII, § 1; X, §§ 2-7; and XIV, § 6); and the use of fixed point theorems
and other basic facts in functional analysis (cf. Chapters XII and XIII).
I should like to acknowledge my deep indebtedness to the late Professor
Aurel Wintner from whom and with whom I learned about differential
equations, first as a student and later as a collaborator. My debt to him
is at once personal, in view of my close collaboration with him, and
impersonal, in view of his contributions to the resurgence of the theory
of ordinary differential equations since the Second World War.
I wish to thank several students at Hopkins, in particular, N. Max,
C. C. Pugh, and J. Wavrik, for checking parts of the manuscript. I also
wish to express my appreciation to Miss Anna Lea Russell for the excellent
typescript created from nearly illegible copy, numerous revisions, and
changes in the revisions.
My work on this book was partially supported by the Air Force Office
of Scientific Research.
Philip Hartman
Baltimore, Maryland
August, 1964
I
Contents
I. Preliminaries
1. Preliminaries, 1
2. Basic theorems, 2
3. Smooth approximations, 6
4. Change of integration variables, 7
Notes, 7
II.
III.
Existence
1. The Picard-Lindelof theorem, 8
2. Peano's existence theorem, 10
3. Extension theorem, 12
4. H. Kneser's theorem, 15
5. Example of nonuniqueness, 18
Notes, 23
Differential inequalities and uniqueness .
1. Gronwall's inequality,-24
2. Maximal and minimal solutions, 25
3. Right derivatives, 26
4. Differential inequalities, 26
5. A theorem of Wintner, 29
6. Uniqueness theorems, 31
7. van Kampen's uniqueness theorem, 35
8. Egress points and Lyapunov functions, 37
9. Successive approximations, 40
Notes, 44
24
IV.
Linear differential equations ....
1. Linear systems, 45 -
2. Variation of constants, 48
3. Reductions to smaller systems, 49
4. Basic inequalities, 54
5. Constant coefficients, 57
6. Floquet theory, 60
IX
45
Contents
Contents
XI
7. Adjoint systems, 62
8. Higher order linear equations, 63
9. Remarks on changes of variables, 68
Appendix. Analytic linear equations, 70
10. Fundamental matrices, 70
11. Simple singularities, 73
12. Higher order equations, 84
13. A nonsimple singularity, 87
Notes, 91
V. Dependence on initial conditions and parameters
1. Preliminaries, 93
2. Continuity, 94
3. Differentiability, 95
4. Higher order differentiability, 100
5. Exterior derivatives, 101
6. Another differentiability theorem, 104
7. »S- and L-Lipschitz continuity, 107
8. Uniqueness theorem, 109
9. A lemma, 110
10. Proof of Theorem 8.1, 111
11. Proof of Theorem 6.1, 113
12. First integrals, 114
Notes, 116
93
VI. Total and partial differential equations
Part I. A theorem of Frobenius, 117
1. Total differential equations, 117
2. Algebra of exterior forms, 120
3. A theorem of Frobenius, 122
4. Proof of Theorem 3.1, 124
5. Proof of Lemma 3.1, 127
6. The system A.1), 128
Part II. Cauchy's method of characteristics, 131
7. A nonlinear partial differential equation, 131
8. Characteristics, 135
9. Existence and uniqueness theorem, 137
10. Haar's lemma and uniqueness, 139
Notes, 142
117
VII. The Poincare-Bendixson theory 144
1. Autonomous systems, 144
2. Umlaufsatz, 146
3. Index of a stationary point, 149
4. The Poincare-Bendixson theorem, 151
5. Stability of periodic solutions, 156
6. Rotation points, 158
7. Foci, nodes, and saddle points, 160
8. Sectors, 161
9. The general stationary point, 166
10. A second order equation, 174
Appendix. Poincare-Bendixson theory on 2-manifolds, 182
11. Preliminaries, 182
12. Analogue of the Poincare-Bendixson theorem, 185
13. Flow on a closed curve, 190
14. Flow on a torus, 195
Notes, 201
VIII. Plane stationary points
1. Existence theorems, 202
2. Characteristic directions, 209
3. Perturbed linear systems, 212
4. More general stationary point, 220
Notes, 227
202
IX. Invariant manifolds and linearizations .
1. Invariant manifolds, 228
2. The maps T*, 231
3. Modification of F(?), 232
4. Normalizations, 233
5. Invariant manifolds of a map, 234
6. Existence of invariant manifolds, 242
7. Linearizations, 244
8. Linearization of a map, 245
9. Proof of Theorem 7.1, 250
10. Periodic solution, 251
11. Limit cycles, 253
Appendix. Smooth equivalence maps, 256
12. Smooth linearizations, 256
13. Proof of Lemma 12.1, 259
14. Proof of Theorem 12.2, 261
Notes, 271
228
Xll
X.
Contents
Perturbed linear systems 273
1. The case ? = 0, 273
2. A topological principle, 278
3. A theorem of Wazewski, 280
4. Preliminary lemmas, 283
5. Proof of Lemma 4.1, 290
6. Proof of Lemma 4.2, 291
7. Proof of Lemma 4.3, 292
8. Asymptotic integrations. Logarithmic scale, 294
9. Proof of Theorem 8.2, 297
10. Proof of Theorem 8.3, 299
11. Logarithmic scale (continued), 300
12. Proof of Theorem 11.2, 303
13. Asymptotic integration, 304
14. Proof of Theorem 13.1, 307
15. Proof of Theorem 13.2, 310
16. Corollaries and refinements, 311
17. Linear higher order equations, 314
Notes, 320
XI. Linear second order equations 322
1. Preliminaries, 322
2. Basic facts, 325
3. Theorems of Sturm, 333
4. Sturm-Liouville boundary value problems, 337
5. Number of zeros, 344
6. Nonoscillatory equations and principal solutions, 350
7. Nonoscillation theorems, 362
8. Asymptotic integrations. Elliptic cases, 369
9. Asymptotic integrations. Nonelliptic cases, 375
Appendix. Disconjugate systems, 384
10. Disconjugate systems, 384
11. Generalizations, 396
Notes, 401
I
XII. Use of implicit function and fixed point theorems
Part I. Periodic solutions, 407
1. Linear equations, 407
2. Nonlinear problems, 412
. 404
Contents xiii
Part II. Second order boundary value problems, 418
3. Linear problems, 418
4. Nonlinear problems, 422
5. A priori bounds, 428
Part III. General theory, 435
6. Basic facts, 435
7. Green's functions, 439
8. Nonlinear equations, 441
9. Asymptotic integration, 445
Notes, 447
XIII. Dichotomies for solutions of linear equations 450
Part I. General theory, 451
1. Notations and definitions, 451
2.
3.
4.
5.
6.
7.
Preliminary lemmas, 455
The operator T, 461
Slices of ||P«/@1|, 465
Estimates for \\y(t)\\, 470
Applications to first order systems, 474
Applications to higher order systems, 478
P(B, ?>)-manifolds, 483
Part II. Adjoint equations, 484
9. Associate spaces, 484
10. The operator T, 486
11. Individual dichotomies, 486
12. P'-admissible spaces for T', 490
13. Applications to differential equations, 493
14. Existence of PD-solutions, 497
Notes, 498
XIV. Miscellany on monotony
Part I. Monotone solutions, 500
1. Small and large solutions, 500
2. Monotone solutions, 506
3. Second order linear equations, 510
4. Second order linear equations (continuation), 515
500
xiv Contents
Part II. A problem in boundary layer theory, 519
5. The problem, 519
6. The case A > 0, 520
7. The case A < 0, 525
8. The case A = 0, 531
9. Asymptotic behavior, 534
Part III. Global asymptotic stability, 537
10. Global asymptotic stability, 537
11. Lyapunov functions, 539
12. Nonconstant G, 540
13. On Corollary 11.2, 545
14. On "J(y)x ¦ x <; 0 if x -f{y) = 0", 548
15. Proof of Theorem 14.2, 550
16. Proof of Theorem 14.1, 554
Notes, 554
Hints for exercises, 557
References, 581
Index, 607
I
Ordinary Differential Equations
Chapter I
Preliminaries
1. Preliminaries
Consider a system of d first order differential equations and an initial
condition
A.1)
y'=f(t,y),
where y = dyjdt, y = (y1,... ,yd) and / = (J1,... ,fd) are rf-dimensional
vectors, and f(t, y) is defined on a (d + l)-dimensional (t, «/)-set E. For
the most part, it will be assumed that/is continuous. In this case, y = y(t)
defined on a ^-interval J containing t = t0 is called a solution of the initial
value problem A.1) if y(t0) = y0, (t, y(t)) e E, y(t) is differentiate, and
y'(t) =f(t,y(t)) for teJ. It is clear that y(t) then has a continuous
derivative. These requirements on y are equivalent to the following:
y(to) = 2/o> ('»2/@) e ?> 2/@ is continuous and
A.2)
2/@ = 2/o +
s, y(s)) ds
for ? e J.
An initial value problem involving a system of equations of mth order,
A.3)
z(m) =
f, z, z
A)
), z(j)(a;0) = zoj) for 7 = 0, . . . , m - 1,
where zU) = d'zjdt', z and F are e-dimensional vectors, and F is defined
on an (me + l)-dimensional set E, can be considered as a special case of
A.1), where y is a d = me-dimensional vector, symbolically, y = (z, zA),
. . ., z<m-D) (or more exactly, y = (z1, . . ., ze, z1',.. ., ze', . .., z^™'));
correspondingly,/(r, y) = (zA», ... , z*™', F(f, y)) and y0 = (z0, z(j»,.. .,
,<».-!)¦) por exampie) if e = l so that z is a scalar, A.3) becomes
2/1' = y\
yW.~\l _ yV
ym> = F(t, y\..., ym),
where yl = z, y* = z ,. . ., ym =
I
2 Ordinary Differential Equations
The first set of questions to be considered will be A) local existence
(does A.1) have a solution y(t) defined for t near ?0?); B) existence in the
large (on what r-ranges does a solution of A.1) exist?); and C) uniqueness
of solutions.
The significance of question B) is clear from the following situation:
Let y,f be scalars, f(t, y) defined for 0 <; t ^ 1, \y\ ^ 1. A solution
y = y{t) of A.1), with (t0, y0) = @, 0), may exist for 0 < t ^ | and
increase from 0 to 1 as t goes from 0 to \, then one cannot expect to have
an extension of y(t) for any t > \. Or consider the following scalar case
where f(t, y) is defined for all (t, y):
A.4)
V = V\ 2/@) = c (> 0).
It is easy to see that y = c/(l — ci) is a solution of A.4), but this solution
exists only on the range — oo < t < 1/c, which depends on the initial
condition.
In order to illustrate the significance of the question of uniqueness, let
y be a scalar and consider the intial value problem
A.5) y' = \y\1A, 2/@) = 0.
This has more than one solution, in fact, it has, e.g., the solution y(t) = 0
y--
tt-cr
Figure 1.
and the 1-parameter family of solutions defined by y(t) = 0 for t ^ c,
y(t) = (t — cJ/4 for t ^ c, where c ^ 0; see Figure 1. This situation is
typical in that if A.1) has more than one solution, then it has a "con-
"continuum" of solutions; cf. Theorem II 4.1.
2. Basic Theorems
This section introduces some conventions, notions and theorems to be
used later. The proofs of most of the theorems will be omitted.
Preliminaries
The symbols O, o will be used from time to time where, e.g., f(t) =
0{g(t)) as t -*¦ oo means that there exists a constant C such that |/@l ^
C \g(t)\ for large t, while f(t) = o(g(t)) as t -*¦ oo means that C > 0 can be
chosen arbitrarily small (so that if g(t) # 0, f(i)lg(t) -»¦ 0 as t -*- oo).
"Function" below generally means a map from some specified set of
a vector space Re into a space Rd, not always of the same dimension. Rd
denotes a normed, real <i-dimensional vector space of elements y =
(j/1, . . ., yd) with norm \y\. Unless otherwise specified, \y\ will be the
norm
B.1)
\y\ = max {\y
and ||y|| the Euclidean norm.
If y0 is a point and E a subset of R", then dist (y0, E), the distance from
//„ to E, is defined to be inf \y0 — y\ for y e E. If Elt E2 are two subsets
of Rd, then dist (?ls E2) is defined to be inf \y^ — y2\ for yx e Elt y2 e E2,
and is called the distance between E1 and E2. If E1 (or E2) is compact and
/-.'„ E2 are closed and disjoint, then dist (Eu E2) > 0.
If ? is an open set or a closed parallelepiped in Rd,/e C"(E), 0 ^ n <
¦f\ means that/(«/) is continuous on E and that the components of/have
continuous partial derivatives of all orders k ^ n with respect to y1, . . ., yd.
A function /(«/, z) = f(y\ . . ., yd, z1, . . . , ze) defined on a («/, z)-set ?,
where y e Rd, is said to be uniformly Lipschitz continuous on E with respect
to y if there exists a constant K satisfying
B.2) |/B/i, z)-
for all (</„ z) e E
withy = 1, 2. Any constant AT satisfying B.1) is called a Lipschitz constant
{tor /on ?). (The admissible values of K depend, of course, on the norms
in the/- and jz-spaces.)
A family F of functions/^) defined on some y-sct E c R* is said to be
(¦(juiamtinuous if, for every e > 0, there exists a E = <5e > 0 such that
1 /<//i) — /Wl ^ « whenever yls j/2 e ?, \yx — y2\ ^ E and /ef. The
point of this definition is that de does not depend on/but is admissible
tor all / e F. The most frequently encountered equicontinuous families F
below will occur when all/efare uniformly Lipschitz continuous on E
.mil there exists a K > 0 which is a Lipschitz constant for all/eF; in
«Inch case, d can be chosen to be d = e/K.
Lemma 2.1. If a sequence of continuous functions on a compact set E
m uniformly convergent on E, then it is uniformly bounded and equicontinuous.
Cantor Selection Theorem 2.1. Let /i(j/),/2B/), • • • be a uniformly
hounded sequence of functions on a y-set E. Then for any countable set
l> • /:', there exists a subsequence /nA ^(y),/nB)(j/),. .. convergent on D.
4 Ordinary Differential Equations
In order to prove Cantor's theorem, let D consist of the points ylt y%, . . ..
Also assume thatfn(y) is real-valued; the proof for the case thatfn(y) =
ifn(y)i • • • ,fnd(y)) *s a ^-dimensional vector is similar. The sequence of
numbers fi(y),My),... is bounded, thus, by the theorem of Bolzano-
Weierstrass, there is a sequence of integers n^l) < ^B) < . .. such that
lim/^^i) exists as k ->¦ oo, where n = n^k). Similarly there is a sub-
subsequence «2A) < «2B) < ... of «!A), «iB),. . . such that lim/n(«/2) exists
as k -»¦ co for n = n2(&). Continuing in this fashion one obtains successive
subsequences of positive integers, such that if n,(l) < n,B) < ... is the
y'th one, then lim/^) exists on k ->¦ co, where n = «,-(&) and j = 1, . . . ,j.
The desired subsequence is the "diagonal sequence" n^l) < n2B) <
«3C) < . . . . Variants of this proof will be referred to as the "standard
diagonal process."
The next two assertions usually have the names Ascoli or Arzela
attached to them.
Propagation Theorem 2.2. On a compact y-set E, Ietfx(y),f2(y),.. . be a
sequence of functions which is equicontinuous and convergent on a dense
subset of E. Thenf1(y),f2(y),. . . converges uniformly on E.
Selection Theorem 2.3. On a compact y-set E <= Rd, Ietf{y),f2{y), . . .
be a sequence of functions which is uniformly bounded and equicontinuous.
Then there exists a subsequence fn{1){y),fn{2){y), . . . which is uniformly
convergent on E.
This last theorem can be obtained as a consequence of the preceding
two. By applying Theorem 2.3 to a suitable subsequence, we obtain the
following:
Remark 1. If, in the last theorem, yoe E and/0 is ajcluster pointfof the
sequence fx(yo),fM,..., then the subsequence/BA)(yO/BB)(y),. . .'in the
assertion can be chosen so that the limit function/(«/) satisfies /(«/0) = fa-
Remark 2. If, in Theorem 2.3, it is known that all (uniformly) con-
convergent subsequences off1(y),f2(y),. .. have the same limit, say/(«/), then
a selection is unnecessary and /(«/) is the uniform limit offx(y),f2{y),....
This follows from Remark 1.
Theorem 2.3 and the following consequences of it will be used
repeatedly.
Theorem 2.4. Lety,fe Rd andfo(t, 2/),/iO, y),f2(t, y),... beasequence
of continuous functions on the parallelepiped R : t0 ^ t ^ t0 + a, \y —
yo\ ^ b such that
B.3) fo(t, y) = lim fn(t, y) uniformly on R.
Let yn(t) be a solution of
B-4n) y' =/„(/, y), y{tn) = yn,
Preliminaries 5
on [t0, t0 + a], where n = 1, 2,..., and
B.5) tn^t0, yn^y0 as n^co.
Then there exists a subsequence «/n(i)@> 2/nB)(O> • • • which is uniformly
convergent on [t0, t0 + a]. For any such subsequence, the limit
B.6)
yo(t) = lim yn(k)(t)
is a solution of B.40) on [t0, t0 + a]. In particular, if B.40) possesses a
unique solution y = yo(t) on [t0, t0 + a], then
B.7) yo(t) = lim yn(t) uniformly on [t0, t0 + a].
n—»-co
Proof. Since/1;/2,... are continuous and B.3) holds uniformly on R,
there is a constant K such that \fn(t, y)\ ^ K for n = 0, 1, . . . and
(t, y) e R; Lemma 2.1. Since |yB'@l = K> it is Clear tnat Ki%z. Lipschitz
constant for yu y2, . . . , so that this sequence is equicontinuous. It is also
uniformly bounded since \yn(t) — yo\ ^ b. Thus the existence of uni-
uniformly convergent subsequences follows from Theorem 2.3. By B.3),
Lemma 2.1, and the uniformity of B.6), it is easy to see that
/*<*)('> 2/n<w@) ~^fo(t, y{t))
uniformly on [t0, t0 + a] as k ->¦ oo. Thus term-by-term integration is
applicable to
2/n@ = Vn + fn(s, 2/n(
Jtn
where n = n(k) and k —*¦ co. It follows that the limit B.6) is a solution of
B.40).
As to the last assertion, note that the assumed uniqueness of the solution
yo(t) of B.40) shows that the limit of every (uniformly) convergent sub-
subsequence of ^@, «/2@> • • ¦ is the solution yo(t). Hence a selection is un-
unnecessary and B.7) holds by Remark 2 above.
Implicit Function Theorem 2.5. Let x, y, f g be d-dimensional vectors
and z an e-dimensional vector. Let f(y, z) be continuous for (y, z) near a
point (y0, z0) and have continuous partial derivatives with respect to the
components of y. Let the Jacobian det {df'jdyk) ^ 0 at (y, z) = (y0, z0).
Let x0 = /(«/(,, z0). Then there exist positive numbers, e and d, such that if
x and z are fixed, \x — xo\ < d and \z — zo\ < d, then the equation x =
f(y, z) has a unique solution y = g(x, z) satisfying \y — yo\ < e. Further-
Furthermore, g(x, z) is continuous for \x — xo\ < d, \z — zo\ < d and has continuous
partial derivatives with respect to the components of x.
For a sharper form of this theorem, see Exercise II 2.3.
6 Ordinary Differential Equations
3. Smooth Approximations
In some situations, it will be convenient to extend the definition of a
function/, say, given continuous on a closed parallelepiped, or to approxi-
approximate it uniformly by functions which are smooth (C1 or C00) with respect
to certain variables. The following devices can be used to obtain such
extensions or approximations (which have the same bounds as/).
Let/0, y) be denned on R:t0 <; t <; tv \y\ ^ b and let I/O, y)\ ^ M.
Let/*0, y) be denned for to<t^tt and all y by placing f*(t, y) = f{t, y)
if \y\ ^ b and/*0, y) = f(t, by/\y\) if \y\ > b. It is clear that/*O, y) is
continuous for *0 = * = *i> ^ arbitrary, and that \f*(t, y)\ ^ M. In some
cases, it is more convenient to replace/* by an extension of/which is 0 for
large \y\. Such an extension is given by f°(t, y) =f*(t,y)<p°(\y\), where
<p°(s) is a continuous function for t ^ 0 satisfying 0 ^ <p°(s) ^ 1 for
s ^ 0, <p°(j) = 1 for 0 ^ j < b, and <p°(j) = 0 for s ^ 6 + 1.
In order to approximate f(t, y) uniformly on R by functions fe(t, y)
which are, say, smooth with respect to the components of y, let <p(s) be a
function of class C° for s ^ 0 satisfying f>(s) > 0 for 0 ^ j < 1 and
cp(s) = 0 for s =i 1. Then there is a constant c > 0 depending only on
(p(s) and the dimension d, such that for every e > 0,
J — 00 J — 00
C.1)
where ||y|| = B |2/&|2)^ is the Euclidean length of y. Put
/• + 00 /" + 00
C.2) /«(/, ») = (*-« ...
•/ — 00 J — 00
where r\ = (rj1, . . ., rf), so that
C-3) /U y) = ce-d f+°°.. . f+"/V. V ~ V)<P(e~2 11*7 II 2> drj1... drf.
J—00 J — 00
Since fe(t, y) is an "average" of the values of/0 in a sphere ||»7 — y|| ^ e
for a fixed /, it is clear that/^ -»-/0 as e -»¦ 0 uniformly on t0 ^ t ^ /l5 y
arbitrary. Note that |/e| ^ M for all e > 0 and that f%t, y) = 0 for
|y| ^ 6 + 1 + e. Furthermore, /eO, y) has continuous partial derivatives
of all orders with respect to y1, . . ., y*.
The last formula can be used to show that if f\t, y) has continuous
partial derivatives of order k with respect to y1, . . ., y*, then the corre-
corresponding partial derivatives of /'(/, y) tend uniformly to those of f°(t, y)
as € -»¦ 0.
Preliminaries
4. Change of Integration Variables
In order to avoid an interruption to some arguments later, it is con-
convenient to mention the following:
Lemma 4.1. Let t, u, U be scalars; U(u) a continuous function on
A < u ^ B; u = u(t) a continuous function of bounded variation on
a ^t <b such that A < u(t) ^ B. Then, for a < t < b,
D.1)
U(u(s)) du(s) = U(u) du,
Ja Ju(a)
where the integral on the left is a Riemann-Stieltjes integral and that on the
right is a Riemann integral.
The point of the lemma is the fact that the change of variables t—*-u
given by u = u(t) is permitted even when u(t) is not monotone (and not
absolutely continuous).
Proof. It is clear that the relation D.1) holds for a < / ^ b if u(t) has a
continuous derivative. For in this case, both integrals in D.1) vanish at
t = a and have the same derivative U(u(t))u'(t). If u(t) does not have a
continuous derivative, let ut{t), u2(t),... be a sequence of continuously
differentiable functions on a ^ ( ? i satisfying A < un(t) ^ B, un(t) -»¦
u(t) as n -*¦ oo uniformly on [a, b], and such that the sequence of total
variations of ut{t), u2(t), . . . over [a, b] is bounded. (The existence of
functions ux{t), u2(t),. .. follows from the last section.) Then D.1) holds
if u(t) is replaced by un(t). Term-by-term integration theorems applied to
both sides of the resulting equation lead to D.1).
Notes
section 2. Theorems 2.2 and 2.3 go back to Ascoli [1] and Arzela [1], [2].
Chapter II
Existence
1. The Picard-Lindelbf Theorem
Various types of existence proofs will be given. One of the most simple
and useful is the following.
Theorem 1.1. Let y,feKd; f{t,y) continuous on a parallelepiped
R:t0 ^ t < f0 + a, \y — yo\ ^ b and uniformly Lipschitz continuous with
respect to y. Let M be a bound for \f(t, y)\ on R; a = min (a, b/M). Then
A.1) y'=f(t,y), y(to) = yo
has a unique solution y = y(t) on [t0, t0 + a].
It is clear that there is a corresponding existence and uniqueness theorem
if R is replaced by f0 — a ^ t ^ t0, \y — yo\ ^ b. It is also clear from
these "right" and "left" existence theorems that if R is replaced by
\t — fo| ^ a, \y — yo\ ^ b, then A.1) has a unique solution on \t — to\ ^ a,
since the solutions on the right and left fit together.
The choice of a = min (a, b/M) in Theorem 1.1 is natural. On the one
hand, the requirement a ^ a is necessary. On the other hand, the require-
requirement a 5; bjM is dictated by the fact that if y = y(t) is a solution of A.1)
on [f0, t0 + a], then \y'(t)\ < M implies \y(t) - yo\ ^ M(t - t0), which
does not exceed b if t — t0 ^ bjM.
Remark 1. Note that in Theorem 1.1, \y\ can be any norm on Rd, not
necessarily the norm B.1) or the Euclidean norm.
For another proof of "uniqueness," see Exercise III 1.1.
Proof by Successive Approximations. Let yo(t) = y0. Suppose that
yk(t) has been defined on [t0, t0 + a], is continuous, and satisfies \yk(t) —
yo\ ^ b for k = 0 n. Put
A.2) yn+M = 2/o + f(s, yn(s)) ds.
Then, since/(f, yn(t)) is defined and continuous on [t0, t0 + a], the same
holds for yn+1(t). It is also clear that
»+i@ - Vo\ ^ P I/O. y-(
^Mx< b.
Existence 9
Hence yo(t), y^t),... are defined and continuous on [t0, f0 + a], and
|y»@ - jol ^ 6.
It will now be verified by induction that
A-3J \yn+1(t) - yn(t)\ ^ MKn{n(t~1tf+1 for to^t^to +a,
n — 0, 1,. .., where K is a Lipschitz constant for/. Clearly, A.30) holds.
Assume A.30) A.3^). By A.2),
- yn(t) = [/(s, y»(s)) - /(s, »_!
for n ^ 1. Thus, the definition of K implies that
\yn+i(t) - y»(OI ^ «? l»»(s) - y«-i(s)\ ds
andso,
- yn(t)\ ^
is- toy ds =
n
This proves A.3J.
In view of A.3 J, it follows that
(n + 1)!
2/o+2 [yn+M - yn(t)] =
tt=0
is uniformly convergent on [t0, f0 + a]; that is,
A-4) t/@ = lim yn(t) exists uniformly.
n—>00
Since f(t, y) is uniformly continuous on R, f(t, yn{t)) -+f(t, 2/@) as
n -> oo uniformly on [f0, f0 + a]. Thus term-by-term integration is
applicable to the integrals in A.2) and gives
A-5)
2/@
= 2/o+f7(s,
y(s)) ds.
Hence A.4) is a solution of A.1).
In order to prove uniqueness, let y = z(t) be any solution of A.1) on
[t0, t0 + a]. Then
A.6)
z(t)
= 2/o + /(s,
An obvious induction using A.2) gives
MKn(t — / 1n+
A.7) \yn(t) - 2@l ^ — U W
(n + 1)!
2E)) ds.
for f0 ^ t ^ f0 + a
10 Ordinary Differential Equations
and n = 0, 1, .... If n -»- oo in A.7), it follows from A.4) that \y(t) —
z(t)\ ^ 0; i.e., y(t) = z{t). This proves the theorem.
Remark 2. Since z = y, A.7) gives an estimate of the error of approxi-
approximation
MKn(t - to)n+1
A-8)
\yn(t) - KOI ^
(n + 1)!
on [t0, t0 + a].
Exercise 1.1. Show that if, in addition to the conditions of the theorem,
f(t, 2/) is analytic on i? (i.e., in a neighborhood of every point (f°, 2/°) ? R,
f(t, y) is representable as a convergent power series in t — t°, y1 — y01, . . . ,
yd _ yoa^ then the solution 2/ = 2/@ °f A-0 is analytic on [?0, f0 + a].
The analogous theorem in which t and the components of y,/are allowed
to be complex-valued is also-valid.
Exercise 1.2. If, in Theorem 1.1, v is near to y0, then the initial value
problem y' = f(t, 2/), 2/Oo) = v has a unique solution y = y(t, v) on some
interval [t0, t0 + ft] independent of v. Show that y(t, v) is uniformly
Lipschitz continuous with respect to (t, v) for t0 ^ t ^ f0 + /S, t> near 2/0-
2. Peano's Existence Theorem
The next theorem to be proved drops the assumption of Lipschitz
continuity and the assertion of uniqueness.
Theorem 2.1. Let y,fs Rd; f(t, y) continuous on R:t0 ^ t < t0 + a,
\y -y«\ ^b; M a bound for \f(t, y)\ on R; a = min (a, bjM). Then A.1)
possesses at least one solution y = y(t) on [t0, t0 + a].
In this theorem, \y\ can be any convenient norm on R*.
Proof. Let d > 0 and yo(t) a C1 (/-dimensional vector-valued function
on [t0 - d, t0] satisfying 2/0(f0) = Z/o> 2/'Oo) =/Oo> 2/o) and \yo(t) - yo\ < b,
|2/0'@l ^ M. For 0 < e ^ d, define a function ye(t) on [f0 - ^, t0 + a]
by putting ye(t) = yo(t) on [f0 - d, t0] and
B.1)
= 2/o
on
a].
Note first that this formula is meaningful and defines y€(t) for t0 ^ t <
to + aD ai = min (a> e)> so that ye(t) is C1 on [t0 — d, t0 + «i] and, on
this interval,
B-2) \ye(t) - 2/0| ^ *•
It then follows that B.1) can be used to extend ye(t) as a C1 function over
[t0 — d, t0 + a2], a2 = min (a, 2e), satisfying B.2). Continuing in this
fashion, B.1) serves to define y((t) over [t0, t0 + a.] so that ye(t) is a C1
function on [/„ — d, to+ a], satisfying B.2).
Existence 11
Since \ye'(t)\ ^ M, it follows that the family of functions, ye(t), 0 <
€ ^ ^, is equicontinuous. Thus, by Theorem I 2.3, there is a sequence
€A) > eB) > . . . , such that e(«) -> 0 as n -+ oo and
2/(f) = lim 2/e(n)@ exists uniformly
n—•¦oo
on [f0 — d, t0 + a]. The uniform continuity of/implies that/(f, yein)(t —
€(«)) tends uniformly to/(f, 2/@) as n -»¦ oo; thus term-by-term integration
of B.1) where e = e(n) gives A.5). Hence y(t) is a solution of A.1). This
proves the theorem.
An important consequence of Peano's existence theorem will often be
used:
Corollary 2.1. Let f(t, y) be continuous on an open (t, y)-set E and
satisfy \f(t, y)\ ^ M. Let Eo be a compact subset of E. Then there exists
an a > 0, depending on E, Eo and M, with the property that if(t0, y0) e Eo,
then A.1) has a solution and every solution exists on \t — to\ < a.
In fact, if a = dist (Eo, dE) > 0, where dE is the boundary of E, then
a = min (a, a/M). In applications, when/is not bounded on E, the set E
in this corollary is replaced by an open subset E° having compact closure
in E and containing Eo.
Exercise 2.1 (Polygonal Approximation). Under the conditions of
Theorem 2.1, define a set of functions 2/^@ as follows: Let S:f0 <
?i < ' ¦ ¦ < tm = 10 + a be a mesh on [t0, t0 + a] with a degree of fineness
EB) = max (tk+1 - tk) for k = 0,. . . , m - 1. On [t0, fj, put 2/2@ =
yo + (t- ?o)/Oo> 2/o)- If 2/2@ has been defined on [t0, tk], k < m, and
Iz/l@ - Z/ol ^ b, put 2/2@ = y^h) +(t- W(h> 2/lO*)) °n ['*> Wl-
This serves to define 2/e@ on [fo> fo + a] as a continuous piece wise linear
function. Prove Theorem 2.1 by obtaining a solution of A.1) as a limit of
a suitable sequence 2/e(i>@> 2/sB)(Oj • • • j where <5B(n)) -»¦ 0 as n -»- 00.
[Note that if the solution of A.1) is not unique, then not all solutions can
be obtained by this procedure; cf. the scalar problem y = \y\l/i, 2/@) = 0.]
Exercise 2.2 (Another Proof). There exists a sequence of continuous
functions/(f, y),fz(t, y), ... on R which tend uniformly to f(t, y) on R,
l/«(f> V)\ tkM for (t,y)eR and n = 1,2,..., and fn(t, y) is uniformly
Lipschitz continuous with respect to y\ cf. § I 3. Consider y' = fn(t, y),
y(to) = 2/o and apply Theorems 1.1 and I 2.4. [In contrast to Exercise 2.1,
all solutions of A.1) can be obtained by the method of this exercise; cf.
the proof of Theorem 4.1.]
Exercise 2.3. [This exercise gives a sharpened form of the Implicit
Function Theorem I 2.5 (with no parameters z). In the statement of this
theorem, the norm \\A\\ of a d x d matrix occurs. Let Rd be the (real)
J-dimensional vector space with any convenient norm \y\. Then ||^4|| is
defined to be \\A\\ = max \Ay\ for \y\ = 1. This norm ||^4|| depends on the
12 Ordinary Differential Equations
choice of the norm \y\ in Rd.] Let x = f{y) be a function of class C1 on
D:\y\^ b, and let/@) = 0. Let the Jacobian matrix, fy(y) = {df^jdy1')
for j, k = 1, . . ., d, be nonsingular on D and put M = max ||/v~%)||,
M1 = max ||/vB/)|| for y e D, where/~x is the inverse of the matrix/v and
||/J, ll/j^1!! denote the norms of the respective matrices. Let X>x: |2/| ^
b\MMx. (Note that MMj ^ 1. Why?) Then there exists a domain Do
such that Dj c: ?H c D and a; = /(?/) is a one-to-one map of [the closure
of] Do onto [the closure of] the ball D°: \x\ < Z>/Af; see Figure 1. Assum-
Assuming the Implicit Function Theorem I 2.5, deduce the result just stated from
ljrl-6/MMi
x-f(y)
x-t-t
Figure 1.
Peano's Existence Theorem 2.1 by writing the equation x = f(y) for y in
the form f ? = f{y), where | ^ 0 is a constant vector, differentiating with
respect to t to obtain the differential equation y = fyl(y)?, and consider-
considering the solution satisfying the initial condition 2/@) = 0. (It is possible to
avoid the use of the Implicit Function Theorem by using the results of
§V6.)
3. Extension Theorem
Let/(/, y) be continuous on a (t, y)-sel E and let y = y(t) be a solution of
C.1) y'=f(t,y)
on an interval /. The interval / is called a right maximal interval of
existence for y if there does not exist an extension of y(t) over an interval
Jx so that y = y(t) remains a solution of C.1); / is a proper subset of Jt;
J, Jx have different right endpoints. A left maximal interval of existence
for y is defined similarly. A maximal interval of existence is an interval
which is both a left and right maximal interval.
Theorem 3.1. Let f(t, y) be continuous on an open (t, y)-set E and let
y(t) be a solution of C.1) on some interval. Then y(t) can be extended (as a
Existence 13
solution) over a maximal interval of existence (&>_, m+). Also, if (u>_, m+) is
a maximal interval of existence, then y(t) tends to the boundary dE of E as
t —*¦ a>_ and t —* co+.
The extension of y(t) need not be unique and, correspondingly, oj±
depends on the extension. To say, e.g., that y(t) tends to dE as t -> a>+ is
interpreted to mean that either co+ = oo or that co+ < oo and if E° is any
compact subset of E, then (t, y(t)) $ E° when t is near co+.
Proof. Let Elt E2,... be open subsets of E such that E = (jEn; the
closures Elt E2,. . . are compact, and En c En+1 (e.g., let En = {(t, y):
(t,y)sE,\t\<n,\y\<n and dist ((t, y), dE)> l/n}). Corollary 2.1
implies that there exists an en > 0 such that if (t0, y0) is any point of En,
then all solutions of C.1) through (t0, y0) exist on \t — fol = e«-
Consider a given solution y = y(t) of C.1) on an interval /. If J is not a
right maximal interval of existence, then y(t) can be extended to an interval
containing the right endpoint of J. Thus, in proving the existence of a
right maximal interval of existence, it can be supposed that y(t) is defined
on a closed interval a ^ t ^ b0 and that y(t) does not have an extension
over a ^ t < oo.
Let n(l) be so large that (b0, y(b0)) e En{1). Then t/(f) can be extended
over an interval [b0, b0 + eniv]. If (b0 + enil), y(b0 + eniv)) e Eniv, then
y(t) can be extended over another interval [b0 + enA), b0 + 2enA)] of
length enA). Continuing this argument, it is seen that there is an integer
/(I) ^ 1 such that y(t) can be extended over a ^ t < bu where Z>x =
^o + ;(lK(i) and Fls t/^j)) ^ ?BA).
Let nB) be so large that (bu y(bj) e Eni2). Then there exists an integer
jB) ^ 1 such that y(t) can be extended over a ^ t ^. b2, where Z>2 =
*i +./B)€nB) and F2s 1/(^2)) ^ ?nB).
Repetitions of this argument lead to sequences of integers n(l) <
nB) < . . . and numbers b0 < Z>x < . . . such that t/@ has an extension
over [a, co+), where co+ = limftj. as ^ -* 00, and that (bk, y(bk)) $ EnM.
Thus (bu y(bj)), (b2, y(b2)), ... is either unbounded or has a cluster point
on the boundary dE of E.
To see that y(t) tends to dE as f -> o>+ on a right maximal interval
[a, m+), it must be shown that no limit point of a sequence (tu y(tj),
(h> 2/(^2))' ¦ ¦ • 5 where f „ -> o)+, can be an interior point of E. This is a
consequence of the following:
Lemma 3.1. Let f(t, y) be continuous on a (t, y)-set E. Let y = y(t) be
a solution of C.1) on an interval [a, d), d < 00, for which there exists a
sequence tx, t2, . . . such that a ^ tn-+ d as n-*oo and y0 = lim y(t J
exists. If f(t, y) is bounded on the intersection of E and a vicinity of the
point (d, y0), then
C.2) yo= lim y(t) as t-*d.
14 Ordinary Differential Equations
If, in addition, f(d, y0) is or can be defined so that f{t, y) is continuous at
(d, y0), then y(t) e O[a, d] and is a solution of C.1) on [a, d].
Proof. Let e > 0 be so small and Me > 1 so large that \f(t,y)\ ^ Me
for (t, y) on the intersection of E and the parallelepiped 0 ^ d — t ^ e,
\V - 2/ol ^ e- If " is S0 lar8e that 0 < d - fn ^ e/2Me and Win) - 2/ol ^
e/2, then
C.3) |2/@ - y(tj <Me(d- tn) <:
for tn<t<
Otherwise, there is smallest t1 such that tn < t1 < d, \y(P) — 2/@1 =
M(d - tn) ^ |e. Hence |2/@ - Jol ^ ie + 12/OJ - 2/ol ^ e for tn ^
f < t1; thus |2/'(OI ^ -^e for tn^t< t1. Consequently, ^(t1) — y(tn)\ ^
Afe(fx — tn) < Me(<5 — ?„). This proves C.3), hence C.2). The last part
of the lemma follows from y\t) = f(t, 2/@) -»-/(<5, 2/o) as f -»¦ d.
Corollary 3.1. Let f(t, y) be continuous on a strip t0^ t ^ f0 + a
«oo), yeRd arbitrary. Let y = y(t) be a solution of A.1) on a right
maximal interval!. Then either J = [t0, t0 + a] or J = [t0, d), d < t0 + a,
and \y(t)\ -»¦ oo as t -»¦ d.
More generally,
Corollary 3.2. Let f(t, y) be continuous on the closure E of an open
(t,y)-set E and let A.1) possess a solution y = 2/@ on a maximal right
interval J. Then either J = [?„, oo), or J = [t0, d] with d < oo and
(d, y(d)) e dE, or J = [t0, d) with 5 < oo and \y(t)\ -»¦ oo as t -+ d.
A somewhat different, but very useful, result is given by the following
theorem.
Theorem 3.2. Let f(t, y) and fx(t, y), f2(t, y),. . . be a sequence of con-
continuous functions defined on an open (t, y)-set E such that
C.4) fn(t, y) -»¦/(?, 2/) as «->co
holds uniformly on every compact subset of E. Let yn(t) be a solution of
C.5) 2/' =/«(*, y), y(tn) = 2/«o>
(tn, yn0) e E, and let («„_, a>n+) be its maximal interval of existence. Let
C.6) (tn, 2/n0) -•> (t0,2/o) e E as n -* oo.
Then there exist a solution y(t) of
C.7) y'=f{t,y),
y(h) =
having a maximal interval of existence (a>_, w+) and a sequence of positive
integers «A) < «B) < . . . with the property that if a>_ < t1 < t2 < a>+,
then mn_ < t1 < t2 < a>n+for n = n(k) and k large, and
C.8)
2/@
oo
Existence
15
uniformly for t1 < t < t2. In particular,
C.9) lim sup con_ < co_ < a>+ < lim inf a>n+ as n = n(k) -»¦ oo.
Proof. Let ?1; E2, . . . be open subsets of ? such that ? = \jEn, the
closures ?1; E2,. . . are compact and ?„ <=¦ En+1. Suppose that (t0, y0) e Ex
and hence that (t„, yn0) e Er for large n.
In the proof, y(t) will be constructed only on a right maximal interval of
existence [t0, a>+). The construction for a left maximal interval is similar.
By Corollary 2.1, there exists an e1; independent of n for large n, such
that any solution of C.5) [or C.7)] for any point (tn, yn0) e Ex [or (t0, y0) e
?J exists on an interval of length 3ex centered at t = tn [or t = ?„]. By
Arzela's theorem, it follows that if n(l) < nB) < ... are suitably chosen,
then the limit C.8) exists uniformly for ?0 < ? < ?0 +ex and is a solution of
C.7). If the point (t0 + euy(t0 + ex)) e Ex the sequence n(l) < nB) < . . .
can be replaced by a subsequence, again called n(l) < nB) < . . . , such
that the limit C.8) exists uniformly for t0 + ex ^ t < t0 + 2ex and is a
solution of C.1). This process can be repeated infinitely often to obtain
2/@ for 0 ^ t < oo or can be repeated only j times, where (f0 + wie1;
2/(?0 + wjej)) e ?x for m = 0,. . . ,j — 1 but not for m = j.
In the latter case, let tx = t0 + je1 and choose the integer r > 1 so that
(?1; 2/(?i)) e ?r- Repeat the procedure above using a suitable er > 0
(depending on r but independent of n for large n) to obtain either y(t) on
?i = t < oo or 2/@ on an interval fx ^ f ^ fx +y!er, where (tx + mer,
y(tx + me,)) sEr for w = 0,. . .j\ — 1 but not for m =f /j. Put t2 =
h + '
Repetitions of these arguments lead to a sequence of f-values
fo < h < • • • and a sequence of successive subsequences of integers:
«2B)
such that C.8) holds uniformly for t0 < t < tm if n(k) = njk). Put
w+ = lim tm (< oo). Since (tm+1, y(tm+1)) $ Em, for m = 1,2,..., [f0, «+)
is the right maximal interval of existence for y(t). The usual diagonal
process supplies the desired sequence n(l) < nB) < . . . . This proves
the theorem.
4. H. Kneser's Theorem
The following theorem concerning the case of nonunique solutions of
initial value problems will be proved in this section.
Theorem 4.1. Letf(t, y) be continuous onR:t0<t<t0 + a,\y — y0\ <
b. Let \f(t, y)\ ^ M, a. = min (a, b/M) and t0 < c < t0 + a. Finally, let
16 Ordinary Differential Equations
Sc be the set of points ycfor which there is a solution y = y(t) of
D.1)
y'=f(t,y),
on [t0, c] such that y(c) = yc; i.e., yc e Sc means that yc is a point reached
at t = c by some solution of D.1). Then Sc is a continuum, i.e., a closed
connected set.
Exercise 4.1. If y is a scalar, Theorem 4.1 has a very simple proof
even without the assumption t0 ^ c ^ t0 + a. The conclusion is that Sc
is either empty, a point, or a closed y-interval. Prove Theorem 4.1 in this
case by first showing that if ylt y2 e Sc, so that D.1) has solutions yt(i) on
[t0, c] such that y^c) = yi for; = 1, 2, and if 2/1 < y° < y2, then y° e Sc.
Proof. Let S denote the set of solutions of D.1). These exist on
[t0, c] and Sc is the set of points y(c), where y(t) e S. To see that the set
Sc is closed, let ync -+yc,n-+ oo, and ync e Sc. Then ync = yn(c) for some
yn(t) e S. By Theorem 12.4, y^t), y2(t),... has a subsequence which is
uniformly convergent to some y{t) e S on [t0, c]. Clearly, yc = y(c) e S.
Suppose that the assertion is false, then Sc is not connected and is
therefore the union of two nonempty, disjoint closed sets S°, S1. Since Sc
is bounded, d = dist (S0, S1) > 0, where dist (S°, S1) = inf \y° - y*\ for
y° e S°, y1 e S1. For any y, put e(y) = dist (y, S°) — dist (y, S1), so that
e(y) ^ 8 > 0 if y e S1 and e(y) <-d<Qi?yeS°. The function eB/) is
continuous and e(y) ^ 0 for y e S1,..
Let e > 0 and 2/@ e S. There exists a continuous function g(t, y)
depending on e and (the fixed) y{t), defined for f0 ^ f < c and all y such
that (i) \g(t, y)\<M+ e; that (ii)
D.2)
\f(t, y) - g(t, y)\ ^
that (iii) g(f, y) is uniformly Lipschitz continuous with respect to y; and
that (iv) y = 2/@ is a solution of
D.3)
y' = git, y), y(to) = 2/0-
In order to see this, let g*(t, y) be a function with the properties (i)-(iii),
but with M + e replaced by M in (i) and e by |e in D.2); cf. § I 3. Let
g{t, y) = g*(t, y) + f(t, 2/@) - g*(U y(t))- Then \g(t, y) - g*(t, y)\ <
I/O, 2/@) — g*(t, 2/@I ^ ie= so that conditions (i), (ii) follow. Condition
(iii) is clear and (iv) follows from g(t, 2/@) = fit, 2/@) = 2/'@-
Let 2/ = 2/0@,2/i@ e S, and 2/0(c) e S1", 2/x(c) e 51. For a given € > 0,
let go(t, y), gt(t, y) be functions with the properties (i)—(iv), when y(t) =
2/o(O> 2/i@, respectively. Consider the 1-parameter family of initial value
problems:
D.4)
2/' = ge(t, y), y(t0) = 2/o,
Existence 17
where 0 ^ 0 < 1 and
, 2/) = 0gl(t, y) + A -
D-5)
Since ^e(f, y) is uniformly Lipschitz continuous with respect to y, D.4)
has a unique solution 2/ = y(t, d); see Theorem 11.1. As a consequence
of Theorem 1.1 (where b > 0 is arbitrary), this solution exists on [t0, c]
since ge is bounded, \ge(t, y)\ < M + e, on the strip t0 ^ f < c and all 2/.
Note that \ge(t, y)\ <L M + e, c <L t0 + * imply that \y(t, 0) - 2/ol ^
(M + e)a for f0 ^ f ^ c. Theorem I 2.4 implies that y(t, 6) -> y(t, 0O),
d -»¦ 0O, uniformly on [f0, c]. In particular, 2/(c, 0), hence eB/(c, 0)), is a
continuous function of 0. Since y(c, 0) = 2/o(c) e ^"j 2/(c, 1) = 2/i(c) e ^S
so that e(y(c, 0)) < -^ < 0, eB/(c, 1)) ^ 5 > 0, there exists a 0-value,
6 = r\, 0 < r\ < 1 such that eB/(c, *?)) = 0.
If 2/0@,2/i@ are fixed, a choice of an r\ depends only on e, say r\ = rj(e).
Let e=l/n,n> 1, and let gn(t, y) = ge(t, y), where 0 = »?(l/n). Thus
D.2) and D.5) show that
D.6) \f(t,y)-gn(t,y)\ ^1/n oni?
and, by the choice of 6 = rj,
D-7) 2/' = ?"(*, 2/), 2/(?o) = 2/0
has a (unique) solution y = yw{i) on [t0, c] such that e(y{n)(c)) = 0. The
sequence 2/A>@> 2/<2>(O> • • • nas a subsequence which is uniformly con-
convergent, say to 2/ = 2/@= on [?0, c]. Since \y{n\i) — yo\ ^b for to-<f^
min (c, f0 + b/(M + 1/n)) and min (c, f0 + 6/(Af + l/«)) -> c as n -» 00,
Theorem 12.4 implies that 2/@ is a solution of D.1) on [t0, c]. Also
e(y(c)) = lim e(y{n\c)) = 0 as n -»¦ 00. But then 2/(c) e 51,. and eB/(c)) = 0.
This contradiction proves the theorem.
Exercise 4.2. Show by an example that Sc need not be a convex set if
d > 1, where y is a (/-dimensional vector; e.g., if d = 2, Sc can be the
boundary of a circle.
Exercise 4.3. (a) Let/(f, 2/) be continuous for f0 < f < f0 + a and all
y. Let t0 < c ^ f0 + a and assume that all solutions y(t) of D.1) exist on
t0 < t ^ c. Then S1,. is a continuum. (Z>) Show, by an example, that Sc
need not be connected if d = 2 and not all solutions of D.1) exist for
t0 ^ t ^ t0 + c.
Exercise 4.4. Let /(?, y), c be as in Theorem 4.1 or in part (a) of
Exercise 4.3 and Te = {(t, y): t0^ t ^ t0 + c,y = y(t) for some solution
of D.1)}. In particular, Sc = {y: (c, y) e 7"c}. (a) Let y* be a boundary
point of Sc. Show that D.1) has a solution y = y*(t) such that (t, y*{t)) is
on the boundary of Tc for t0 ^ f ^ f0 + c and that 2/*(c) = y*. This is a
theorem of Fukuhara; see Kamke [2]. (b) Let (tlt 2/1) be a point of the
18 Ordinary Differential Equations
boundary of Tc, where t0 < t1 < c. Show that there need not exist a
solution y(t) of D.1) such that y(tj) = yl and (t, y{t)) is on the boundary
of Tc on any interval t0 ^ t ^ tx + e, e > 0. This is a result of Fukuhara
and Nagumo; see Digel [1].
5. Example of Nonuniqueness
In order to illustrate how bad the situation as to uniqueness can become,
it will be shown that there exists a (scalar) function U(t, u) continuous on
the (?, «)-plane such that for every choice of initial point (t0, u0), the initial
value problem
E.1)
u = U{t, u), u(t0) = u0
has more than one solution on every interval [t0, t0 + e] and [t0 — e, t0]
for arbitrary e > 0.
Let So be the set of arcs,
E.2) u = 4i + cos 77? and u = 4/ + 2 — cos nt for — oo < t < oo,
i = 0, ± 1, . . . , considered to be made up of subarcs defined on the
intervals of length 1, fc < f < fc + 1, and A; = 0, ± 1, ....
For every n = 0, 1,2,. . ., there will be constructed a set Sn of twice
continuously differentiable arcs
E.3) u = ujk(t), ±<t<-
and
j, k = 0, ±1,. . . .
The symbol Sn will denote either the set of arcs E.3) or the set of points on
these arcs. The set Sn of arcs E.3) will have the properties that (i)
E.4)
{otjL<t<a±i.t
2n 1n
(ii) the arcs u = uik(t) and u = uj+1 k(t) have exactly one endpoint in
common; (iii) for any pair j, k, there is at least one index h such that
uKh_x = «A+iji;_i = ujk at t = k\2n and an index / such that uik+1 =
ui+lk+1 = ujk at t = (k + l)/2"; (iv) any two arcs of Sn which have a
point in common have the same tangent at that point; hence (v) any
continuous arc u = u(t), say, on a ^ t ^ b, which is made up of arcs of Sn
can be continued over — oo < t < oo, not uniquely, so as to have the
same property and any such continuation is of class C1 (and piecewise of
class C2); also, (vi) if Un(t, u) is defined on the point set Sn to be the
slope of the tangent at the point (?, u) e Sn, then Un(t, u) is uniformly
continuous on Sn and arcs of (v) constitute the set of solutions of
E.5)
u' = un(t,«);
Existence 19
Figure 2. The heavy curved lines represent arcs of So. Heavy and light curved lines
represent arcs of Si if m0 = 3. The construction of the arcs of Sls notin So, is indicated
above: the arcs «(f) = «0(f), u^f), «2(/). «3(O = v(t) are defined on [a, b] = [0,1], the
arcs uo(/). *>i(O. v2(t) on [c, b] = [i, 1]. The sketch makes it clear how So or S, divides
the plane into sets G.
(vii) the sets So, Su . .. satisfy Sn <= Sn+1, so that Un+1(t, u) is an extension
of UJt, u); (viii) S = \JSn and, in fact, the set of end points (k/2n,
ujk{kj2n)), for;, k = 0, ±1,. .. and n = 0, 1,. . ., is dense in the plane;
finally, (ix)
E.6)
U(t, u) = lim Un(t, u)
which is defined on S = \JSn has a (unique) continuous extension over
the plane. Condition (ix) is the only nontrivial condition. (The con-
construction of S is indicated in Figure 2.)
Let Tr2 = e0 > ?l > ...,
E.7) Mn=%ek and M = f ek < oo.
Suppose that Sn has already been constructed so that the functions E.3)
20 Ordinary Differential Equations
satisfy
E.8) \u'jk(t)\, \u"ik(t)\ ^ Mn;
and that if
E.9) dn = sup max(\ui+Uk(t) - uUk(t)\,
i.k.t
then
E-10) dn <, €„;
and, if n > 0 and no arc of 5'TC_1 lies between u = uik(t), u = uhk{t), then
E.H) K(t) - <@l ^ d-i + «„•
The set of arcs .STC+1 will be obtained from those of Sn by inserting on
each interval, [k/2n, Bk + l)^1] and [Bk + 1)/2TC+1, (k + 1)/2B], a
finite number of arcs between the arcs u = ujk(t), u = u1+lk(t) of Sn. The
arcs of Sn and these inserted arcs will constitute the set Sn+1.
For convenience, let u(t) = ujk(t), v(t) = uj+lk(t), a = k/2n, b =
(k + l)/2", and c = J(a + b). Suppose that u(d) = v(a); the construction
in the case u(b) = v(b) is similar. Then u(t), v(t) are defined on [a, b],
b — a = 2~n;
E.12) u(t) < v(t) on (a, b], u(a) = v(a), u'(a) = v'(a);
E.13) \u'(t)\,\u"(t)\,\v'(t)\,\v"(t)\^Mn;
E.14) \u(t)-v(t)\, \u'(t)-v'(t)\^dn<:en.
Let m = mn > 0 be an integer to be specified below. For i = 0, 1, .. .,
m, put
(m —
i
..... ,.
E.15) u((t) =
m
[v(t) - u(t)] -
m
on [a, b]. Then uo(t) = u(t), um(t) = v(t), and
E.16) u(t) ^ ut(t) < ul+1(t) ^ v(t) on (a, b],
E.17) ut(a) = u(a), Ui'(a) = u'(a) for i = 0, 1, . . ., m.
It is clear from E.14) that
E-18) K - "il ^ \h
m
E.19)
E.20)
k'l ^ Mn, \uh' - m/I ^ \h - i\ ^ ^ dn,
m
Existence 21
For i = 0, 1, . . ., m — 1 and c = %(a + b), put
E.21) vt(t) = ut(t) sin2 2n+1n(t - c) + ui+1(t) cos2 2n+1n(t - c)
on [c, b], so that c — a = b — c = l/2"+1 implies that
E.22) Ui(t) < Vi(t) < ui+1(t) on (c, b),
E.23) Vi = ui+1, v- = u'i+1 att = c, and v{ = u{, v{' = m/ at t = b.
The relations in E.23) involving derivatives follow from
E.24) v- = ut' sin2 2n+\(t - c) + w,'+1 cos2 2n+\(t - c)
+ 2"+V(m,- - wj+1) sin 2TC+27r(/ - c).
From E.24) and E.18)-E.20),
E.25) |d/| < Mn + 2n+1n ^, \v{"\ ^ Mn + Bn+2 + 22n+\)n ^ .
m m
Also, by E.21),
vi — ui = ("t+i — "t) cos2 2TC+17r(/ — c),
v{ — ui+1 = (u; — ui+1) sin2 2TC+17r(? — c),
sothatE.18)-E.19)give
E.26) |t)i -uh\^^n, \v(' - uh'\ ^ A + 2"+V) ^ for h = i, i + 1.
Finally, let m = mn be chosen so large that
E.27)
m
In order to obtain Sn+1 from 5B> let the arcs u = u{(t), i = 0, . . ., m,
on [a, c] and the arcs u = u{(t), i = 0, . . ., m, and u = t>A(/), A = 0,. . .,
m — 1, on [c, b] be inserted between u = m(/), m = v(t). It is clear from
E.19)-E.20), E.25)-E.26), and E.27) that the analogues of E.8) and
E.10) hold if n is replaced by n + 1. Also the analogue of E.11) follows
from E.19), E.26), E.27).
This completes the construction of the sequence So <= St <= . . .. It is
clear that S = \JSn is dense in the (t, w)-plane.
The continuity of U(t, m), given by E.6), will now be considered. Let
p ^ n ^ 0. The set of arcs Sn divide the plane into closed sets G of the
form G = {(t, u):y <: t ^,6, un(t) ^ u ^ vn(t)}, where no point of Sn is
interior to G; m = m"(?) and u = vn(t) on [y, 5], <5 — y = 2/2", are arcs
each made up of two arcs of Sn; un — vn at t = y, d; and un < vn on
(y, «)•
22 Ordinary Differential Equations
Let (/„, uv)eG n Sv and let (t1, u1) be any point of the boundary of G.
The difference
will be estimated. Consider first the case that p = n. Then (/„, uv) is on
the boundary of G, say uv = un(t0). Since Un(t, un{t)) = un'(t), it is seen
by E.8) that
Wn{t0, un) - Un{y, u{y))\ < Mn\t0 - y\ < ^ .
Thus, in the case that/? = n, ATC < 4M/2".
Let p > n. It can be supposed that (/„, up) e Sp — S^. Let un =
un(t0), and un <L un+1 ^ • • • ^ Wj,, where (/„, w,) is the highest point of the
segment / = /„, «,-_! ^u^uv which in S1,-,; = n + 1, ...,/>. Then, by
E.11),
|tf,('o> «,+i) ~ fp('o, «,)l ^ ^ + «,-+! ^ 2e,..
Hence
If this is combined with AB < 4JI//2", it follows that
K ^ »?„ where Vn = — + 2 ? e,..
Consider now two points (/,., mJ, / = 0, 1, in S'j,,/? ^ n. Each of these
points (/j, «j) is contained in a region G = Gt of the type just considered.
There exist points {t\ «') on the boundary of Gt such that
|/0 _ ,1|2 + |WO _ Ml|2 < |/o _ ,x|2
(where, e.g., (/°, m°) = (t1, u1) if Go = GJ. Thus the above estimate for
A, implies that
\U(t0, u0) - U(tlt Ml)| < 2Vn + \Un(t\ ifi) - Un{t\ «%
Since [/„(/, u) is uniformly continuous on Sn, it follows from the last three
formula lines that U(t, u) is uniformly continuous on S. Hence U(t, u) has
a continuous extension, denoted also by U(t, u), on the (/, «)-plane.
It will now be verified that E.1) has the asserted property. It is clear
that any continuous arc u = u{t) on an interval [c, d] made up of subarcs
of Sis a solution of E.1). The Extension Theorem 3.1 or the case d = 1 of
Theorem 4.1 shows that if (/„, w0) is any point of a set G of the type just
considered, then E.1) has a solution u = u(t) over [y, d] satisfying
u = un = vn at / = y, d. Such a solution can be continued to the left of
/ = y [right of / = d] in a nonunique manner by using arcs of Sn. If n is
Existence 23
sufficiently large, the interval [y, d] containing /„ can be made arbitrarily
small. This completes the verification.
Notes
For references and discussion of the history of existence theorems, see Painleve [1],
Vessiot [1], Miiller [3], and Kamke [4,1, pp. 2, 33].
section 1. Theorem 1.1 goes back to Cauchy (see Moigno [1]) and Lipschitz [1].
Its proof by successive approximations is due to Picard [1] and Lindelof [1], although
this method had been used earlier in special cases by Liouville and by Cauchy. The
existence theorem in Exercise 1.1 is often associated with the names of Cauchy and
Poincare who used the method of majorants. A proof by successive approximations
was given by von Escherich [2].
section 2. Theorem 2.1 is due to Peano [2]. Simplifications in the proof have been
made by Mie, de la Vallee Poussin, Arzela, Montel, and Perron. The proof in the text
utilizes a device of Tonelli [1]. The polygonal approximations in Exercise 2.1 go back
to Cauchy (and the method of Cauchy-Lipschitz).
The result of Exercise 2.3 is due to Wazewski [4] but the proof in the Hints may be
new; cf. Nevanlinna [1] and Heinz [3]. Results of this type often involve a "degree
of continuity" for /„(«/) at y = 0; cf. Yamabe [1], Bartle [1], and Sternberg and
Wintner [1].
section 4. Theorem 4.1 is a theorem of H. Kneser [1 ]; the proof in the text is that of
Miiller [2]; for related results, see Pugh [1]. The suggestion given in Hints for Exercise
4.2 is due to C. C. Pugh (unpublished); cf. also Fukuhara and Nagumo [1]. Exercise
4.3(a) is a result of Kamke [2]. Exercise 4.4 is due to Fukuhara [1 ]; see also Kamke [2],
Fukuhara [2], and Fukuhara and Nagumo [1]. A related example is given in Digel [1].
section 5. The first example of this type was given by Lavrentieff [1]; the example
in the text is that of Hartman [27].
Chapter III
Differential Inequalities and Uniqueness
The most important techniques in the theory of differential equations
involve the "integration" of differential inequalities. The first part of this
chapter deals with basic results of this type which will be used throughout
the book. In the second part of this chapter immediate applications are
given, including the derivation of some uniqueness theorems.
In this chapter u, v, U, Fare scalars; y, z,f g are (/-dimensional vectors.
1. Gronwall's Inequality
One of the simplest and most useful results involving an integral
inequality is the following.
Theorem 1.1. Let u{i), v{t) be non-negative, continuous functions on
[a, b]; C ^ 0 a constant; and
A.1)
Then
A.2)
v{t) < C +
f't<*)«(
Ja
ds for a ^ t ^ b.
v@ ^ C exp u(s) ds for a ^ t ^ b;
Ja
in particular, if C = 0, then v(t) = 0.
For a generalization, see Corollary 4.4.
Proof. Case (i), C > 0. Let V(t) denote the right side of A.1), so that
*>@ ^ V(t), V(t) ^ C> 0 on [a, b]. Also, V'{t) = u{t)v{t) < u(t)V(t).
Since V > 0, V'\ V ^ u, and V(a) = C, an integration over [a, t] gives
V(t) < C exp f k(s) <fc. Thus A.2) follows from v(t) < V(t).
Ja
Case (ii), C = 0. If A.1) holds with C = 0, then Case (i) implies A.2)
for every C > 0. The desired result follows by letting C tend to 0.
Exercise 1.1. Show that Theorem 1.1 implies the uniqueness assertion
of Theorem II 1.1.
24
Differential Inequalities and Uniqueness 25
2. Maximal and Minimal Solutions
Let U(t, u) be a continuous function on a plane (t, w)-set E. By a
maximal solution u = w°(?) of
B.1)
u = l/(f, w), m(?0) = u0
is meant a solution of B.1) on a maximal interval of existence such that
if u(t) is any solution of B.1), then
B.2)
u(t) ^ u\t)
holds on the common interval of existence of u, u°. A minimal solution
is similarly defined.
Lemma2.1. Let U(t, u) be continuous on a rectangle R:t0 ^ t ^ t0 + a,
\V — Z/ol = *>' let \u(t, ")l ^ M and a = min («. */^)- r^e« B-1) Aai
a solution u = u°(t) on [t0, t0 + a] with the property that every solution
u = u(t) ofu' = U(t, u), u(t0) ^ u0 satisfies B.2) on [t0, t0 + a].
In view of the proof of the Extension Theorem II 3.1, this lemma implies
existence theorems for maximal and minimal solutions (which will be
stated only for an open set E):
Theorem 2.1. Let U(t, u) be continuous on an open set E and (t0, u0) e E.
Then B.1) has a maximal and a minimal solution.
Proof of Lemma 2.1. Let 0 < a' < a. Then, by Theorem II 2.1,
B.3)
u' = U(t,u) + I/n, u(to) = u0
has a solution u = un(t) on an interval [t0, t0 + a'] if n is sufficiently large.
By Theorem I 2.4, there is a sequence n(l) < nB) < • • • such that
B.4)
u\t) = lim unii
ft->00
exists uniformly on [t0, t0 + a'] and is a solution of B.1).
It will be verified that B.2) holds on [t0, t0 + a']. To this end, it is
sufficient to verify
B.5)
u(t) < un(t) on [t0, t0 + a']
for all large fixed n. If B.5) does not hold, there is a t = tu t0 < t1 <
t0 + a' such that u{t^) > un(tj). Hence there is a largest t2 on [t0, fx),
where u(t2) = un(t2), so that u(t) > un(t) on (f2, fj. But B.3) implies that
un{'i) = u'(t2) + I/n, so that un(t) > u(t) for t(> t2) near t2. This con-
tradication proves B.5). Since a' < a is arbitrary, the lemma follows.
Remark. The uniqueness of the solution u = u°(t) shows that un(t) -»
u°(t) uniformly on [t0, t0 + a'] as n -»¦ oo continuously.
26 Ordinary Differential Equations
3. Right Derivatives
The following simple lemmas will be needed subsequently.
Lemma 3.1. Let u(f)e C*[a, b]. Then \u(t)\ has a right derivative
DR \u(t)\for a ^ t < b, where
C.1) DR \u{t)\ = lim h-\\u{t + h)\ - \u{i)\) as 0 < h -* 0,
and DR \u{t)\ = u'(t) sgn u{t) if u(t) ^ 0 and DRu(t) = |m'@I if «@ = 0.
In particular, \DR \u{t)\ | = |w'@l-
The assertion concerning DR \u{t)\ is clear if u{t) ^ 0. The case when
m@ = 0 follows from u(t + h) = h(u'(t) + o(l)) as h-+0, so that
\u(t + h)\ = h(\u'(t)\ + o(l)) as 0 < h -* 0.
Lemma 3.2. Let y = y(t) e Cl[a, b]. Then \y(t)\ has a right derivative
DR \y{t)\ and \DR \y{t)\ \ ^ \y\t)\for a <: t < b.
Since \y(t)\ = max A/@1, • ¦ •, 1/@1), there are indices k such that
\y\t)\ = \y(t)\. In the following, k denotes any such index. By the last
lemma, \y\t)\ has a right derivative, so that
\y\t + h)\ = \y(t)\ + h(DR |
as 0
0.
For small h > 0, \y(t + h)\ = max* \y\t + h)\, so that by taking the
max* in the last formula line,
h)\ = \y(t)\
DR \y*U)\
as 0 < h -* 0.
Thus DR \y{t)\ exists and is max* DR \y\t)\. Since \DR \y\t)\ | = \yk'(t)\ ^
\y'(t)\, Lemma 3.2 follows.
Exercise 3.1. Show that Lemma 3.2 is correct if \y\ is replaced by the
Euclidean length of y.
4. Differential Inequalities
The next theorem concerns the integration of a differential inequality.
It is one of the results which is used most often in the theory of differential
equations.
Theorem 4.1. Let U(t, u) be continuous on an open (t, u)-set E and
u = m°@ the maximal solution of B.1). Let v(t) be a continuous function on
[t0, t0 + a] satisfying the conditions v(t0) ^ u0, (t, v(tj) e E, and v(t) has
a right derivative DRv(t) on t0 ^ t < t0 + a such that
D.1) DRv(t)^U(t,vit))-
Then, on a common interval of existence ofu°(t) and v(t),
D.2) v(t) <: u\t).
Remark 1. If the inequality D.1) is reversed and v(t0) ^ u0, then the
Differential Inequalities and Uniqueness 27
conclusion D.2) must be replaced by v(t) ^ mo(O, where u = uo(t) is the
minimal solution of B.1). Correspondingly, if in Theorem 4.1 the function
r@ is continuous on an interval t0 — a ^ t ^ t0 with a left derivative
DLv{i) on (t0 — a, t0] satisfying DLv(t) ^ U(t, v(t)) and v(t0) ^ u0, then
again D.2) must be replaced by v(t) ^ mo(O-
Remark 2. It will be clear from the proof that Theorem 4.1 holds if
the "right derivative" DR is replaced by the "upper right derivative"
where the latter is denned by replacing "lim" by "lim sup" in C.1).
Proof of Theorem 4.1. It is sufficient to show that there exists a 6 > 0
such that D.2) holds for [t0, t0 + d]. For if this is the case and m°@, v(t)
are denned on [?0, t0 + /?], it follows that the set of f-values where D.2)
holds cannot have an upper bound different from /?.
Let n > 0 be large and let 6 > 0 be chosen independent of n such that
B.3) has a solution u = un(t) on [t0, t0 + d]. In view of the proof of
Lemma 2.1, it is sufficient to verify that v(t) ^ un(t) on [t0, t0 + d], but
the proof of this is identical to the proof of B.5) in § 2.
Corollary 4.1. Let v(t) be continuous on [a, b] and possess a right
derivative DRv(t) ^ 0 on [a, b]. Then v(t) ^ v(a).
Corollary 4.2. Let U(t, u), u\i) be as in Theorem 4.1. Let V(t, u) be
continuous on E and satisfy
D.3)
V(t, u) ^ U(t, u).
Let v = v(t) be a solution of
D.4) v' = V(t, v), v(t0) = vo(^ u0)
on an interval [t0, t0 + a]. Then D.2) holds on any common interval of
existence ofv(t) and u°(t) to the right of t = t0.
It is clear from Remark 2 that if v(t) is extended to an interval to the
left of t = t0, then, on such an interval, D.2) must be replaced by i'@ ^
mo(O where mo(O is a minimal solution of B.1) with u0 ^ v(t0).
Corollary 4.3. Let U(t, u) ^ 0, m°@ be as in Theorem 4.1; u = uo(t)
the minimal solution of
D.5)
u' = -U(t,u),
Let y = y@ be a C1 vector-valued function on [t0, t0 + a] such that u° ^
\y(to)\^uo,(t,\y(t)\)eEand
D.6) \y\t)\ <: U(t, \y(t)\)
on [t, t0 + a]. Then the first [second] of the two inequalities
D-7) uo(t) ^ \y(t)\ ^ u\t)
holds on any common interval of existence ofu0(t) and y [u°(t) and y].
28 Ordinary Differential Equations
This is an immediate consequence of Theorem 4.1 and Remark 1
following it, since \y(t)\ has a right derivative satisfying — \y'(t)\ ^
Dr \y(t)\ ^ 12/@1 by Lemma 3.2. (In view of Exercise 3.1, this corollary
remains valid if \y\ denotes the Euclidean norm.)
Exercise 4.1. (a) Let/(f, y) be continuous on the strip S:a ^ t ^ b,
y arbitrary, and let/*(f, y1, . . . , ya) be nondecreasing with respect to each
of the components y\ i ^ k, of y. Assume that the solution of the initial
value problem y = f(t, y), y(a) = y0 is unique for a fixed y0, and that this
solution y = y(t) exists on [a, b]. Let z(t) = (zx@> . . ., z\t)) be contin-
continuous on [a, b] such that z\t) has a right derivative for k = 1, . . . , d,
z\d) <; j/0* and DRz\t) ^fk(t, z(t)) for a^t^b [or z*(a) ^ yok and
Ar^O) ^/*0, 2@) for a ^ f <| b]. Then z*(f) ^ yk(t) [or z*(f) ^ j/*(f)]
for a ^ f ^ b. (This is applicable if g(t, y) is continuous on S, z(t) is a
solution of z = g(f, z) and zk(a) ^ j/0*, g*(f, y) ^ /*(f, j/) on S [or z*(a) ^
y<F,g\U y) ^fk(t, y) on S].) See Remark in Exercise 4.3.
(b) If, in part (a), all initial value problems associated with y =f(t, y)
have unique solutions, fk(t, y) is increasing with respect to y\ i ^ k and
k = 1, . . . , d, and z\a) < y0} [or zj(a) > yo>] for at least one index j,
then z*(f) < yok(t) [or z*(f) > j/0*(f)] for a < f ^ b, k = 1, ..., d.
(c) If, in addition to the assumptions of (a), there is an index h such that
f\t, y) is nondecreasing with respect to yh, then yoh(t) — z\t) is non-
decreasing [or nonincreasing] on a ^ f ^ 6.
(d) If the assumptions of (b) and (c) hold, then yo\t) — z\t) is increasing
[or decreasing] on a ^ t ^ 6.
(e) Let m, ?/ denote real-valued scalars and y = (j/1, . . ., if) a real J-
dimensional vector. Let U(t, y) be continuous for a ^ f ^ b and arbitrary
j/ such that solutions of wM) = ?/(f, u,u',..., i/d~u) are uniquely deter-
determined by initial conditions and that U(t, y1, . . . , j/d) is nondecreasing
with respect to each of the first d — 1 components y\ j = 1,. .., d — 1,
of y. Let MjO), m2@ be two solutions of um = U on [a, 6] satisfying
M«'(a) ^ M</'(a) for ; = 0, . . ., d - 1. Then u{j\t) ^ u(j\t) for y =
0,... , d — 1 and a^(^i; furthermore, u^\t) — m^'@ is non-
decreasing for y = 0, . . ., d — 2 and a ^ t ^ b.
Exercise 4.2. Let/(f, y), g(t, y) be continuous on a strip, a ^ t ^ b,
and j/ arbitrary, such that f\t, y) < g\t, y) for k = 1, . .., d and that,
for each k= \,...,d, either f\t, y\ . .., y") or g\t, y\ . .., j/d) is
nondecreasing with respect to y\ i ¦? k. On a ^ f ^ 6, let j/ = j/(f) be a
solution of y' =f(t, y), y(a) = y0 and z = z{t) a solution of z' = g(t, z),
z(a) = z0, where yf^z* for /c=l,...,rf. Then y\t)^z\t) for
Exercise 4.3. Let/(f, j/) be continuous for f0 ^ t ^ f0 + a, \y — j/0| ^
A such that f\t, yl,..., y*) is nondecreasing with respect to each y\
Differential Inequalities and Uniqueness 29
i 5^ k. Show that y' = f(t, y), y(t0) = y0 has a maximal [minimal] solution
yo(t) with the property that if y = y(t) is any other solution, then yk(t) ^
2/o*(O [j/*@ ^ 2/o*(O] holds on the common interval of existence. Remark:
The assumption in Exercise 4.1 (a) that the solution of the initial value
problem y' =f(t, y), y(a) = y0, is unique can be dropped if y{t) is replaced
by the maximal solution [or minimal solution] yo(t).
Exercise 4.4. Let/(f, y), g(t, y) be linear in y, say/*(f, y) = Saw(fJ/3 +
f\t) and g\t, y) = 2^/fV + g\0, where akj(t), bkj(t), f\t), g\t) are
continuous for a ^ t ^ b. Let y(t), z{t) be solutions of y =f(t,y),
y(a) = y0 and z = g(t, z), z{a) = z0, respectively. (These solutions exist on
[a,b]; cf. Corollary 5.1.) What conditions on ajk(t), bjk(t), f\t), g\t),
j/0, z0 imply that \z\t)\ ^ y\t) On [a, b] for k = 1, . . . , di
Theorem 4.1 has an "integrated" analogue which, however, requires
the monotony of U with respect to u. This theorem is a generalization of
Theorem 1.1:
Corollary 4.4. Let U(t, u) be continuous and nondecreasing with respect
to u for t0 ^ t ^ t0 + a, u arbitrary. Let the maximal solution u = u°(t)
o/B.1) exist on [t0, t0 + a]. On [t0, t0 + a], let v(t) be a continuous function
satisfying
D.8)
U(s, v(s)) ds,
where v0 ^ u0. Then v(t) ^ u°(t) holds on [t0, t0 + a].
Proof. Let V(t) be the right side of D.8), so that v(t) ^ V(t), and
V\t) = U(t, v(t)). By the monotony of U, V'(t) ^ U(t, V(t)). Hence
Theorem 4.1 implies that V(t) ^ M°(f) on [t0, t0 + a]; thus v(t) ^ u°(t)
holds.
Exercise 4.5. State the analogue of Corollary 4.4 for the case that the
constant v0 in D.8) is replaced by a continuous function ^0@-
Exercise 4.6. Let y,f z be rf-dimensional vectors; f(t, y) continuous
for t0 ^ t ^ t0 + a and y arbitrary such that fk(t, y1, . . . ,yd) is non-
decreasing with respect to each y\ j = 1, .. ., d. Let the maximal solu-
solution yo(t) of 2/' = f(t, y), y(t0) = y0 exist on [f0, t0 + a]; cf. Exercise 4.3.
Let z{t) be a continuous vector-valued function such that z\t) ^ 2/0* +
f\s, z(s)) ds for t0 <: t ^ t0 + a. Then z\t) ^ yk{t) on [t0, t0 + a].
-It,
5. A Theorem of Wintner
Theorem 4.1 and its corollaries can be used to help find intervals of
existence of solutions of some differential equations.
Theorem 5.1. Let U{t, u) be continuous for t0^ t ^ t0 + a, u^.0, and
let the maximal solution o/B.1), where u0 ^ 0, exist on [t0, t0 + a], e.g.,
30 Ordinary Differential Equations
let U(t, u) = f(u), where y(u) is a positive, continuous function on u ^ 0
such that
E.1) du/rp(u) = oo.
Let f(t, y) be continuous on the strip t0 ^ t ^ t0 + a, y arbitrary, and
satisfy
E.2) \f(t, y)\ ^ U(t, \y\).
Then the maximal interval of existence of solutions of
E.3) y'=f{t,y), y(to) = yo,
where \yo\ ^ u0, is [t0, t0 + a].
Remark 1. It is clear that E.2) is only required for large \y\. Admis-
Admissible choices of y>(u) are, for example, \p(u) = Cu, Cu log u,... for large
u and a constant C.
Proof. E.2) implies the inequality D.6) on any interval on which y(t)
exists. Hence, by Corollary 4.3, the second inequality in D.7) holds on
such an interval and so the main assertion follows from Corollary II 3.1.
In order to complete the proof, it has to be shown that the function
U(t, u) = y>(u) satisfies the condition that the maximal solution of
E.4)
u' = y,(u), u(t0) = uo(^ 0)
exists on [t0, t0 + a] by virtue of E.1). Since y> > 0, E.4) implies that for
any solution u = u(t),
rt r u{t)
E.5) t - t0 = u'(t) dt/y>(u(t)) = du/y>(u).
Jto Juo
Note that f > 0 implies that u'(t) > 0 and u(t) > 0 for t > t0. By
Corollary II 3.1, the solution u(t) can fail to exist on [t0, t0 + a] only if it
exists on some interval [t0, d) and satisfies u(t) —- oo as t -*¦ d (< a). If
this is the case, however, t -»¦ d in E.5) gives a contradiction for the left side
tends to d — t0 and the right side to oo by E.1). This completes the proof.
Remark 2. The type of argument in the proof of Theorem 5.1 supplies
a priori estimates for solutions y(t) of E.3). For example, if y>(u) is the same
as in the last part of Theorem 5.1, let
(u) = f "ds/K
J Wo
s) for m ^ m0
and let u = E>(v) be the function inverse too= Y(m). Then \f(t, y)\ ^
y>{\y\) implies that a solution y(t) of E.3) satisfies
tf. E.5).
\y(t)\ ^ <D(f - t0) for f0 ^ t ^ t0 + a;
Differential Inequalities and Uniqueness 31
Exercise 5.1. Let f(t, y) be continuous on the strip t0 ^ t ^ t0 + a,
y arbitrary. Let \f(t, y)\ ^ <p(t)f(\y\), where <p(t) ^ 0 is integrable on
[fo> h + a] and y(u) is a positive continuous function on u ^ 0 satisfying
E.1). Show that the assertion of Theorem 5.1 and an analogue of Remark
2 are valid.
Corollary 5.1. If A(t) is a continuous d x d matrix function and g(t) a
continuous vector function for t0 ^ t ^ t0 + a, then the (linear) initial value
problem
E.6) y' = A(t)y + g(t), y(t0) = y0
has a unique solution y = y(i), and y(t) exists on t0 ^ t ^ t0 + a.
This is a consequence of Theorem II 1.1 and Theorem 5.1 with the
choice of y>(u) = C(l + u) for some large C.
In a scalar case, Theorem 5.1 can be "read backwards":
Corollary 5.2. Let U(t, u), V(t, u) be continuous functions satisfying D.3)
on t0 ^ t ^ f0 + a, u arbitrary. Let some solution v = v(t) of D.4) on
[t0, d), 6 ^ t0 + a, satisfy v(t) —>¦ oo as t —>- 6. Then the maximal solution
u = u°(t) of BA) has a maximal interval of existence [a, a>+), where co+ ^ 6,
and u°(t) -»¦ oo as t -*¦ a>+.
6. Uniqueness Theorems
One of the principal uses of Theorem 4.1 and its corollaries is to obtain
uniqueness theorems. The following result is often called Kamke's general
uniqueness theorem.
Theorem 6.1. Letf(t, y) be continuous on the parallelepiped R: t0 ^ t <
t0 + a, \y — yo\ ^ b. Let co(t, u) be a continuous (scalar) function on
Ro'to < * = h + a> 0 = u = 1b, with the properties that a>(t, 0) = 0 and
that the only solution u = u(t) of the differential equation
F.1) u' = w(t,u)
on any interval (t0, t0 + e] satisfying
F.2)
u(t) -»¦ 0 and
u(t)
• 0 as t ->¦ to + 0
t- t0
is u(t) = 0. For (t, 2/x), (t, yj e R with t > t0, let
F.3) \f(t, Vl) -f(t, y,)| ^ co(t, \yi - y2\).
Then the initial value problem
F.4) y'=f(t,y), y(to) = yo
has at most one solution on any interval [t0, t0 + e].
u=\y(t)\
32 Ordinary Differential Equations
In Theorem 6.1, we can also conclude uniqueness for initial value
problems y' =f(t, y), 2/@ = 2/i for tt ^ t0. Theorem 6.1 remains valid
if Euclidean norms are employed.
Exercise 6.1. Show that Theorem 6.1 is false if F.2) is replaced by
u(t),u'(t)^Oast-+to + O.
Proof. The fact that
F.5) co(t, 0) = 0 for t0 < t ^ t0 + a
implies of course that u(t) = 0 is a solution of F.1).
Suppose that, for some e > 0, F.4) has two distinct solutions y = 2/i@>
2/2(f) on t0 ^ t ^ t0 + e. Let y(t) = 2/i@ - y2{t). By decreasing e, if
necessary, it can be supposed that
y(t0 + e) ^ 0 and \y(t0 + e)| < 2b.
Also y(t0) = y'(t0) = 0. By F.3),
\y'(t)\ ^ co(t, \y(t)\) on (f0, t0 + e]. It
follows from Corollary 4.3 (and the
Remark 1 following Theorem 4.1) that
if u = uo(t) is the minimal solution of
the initial value problem u = a>(t, u),
«('o + 0 = \y(t0 + e)|, where 0 <
\y(t0 + e)| < 2b, then
F.6) \y(t)\ ^ uo(t)
on any subinterval of (f0, t0 + e] on
which uo(t) exists; see Figure 1.
By the proofs of the Extension Theorem II 3.1 and Lemma 2.1, uo(t) can
be extended, as the minimal solution, to the left until (t, uo(t)) approaches
arbitrarily close to a point of dR0 for some f-values. During the extension
F.6) holds, so that (?, u°(tj) comes arbitrarily close to some point (d, 0) e
dR0 for certain f-values, where d ^ t0. If d > t0, then F.5) shows that
uo(t) has an extension over (t0, t0 + e] with uo(t) = 0 for (t0, d]. Thus,
in any case, the left maximum interval of existence of uo(t) is (f0, t0 + «].
It follows from F.5) and F.6) that uo(t) ->¦ 0 and uo(t)l(t — t0) -»¦ 0 as
t->-10 + 0. By the assumption concerning F.1), uo(t) = 0. Since this
contradicts uo(to + e) = \y(t0 + e)| ^ 0, the theorem follows.
Corollary 6.1 (Nagumo's Criterion). Ifto = O, then co(t,u) = u/t is
admissible in Theorem 6.1 (i.e., the conclusion of Theorem 6.1 holds if
F.3) is replaced by
F.7)
t-
for (U yd, (t, yJeR with t > t0).
Exercise 6.2. The function w(t, u) = ujt in Corollary 6.1 cannot be
replaced by w(t, u) = Cujt for any constant C > 1. Show that if C > 1,
Differential Inequalities and Uniqueness 33
then there exist continuous real-valued functions f(t, y) on 0?fl 1,
12/1 ^ 1 with the properties that
\f(t,yi)-f(t, y2)\^
for t > 0,
but that y' =f(t, y), 2/@) = 0 has more than one solution.
Corollary 6.2 (Osgood's Criterion). If t0 = 0, then co(t, u) = (p(t)y>(u)
is admissible in Theorem 6.1 if <p(t) ^ 0 is continuous for 0 < t ^ a; y>(u)
is continuous for u ^ 0 and y@) = 0, y>(u) > 0 ifu > 0; and rp{t) dt <
r J+o
oo, dujy>{u) = oo.
J+o
Actually, the continuity condition on <p(t) in this corollary can be
weakened. The analogous uniqueness theorem can be proved directly if
(p(t) is only assumed to be integrable over 0 < t ^ a.
Exercise 6.3 [Generalization of Corollaries 6.1 and F.2)]. Let t0 = 0.
(a) If cp{t) ^ 0 is continuous for 0 < t ^ a, show that a>(t, u) = (p{f)u
r /"« "I
is admissible in Theorem 6.1 if and only if lim inf <p(s)ds + log? < oo
Ut J
as t -» +0. (b) Let cp(t) ^ 0 be continuous for 0 < t ^ a; y>(u) continuous
for 0 ^ u ^ 2b, y@) = 0, y>(u) > 0 for 0 < u ^ b, and ^m/v(m) = oo.
J+o
Show that w(f, u) = <p(t)y>(u) is admissible in Theorem 6.1 if, for every
C> 0, lim sup t~l<&{c + <p(s) ds) > 0 as t -> 0, where u = <E>(r) is the
function inverse to Y(m) = dsjy>(s).
Ju
Exercise 6.4. Let y>(u) be continuous for |m| ^ 1, y>@) = 0. Show that
the initial value problem u = y>(u), m@) = 0 has a unique solution
u(t) = 0 unless there exists an e, 0 < e ^ 1, such that either rp(u) ^ 0
for 0 ^ m ?! e and l/v(«) is (Lebesgue) integrable over [0, e] or y>(u) ^ 0
for — e ^ m ^ 0 and l/y(«) is (Lebesgue) integrable over [—e, 0].
Exercise 6.5. Let/; co be as in Theorem 6.1. Show that there exists
a function co0(t, u) which is continuous on the closure of Ro, is nondecreas-
ing with respect to u for fixed t, and satisfies the conditions on co(t, u);
thus co0(t, 0) = 0; the only solution of u' = co0(t, u) and u(t0) = 0 on any
interval [t0, t0 + e] is u(t) = 0; and \f(t, 2/x) -f(t, y^)\ ^ wo(t, \yt - y2\).
(Note that, since co0 is continuous on the closure of Ro, any solution of
u' = a>0(t, u) on (t0, t0 + e] satisfying F.2) is necessarily continuously
differentiable and is the usual type of solution on [t0, t0 + «].)
Exercise 6.6. (a) Let e0,..., ea_x be non-negative constants such that
eo + ¦ ¦ ¦ + ed_! =1. Let U(t, y) = U{t, y\ ..., ya) be a real-valued
continuous function on ^?:0 ^ t ^ a and |2/*| ^ 6 for fe = 1,.. ., d
34 Ordinary Differential Equations
such that | U(t, Vl) - U(t, yj\ ^ f ek^(d -k + l)!r <*-*«> \Vlk - y2k\ if
k = \
t > 0. Show that the dth order (scalar) equation um = U(t, u, u, . . . ,
M(<*-D) has at most one solution (on any interval 0 ^ t _ e ^ a) satisfying
given initial conditions m@) = u0, u = u0', . . ., m'*'^) = u(q~1), where
u0, u0', . . ., m'"*' are d given numbers on the range |w| _ b. (b) Note that
part (a) remains correct if the constants e0,. . ., ta_1 are replaced by
continuous non-negative functions eo(f),. • •, e<j-i(O such that eo(f) +
Exercise 6.7. (a) Let/(f, y) be continuous for R:0 _ t _ a, \y\ _ 6.
On R0:0 < f _ a, \u\ ^ 26, let coj(f, m), «2(f, m) be continuous non-nega-
non-negative functions which are nondecreasing in u for fixed t, satisfy co3(f, 0) = 0,
and
\f(t, 2/i) -/(*, y2)\ ^ Ht, \Vl - yt\) for; =1,2.
Let there exist continuous non-negative functions a(f), jS(f) for 0 _ t ^ a
satisfying a@) = C@) = 0, fi(t) > 0 for 0 < t < a, and «@/|8@ -> 0 as
f -»¦ 0. Suppose that each solution m(?) of m' = co^f, u) for small f > 0 with
the property that u(t) -»¦ 0 as f —>¦ 0 satisfies u{t) ^ a(f) on its interval of
existence. Finally, suppose that the only solution of v' = co2{t, v) for small
t > 0 satisfying v(t)lfi(t) -» 0 as t ->¦ 0 is v(t) = 0. Then the initial value
problem y' = f(t, y), y@) = 0 has exactly one solution, (b) Prove that
coj(f, u) = Cux, co2(f, u) = ytw/f are admissible if k > 0, 0 < A < 1,
k(\ - A)< 1 with a(f) = C(l - A)f1/A-A>, p(t) = r*.
The following involves a "one-sided inequality" and gives "one-sided
uniqueness."
Theorem 6.2 Letf(t, y) be continuous for t0S t _ t0 + a,\y — 2/0| ^ 6.
Considering y, f to be Euclidean vectors, suppose that
F.8) [/(f,2/2)-/(f,2/i)]'B/2-2/i) = O
for I, | ( ^ (, | d a«d |2/s — yo\ _ i, / = 1, 2, wAere fAe dot denotes
scalar multiplication. Then F.4) has at most one solution on any interval
[t^ t0 + e], e > 0.
When it is desired to obtain uniqueness theorems for intervals [f0 — e, /„].
it is necessary to assume the reverse inequality in F.8).
Corollary 6.3. Let U(t, u) be a continuous real-valued function for
t0 ^ t ^ f0 + a, \u — mo| _ b which is nonincreasing with respect to u
{for fixed t). Then the initial value problem u' = U(t, u), u(t0) = m0 has at
most one solution on any interval [t0, t0 + e], e > 0.
Proof of Theorem 6.2. Let y = yx{t), y2(t) be solutions of F.4) on
[t0, t0 + €]. Let d(t) = ||ys(r) - yi(r)||2 = B/2 - *) • (y, - 2/x) be the
square of the Euclidean length of y2(t) — yi(t), so that d(t0) = 0, d(t) _ 0.
Differential Inequalities and Uniqueness 35
But d'(t) = 2(j/2' - j//) • (j/2 - j/i) = 0 by F.8). Hence d(t) = 0 on
[t0, t0 + e] as was to be proved.
Exercise 6.8 (One-sided Generalization of Nagumo's Criterion and of
Theorem 6.2). Theorem 6.2 remains valid if condition F.8) is relaxed to
[At, yd -f(t,yi)]- B/2-2/1)^
for t0 < t = t0 + a.
II2/2-2/1II2
t- t0
7. van Kampen's Uniqueness Theorem
In the following uniqueness theorem, conditions are imposed on a
family of solutions rather than on f(t, y) in
G.1) y'=f(t,y), y(to) = yo.
Theorem 7.1. Letf(t, y) be continuous on a parallelepiped R: t0 _ t _
to + a,\y — 2/ol = b. Let there exist a function r](t, tlt 2/1) on t0 ^ t,t1 ^
to + a> I2/1 — 2/ol ^ P(< b) w'th the properties (i) that, for afixed(tx, j/j),
y = r](t, tlt 2/1) is a solution of
G.2) y'=f(t,y), y(t1) = y1;
(ii) that »?(?, ?i, 2/1) w uniformly Lipschitz continuous with respect to j^;
finally, (iii) fAaf «o two solution arcs y = r\(t, tlt 2/1), 2/ = V(t> t2,2/2) /«"*
through the same point (t, y) unless r/(t, tlt 2/1) = r](t, t2, y2) for f0 _ f _
t0 + a. Then y = r](t, t0,2/0) 's the only solution of G.1) for f0 = ?i =
to + a, I2/1 - 2/0I ^ /?¦
Exercise 7.1. Show that the existence of a continuous ??(f, tu 2/1)
satisfying (i) and (iii) [but not (ii)] does not imply the uniqueness of the
solution of G.1).
Exercise 7.2. When f(t, y) is uniformly Lipschitz continuous with
respect to y, it can be shown that a function y = rj(t, tlt 2/1) satisfying the
conditions of the theorem exists (for small jS > 0); e.g., cf. Exercise II 1.2.
Show that the converse is not correct, i.e., the existence of r](t, tx, j/J
satisfying (i)-(iii) does not imply that f(t, y) is uniformly Lipschitz con-
continuous with respect to y (for y near j/0).
Proof. Let y(i) be any solution of G.1). It will be shown that y(t) =
»/('. '0,2/o) for small f - f0 = 0.
Condition (ii) means that there exists a constant K such that
G.3) \rj(t, rlf 2/J - rj(t, tlt 2/2I ^ K |2/i - 2/2I
for to^t, t^to+a and |yi - 2/0I ^ ft I2/2 - 2/ol = C-
Let |/(/, 2/)| ^ M on R. Then any solution j/ = y(t) of G.1) satisfies
\y(t) - 2/ol ^ A/(? - to) = iC if t0 ^ t = r0 + ,3/2M. Thus »j(r, 5, y(s)) is
36 Ordinary Differential Equations
defined and \rfit, s, y(s)) - y(s)\ ^ M \t - s\ < tf if t0 ^ t, s ^ t0 +
$\2M. Hence
G.4) Mr, s, y(s)) - yo\ ^ P if to^t,s^to + y,
where y = min (a, jS/2M). Condition (iii) means that any point on any
of the arcs y = rj(t, tx, yt) can be used to determine this arc. Thus G.3),
,y-n(t,to,yo)
\(h,s,yfs))-y(t1)
~T0 s h t t
Figure 2. The case d = dim y is 1.
with 2/j = 2/(fx) and y2 = ^(t^ s, y(s)), implies that
G.5) \rfit, tlt y{h)) - rfif, s, y(s))\ ^ K \y(tj - rj(tv s, y(s))\
if t0 ^ t, tlt s ^ t0 + y; cf. Figure 2.
Let t be fixed on t0 5j t ^ f0 + y. It will be shown that
G.6) r(f) = »j(r, t0, y0) - y(t) = 0.
To this end, put
G.7) a(s) = rj(t, r0,j/0) - »?(', *, sfa)) for to^s<t (^ r0 + y),
so that a(t0) = 0 and a(t) = r(f). Then G.5) and G.7) imply that
G.8) 1G@ - a(s)\ ^ K \yitj - r,(tl, s, y(s))\.
Since y = r]{t, s, y(s)) is a solution of y' =/through the point (s, y(s)),
Differential Inequalities and Uniqueness 37
it is seen that ^(f1; s, y(s)) = y(s) + (fx — s) [f(s, y(s)) + o(\)] as tx -+s.
Also, 2/@ = y(s) + (t, - s) [f(s, y(s)) + o(l)] as t, -* s. Hence G.8)
gives CT(fj) — a(s) = Ko(\) ]rx — j| as tx -+s; i.e., ^/a/dy exists and is 0.
Thus a(s) is the constant a(t0) = 0 for t0 ^ s ^ f. In particular, r(f) =
a(f) satisfies G.6), as was to be proved.
Exercise 7.3 (One-sided Analogue of Theorem 7.1). Let f(t, y) be
continuous on R:t0 ^ t < f0 + a, |j/ — j/0| < 6. Let there exist a function
??(f, tu j/j) on f0 ^ fx ^ f ^ f0 + a, Ij/j - j/0| < ,3« b) with the proper-
properties (i) that, for fixed (f x, j/j), j/ = r)(t, tu yt) is a solution of G.2) and (ii)
that there exists a constant AT such that for max (tv t2) ^ t* ^ t ^ t0 + a.
HU h, J/x) - rj(t, t2, j/2)| < K \r}(t*, h, Vl) - i](t*, t2, j/2)|.
Then y = rj(t, tu j/x) is the only solution of G.2) for sufficiently small
intervals [tv tx + e], e > 0, to the right of fx (but not necessarily to the
left of O-
8. Egress Points and Lyapunov Functions
Let/(f, y) be continuous on an open (t, j/)-set Q and let Qo be an open
subset of Q. Let 3Q0 and 0.0 denote the boundary and closure of Qo,
respectively. A point (t0, y0) e 3Q0 n Q is called an egress point [or an
ingress point] of Qo with respect to the system
(8.1)
y'=f{t,y)
if, for every solution y = y(t) of (8.1) satisfying y(t0) = y0, there exists an
e > 0 such that (t, y(t)) e Qo for t0 — e < t < t0 [or for t0 < t < t0 + «].
If, in addition, (t, y(t)) $ Do for t0 < t < t0 + e [or for t0 — e < t < t0]
for a small e > 0, then (f0, y0) is called a strict egress point [or strict
ingress point]. A point (f0, y0) e 3Q0 n Q will be referred to as a nonegress
point if it is not an egress point.
Lemma 8.1. Let fit, y) be continuous on an open set Q and Qo an open
subset of Q such that 3Q0 n Q is either empty or consists of nonegress
points. Let y(t) be a solution of (S.I) satisfying (t°, y(t0)) e Qo for some
t". Then (t, y(t)) e Qo on a right maximal interval of existence [t°, a>+).
If the conclusion is false, there is a least value fo(> t°) of t, where
Co» y(to)) e 9^o n ^- But then (t0, y(toj) is an egress point, which contra-
contradicts the assumption and proves the lemma.
Let u(t, y) be a real-valued function defined in a vicinity of a point
(/,,!/,)eQ. Let y(t) be a solution of (8.1) satisfying y(tt) = yx. lfu(t,y(t))
is differentiate at t = tu this derivative is called the trajectory derivative
of u at (?!, yx) along j/ = y(t) and is denoted by u(tlt j/j). When w(f, j/) has
continuous partial derivatives, its trajectory derivative exists and can be
38 Ordinary Differential Equations
calculated without finding solutions of (8.1). In fact,
(8.2) u(t, y) = dujdt + (grad u) -f(t, y),
where the dot denotes scalar multiplication and grad u = (dujdy1,. ..,
dujdy*) is the gradient of u with respect to y.
Let (t0,y0) e dQ0 ^ Q and let w(f, y) be a function of class C1 on a
neighborhood JV of (t0, y0) in Q such that (t, y) e Qo O TV if and only if
u(t> V) < 0. Then a necessary condition for (f0, y0) to be an egress point
is that u(t0, y0) ^ 0 and a sufficient condition for (f0, y0) to be a strict
egress point is that u(t0, y0) > 0. Further, a sufficient condition for
(?o. 2/o) to t>e a nonegress point is that u(t, y) ^ 0 for (f, j/) e Qo.
When the system under consideration
(8.3) y' = f(y),
is autonomous (i.e., when the right side does not depend on t), definitions
are similar. For example, let/(j/) be continuous on an open j/-set Q, Qo
an open subset of Q, and y0 e 9Q0 O Q. The point y0 is called an egress
point of Qo with respect to (8.3) if, for every solution y(t) of (8.3) satisfying
2/@) = 2/o> there exists an e > 0 such that y(t) e Qo for — e < t < 0. If,
in addition, y(t) $ &0 for 0 < t < e for some e > 0, then y0 is called a
strict egress point. A lemma analogous to Lemma 8.1 is clearly valid here.
For an application of these notions, consider a function f{y) defined
on an open set containing y = 0. A function V(y) defined on a neighbor-
neighborhood of y = 0 is called a Lyapunov function if (i) it has continuous partial
derivatives; (ii) V(y) ^ 0 according as \y\ ^ 0; and (iii) the trajectory
derivative of V satisfies V(y) ^ 0.
Theorem 8.1. Let f(y) be continuous on an open set containing y = 0,
f@) = 0, and let there exist a Lyapunov function V(y). Then the solution
y = 0 o/(8.3) is stable (in the sense of Lyapunov).
Lyapunov stability of the solution y = 0 means that if e > 0 is arbitrary,
then there exists a de > 0 such that if \yo\ < de, then a solution y(t) of
(8.3) satisfying the initial condition j/@) = y0 exists and satisfies \y(t)\ < e
for t ^ 0. If in addition, y(t) -*¦ 0 as t -*¦ oo, then the solution y = 0 of
(8.3) is called asymptotically stable (in the sense of Lyapunov). Roughly
speaking, Lyapunov stability of y = 0 means that if a solution y(t) starts
near y = 0 it remains near y = 0 in the future (t ^ 0); and Lyapunov
asymptotic stability of y = 0 means that, in addition, y(t) -*¦ 0 as t -*¦ oo.
Proof. Let e > 0 be any number such that the set |y\ ^ e is in the open
set on which/and V are defined. For any r\ > 0, let d(rj) be chosen so
that 0 < d(rj) < e and V(y) < r\ if \y\ < d(rj).
Reference to Figure 3 will clarify the following arguments. Since V(y)
is continuous and positive on \y\ = e, there is an r\ = r]f > 0 such that
Differential Inequalities and Uniqueness 39
V(y) > rj for \y\ = e. Let O0 be the open set {y: \y\ < e, V(y) < r{\. The
boundary 3Q0 is contained in the set {y: \y\ < e, V{y) = rj]. The function
u(y) = V{y) — rj satisfies u(y) < 0 at a point y, \y\ < e, if and only if
y e Qo. Clearly u = V ^ 0. Hence, no point of 3Q0 is an egress point.
Consequently, by the analogue of Lemma 8.1, a solution y(i) of (8.3)
satisfying y@) e Qo remains in Qo on its right maximal interval of existence
[0, co+). Since Qo is contained in the sphere \y\ ? e in O, it follows that
= oo; Corollary II 3.2.
co
e, where V> 77
3fi0, where V=t)
Figure 3.
Finally, put <5e = d(rje) > 0, so that V(y) < rj if \y\ < de < e. Thus
\y@)\ < Ee implies that j/@) e Qo, hence y(t) exists and j/(i) e Qo for t ^ 0.
In particular, \y(t)\ < e for t ^ 0. This proves the theorem.
Exercise 8.1. Let/(y) be continuous on an open set containing j/ = 0
and let/(O) = 0. Let (8.3) possess a continuous first integral V(y) [i.e., a
function which is constant along solutions y = y(t) of (8.3)] such that
V(y) has a strict extremum (maximum or minimum) at y = 0. Then the
solution y = 0 of (8.3) is stable.
Theorem 8.2. If in Theorem 8.1, F(j/) ^ 0 according as \y\ ^ 0, f/ie/j
?/ze solution y = 0 o/"(8.3) is asymptotically stable (in the sense of Lyapunov).
Proof. Use the notation of the last proof. Let y(i) be a solution of
(8.3) with |j/@)| < Ee. Since V ^ 0, it follows that *¦%(?)) is nonincreasing
and tends monotonically to a limit, say c ^ 0, as t -*¦ oo.
Suppose first that c = 0. Then j(f)—>-0 as f—>- oo. For otherwise,
there is an e0 > 0 such that e0 ^ \y(t)\ ^ e for certain large f-values. But
there exists a constant m0 > 0 such that F(j/) > w0 for e0 ^ |j/| ^ e; thus
K2/@) > w0 > 0 for certain large f-values. This is impossible; hence,
7/@ ->• 0 as t -*¦ oo.
Suppose, if possible, that c > 0, so that 0 < c < ?? and F(y) < \c if
l?/l < <KJf) < e- Hence |«/(?)l ^ ^(if) for large f. But the assumption on
40 Ordinary Differential Equations
F implies that there exists anm>0 such that V(y) ^ — m < 0 if <5(Jc) <
\y\ ^ e. In particular, F(j/(?)) ^ — m < 0, for all large f. This is impos-
impossible. Hence c = 0 and j/(f)->-0 as t-*¦ oo. This proves the theorem.
A result analogous to Theorem 8.1 in which the conclusion is that the
solution y = 0 is not stable is given by the following:
Exercise 8.2. Let /(j/) be continuous on an open set E containing
y = 0 and let /@) = 0. Let there exist a function Viy) on ? satisfying
F@) = 0, having continuous partial derivatives and a trajectory derivative
such that Viy) ^ 0 according as \y\ ^ 0 on ?. Let Viy) assume negative
values for some y arbitrarily near y = 0. Then the solution y = 0 is not
(Lyapunov) stable.
Theorems 8.1 and 8.2 have analogues for nonautonomous systems
which depend on a suitable modification of the definition of Lyapunov
function: Let/(f, y) be continuous for t ^ T, \y\ ^ b and satisfy
(8.4)
fit, 0) = 0 for t ^ T.
A function F(f, y) denned for t ^ T, \y\ ^ b is called a Lyapunov Junction
if (i) F(f, y) has continuous partial derivatives; (ii) F(f, 0) = 0 for t ^ T
and there exists a continuous function W(j/) on \y\ ^ 6 such that Wiy) ^ 0
according as \y\ ^ 0, and F(f, 2/) ^ W(j/) for t ^ 7; (iii) the trajectory
derivative of V satisfies F(f, y) ^ 0.
Theorem 8.3. Let fit, y) be continuous for t ^ T, \y\ ^ b and satisfy
(8.4). Lef fAere exist a Lyapunov function F(f, y). Then the solution
y = 0 0/(8.1) is uniformly stable (in the sense of Lyapunov).
Here, Lyapunov stability means that if e > 0 is arbitrary, then there
exists a d€ > 0 and a/e^ 7such that if j/(f) is a solution of (8.1) satisfying
\yit°)\ < d€ for some t° ^ fe, then j/(f) exists and \y(t)\ < e for all t ^ f°.
If, in addition, y(t)~>-0 as f —>• 00, then the solution j/ = 0 is called
Lyapunov asymptotically stable. The modifier "uniform" for "stability"
or "asymptotic stability" means that te can be chosen to be T for all
e>0.
Theorem 8.4. Let fit, y), Vit,y) be as in Theorem 8.3. In addition,
assume that there exists a continuous Wxiy)for \y\ ^ b such that W^y) ^ 0
according as \y\ ^ 0 and that V{t, y) ^ — W?y) for t ^ T. Then the solu-
solution y = 0 o/(8.1) is uniformly asymptotic stable {in the sense of Lyapunov).
Exercise 8.3. (a) Prove Theorem 8.3. F) Prove Theorem 8.4.
9. Successive Approximations
The proof of Theorem II 1.1 suggests the question as to whether or not
a solution of
(9.1)
y'=fit,y),
Differential Inequalities and Uniqueness 41
can always be obtained as the limit of the sequence (or a subsequence) of
the successive approximations defined in § II 1. That the answer is in the
negative is shown by the following example for a scalar initial value
problem
(9.2)
u' = Uit, u), M@) = 0,
where ?/(f, u) will be denned for t ^ 0 and all u.
Consider the approximations M0(f) = 0 and
*n+i
@ =
s, Mn(s)) ds if n ^ 0.
Let Uit, 0) = It, hence u^t) = t2; put Uit, t2) = -It, hence M2(f) = -t2.
Finally, put ?/(/, — t2) = It, so that M3(f) = t2. Then M2n(f) = — t2 for
n > 0 and w2n+i@ = t2 for n ^ 0. It only remains to complete the
definition of Uit, u) as a continuous function to obtain the desired example.
One possible completion of this definition is to let Uit, u) = 2t if
u ^ 0, Uit, u) = — It if u ^ f2, and to be a linear function of u when
0 ^ u ^ f2, f > 0 fixed. In this way, we obtain an example in which
?/(/, u) is nonincreasing with respect to u (for fixed t ^ 0). In this case, the
solution of (9.2) is unique (Corollary 6.3) although no subsequence of the
successive approximations converge to a solution.
It turns out, however, that if the solutions of (9.1) are unique by virtue
of Theorem 6.1, then successive approximations converge to a solution.
Theorem 9.1. Let R, R0,f co be as in Theorem 6.1. Let |/(f, y)\ ^ M
on R and a = min (a, bjM). Then the functions j/0(f) = y0,
(9.3)
2/n@ = 2/0 +
Jto
/n_x(S))dS ifn^l,
are defined and converge uniformly on [t0, t0 + a] to the solution y = yit)
o/(9.1).
Proof. By Exercise 6.5, it can be supposed that co(f, u) is continuous
on the closure of Ro and is nondecreasing with respect to u for fixed t.
The sequence of approximations (9.3) are uniformly bounded and equi-
continuous on [t0, t0 + a] and hence possesses uniformly convergent
subsequences. If it is known that yjj) — j/n_i(f) -»¦ 0 as n -»¦ oo, then
(9.3) implies that the limit of any such subsequence is the unique solution
y(t) of (9.1). It then follows that the full sequence y0, yu . . . converges
uniformly to y(t); cf. Remark 2 following Theorem I 2.3. Thus, in order
to prove Theorem 9.1, it suffices to verify that k(t) = 0, where
(9.4)
= lim sup \yn(t) - yn-i(t)\ as n ->¦ oo.
42 Ordinary Differential Equations
Since \f\<:MonR,
\yn(h) - 2/«-i('i)l ^ \Vni.h) - y«-iWI + 2M \h - '*l-
The right side is at most A(f2) + e + 2M \h - t2\ for large n if e > 0.
Hence A(fx) ^ A(r.) + e + 2M \tt - tt\. Since e > 0 is arbitrary and
tu U can be interchanged, |A(fx) ~ *(h)\ ^ 2M\tt - t2\. In particular,
() is continuous for f0 = f = fo + a-
By the relation (9.3),
= P
Jio
Hence, by F.3),
J to
ds-
For a fixed t on the range t0 < t <j t0 + a, there is a sequence of integers
«A) < «B) < . .. such that \yn+l(t) - yn(t)\ — A(f) as « = «(/c) — oo and
that AxE) = lim \yjs) - yn-i(s)\ exists uniformly on (,Ss^fo + «as
n = n(jfc)-». oo. Thus,
^ P
J*o
ds.
Since ^(s) ^ lim sup \yn(s) - yn^(s)\ = A(j) and <w(f, m) is monotone in
By Corollary 4.4, A(f) ^ uo(t), where M0(f) is the maximal solution of
u' = a)(t, u), u(t0) = 0.
Since this initial value problem has the unique solution uo(t) = 0, it
follows that A(f) = 0. This proves the theorem.
Exercise 9.1. Show that under the conditions of Exercise 6.7(a), the
successive approximations yo(t) = 0 and (9.3), where t0 = 0 and y0 = 0,
converge uniformly on 0 ^ ( ? min (a, bjM) to the solution of y' =
fit, y), 2/@) = 0.
Exercise 9.2. For two vectors, y = (y1,..., if) and z = (z1,... , z ),
use the notation y ^ z if y* ^ 2* for * = 1,..., d. Let/= (/S . .. ,/")
and j/ = (j/\ ...,/)¦ Assume that /(f, j/) is continuous on R:0 ^
f ^ a, 12/1 ^ * and that /(f, y,) ^/(f, 2/2) if 2/1 ^ 2/2- (a) Define two
sequences of successive approximations 2/o±(O>2/i±(O> . ¦ ¦ on 0 ^ f ^ a =
min (a, b/M), where yo±(t) = ±MA, ..., l)t and
2/n±@ = f(f, yn-i±(s)) ds torn =1,2,....
Jo
Differential Inequalities and Uniqueness 43
Show that 2/o+(O ^ 2/i+@ ^ ¦ ¦ ¦ and j/0-@ ^ 2/i-@ ^ ¦ ¦ ¦ and that
both sequences converge uniformly to solutions of y' = f(t, y), y@) = 0.
(b) Show that 2/o±(O can be replaced by continuous functions
2/o ±@ on 0 ^ f ^ a satisfying |2/0±@l ^ * and
2/o+(O ^
2/o-(O ^
(e.g., 2/o-(O = 2/o is admissible if/(f, j/0) ^ 0).
Exercise 9.3. (a) Using the notation y ^ 2 introduced in Exercise 9.2,
let/(f, y) be continuous for f ^ 0 and all 2/ and satisfy/(f, j/x) ^/(f, 2/2)
if 2/i ^ 2/2- Let 2/@ be a solution of y' = —/(f, j/) satisfying y(t) ^ 2/@)
for f ^ 0; cf., e.g., §XIV2. Consider the successive approximations
2/o(O, 2/i@, ¦ ¦ ¦ defined by yo(t) = j/@), yn(t) = j/@) - ('f(s, y^s)) ds
Jo
for n = 1, 2, Let zn@ denote the "error" zn(t) = 2/n(f) - 2/@- Show
that(-l)"zn(r)^ Ofor« = 0, 1,. . . and (-l)X'(O ^ Ofor« = 1, 2,. ..
and t ^ 0. (Convergence of the successive approximations is not asserted.)
n
(b) Let En(t) = J {-\)mtmlm\ be the «th partial sum of the MacLaurin
0
TO=0
series for e-f. Show that (-1 )"(?„(/) - er*) ^ 0 for n = 0, 1,.. . and
t ^0.
Exercise 9.4. Let U(t, u) be real-valued and continuous for t ^ 0 and
arbitrary u and U(t, u) nondecreasing with respect to u for fixed t. Let
m0, m0' be fixed numbers and u(t) a solution of u" = -•- U(t, u). Define
successive approximations for u{t) by putting uo(t) = u0 + uo't and
"«@ = «o(O - (? - s)?/(s, «,_i(s))
Jo
for « = 1, 2,....
Then uo(t), u^t),... are defined for t ^ 0. (a) Suppose that u{i) satisfies
u(t) ^ m0 + wo'f on its right maximal interval of existence [0, a>+). Show that
w+ = co and that the "error" vn(t) = un(t) — u(t) satisfies (— 1)X(O ^
0, (-l)X'(O ^0 for n = 1, 2,. . . and t ^ 0. (Convergence of the
successive approximations is not asserted.) (b) Let Cn(f) = JT (—\)mt2m/
» m = 0
Bm)! and 5n(r) = 2 (-\)mt2m+1lBm + 1)! be the nth partial sums of
TO=0
the Maclaurin series for cos t and sin f, respectively. Show that (—1)"
[Cn(t) - cos t] ^ 0 and (-1)" [Sn(f) - sin f] ^ 0 for « = 0, 1,... and
/ ^ 0. (c) Let ?/(f, m) = q{i)u, where ^(f) ^ 0 is continuous and non-
decreasing for t ^ 0. Using Theorem XIV 3.1 x and the remarks following
it, show that (a) is applicable if u0 ^ 0 and u0' ^ 0 [i.e., show that u(t) ^
"o + uo't for t ^ 0].
44 Ordinary Differential Equations
Notes
section 1. Theorem 1.1 goes back essentially to Peano [1]. A special case was
stated and proved by Gronwall [1]; a slightly more general form of the theorem (which
is contained in Corollary 4.4) is given by Reid [1, p. 290]. The proof in the text is that
of Titchmarsh [1, pp. 97-98].
section 2. Maximal and minimal solutions were considered by Peano [1]; see
Perron [4].
section 4. Differential inequalities of the type D.1) occur in the work of Peano [1]
and of Perron [4]. Theorem 4.1 and its proof are taken from Kamke [1] and are
essentially due to Peano. Exercises 4.2 and 4.3 are results of Kamke [2]; see Wazewski
[7]. A special case of Corollary 4.4 is given by Bihari [1]. Exercise 4.6 is a result of
Opial [1].
section 5. Results of the type in Theorem 5.1 and Exercise 5.1 were first given by
Wintner [1], [4].
section 6. Theorem 6.1 is due to Kamke [1]. An earlier version, in which it is
assumed that m{t, u) is continuous also for t = 0, was given by Perron [6]. (Exercise
6.5, due to Olech [2], shows that, in a certain sense, Perron's theorem is not less general
than Kamke's.) For the case d = 1, earlier results of the type of Perron's were given
by Bompiani [1] and Iyanaga [1]. For Exercise 6.1, see Szarski [1]. For Corollary 6.1,
seeNagumo [1]; a less sharp form was first proved by Rosenblatt [1] with io(t, u) = Cult
and 0 < C < 1. An example of the type required in Exercise 6.2 was given by Perron
[8]. For Corollary 6.2, see Osgood [1]. For Exercise 6.3(a), see Levy [1, pp. 46-47].
For Exercise 6.4, see Wallach [1]. For Exercise 6.5, see Olech [2]. For a particular case
of Exercise 6.6, see Wintner [22]. For Exercise 6.8, part (a), see F. Brauer [1], who
generalized the result of part F) due to KrasnoseFskii and S. G. Krein [1].
For other uniqueness theorems related to those of this section, see F. Brauer and
S. Sternberg [1]. These involve estimates for a function Vit, |j/2(f) — yi(t)\) instead of
lz/i@ ~ 2/a@|. F°r earlier references on the subject of uniqueness theorems, see Miiller
[3] and Kamke [4, pp. 2 and 33].
section 7. Theorem 7.1 is a result of van Kampen [2].
section 8. The terminology "egress point" and "ingress point" is that of Wazewski
[5]. Exercise 8.1 is due to Dirichlet [1]; it was first given by Lagrange [1, pp. 36^14]
under the assumption that V(y) is analytic and that the Hessian matrix E2K/9j/* dy')
of V at y = 0 is definite. This result is the forerunner of Lyapunov's Theorem 8.1.
Theorems 8.1 and 8.2, Exercise 8.2, and Theorems 8.3 and 8.4 are due to Lyapunov [2]
(and constitute the basis for his "direct" or "second" method); cf. LaSalle and
Lefschetz [1]. For references and recent developments on this subject, see W. Hahn [1],
Antosiewicz [1], Massera [2], and Krasovskil [4].
section 9. The example of nonconvergent successive approximations is due to
Miiller [1]. Theorem 9.1, as stated, is due to Olech [2] and avoids an assumption of
monotony on <o(t, u) occurring in earlier versions of this result. Earlier versions and
special cases are to be found in Rosenblatt [1], van Kampen [3] (cf. also Haviland [1]),
DieudonnS [1], Wintner [2], LaSalle [1], Coddington and Levinson [1], Viswanatham
[1], and Wazewski [8]. The reduction of the proof of Theorem 9.1 to the verification
that l{t) = 0 is due to Dieudonn6 (and independently to Wintner) and is used by the
authors following them. Exercise 9.1 is a result of F. Brauer [1] and generalizes
Luxemburg [1]. Exercise 9.2(a) is due to Miiller [1]; cf. LaSalle [1] for part F). For
Exercise 9.3(a), cf. Hartman and Wintner [16], For Exercise 9.4, cf. Wintner [16].
Chapter IV
Linear Differential Equations
In this chapter, u, v,p are scalars; c, y, z,f, g are (column) ^-dimensional
vectors; and A, B, Y, Z are matrices. The scalars, components of the
vectors, and elements of the matrices will be supposed to be complex-
valued.
1. Linear Systems
This chapter will be concerned with some elementary facts about linear
systems of differential equations in the homogeneous case,
A.1) V' = A(t)y,
and in the inhomogeneous case,
A.2) y' = A(t)y+f(t).
Throughout this chapter, A(t) is a continuous d X d matrix and f(t) a
continuous vector on a f-interval [a, b]. Recall the following fundamental
fact stated as Corollary III 5.1.
Lemma 1.1. The initial value problem A.2) and
A-3) y(t0) = 2/o,
a ^ t0 ^ b, has a unique solution y = y(t) and y(t) exists on a ^ t ^ b.
The fact that the elements of A(t) and components of y are complex-
valued does not affect the applicability of Corollary III 5.1. For example,
A.2) is equivalent to a real linear system for a 2J-dimensional vector made
up of the real and imaginary parts of the components of y. Actually, the
simplest proof of Lemma 1.1 is a direct one employing the standard
successive approximations:
Exercise 1.1. Prove Lemma 1.1 by using successive approximations.
(This proof also gives the majorization \y(t)\ ^ e-KI<—<ol |^0| if x denotes
a constant such that |^4(f)j/| = K \y\ for all vectors y and a ^ t ^ b; cf.
D.2) below.)
45
46 Ordinary Differential Equations
Exercise 1.2. Let A(t) = (ajk(t)) be (not necessarily continuous but)
integrable over [a, b]; i.e., let the entries ajk(t) be Lebesgue integrable
over [a, b]. Show by successive approximations that Lemma 1.1 remains
correct. Here a solution y(t) is interpreted as a continuous solution of the
integral equation
2/@ =
-J
Jto
A(s)y(s) ds
or, equivalently, y{t) satisfies A.3) and is absolutely continuous on [a, b]
with its derivative y'(t) satisfying A.1) except on a null set (i.e., a set of
Lebesgue measure 0).
The uniqueness of solutions of A.1), A.3) implies that
Corollary 1.1. If' y = y(t) is a solution of (I.I) and y(t^) = Ofor some
t0, a ^to^b, then y(t) = 0.
For the solutions of A.1) and A.2), there is the obvious theorem:
Theorem 1.1 (Principles of Superposition) (i) Let y = 2/i@> 2/2@ be
solutions o/(l.l), then any linear combination y = c1y1(/) + C02(t) with
constant coefficients clt c2 is a solution of A.1). (ii) If y = y^t) and y =
yo(t) are solutions of A.1) and A.2), respectively, then y = yo(t) + yx(t)
is a solution of A.2); conversely, ify = yo(t),y°(t) are solutions of A.2), then
V = 2/o(O — 2/°@ is a solution of (I.I).
The vector equation A.1) can be replaced by a matrix differential
equation,
A.4) 7' = A(t)Y,
where 7 is matrix with d rows and k (arbitrary) columns. It is clear that a
matrix Y = Y(t) is a solution of A.4) if and only if each column of Y(t),
when considered as a column vector, is a solution of A.1).
Corollary 1.1 and the principle of superposition imply that if Y = Y(t)
is a d X k matrix solution of A.4), then rank Y(t) does not depend on t.
That is, if 2/i@» ¦ ¦ •»2/r@ are r solutions of A.1), then the constant vectors
2/i(^o)> • • ¦ > 2/r(O are linearly independent for some t0 if and only if they are
linearly independent for every @ on a ^ @ ^ b.
Below, unless otherwise specified only d X d matrix solutions Y = Y(t)
of A.4) will be considered. In this case, either det Y(t) = Oordet Y(t) 5* 0
for all t. This fact can be strengthened as follows:
Theorem 1.2 (Liouville). Let Y = Y(t) be a d X d matrix solution of
A.4), A(r) = det Y(t), and a < t0 ^ b. Then, on [a, b],
A(f) = A(/o) exp tr A(s) ds.
Jto
A.5)
For a square matrix A = (ajk), the trace of A is defined to be the sum of
its diagonal elements, tr A = 2 ait.
Linear Differential Equations 47
Proof. Let (At) = (ajk(t)), j,k = 1, . . ., d. The usual expansion for
the determinant A@ = det Y(t), where 7@ = (yk%t)), and the rule for
differentiating the product of d scalar functions show that
A'@ = 2 det 7/0,
where 7,@ is the matrix obtained by replacing the/th row (yi(t),. ..,
yd\t)) of Y(t) by its derivative (y{'(t), ..., yJd'(t)). Since yl'(t) = 2 aiiVj
by A.1), it is seen that theyth row of Yj(t) is the sum of ajt(t) times the/th
row of Y(t) and a linear combination of the other rows of Y5(t). Hence
det Y5(t) = an(t) det Y(t) and so, A'@ = (tr A(t)) A(/). This gives A.5).
By a fundamental matrix Y(t) of A.1) or A.4) is meant a solution of A.4)
such that det Y(t) j± 0. In order to obtain a fundamental matrix Y(t), let
Y(t) be a matrix with columns 2/i@» • • • »2/<j@» where y = ys(t) is a solution
of A.1) belonging to a given initial condition «///„) = yj0, where y10,. . .,
yd0 are (constant) linearly independent vectors. It is clear that all funda-
fundamental matrices Y(t) can be obtained in this fashion. Let Y(t) = Y(t, t0)
denote the particular fundamental matrix satisfying
A.6)
Y(t0, t0) = I.
Exercise 1.3. Let A(t) be a continuous d x d matrix for t _ 0 such
that every solution y(t) of A.1) is bounded for t ^ 0. Let Y(t) be a
fundamental matrix of A.1). Show that Y~\t) is bounded if and only if
Re tr A(s) ds is bounded from below.
If 7@ is a solution of A.4) and c is a constant vector, the principle
of superposition states that
A.7) y(t)=Y(t)c
is a solution of A.1). Furthermore, if 7@ is a fundamental solution of
A.4), then every solution of A.1) is of the form A.7) with c = 7-1(/0)«/(/0);
that is,
A-8) 2/@= Y(t)Y-\to)y(to)-
In particular, if 7@ = Y(t, t0), then
A.9) y@ = Y(t, to)y(to).
More generally, if 7= 70@ is a matrix solution of A.4) and C is a
constant d X d matrix, then 7@ = 70@C is a solution of A.4). When
K0@ is a fundamental solution of A.4), all d x d matrix solutions of A.4)
are obtained in this fashion and all fundamental solutions are obtained in
this way with a choice of C, det C ^ 0.
0,1
48 Ordinary Differential Equations
Lemma 1.2. Let Y(t)= Y(t, t0) be the fundamental solution o/A.4)
satisfying A.6). Then, for t0, t e [a, b],
A.10) Y(t,to)= Y(t,s)Y(s,t0).
Proof. By the remarks just made, where C = Y(s, t0), the right side is a
fundamental matrix and reduces to Y(s, t0) at t = s. Since the left side of
A.10) is a matrix with the same properties, the relation A.10) is clear
from uniqueness (Lemma 1.1).
2. Variation of Constants
A linear change of dependent variables in A.1) or A.2) will often be
used.
Theorem 2.1. Let Z(t) be a continuously differentiable, nonsingular
d x d matrix for a ^ t ^ b. Under the linear change of variables y —>• z,
where
B.1) y = Z(t)z,
A.2) is transformed into
B.2) z' = Z-\t)[A(t)Z(t) - Z'(t)]z + Z-\t)f(f).
In particular, ifZ(t) is a fundamental matrix for
B.3) z = B(t)z,
where B(t) = Z\t)Z-\t) is continuous for a < t ^ b, then B.2) becomes
B.4) z' = Z-\t)[A{t) - B(t)]Z(t)z + Z-\t)f(t).
The equation B.2) is clear, for B.1) implies that y = Z'z + Zz , so
that z' = Z-\y - Z'z) and y = Ay +/from A.2).
In the particular case where A(t) = B(t), Z(t) = Y(t), and the latter is
a fundamental matrix for A.1), the change of variables B.1) is called a
"variation of constants," i.e., the replacement of the constant vector c in
A.7) by a variable vector z. In this case, B.4) reduces to z = Y-\t)f(t),
so that its solutions are given by a quadrature
z{t)
= c + f' Y-1*
Jt0
(s)f(s) ds,
where c = z(t0) is a constant vector. In view of B.1), this gives the first
part of the following corollary. The last part follows from A.10).
Corollary 2.1. Let Y(t) be a fundamental matrix o/ A.1). Then the
solutions of A.2) are given by
Linear Differential Equations 49
In particular, if Y(t) = Y(t, t0), B.5) becomes
B.6) y = Y(t, to)c + f' Y(t, s)f(s) ds.
Formula B.5) or B.6) shows that the solutions of A.2) are determined
by a quadrature if the solutions of A.1) are known. For arbitrary c,
Y(t)c in B.5) is merely an arbitrary solution of A.1).
Exercise 2.1. Let A(t) be continuous on some /-interval (not necessarily
closed or bounded), Y(t) a fundamental matrix for A.1), and Z(t) a
continuously differentiate, nonsingular matrix. Under the change of
variables, y = Z(t)z, let A.1) become z = C(t)z; cf. B.2) where f(t) = 0.
For any matrix A, let A* denote the complex conjugate transpose of A
and AH = \{A + A*), the Hermitian part of A. (a) Show that if Z(t) is
unitary [i.e., Z*(t) = Z~\t)], then CH(t) = Z*(t)AH(t)Z(t) [since the
derivative of Z*(t)Z(t) = / is 0]. (b) Let Z(t) = Y(t)Q(t), so that Q =
Y~XZ is continuously differentiable and nonsingular. Show that C(t) =
— Q~1{t)Q\i). In particular, C{t) is triangular [or diagonal] if Q(t) is
triangular [or diagonal].
Exercise 2.2 (Continuation), (a) Show that there exists a unitary Z(t)
such that C(t) is triangular; in this case, C(t) is bounded if A(t) is bounded.
(b) Show that there exists a bounded Z(t) such that C(t) is diagonal. It is
not claimed that Z(t) can be chosen so that Z~\t) is bounded.
3. Reductions to Smaller Systems
If a set of r linearly independent solutions of A.1) is known, the deter-
determination of all solutions of A.1) can be reduced essentially to the problem
of determining the solutions of a linear homogeneous system of d-r
differential equations. The simplest formulae giving this reduction are,
however, "local," i.e., applicable only on subintervals of [a, b] and vary
from subinterval to subinterval.
Let Y = Yr(t) be a d x r matrix solution of A.4). Corresponding to a
given point t = t0 of [a, b], renumber the components of y so that if
Yr(t) = (ykj(t)), j = 1, ..., d and k = 1, . . ., r, and
Yrl(t)
B.5)
y = Y(t)[c
c + ? Y-\s)f(s)
C.1)
where Yrl(t) is an r x r matrix, then det 7rl(/0) 5^ 0. Let [y, d] be any
subinterval of [a, b] containing /„ on which det Yrl(t) j^ 0. Define the
d x d matrix
(Yrl(t) 0
C-2) Z(t) = (
\Yri(t) ij
50 Ordinary Differential Equations
where Id_r is the unit (d — r) x (d — r) matrix. Then det Z(t) =
det Yrl(t) ?s0on [y, <5]. A simple calculation shows that Z~\t) is a
matrix of the form
(Y^\t) 0
C.3) Z-\t) =
\Zr2@
where Zri(t) is the (d — r) x r matrix
C.4) zr2« = -yr2
W{t)
Note that Z'(t) = G/@,0) = A(t)( Yr(t), 0); i.e.,
Introduce the change of variables y = Z(t)z into A.1), then the case
f(t) = 0 of B.2) gives the resulting differential equation for z. Writing
the right side of B.2), with/= 0, as Z~\t)[A{t)Z(t) - Z\t)]z gives, by
virtue of C.2) and C.5),
/0 0 \
C.6) z'= Z-\t)A{t)[ z.
W d—rf
Let Au(t), Ai2(t) be square r x r, (d — r) x (d — r) matrices such that
C.7)
and let
C.8)
where z1 is an r-dimensional vector, z2 a (d — r)-dimensional vector. Then
C.3) and C.8) show that C.6) splits into
C.9) z/
C.10) 2,' = [
Note that C.10) is a linear homogeneous system for the (d — ^-dimen-
^-dimensional vector z2 and that z1 is given by a quadrature when z2 is known. In
C.10), Zr2@ is given by C.4). The reduction of A.1) to C.10) is of course
only valid on an interval [y, <5] where Yrl(t) is nonsingular. The result can
be summarized as follows:
Lemma 3.1. Let Y= Yr{t) be a d x r matrix solution o/A.4) on [a, b]
such that if Yr(t) is written as C.1), then det Yrl(t) = det (yk'(t)), where
k,j = 1, . . . , r, does not vanish on [y, d] <= [a, b]. Then the change of
Linear Differential Equations 51
variables B.1) in terms of C.2) reduces A.1) to C.9)-C.10) on [y, <5] in
which A12, Ai% and Zr2 are defined by C.7) and C.4), respectively.
An application. Consider a system A.1) in which A(t) = (ajk(t)) satisfies
C.11) «i,i+i@ ^ 0 and ajk(t) = 0 for k ^ ; + 2
on [a, b]. It is readily verified that a solution y = y(t) of A.1) is known as
soon as its first component u = y\i) is known.
Corollary 3.1. Let A(t) satisfy C.11) on [a, b] and let A.1) possess d
solutions y^t),. . ., yd(t) with the property that, for k = 1, .. ., d and
C.12)
Wk(t) = det
0 for i,j = 1 k.
Then A.1) is equivalent to the single differential equation of dth order for
u = y1 of the form
C.13)
where
C.14)
(ad^ .. .{ala^u)'}'}' ...)' = 0,
a, =
fOTJ = 0, ...,d- 1
and W_x =W^=\, a01 = 1.
Proof. This will be seen to follow by d — 1 successive applications of
the reduction process described in Lemma 3.1 with r = 1. Introduce the
notation
C.15) Wtl.. Atc. h ... h = det (y)$ for m, n = 1,. . . , k,
C.16) W?= W'i,...«_!,,;«..._!.» forj,fc = a+ 1, . . . , d.
Then, by a standard formula for minors of the "adjoint determinant"
In order to verify this, let a symbol (e.g., W^ or W*k) denote either a
matrix or its determinant. Let ymn denote the cofactor of the (m, n)th
element in the (a + 1) X (a + 1) determinant Wjk+1. In particular,
yaa = W%, ya_a+1 = - Wfa, ya+1>a = - WZk, and ya+1,a+1 = W^. Con-
Consider the product of the determinants W-k+lY, where
Yla
Yla+1
= det
\
0
0
Yx.x+l
Y XX
Yx+l.x
Yx.x+1
52 Ordinary Differential Equations
On the other hand, by matrix multiplication,
0 0 \
= det
Vi
V •••
Vx-1
vik
0
The last two displays give C.17).
In order to systematize notation, write t/A), A1 = (ajk) fort/, A, respec-
respectively. Thus W1 = 2/a)i@ ?* 0- Introduce new variables t/<2) by the
variation of constants associated with C.2), r = 1; namely,
C.18)
for k = 2 d.
Consider yw to be a (d — l)-dimensional vector yw = (t/B2), . . . , t/f2)),
so that y\2) is not considered a component of t/B). Then A.1) reduces, by
C.9)-C.10)andC.4), to
(D2°)
(Da)
Van
a)k — aik
Van
^a\k for j,k = 2 d;
so that solutions of (D2°) are determined by quadratures when solutions
of (D2) are known. Using C.18) and the known solutions yim,. . . , yil)d
of A.1), we obtain d-\ solutions ymj = (yfiyj(t), . .., yfx)j(t)) of
(D2) for j = 2, . . ., d; namely,
C.19)
In particular, yflM^) = W2/ W1 j^ 0, and this procedure can be repeated
to reduce the order of (D2). Suppose that the successive changes of
variables Tu . . ., 7^ have been denned, each of the form
C.20) Tx:
k = a + 1 d,
where y(ai+1) = (tf?»,..., yfa+1)) is a (d - a)-dimensional vector
and yfa+D is not considered a component of t/(a+1). Suppose that there
Linear Differential Equations 53
results a system of differential equations for y"a+1), y<a+i) of the form
V±a)
Vlx+1) — 2, ax
A*+\
y\x)x
where j, k = a + 1 d, and suppose that d — a + 1 solutions t/(a)r =
(yfa)r(O. • • •. y?«),@) of (AJ are given by
C.21J
for j, k = a.,. . . , d.
In particular, since W^ = W^,
C-22) y('x)
It is readily verified from Tx and C.17) that (Dx+1) has d —a solutions
y<«+i>i = (^?«+i)j@ »f«+i)j(O) for y = a + 1 rf given by
C.21a+1). For, by C.21J and C.22), the relations in C.20) show that
for fc = a + 1, . . . , d.
The replacement of yfa), t/(aa) by t/faW, y("a)j given in C.21J and the use of
Wx = PC and C.17) give C.21a+1).
Note that yld) is a 1-dimensional vector yw) = yfd) and that (Dd) which
is a linear homogeneous equation has, by C.22), the general solution
yfd)(t) = cWd\Wd_x, where c is an arbitrary constant. Thus (Dd) is
equivalent to
C.23)
The assumption C.11), which has not been used so far, and an induction
on a show that afk = ajk if d ^ k > j = a, .. ., d — 1. Hence
C.24) alx+1 = ax.x+1 and axxk = 0 for k ^ a + 2,
so that, by C.22), (?°+1) reduces to
C.25)
w.
Note that, by virtue of C.22), the first equation for Tx in C.20) gives
C.26) y*x) = W^±H fora = l d - 1.
54 Ordinary Differential Equations
Hence, by C.25),
2/(o+i) = — tor a = 1, . . ., d — I,
or, by C.14),
C.27) ?„„ = & fora=l d - 2.
By C.18), u = y[X) determines y\2) = WoujW1 = aou which satisfies
(aou)' = yfS)/a1 by C.27). Thus yfS) = a^u)' satisfies [a^u)']' =
*/f4)/a2 by C.27). Repetitions of this argument give
Note that C.25), with a = d — 1, shows that yf^1' is
Hence the desired result C.13) follows from C.23).
4. Basic Inequalities
Let the norm \\A\\ of the matrix A be denned by
D.1) M|| = sup \Ay\ for M = 1.
This norm of A depends on the norm \y\ of the vector y. For the choice
of either the norm \y\ = max (\y*\ \y\d) or the Euclidean norm, there
is the following estimate for solutions of A.2):
Lemma 4.1. Let y = y(t) be a solution of A.2) and t, toe [a, b]. Then
D.2) \y(t)\ ^ \\y{to)\
\f(s)\ ds | j exp | j\\A(s)\\ ds
Proof. By A.2), the inequality \y'\ ^ \\A(t)\\ ¦ \y\ + \f(t)\ holds and is
an analogue of (III 4.6). Thus, if u°(t) is the (unique) solution of
D.3) «'
V(s)|| ds,
satisfying u(t0) = u0 with u0 = \y(to)\, i.e.,
u\t) = j«0 +JV(»)I (exp -j'\\A(r)\\ </r) ds} expJ
Corollary III 4.3 and the remark following it imply that \y(t)\ ^ u°@ for
to<t^b. This gives D.2) for t0 ^ t. Similarly, if u = uo(t) is the
solution of
u'= -11^@11«-1/@1
Linear Differential Equations 55
satisfying u(t0) = mo(= \y(to)\), i.e.,
«0@ = {«o - j'\f(s)\ (exp|'M(r)|| dr} ds} exp (-J'
then |«/@l ^ «0@ f°r {o = ?- Interchanging / and /0 in the inequality so
obtained gives D.2) for t ^ /„.
Corollary 4.1. Let A0(t); A^t), A2(t),. . . be a sequence of continuous
d X d matrices and fo(t); fi(t),f2(t),. . . a sequence of continuous vectors
on [a, b] such that An(t) -*¦ A0(t) and fn(t) —*-/0@ as n -*¦ oo uniformly on
[a, b]. Let y = yn(t) be the solution of
D.40 y' = An(t)y +/„«. y(Q = ».,
where a ^ tn ^ b and (tn, yj -> (/0) y0) as n-+co. Then yn(t) -* yo(t)
uniformly on [a, b] as n -*¦ oo.
Proof. It is clear that 11^@11, P2@ll, • • • and 1/@1,1/2@1, • • • are
uniformly bounded on [a, b]. Thus Lemma 4.1 implies that ^(Ol,
|«/2@l,... is uniformly bounded, say \yn{t)\ ^ c for n = 1,2,..., and
a < t ^ b. The right side fn(t, y) = An(t)y +fn(t) of the differential
equation in D.4J tends uniformly to f(t, y) = A0(t)y +fo(t) as n -*¦ oo
for a ^ t ^ b, \y\ ^ c. Thus Corollary 4.1 follows from Theorem I 2.4.
For a matrix A, let A* denote the complex conjugate transpose of A
and AH = \{A + A*), the Hermitian part of A. Let
d
y ¦ * = 2 yk*k
denote the scalar product of the pair of vectors y, z (so that in the case of
vectors with complex-valued components, y ¦ z is the complex conjugate
of z ¦ y). In particular,
D.5) Ay ¦ z = y ¦ A*z.
Finally, let ^0, fi° be the least and greatest eigenvalue of AH, i.e., the least
and greatest zero of the polynomial det (AH — XI) in A. (The fact that AH
is Hermitian implies that its eigenvalues are real.) Equivalently, if ||«/||
denotes the Euclidean norm, then fx0, ft0 are given by
D.6) [io = 'mfAHyy and fi° = sup AHy ¦ y for \\y\\ = 1
= inf Re (Ay ¦ y) and ft0 = sup Re (Ay ¦ y) for ||y|| = 1,
or
D.7)
where "Re a" denotes the real part of the complex number a.
When «/@, z(t) are differentiable vector-valued functions, then
0/@• *@)' = y'(t)¦ z(t) + 2/@• z'(t); inparticular,(||2/(OH2)' = y'(t}-y(t) +
y(t) ¦ y'(t) = 2 Re [y'(t) ¦ y(t)]. This implies
56 Ordinary Differential Equations
Lemma 4.2. Let \\y\\ denote the Euclidean norm of y; fio(t), fi\i) the
least, greatest eigenvalue of the Hermitian part AH(t) of A(t); and y{t) a
solution of A.2). Then
D.8) ^@ II2/H - 11/@11 ^ H»H' ^ /"°@ 112/11 + 11/@11
(where \\y\\' denotes either a left or right derivate of \\y\\). Hence, for
a<^to<t<b,
D.9) \\y(t)\\ ^ \\y(to)\\ exp P /(s) ds + P||/(s)|| (exp f /W dr) ds,
D.10) \\y(i)\\ ^ \\y(tj)\\ exp
j' ^(s) </s -J
V(s)ll
r)
dr) ds.
Proof. The inequalities D.8) follow from A.2). For, on the one hand,
y(t) * 0 implies that \\y(t)\\' = Kll2/(OII2Oll2/(OII = Re [2/'@" 2/@1/112/@115
on the other hand, y(t) = 0 implies that \D \\y(t)\\ | = ||/(/)|| for D = DR
ox D = DL. The continuity of A(t) implies that of (io(t) and /j,°(t), so that
D.9) and D.10) are consequences of D.8) by Theorem III 4.1 and Remark 1
following it.
Consider a function y(t) for t ^ 0. A number t is called a (Lyapunov)
order number for «/(/) if, for every e > 0, there exist positive constants
C0(e), C(e) such that
D.11) |i 2/(f) || ^ C(e)e(r+f)( for all large t.
D.12) \\y(t)\\ ^ C0(e)e(r"f)( for some arbitrarily large /.
When «/(/) ?? 0 for large /, an equivalent formulation of D.11) and D.12)
is
D.13)
lim sup t x log 11X011 = T-
Lemmas 4.1 and 4.2 show that if A(t) is continuous for t ^ 0, then a
sufficient condition for every solution y(t) ^ 0 of the homogeneous
system A.1) to possess an order number t is that
in which case
-1 \\A(s)\\ ds be bounded,
Jo
or, more generally, that
r1 fi°(s) ds, — r1 /uo(s) ds be bounded from above,
Jo Jo
in which case
Linear Differential Equations 57
lim inf r1 fio(s) ds ^ t ^ lim sup r1 (t°(s) ds.
(->OD Jo t—X JO
5. Constant Coefficients
Let R be a constant d x d matrix and consider the system of differential
equations
E.1) y' = Ry.
Let y1 5^ 0 be a constant vector, A a complex number. By substituting
E-2) y = yieu
into E.1), it is seen that a necessary and sufficient condition, in order that
E.2) be a solution of E.1), is that
E-3) Ry1 = Xy1;
i.e., that A be an eigenvalue and that yx ^ 0 be a corresponding eigen-
eigenvector of R. Thus to each eigenvalue A of R, there corresponds at least one
solution of E.1) of the form E.2). If R has simple elementary divisors
(i.e., if R has linearly independent eigenvectors yx,..., yd belonging to the
respective eigenvalues Al5..., Ad), then
E-4) Y= (yieXl\ ..., ydeXdt)
is a fundamental matrix for E.1).
In the general case, a fundamental matrix can be found as follows:
Successive approximations for a solution of the initial value problem
E.1) and «/@) = c are
E.5) 2/o(O = c, yn(i) = c + \ Ryn^(t) dt for n ^ 1,
Jo
so that an induction shows that
E.6)
2!
n\
This suggests the following definition: For any d x d matrix B, let eB
denote the matrix defined by
E.7)
2!
^
where the matrix series on the right can be considered as d2 (scalar) series,
one for each of the elements of eB. If B = (bjk), then D.1) shows that
58 Ordinary Differential Equations
\bjk\ ^ ||?|| for;, k = 1,..., d. Since D.1) clearly implies \\Bn\\ < \\B\\n,
it follows that if Bn = (Z>j?>), then \b$\ < \\B\\n. Thus each of the d2
series for the elements of eB is convergent. The standard proof for the
functional equation of the exponential function shows that
E.8) eB+Cf = eBec if BC = CB.
This makes it clear that E.6) converges uniformly on any bounded
/-interval to y = emc, which is then a solution of E.1) by E.5). In other
words,
E.9) Y{t) = em
is a fundamental matrix for E.1) and Y@) = /.
Consider the inhomogeneous equation
E.10)
y' = Ry +/(/)
corresponding to E.1). In this case formula B.6) for the general solution
of A.2) reduces to
E.11)
y =
+ fV
Jto
ds
for the general solution of E.19).
Let Q be a constant, nonsingular matrix. The change of variables
transforms E.1) into
z = Jz,
y=Qz
where J= Q^RQ.
This has the fundamental matrix Z = c7'. It is readily verified from
R = QJQ-1 and the definition E.7) that
E-12)
eRt =
where J = Q~lRQ.
Since a fundamental matrix, say em, can be multiplied on the right by a
constant nonsingular matrix, say Q, to obtain another fundamental
matrix, it follows that
E.13) Y(t) = Qe"
is also a fundamental matrix for E.1).
Let Q be chosen so that J is in a Jordan normal form, i.e.,
E.14)
= dmg[J(\),...,J(g)],
where J(j) is a square h(J) x h(j) matrix with all of its diagonal elements
equal to a number X = k(j) and, if h(j) > 1, its subdiagonal elements
equal to
E.15)
1 and its other elements equal
J(J) =
1
0
0
X
1
0
0
X
... 0
... 0
... 0
Linear Differential
to
0
0
0
0:
°\
0
0
= XIh +
Equations
Kh,
59
\0 0 0 ... 0 1 .
where X = X(j), h = h(j), and Kh is the nilpotent square matrix
/0 0 0 ... 0 0 ON
.000
.000
E.16)
1 0 0
0 1 0
\0 0 0
0 1 0y
and h{\) -\ + %) = d. J(J) is just the scalar X = X(j) and Kh = 0 if
/*(/)= 1.
From J = diag [7A), . .., J(g)] follows Jn = diag [/"(I), .. ., j*(g)];
hence
E.17)
expJt = diag [exp/(l)/,. . ., expJ(g)t].
In view of E.8) and E.15), cxpj(j)t = ext cxp Kht where X = X(j), h =
h(j). Note that Kh2 is obtained from Kh by moving the l's from the sub-
diagonal to the diagonal below this; Khs is obtained by moving the l's to
the next lower diagonal; etc. In particular, Khh = 0. Hence
E.18)
exp J{j)t =
1
t
2
0
1
t
. . . 0
.. . 0
2!
-l)l (h-2)\
,h = h{j) and X = X(j).
The formulae E.12), E.17) and E.18) completely determine the funda-
fundamental matrix E.9) of E.1).
60 Ordinary Differential Equations
It follows from these formulae that if y(t) is a solution of E.1), then its
components are linear combinations of the exponentials eX(l)t,.. ., ew
with polynomials in t as coefficients. These polynomial coefficients
cannot, of course, be chosen arbitrarily.
Thus the problem of determining the solutions of E.1) is the algebraic
one of determining the Jordan normal form J of R and a matrix Q such
that J = Q~lRQ. The simple case, at the beginning of this section,
leading to E.4) corresponds to a situation where h(j) = 1 for/ = 1,. . ., g
and g = d, so that cxpj(j)t is the 1 X 1 matrix (scalar) eu, X = X(J).
Note that, in any case, if the eigenvalues of R are XlT X2,..., Xd, then
E.1) has d linearly independent solutions y^t), . . ., yd(t) such that the
order number of yfa) is t = Re X} for/ = 1,.. ., d; cf. D.13).
6. Floquet Theory
The case of variable, but periodic, coefficients can theoretically be
reduced to the case of constant coefficients. This is the essence of the
following.
Theorem 6.1. In the system
F.1)
y' = P{t)y,
let P(t) be a continuous d X d matrix for — oo < t < oo which is periodic
of period p,
F.2)
P(t+p) = P(t).
Then any fundamental matrix Y(t) of F.1) has a representation of the form
F.3) Y(t) = Z(t)eRt, where Z(t + p) = Z(t)
and R is a constant matrix (and Z(t), Rare d x d matrices).
lfy0 is an eigenvector of R belonging to an eigenvalue X, so that eRty0 =
«/</', then the solution y = Y(t)y0 of F.1) is of the form z^e* where the
vector zx@ = Z(t)y0 has the period p. More generally, it follows, from
the structure of em discussed in the last section, that if y(t) is any solution
of F.1), then the components of y(t) are linear combinations of terms of
the type a.(t)tlceu, where cn(t + p) = cn(t), k is an integer 0 ^ k ^ d — 1,
and X is an eigenvalue of R.
Neither the matrix R nor its eigenvalues are uniquely determined by the
system F.1). For example, the representation F.3) can be replaced by
Y(t) = Z@e~2"'VR+2W/)', i-e., Z(t) by Z(t)e~2wit and R by R + 2-rriI.
On the other hand, the eigenvalues of eR are uniquely determined by the
system F.1). This will be clear from the argument below which shows
Linear Differential Equations 61
that eR is determined by the fundamental matrix Y(t), while if Y(t) is
replaced by the arbitrary fundamental matrix Y(t)C0, where Co is a
nonsingular constant matrix, then eR is replaced by CoeRC^1. The
eigenvalues ax, .. . , ad of C = eRp are called the characteristic roots of
the system F.1). If Xu . . ., Xd are eigenvalues of R, then ex\ . . ., ex* are
the eigenvalues of eR so that if the numeration is chosen correctly, a1 =
e^p,..., ad = ex*p. The numbers X1,...,Xd which are determined by
F.1) only modulo 2rri are called characteristic exponents of F.1). It is seen
from F.3) that F.1) has d linearly independent solutions y^t), . . ., yd(t)
such that the order number of yJ[t) is Re Xi = p-1 Re log af for j =
1, . . . ,d.
Proof of Theorem 6.1. Since Y(t) is a fundamental matrix of F.1), it
follows from F.2) that Y(t + p) is also a fundamental matrix of F.1).
Hence, by the remark preceding Lemma 1.2, there exists a constant
nonsingular matrix C such that
F.4)
Y(t+p)= Y(t)C.
It will be shown that det C ^ 0 implies that there is a (nonunique)
matrix R such that
F.5)
C=eRp;
i.e., C has a logarithm Rp. If this is granted, F.4) can be written as
F.6) Y(t+p)= Y(t)eRp.
Define Z(t) by F.3), i.e., by Z(t) = Y(t)e-Rt; cf. E.8). Then Z(t + p) =
Y(t+p)e-R«+p>=[Y(t+p)e-Rp]e-Rt=Y(t)e-Rt by F.6). Thus
Z(t + p) = Z(t), as claimed.
In order to complete the proof, it is necessary to verify the existence of
an R satisfying F.5). Since F.5) is equivalent to QCQ'1 = exppQRQ-1,
it is sufficient to suppose that C is in a Jordan normal form. In fact, the
considerations of the last section show that it is sufficient to consider the
case that C is a matrix of the form J = XJ + K, where the elements of K
are 0 except for those on the subdiagonal which are 1; cf. E.14) and
E.17). Also det/ = Xd ^ 0 implies that X ^ 0. Writing J = X(I + KjX)
and noting that log A +/) = /- t*/2 + /3/3 , we are led to
expect that a logarithm of J is given by
F.7)
log J = (log /)/ + S, where S = - f
i=l
i.e., J = Xes or equivalently, I + K/X = es. This can be readily verified
as follows:
62 Ordinary Differential Equations
Note that the series in F.7) is, in fact, a finite sum since Kd = 0. The
formal rearrangement of power series to obtain
h j
nl
i.e., exp [log A +/)] = 1 + /, is valid for |/| < 1. It is clear that the same
formal calculation gives e8 = 1 + KjX. Furthermore, this formal
calculation is permissible, since the powers of K commute and there is
obviously no question of convergence to be considered.
7. Adjoint Systems
Consider again the system A.1). If A* is the complex conjugate trans-
transpose of A, the system
G.1) a' A*{t)z
is called the system adjoint to A.1). The corresponding inhomogeneous
system is
G.2) z' = -A*(t)z -g(t).
There are several results relating A.1) and G.1). The first of these is
Lemma 7.1. A nonsingular, d x d matrix Y(t) is a fundamental matrix
for (I A) if and only ifiY*^))-1 = (Y~\t))* is a fundamental matrix for
G.1).
This follows from the fact that if Y(t) has a continuous derivative Y',
then (F-KO)' = - Y-\t) Y'(t) Y-^t) as can be seen by differentiating
YtyY-^t) = I. Thus, if Y(t) is a fundamental matrix for A.1), so that
Y' = A Y, then (F)' = — Y~lA and taking the complex conjugate
transpose of this relation gives (Y*-1)' = — A*(Y*~1). The converse is
proved similarly.
Exercise 7.1. Show that A.1) has a fundamental matrix Y(t) which is
unitary, Y = Y*-1, if and only if A.1) is self-adjoint; i.e., if and only if
A(t) is skew Hermitian, A = —A*. In this case, if y = y{t) is a solution of
A.1), then \\y{t)\\ is a constant.
Lemma 7.2 (Green's Formula). Let A(t), f(t), g(t) be continuous for
a ^ t ^ b; y(t) a solution of A.2); z(t) a solution of G.2). Then, for
G.3)
f
Ja
\f(s) ¦ z(s) - y(s) • g(s)] ds = «/@ • 2@ - y(a) ¦ z(a),
where the dot denotes scalar multiplication.
This relation is proved by snowing that both sides have the same
derivative, since Ay • z = y -A *z.
Linear Differential Equations 63
8. Higher Order Linear Equations
In this section let po(O,Pi(t),... ,pd-i{t),h{t) be continuous, real- or
complex-valued functions for a ^ t ^ b. The linear homogeneous
differential equation,
(8.1) «<*) + /»d_1(/)«<<|-1) + ' • • + fi(t)u' +po(t)u = 0,
and the corresponding inhomogeneous equation,
(8.2) «<*) + p^it
will be considered. The treatment of these equations reduces to that of
A.1) and A.2) by letting y = («<°>, u™, ..., u^"), where « = ««»,
0 \
. . 0
(8.3) ^(/) =
0
1
0
0
1
0 0 0
\ —Po —P\ —Pi
0
0
0
-Pz
1
-Pd-J
and /(/) = @,. . ., 0, h(t)). It seems worthwhile, however, to summarize
the essential facts for this important special case:
(i) The initial value problem u(t0) = u0, u'(t0) = u0', . . . , m*"-11^) =
utf~l) belonging to (8.2), where u0, u0', . . ., u^~X) are darbitrary numbers,
has a unique solution u = u(t) which exists on a < t ^ b.' In particular,
if (8.2) is replaced by (8.1) and «„ = «„' = ••• = m^-d = 0, then u(t) =
0; hence no solution u(t) ^0 of (8.1) has infinitely many zeros ona<
/ <b.
(ii) (Principle of superposition) (a) Let u = u^t), u2(t) be two solutions
of (8.1), then any linear combination u = CfU-^t) + c2u(t) with constant
coefficients c1; c2 is also a solution of (8.1); (b) if u = u(t) and u = ux{t)
are solutions of (8.1) and (8.2), respectively, then u = u(t) + u^t) is a
solution of (8.2); conversely, if u = uo(t), ui{t) are solutions of (8.2), then
u = uo(t) — Ui(t) is a solution of (8.1).
When the functions u^t),..., uk(t) possess continuous derivatives of
order k — 1, their Wronskian or Wronskian determinant, W(t) =
W(t, «!,..., uk), is defined to be det {u^~l\t)) for i,j = 1, . . ., k,
W(t) = det
„(*-!)
.(fc-l)J
64 Ordinary Differential Equations
A set of k continuous functions «i@> ¦ ¦ ¦, ^k(t) on a < t ^ b is said to
be linearly dependent if there exists constants clt . . ., ck, not all 0, such
that Cit/jCO + ¦ ¦ • + ckuk(t) = 0 for a ^ t < b. Otherwise, the functions
u^t),.. ., uk(t) are called linearly independent. It is clear that if ux,. . ., uk
have continuous derivatives of order k — 1, then a necessary condition for
«!,..., uk to be linearly dependent is that W(t; ux, . . ., «*) = 0 for
a ^ / < ft. It is also clear that the converse is false (e.g., u^t), u2(t) can be
linearly independent on 0 < t ^ 1 with u^t) = 0 for 0 ^ t ^ Vz and
«2@ = 0 for H ^ / < 1 so that W(t\ uu «2) = 0 for 0 ^ t < 1). For
A: = rfsolutions of (8.1), however, the following holds:
(iii) Let u^t),..., ujj) be solutions of (8.1) and W(t) = W(t; uv . . .,
u^. Then
(8.4)
W(t) exp
s) ds = W(t0) for a ^ t, t0 ^
= 0
and ult. . . ,ud are linearly dependent if and only if W(t) vanishes at one
point, in which case W{t) = 0 for a < t ^b.
Formula (8.4) is a particular case of A.5) in view of (8.3). The last part
of (iii) is a consequence of uniqueness (i) and the superposition principle
(ii).
(iv) Let u^i),...., «d_i@ be d — 1 linearly independent solutions of
(8.1). Then equation (8.1) is equivalent to the equation
(8.5) \jV(t; «,«!,..., ud_,) e
for the unknown u. When d = 2, this equation is equivalent to
(u/Ul)' = uT\t) exp ( - J p1(s) ds\ where u^t) ¦? 0.
If d linearly independent solutions uu .. ., ud of (8.1) are known, we can
particularize B.5) to find a formula giving solutions of (8.2) in terms of a
quadrature. It is easier to verify the following directly:
(v) For a fixed s,a<s^b, let u = u(t; s) be the solution of (8.1)
determined by the initial conditions
(8.6) u = u' = ¦¦¦ = «<d-2) = 0, u"*-1' = 1 at t = s,
and let uo(t) be an arbitrary solution of (8.1). Then
(8.7)
"@ = «0@ +
(s)ds
is the solution of (8.2) satisfying um{a) = u(ok)(a) for k = 0,. . ., d - 1.
It is easy to deduce from A.7) and Lemma 1.2 that u{t;s) and its d
derivatives «',..., uld) with respect to t are continuous functions of (t, s)
Linear Differential Equations 65
for a < t, s ^ b. In particular, the integral in (8.7) exists and is a function
possessing d continuous derivatives with respect to t which can be calcu-
calculated formally. A direct verification substituting (8.7) into (8.2), shows
that (8.7) is the solution of (8.2) satisfying the specified initial conditions,
(vi) Consider a differential equation,
(8.8)
alU' + aou = 0,
where a0, au . . ., ad_1 are constants. The associated characteristic
equation is defined to be
(8.9)
a0 = 0.
If y is the vector y = {u{d-X),.. ., u', u), then (8.8) is equivalent to the
system E.1) where R is the constant matrix
(8.10) R =
1
0
\ 0
0
1
0
0
0
0
0
0
0
Note that the components of y are written in the order reverse to that
considered in connection with (8.1).
It is readily verified that u = eu is a solution of (8.8) if and only if (8.9)
holds. Actually, (8.9) is identical with the characteristic equation
det (XI — R) = 0 for R. In order to see this, consider the relation Ry = Ay
where y j? 0. It is seen that this relation holds if and only if A satisfies
(8.9) and y = ciX^1, . . ., A2, A, 1) for some constant c. Thus A is an
eigenvalue of R if and only if A satisfies (8.9). It follows that (8.9) is the
characteristic equation of R when the roots of (8.9) are distinct. If the
roots are not distinct, the coefficients a0,. . ., ad_1 in (8.9) [and correspond-
correspondingly in (8.10)] can be altered by arbitrarily small amounts so that the
resulting polynomial has distinct roots and hence is the characteristic
polynomial of the altered R. The desired conclusion follows by letting
the arbitrarily small alterations tend to 0.
Exercise 8.1. By induction on d, give another proof that (8.9) is the
same as the equation det (A/ — R) = 0. (Still another proof follows from
Exercise 8.2.)
If A is a root of (8.9) of multiplicity m, 1 ^ m ^ d, then u = ext
teu,. . ., tm~xeu are solutions of (8.8). In order to see this, let L[u]
denote the expression on the left of (8.8) if u(t) is any function with d
continuous derivatives. Thus (8.8) is equivalent to L[u] = 0. Let f(A) be
66 Ordinary Differential Equations
the polynomial on the left of (8.9), so that F = dF/dX = ¦•¦ = dm'1FI
dX™-1 = 0 at the given value of X. Note that L[eA(] = F(X)eu and, since
the coefficients of (8.8) are constant, L[tkeu] = L[dkeu/dXk] = dkL[eu]/
dXk = dk(F(X)eu)jdXk = 0 for k = 0, ..., m - 1 at the given value of X.
Thus if A(l), . . ., X(g) are the distinct roots of (8.9) and if h(j) is the
multiplicity of the root X(j), then d solutions of (8.8) are given by u =
tkem)t for y = 1,..., g, and A; = 0,..., h(j) - 1, where h(l) + • • • +
h(g) = d.
Exercise 8.2. (a) Show that the functions u = tkex(i)t, where j = 1, . . .,
g, and k = 0, .. ., h(j) — 1 are linearly independent, (b) Let Y(t) be the
fundamental matrix for y = Ry in which the successive columns are
solution vectors y = (k'*', . . . , «@)) corresponding to u (or w@)) =
tkeHi)tjk\ in the following order: first, j = 1 and k = h(l) - l,h(l) -
2,..., 0; then j = 2 and k = hB) — 1,. .. , 0; etc. Let 7 =
diag [7A), . . . ,J(g)] in the notation of § 5. Show that Y'\t)RY(t) = J;
i.e., Y(t)J = RY(t) or, equivalently, Y(i)J = Y'(t). In particular, 7 =
Y'^QfyR Y@) is a Jordan normal form for R. (c) For another proof of (a)
and for use in the proof of Theorem X 17.5, show that
(8.11) det 7@) = II WO - Ki)fmU)-
(vii) When the coefficients po(t),. .. ,pd-i(t) of (8.1) are periodic of
period p, the corresponding linear system A.1) of first order has periodic
coefficients of period p, by (8.3), and § 6 is applicable.
(viii) (Adjoint equations) Consider a cfth order differential equation
(8.12) Pd(t)u^ + /^(/y *-» + • • • + Pl(t)u' + po(t)u = 0
with complex-valued coefficients on an interval a ^ / ^ b, where pk(t)
has k continuous derivatives for k = 0, 1,..., d. The adjoint equation
of (8.12) is defined to be
(8.i3) (- mpdvyv + (- iy-i(pd_lVy-i + ••• + (- w^y + pov = o.
Note that an integration by parts gives
'dt,
kuMv dt = [p^
and repetitions of this give
\\ku™v dt = \i(-iyu^-
Ja Li=0
+ (-if f\(pkc
Ja
Thus if «(/), v(t) have continuous dth order derivatives and
(8.14) L[u]= lPk(t)uM(t), L*M=2(-1
lc=0 *-<>
Linear Differential Equations 67
then
(8.15) !\l[u]v - uL*[v]} dt = \t xViyV*-'-11^
Ja Lt=0 i=0
This is called Green's formula. The differentiated form
(8.16) L[u]v - uL*[v] = ( 2 5\- lyV*-'-1 W)
U=o i=o
is called the Lagrange identity.
A corollary of (8.15) is the fact that if w, v are solutions of (8.12) and
(8.13), respectively, then
(8.17)
=const-
,.=0 1=0
(ix) (Frobenius factorization) Suppose that (8.1) has d solutions
«!(/),..., ud(t) such that
(8.18) Wk(t)=W(t;u1,...,uk)*0 for k = 1, .. ., d.
Then (8.2) can be written as
(8.19)
where at =
sequence of Corollary 4.1.
Exercise 8.3 (Polya). Let po(t),. .. ,pd-i(t) in (8.1) be, continuous for
a ^ / ^ b. If v(t) has d continuous derivatives on [a, b], put
and Wo = W_x = Wd+1 = 1. This is a con-
conL[v](t) s v
+
+Po(t)v(t),
so that (8.2) takes the form L[u] = h(t). If a function v(t) has k — 1
(^ 0) continuous derivatives, a point / = /„ will be called a zero of v(t) of a
multiplicity at least k if v(t0) = v'(t0) = ¦ ¦ ¦ = u(*-1)(/0) = 0. Equation
(8.1) will be said to have property (W) on (a, b) if (8.1) has d solutions
iii{t), .. ., ud(t) satisfying (8.18) for a < / < b; actually this condition for
A = d is trivial by (iii). (a) Zeros and property (IV). Show that if no
solution u(t) ^0 of (8.1) has d zeros counting multiplicities on [a, b)
then (8.1) has property (W) on (a, b). In the remainder of this exercise,
assume that (8.1) has property (W) on (a, b). (b) Generalization of Rollers
theorem. Let v(t) have d continuous derivatives on (a, b) and at least
(I + 1 zeros counting multiplicities on (a, b). Then there exists at least
one point / = 6 of (a, b) where L[v]F) = 0. (c) Partial converse for (a).
Show that no solution u(t) ^ 0 of (8.1) has d zeros counting multiplicities
on («, b). (d) Interpolation. Let k ^ 1; m1 + • • • + mk = d, where m}
(^ 1) is an integer; tx < • • • < tk points of (a, b) and uf\ i = \, ... ,mj
68 Ordinary Differential Equations
andy = 1, . . . , k, arbitrary numbers. Then (8.1) has a unique solution
u = u(t) satisfying u(i)(t}) = uf for / = 1, . . . , m,- and j = 1,. . . , k.
(e) Equation L[u] = 1. Same as (d) with (8.1) replaced by L[u] = 1.
(/) Mean value theorem. Let k; w1;. . ., mk and tx, . . . , tk be as in (d).
Let v(t) have dcontinuous derivatives on (a, b); u(t) the unique solution of
(8.1) satisfying u(i\t,) = v{i\t^ for / = 1,. . ., m, and j = 1,. . . , k;
u = wo(/) the unique solution of L[u] = 1 satisfying u^\tj) = 0 for
/ = 1, . . . , rrij and j = 1,. . . , k; and a < / < b. If [y, d] c (a, b)
contains tlt tk, and t0, then there exists at least one point / = 6 of (y, 6)
such that v(t0) = u(t0) + uo(to)L[v]F). (The assertions (b)— (/) reduce
to standard theorems if (8.1) reduces to the trivial equation u{d) = 0 with
the solutions u = 1, /,..., t"-1.)
Exercise 8.4. Let/?„(/),. . . ,pd-iU) in (8.1) be continuous for a < t <
b. Show that no solution u(t) ^ 0 of (8.1) has d zeros counting multi-
multiplicities on (a, b) if and only if no solution «(/) ^ 0 has d distinct zeros on
(a, b). See Hartman [15].
9. Remarks on Changes of Variables
This section contains remarks which will be referred to in later chapters.
(i) If R is a constant d X d matrix, the usual Jordan normal form
J = Q~*RQ for R under similarity transformations is described by
E.14)—E.16). It will often be convenient to note that the l's on the
subdiagonal of Kh in E.16) can be replaced by an arbitrary e^O. This
is a consequence of the following formula in which J(j) is given in E.15):
if ge = diag A, c-\ ..., e1-").
(ii) Consider a real nonlinear system of differential equations of the
form
(9.1)
y = Ry + f(t, y),
where R is a constant matrix. Under a linear change of variables with
constant coefficients,
(9.2)
(9.1) becomes
(9.3)
y = Qz, det 2 5* 0,
z'=Jz+ Q~lf(t, Qz), J = Q-'RQ-
Although R is a matrix with real entries, its eigenvalues need not be real.
Correspondingly, there need not exist a real Q such that / is in a Jordan
normal form. For a Q with complex entries,/(/, Qz) may not be defined.
Linear Differential Equations 69
The point to be made, however, is that for many purposes the formal
change of variables (9.2) using a Q with complex entries is permissible if
(9.2) and (9.3) are suitably interpreted. Formal operations with (9.3)
are then legitimate.
If Q is chosen so that / = Q~XRQ is in a Jordan normal form, then the
columns of Q are eigenvectors of R or a power of R. Thus, if R has a real
eigenvalues (counting multiplicities), 0 ^ a ^ d, then the other eigenvalues
of R occur in pairs of complex conjugate numbers. Let d — a = 2/3.
Correspondingly, it can be supposed that the first ft columns of Q are,
respectively, complex conjugates of the next ft columns and that the last a
columns are real. Thus, if Qo is the matrix
where Ih is the unit h X h matrix, then QQ0 is a matrix with real entries.
The change of variables
+
t, QQow)
(9.4)
transforms (9.1) into
(9.5) W = QoV
which is equivalent to
(9.6) (Qow)' = J(Qow) + Qf(t, QQow).
The differential equations (9.5) are real equations; the differential
equations in (9.6) result by taking linear combinations of one or two
equations in (9.5) with constant coefficients 1 or ±/.
Below, equation (9.3) is to be interpreted as (9.6). This is equivalent to
saying that, in (9.3), 3*+^ = zk for k = 1, . . . , 0 and zA+2/s is real for k = 1,
. . . , a. Thus we can consider the variables in (9.3) to be w = (w1,. . ., W),
where wk = |(z* + zk) and w*+" = -\i{zk - zk) for k = 1, . . ., ft and
wk = zk for ip^k^d.
Exercise 9.1. Let J? be a constant d x d matrix with eigenvalues
/,,..., Xd such that A1;.. ., lk are simple eigenvalues for some k,
I ^ k ^ d. Let G(t) be a continuously differentiable d X d matrix for
f °°
/ > 0 such that G(t) -»- 0 as t -»- oo and || G'(/)|| dt < oo. (a) For large t,
show that R + G(t) has k simple eigenvalues X^t) such that Ay(/) ->¦ Xs as
/ > oo; Xj(t) is continuously differentiable and \X/(t)\dt < oo for
70 Ordinary Differential Equations
/= 1,. . . , k. (b) Let g0 be a constant, nonsingular matrix such that
Qo^RQo = diag [A1; ..., lk, ?0], where Eo is a (d - k) X (d - k)
matrix (e.g., suppose that Q^RQ0 is a Jordan normal form for R). Let
er = (e/, . . ., erd), where er' is 1 or 0 according asj=r or j ?± r. Show
that, for large t, R + G(t) has an eigenvector ys(t), [R + G@]«//0 =
A/02//0, such that yfc)^- Qoej as t —¦ oo, and y/f) has a continuous deriv-
ative satisfying ||«//@ll dt < oo for; = 1,..., fc. (c) Show that, for large
t, there exists a continuously differentiable, nonsingular matrix Q(t) such
that 2(oo) = lim 2@ as t-*- oo exists and is nonsingular, ||B'@ll dt <
oo, and QrKi)[R + G(i)]Q(t) has the form diag [^(t), ..., 4@, ?@1,
where E(t) is a (d — k) x (d — k) matrix.
APPENDIX: ANALYTIC LINEAR EQUATIONS
10. Fundamental Matrices
This appendix deals with a linear system of differential equations
A0.1) y' = A(t)y
in which Ms a complex variable and A{t) is a matrix of single-valued,
analytic functions on some open set E in the ;-plane. "Analytic" is used
here in the sense of "regular analytic." In a small neighborhood of a
point ?0 ? E, A0.1) has a fundamental matrix 7@ which is an analytic
function of t (i.e., which has elements that are analytic functions of t).
This follows from a modification of the proof by successive approximations
of the existence theorem, Lemma 1.1; cf. Exercise II 1.1. Hence, if E is
simply connected, the monodromy theorem implies that 7@ exists and is
a single-valued, analytic function of t on E.
Most of this appendix deals with the case when E is not simply connected
but is a punctured disc 0 < \t\ < a.
Lemma 10.1. Let A(t) be a matrix of single-valued, analytic functions
on the disc 0 < \t\ < a and suppose that A(t) [i.e., at least one of the
elements of A(t)] is not analytic at t = 0. Then A0.1) cannot have a funda-
fundamental matrix Y(t) which is single-valued, analytic on 0 < \t\ < a and
continuous at t = 0 with det y@) ^ 0.
Proof. If this is false, then 7@, 7~x@ are analytic on \t\ < a and so is
A(t) = 7'@ Y~Kt) by Riemann's theorem on removable singularities.
Theorem 10.1. Let A(t) be a matrix of single-valued, analytic functions
on 0 < \t\ < a. Then any fundamental matrix Y(t) o/A0.1) (which need
Linear Differential Equations 71
not be single-valued) has a representation of the form
A0.2) 7@ = Z(t)tR,
where Z(t) is a matrix of single-valued, analytic functions on 0 < \t\ < a,
R is a constant matrix, and
A0.3)
tR =
n\
If T is a constant nonsingular matrix, then 7@7" is a fundamental
matrix and
7@ T = Z(t)T(T-HRT) = [Z(t)T]tT'lRT.
T can be chosen so that T~*RT is in a Jordan normal form. The form of
the matrix tT'lRT = exp (r-^riog t) can be seen by replacing t by log t
in E.18).
Proof. Let 7@ be a fundamental solution of A0.1) determined locally
near a point t = t0, 0 < \tQ\ < a, and continued analytically, possibly
multiply valued. If the point t makes a circuit around t = 0 in 0 < \t\ < a,
then the matrix returns with values, say 70@, for t near t0. Since A(t) is
single-valued, T0(t) is also a fundamental matrix for A.1), thus, there
exists a nonsingular constant matrix C such that
A0.4)
70@= 7@C.
By analyticity, A0.4) holds for analytic continuations of 7@, 70@-
For a fixed r, 0 < r < a, consider the matrix function 7@, r) = Y(reie)
of 0 for - oo < 0 < oo. Then A0.4) means that 7@ + 2n, r) = 7@, r)C
or that 7@ + 2n, r) = 7@, r)e2tHR if 2-niR is a logarithm of C; cf. § 6.
It is readily seen that Zo@, r) = 7@, r)e~iRe is of period 2n in 0; see the
argument following F.6).
Note that Zo@, r)r~R = 7@, r)e~iRer-R = Y(t)rR is an analytic func-
function of t, say Z@, and is single-valued since Zo@, r) is of period 2n in 0.
This proves Theorem 10.1.
If A0.2) is a fundamental matrix for A0.1), properties of Z(t) and R
will be investigated. To this end, a differential equation satisfied by Z will
be calculated. Since R commutes with tR and (;R)' = RtR/t, it follows
from A0.1) and A0.2) that
ZRtR
Z't
'tR
= AZtB.
Hence
A0.5)
Z'=-— + A(t)Z.
t
72 Ordinary Differential Equations
The equation A0.5) is not the type of matrix differential equation con-
considered above since Z occurs as a factor on the left of ZR and on the right
of A(t)Z. In order to deal with A0.5), it is convenient to arrange the d2
elements zjk of the matrix Z = (zik) in some arbitrarily fixed order and
to consider A0.5) as a linear, homogeneous system for a cP-dimensional
vector Z; say
A0.6) Z' = A(t)Z,
where A(t) is ad2 X d2 matrix. An element of A(t) is a linear combination
of elements of A(t) and Rjt with constant coefficients. Hence A(t) is
single-valued and analytic for 0 < |/| < a. In particular, if the elements of
A(t) are analytic or have a simple pole at t = 0, then the same is true of
At)-
In order to avoid an interruption to the arguments below, a simple
algebraic lemma will be stated and proved here. Let B be a constant
d X d matrix, X a variable d X d matrix and Y the commutator
A0.7)
Y = BX - XB.
Consider the d2 elements of X and Y arranged in a fixed (arbitrary)
manner and A0.7) as a linear transformation from the cP-dimensional
A'-space into itself. Thus A0.7) can be written as
A0.8) Y = BX,
where B is a d2 X d2 matrix and X, Y are ^dimensional vectors.
Lemma 10.2. Let the d eigenvalues of B be ^ Xd counting multi-
multiplicities. Then the d2 eigenvalues of B are X, — Xk for j,k = 1, . . ., d.
Proof. Let T be a nonsingular d x d matrix and let C = T~1BT, so
that B, C have the same eigenvalues. Let C be related to C as 3 is to B.
The matrices C and B have the same eigenvalues. In order to see this,
note that A0.8) is equivalent to T^YT = CfJ-^XT), for A0.7) can be
written as
{T-lBT){T-lXT) - {T-1XT){T-1BT) = T~1YT.
Thus, e.g., EX = XX implies that (CT^XT) = X^T^XT).
Suppose first that the eigenvalues of B are distinct and choose T so that
C = diag [Xlt . . ., Xd]. Then if X = (xjk) and Y = (yik), it is seen that
Y = CX — XC is equivalent to yjk = {X} — Xk)xik for j, k = 1 d.
Thus C is a diagonal matrix with the d2 diagonal elements X, — Xk for
j,k = 1, . . ., n and the lemma is proved in this case.
If the eigenvalues of B are not distinct, let Blt B2, ... be a sequence of
matrices, each having distinct eigenvalues such that Bn -*¦ B, n -> oo. (The
existence of Blt B2,. .. is clear; it is sufficient to suppose that ? is in a
Jordan normal form and to change the diagonal elements by a small
Linear Differential Equations 73
amount to obtain Bn.) Then the eigenvalues of Bn can be ordered
h\m ¦ ¦ ¦ > ^anso tnat % in ~* ^t>n ~* c0> f°r7 = I, ¦ ¦ • , d. Correspondingly,
the eigenvalues of Bn tend to those of S. Hence the general case of the
lemma follows from the special case treated above.
11. Simple Singularities
In A0.2), Z(t) has a Laurent expansion about / = 0. The point / = 0
is called a regular singular point for A0.1) if A0.1) has a fundamental
matrix A0.2) in which the elements of Z(t) do not have an essential
singularity at / = 0 (i.e., are analytic or have a pole at / = 0). In this case,
we have
Corollary 11.1. Let A(t) be analytic and single-valued for 0 < |/| < a
and let t = 0 be a regular singular point for A.1). Then A.1) has a funda-
fundamental matrix Y (t) of the form
Y(t) = Z0(t)tc,
where C is a constant matrix and Z0(t) is analytic for \t\ < a.
In fact, if Z(T) in A0.2) has at most a pole at t = 0, then A0.2) can be
written as Y(t) = Z(t)tntR~nI = Z0(t)tc, where Zo(/) = Z(t)tn is analytic
for some choice of the integer n ^ 0 and C = R — nl.
If the singularity of A(t) [i.e., of each element of A(t)] at / = 0 is
at most a pole of order one, then / = 0 will be called a simple singularity
of A0.1). In this case A0.1) can be written as
(ll.l)
ty' =
where B, Alt A2, . . . are constant matrices and
k=l
A1-2)
is convergent for |/| < a.
If a new independent variable defined by s = log / is introduced, A1.1)
becomes
A1.3)
ds
y,
Re s < a.
(For a treatment of this system for real s, see Chapter X.) If A1 = A2 =
• • • = 0, equation A1.3) is dyjds = By and has the solution y = eB% =
/"; cf. § 5 and Corollary 11.1.
Theorem 11.1 (Sauvage). Let t = 0 be a simple singularity for A0.1),
so that A0.1) is of the form A1.1) where A1.2) is convergent for \t\ < a.
Then i = 0 is a regular singular point for A1.1).
This is an immediate consequence of the following lemma.
74 Ordinary Differential Equations
Lemma 11.1. Let t = 0 be a simple singularity for A0.1) and let y(t)
be a single-valued, analytic solution of A0.1) for 0 < |/| < a. Then y(t) is
analytic or has a pole at t = 0.
For if this lemma is applied to the system A0.6), it follows that Z(t) in
A0.2) is analytic or has a pole at t = 0.
Proof of Lemma 11.1. Let 0 be fixed, 0 < 0 < 2n, and t = reie, so
that dt = eid dr and y(reie) is a solution of
^ = eteA(reie)y
dr
for 0 < r < a. The assumption on A (/) implies that there exists a constant
c such that ||^(/-ei9)?/ll = c \\y\\/r for 0 < r _ \a, where \y\ is the
Euclidean norm of?/. It follows that any solution y ^ 0 satisfies
dr = r
for 0 < r < |a. Hence ||?/(/-ei9)|| < C//-c for 0 < r ^ |a and a suitable
constant C Thus if n is a positive integer, n _^ c, then tny(t) is bounded
for small |/| and hence is analytic at / = 0. Thus y(t) has at most a pole of
order n. This proves the lemma.
The converse of Theorem 11.1 is false. For it is readily verified that the
binary system
yV =
has the fundamental matrix
Y(t) =
yV =
2t -r2,
which is of the form A0.2) with Z(t) = Y(t) and R = 0. Thus / = 0 is a
regular singular point but is not a simple singularity. However, Theorem
11.1 has a partial converse.
Theorem 11.2. Let Q(t) be a d x d matrix of functions which are single-
valued and analytic for 0 < |/| < a and such that / = 0 is a regular
singular point for
A1.4) w' = Q(t)w.
Then there exists a matrix P(t), which is a polynomial in t and satisfies
det P(t) = 1, and a diagonal matrix D = diag [a(l),. .., a(</)], where
a(j) = 0 '•* an integer, such that the change of variables
A1.5) w=T(t)y, T(t) = P(t)tD,
transforms A1.4) into the form A1.1)—A1.2) for which t = 0 is a simple
singularity.
Linear Differential Equations 75
Proof. Since / = 0 is a regular singularity of A1.4), there exists a
fundamental matrix W{t) of the form
A1.6)
W(t) = X(t)tR,
where R is a constant matrix and X(t) is analytic for |/| < a.
Suppose first that det ^@) ^ 0, then X~\t) is analytic for |/| < a and
g(/) = W'{i)W-\t) is given by
A1.7) Q (/) = (A" + r1XR)X-\
Hence Q (/) has at most a simple pole at / = 0, so that / = 0 is a simple
singularity for A1.4) and the theorem follows with P(t) = I, D = 0.
Consider the case that det A'(O) = 0. Suppose, for a moment, that there
exist matrices P(t), D of the type specified, such that
A1.8) X{t) = P{t)tDZ{t\
where Z(t) is analytic for |/| < a and det Z@) ^ 0. Thus
W(t) = T(t)Z(t)tR if T(t) = P{t)tD,
and so A1.5) transforms A1.4) into a system y' = Q0(t)y for which
Y(t) = Z(t)tR is a fundamental matrix. By the analogue of A1.7),
Q0(t) = (Z' + t-lZR)Z~\
Consequently, / = 0 is a simple singularity for the new system y' =
QoiOy, so that Theorem 11.2 will be proved if the following lemma is
verified.
Lemma 11.2. Let X(t) be a matrix of functions analytic for \t\ < a such
that det X(t) ^ 0. Then X(t) has a representation of the form A1.8), where
P(t) is a matrix which is a polynomial in t and detf(/) = 1, D =
diag [a(l),. . ., a(d)] with a(y) _ 0 an integer, and the matrix Z(t) is
analytic for \t\ < a with detZ@) ^ 0.
Remark 1. The relations A1.8), det P(t) = 1, and detZ@) ^ 0 show
that
A1.9) a(l) + ¦ ¦ ¦ + x(d) = a,
where a > 0 is the order of the zero of det X(t) at / = 0.
Proof. The equation A1.8) will be considered in the form
A1.10)
p-\t)X(t) = tDZ{t).
From the usual construction of the inverse of a matrix in terms of minors,
divided by the determinant, it is clear that both P(t) and P^) have the
76 Ordinary Differential Equations
properties just specified. Instead of constructing P(t), it will be simpler to
obtain P~\t). Also, the proof will give a matrix p-\t) such that det P-\t)
is a constant E^ 0), not necessarily 1. The normalization deti>@ = 1 is
then obtained in a trivial way.
The matrix P~\t) will be constructed as a product of a finite number of
elementary matrices N of one of the following three types: (i) multipli-
multiplication of a matrix on the left by N interchanges the yth and kth rows,
e.g., multiplication by the matrix
/0 1 0 ...
1 0 0 ...
N =
0 0 1
\o 0 0
1.
on the left interchanges the first and second rows; (ii) multiplication by
N on the left multiplies the y'th row by a number 1^0, e.g., N =
diag [1,. . . , 1,1, 1,. . ., 1]; and (iii) multiplication by N on the left
replaces theyth row by the sum of theyth row and p(t) times the kth row,
wherep{t) is a polynomial and k <y"; e.g., for; = 2 and k = 1,
/I 0 0 ... 0\
i 1 0 ... 0
N =
0 0 1
0
\o 0 0 ... 1/
Each of these elementary matrices N satisfies det N = const. 5* 0.
Let X(t) = (xjk(t)) and let a ^ 0 be the order of the zero of det X(t)
aU = 0. With theyth row of X(t), associate an integer a(J) ^ 0 such that
Xjk{t) = ta(i)yjk(t), where yjk(t) is analytic at / = 0 and at least one of the
functions yn(t),..., yjd(t) does not vanish at t = 0. In particular,
X(t) = tE 7@ where E = diag [a(l), . . . , a(d)];
so that det X(t) = tn det 7@, where n = o(l) + • • • + a(d). Hence
A1.11) a(l) + • • • + a(d) < a.
If equality holds in A1.11), then det 7@) 5* 0 and the lemma is trivial
with P(t) = I and D = E.
If inequality holds in A1.11), it will be shown that X(t) can be multi-
multiplied on the left by a finite number of the matrices of the type Nto obtain
a matrix X0(t), such that if a(l), . . ., a(rf) belong to X0(t) as a(l),. . . , a(d)
Linear Differential Equations 77
belong to X(t), then a(l) + • • • + a(d) = a. Thus ^0@ = P-\t)X(t)
has the form tDZ(t), where detZ(O) ^ 0.
After multiplying X(t) on the left by matrices of type (i), it can be
supposed that a(l) ^ • • • ^ a(d). Let ejk = yjk@), so that xjk{t) =
ta(i)(ejk +""")• Then not all of the numbers «u, . . . , eld are zero.
Suppose that elm, 1 ^ m ^ d, is the first of these elements which is not
zero. Then after multiplying X(t) on the left by a matrix of type (ii), it
can be supposed that elm = 1. Also, by replacing theyth row of X(t) by
the sum of theyth row andp{t) = — *}mtaU)~aa) times the first row, it can
be supposed that eim = 0 fory" = 2, . . . , d. Thus the matrix of the new
elements eJk = yik@) is of the form
...\
0
if, e.g., m = 2. This procedure does not decrease the integers a(l), . . .,
aid).
If not all elements e2i» •¦•>«&( of the second row are zero, it can be
supposed that if e2n is the first different from 0, then e2n = 1 and ejn = 0
for y" = 3, . . . , d.
This procedure can be applied to the second row, then the third row,... ,
unless all of the elements en,.. . , ejd in theyth row are zero. In this case,
a(j) can be replaced by a larger number, again called a(j). If this occurs,
the rows are again permuted to obtain the order a(\) ^ • • • ^ a(d) and
the entire procedure repeated.
After a finite number of repetitions, the procedure of introducing a one
in each of the d rows with zeros to the left and below it succeeds because,
in view of the limitation a(l) + ¦ ¦ ¦ + a(d) ^ a, it is only possible to
increase an index a(j) a finite number of times.
The first column en, e2i, • • •, e<ji contains at least one element different
from zero. Otherwise, the ones occur in the last d — 1 columns, with
only zeros below the ones. This implies that there is a row en,. . . , e}d
with all zeros, contradicting the construction. If enl is the first element of
the first column such that enl ^ 0, then, by the construction, enl = 1 and
1 mi = 0 for n ^ m. Move the nth row to the first position without
disturbing the order of the other rows.
In the new matrix, the d — 1 elements e22, e32,. . . , td2 of the second
column are not all zero, in fact, exactly one (say, the wth) is one and the
others are zero. This follows by the argument of the last paragraph.
78 Ordinary Differential Equations
Move the mih row to the second position. Continuing this procedure
leads to a matrix X0(t) for which the corresponding (eft) has ones on the
principal diagonal and zeros below it, thus det (e,,.) = 1.
Thus the construction gives an X0(t) = P~\t)X{t) of the form tDZ{t),
with Z@) = (ert) having detZ(O) = 1 and Z) = diag [a(l),..., a(cT)]
where a(l), . . ., a«) belong to X0(t) as a(l), . .., a(d) belong to X(t).
This proves the lemma.
Exercise 11.1. Prove that if Q (/) is a d X d matrix of single-valued,
analytic functions on 0 < |/| < a, then a necessary condition for / = 0
to be a regular singularity for A1.4) is that the elements of Q(t) have at
most a pole (not an essential singularity) at / = 0.
Exercise 11.2. Let A0(s) be a matrix of functions analytic for a < \s\ <
oo. The point s = oo is called a simple singularity [or a regular singular
point] for the system
A1.12)
ds
if / = 0, where / = I Is, is a simple singularity [or a regular singular point]
for y = A (t)y, where A(t) = —/~MOA//). (a) Necessary and sufficient
that s = oo be a simple singularity for A1.12) is that A0(s) ->¦ 0 as \s\ ->- oo.
(b) Let A (/) be analytic for all / ^ h, . .., tn, oo. Necessary and sufficient
conditions that / = tu ..., tn, oo be simple singularities is that A (t) be
of the form .4@ = (/ - /i)^ H + (/ - /J^™. where Ru
R2, . . ., Rn are constant matrices.
Theorem 11.3. Let A1.2) be convergent for \t\ < a and let
00
(n.13) y(t) = J,yntn
ra=O
be a "formal" power series which satisfies A1-1) in the sense that the
formulae
A1.14) Byo = O
(H.15J
nyn =
n = 1, 2, . . ., hold. Then A1.13) is convergent for \t\ < a.
Proof. Since A1.2) is convergent for |/| < a, there exist constants
c > 0, p > 0 such that
A1.16) \\Ak\\<:CPk, k=\,2,...
in the sense that \Aky\ ^ cpk \y\ for all vectors y. (For example, p can be
chosen to be any number such that p > I/a.) Choose r > 0 so large that
A1.17)
Linear Differential Equations 79
Let m be an integer such that \By\ ^ m \y\ for all vectors y. Then
B — nl is nonsingular for n > m, in fact, \By — ny\ ^ (n — m) \y\ ^ 0
for y 9± 0. Thus if n > /w, the equation By — ny = z has a solution ?/,
for any given z, and
A1.18) \y\ ^
(n - m)
Let y > 0 be chosen so large that
in particular, \y\ ^ |z| if « > m.
holds for « = 0, . . ., m. It will be verified by induction that A1.19)
holds for all n. Thus assume A1.19J for n = 0,...,/— 1 and; — 1 >
m. Then
- )Vi =
where z, = -
By A1.16) and the induction hypotheses,
^
k=i\r
so that |z,| ^ yf by A1.17). Hence A1.18) implies that |y,| ^ |z^ ^ yr».
This completes the induction.
Consequently A1.13) is convergent for |/| < \\r and is a solution of
A1.1) for small |/| > 0. But then it is convergent for |/| < a and is a
solution of A1.1) for 0 < |/| < a as no solution of A1.1) has a singularity
on 0 < |/| < a.
Exercise 11.3. Show that Theorem 11.1 is false if A1.1) is replaced by
A0.1) and it is not supposed that / = 0 is a simple singularity.
Corollary 11.2. Let A1.2) be convergent for \t\ < a and suppose that if
Xlt . . ., Xd are the eigenvalues of B, then X} — lk = 0 or Xt — lk is not
an integer for j, k = 1, . . ., d. Then A1.1) has a fundamental matrix of
the form
A1.20) Y(t) = Z(t)tB, where Z(t) = I + Zxt + Z2/2 + • • •
is convergent for \t\ < a.
Proof. The matrix A1.20) is a solution matrix for A1.1) if and only if
7. satisfies the differential equation
A1.21)
tZ'= (BZ-ZB)+
cf. A0.5). A formal series
A1.22)
Jfc-0
80 Ordinary Differential Equations
satisfies A1.21) if and only if the formulae
A1.23) BZo-ZoB
A1.24) nZn = BZn - ZnB +
hold for n = 1,2,....
Consider A1.21) and A1.23)—A1.24) as systems of equations for d2-
dimensional vectors, rather than for d X d matrices. For any constant
matrix D,
lxZn = BZn -ZnB- D
has a unique solution if /j, ^ X} — Xk for /, k = 1, . . . , d. For this set of
equations can be viewed as BZn — fxZn = D, where B is a d2 x d2
matrix with the d2 eigenvalues Xt — Xk for/, k = 1, . . . , dby Lemma 10.2.
Thus Zo = I satisfies A1.23) and Z1( Z2,. .. can be obtained recursively
from A1.24). The resulting series A1.22) is convergent for |/| < a by
Theorem 11.3. This proves Corollary 11.2.
When the eigenvalues Xx,. . . , Xd of B do not satisfy the conditions of
Corollary 11.2, it is possible to change the dependent variable y,
A1.25)
y=U(t)rj, where det U(t) ^ 0 for / ^ 0,
so that A1.1) becomes a system
A1.26) tr]'=
f DkAr,
l I
k=l
to which Corollary 11.2 is applicable. In this case, A1.1) has a funda-
fundamental matrix of the form
A1.27) Y(t) = U(t)Z(t)tc,
where U(t) is a polynomial in / and Z(/) is of the same form as in A1.20).
The precise result to be proved is the following:
Lemma 11.3. Let A1.2) be convergent for \t\ < a. Then there exists a
matrix U(t) with the properties that U(t) is a polynomial in t; that
det U(t) 9± 0 for t 9± 0; and that A1.25) transforms A1.1) into a system
A1.26) in which the factor ofr\ is a convergent power series for \t\ < a and
if fa,. . ., [td are the eigenvalues of C, then ^ — fik = 0 or Hj — nk is
not an integer for j, k = 1, . . . , d.
Remark 2. It will be clear from the proof that if Xx,.. ., Xd are the
eigenvalues of B in A1.1), then fa,..., fid can be ordered so that
X. — ns = ns ^ 0 is an integer.
The proof below will not involve a knowledge of the solutions of A1.1)
[and gives an algorism for the determination of a C in A1.26), i.e., of a C
in Corollary 11.1]. Another proof of Lemma 11.3, but not of Remark 2
Linear Differential Equations 81
following it, can be obtained if one knows a fundamental matrix for A1.1)
as in Corollary 11.1:
Exercise 11.4. Deduce Lemma 11.3 from Lemma 11.2.
Proof of Lemma 11.3. Suppose first that ? is in a Jordan normal form
B = diag[J(l), . . . , J(g)], where J(j) is a Jordan block of the form E.15)
with X = X{j), h = h{j) for/ = 1, . . . , g. Let B2 = diag [7B),. . ., J(g)]
so that B2 is an e x e matrix, where e = d — h(\).
Make the change of variables
A1.28) y = V(t)r,, where V(t) = diag [//wl), /J,
and /W), /„ are the unit matrices of the specified dimensions. Then A1.1)
becomes
tv' = \v-1BV- tV~xV +1tV~1AkVtkU.
Rearranging the coefficient matrix according to the powers of /, this is of
the form
A1-29)
tV'=
k=l
where C1, Akx are constant matrices and C1 is given by
A1.30) C1 =
7A) - /,
Ml)
0
if Ax =
and Au is an h(l) x h(l) matrix and A22 is an e X e matrix. Thus if the
eigenvalues of B are X{\),..., X{\) and XhW+1,. . . ,Xd, then those of C1
are X(l) - 1, . . . , X(\) - 1 and XhW+1,. . . , Xd.
If B is not in a Jordan normal form, the same result is achieved by a
variation of constants y = T\Vx(t)r\, where T~XBTX is in a Jordan normal
form. It is clear that the lemma follows with U(t) of the form U(t) =
^1^1@^2^2@ • • • T}Vj(t), where Tk is a constant nonsingular matrix and
Vk(t) is of the same type as V(t) for k = 1, . . .,/.
Theorem 11.4. Let A1.2) be convergent for \t\ < a. Let X be a fixed
complex constant and nx (^ 0) the number of linearly independent vectors y
satisfying By = Xy. Then the number Nx (^ 0) of linearly independent
solutions of A1.1) of the form
00
A131)
satisfies
A1.32)
max
^ Nx<: nx
82 Ordinary Differential Equations
In particular, the number No of linearly independent solutions y(t) o/(ll.l)
which are analytic at t = 0 satisfies
A1.33)
n0 < max (n0, nu ...) ^ JV0 ^ n0 + nx + ¦
It is not assumed that y0 ^ 0 in A1.31). Fora generalization of Theorem
11.4, see Theorem 13.1.
The proof of this theorem will depend on the proof of Lemma 11.3 and
on the following lemma.
Lemma 11.4. Let A1.2) be convergent for \t\ < a. Then the number
^x(= 0) of linearly independent solutions of A1.1) of the form A1.31)
satisfies
A1.34) Nx <: nx + nx+1 + ¦¦¦ .
If X is an eigenvalue of B and X + 1, X + 2,. . . are not eigenvalues, then
A1.35) Nx = nx
andy0 ^ 0 in any solution A1.31) o/(ll.l).
Proof of Lemma 11.4. It can be supposed that X = 0, otherwise the
change of variables y = txr\ replaces A1.1) by
tv'= (B-
\
k=l
and X + k, nx+k by k, nk.
A function A1.13) is a solution of A1.1) if and only if A1.14) and
A1.15J, n = l,2,..., hold. The equation A1.14) has n0 linearly
independent solutions and if A1.14), and A1.15^, . . ., A1.15&_1) hold,
then the solutions of A1.15,.) are of the form z0 + zk, where yk = z0 is a
particular solution of A1.15&) and zk varies over the w^-dimensional linear
manifold of solutions of Bz — kz = 0. This proves A1.34).
In the last part of the lemma, it is supposed that X = 0 is an eigenvalue
of B, but X = 1, 2,... are not. Then A1.14) has n0 linearly independent
solutions 2/0 and if A1.14) and A1.15!), . . ., A1.15^) hold, then A1.15*)
has a unique solution. Thus for a given y0, the vectors yu y2, . . . are
uniquely determined. The corresponding series A1.13) is convergent for
|/| < a and is a solution of A1.1) by Theorem 11.3. This proves A1.35)
and completes the proof of Lemma 11.4.
Proof of Theorem 11.4. In view of A1.34) in Lemma 11.4, only the
first inequality in A1.32) remains to be proved. Since Nx ^ Nx+1 ^ • • •,
this inequality will be proved if it is shown that
A1.36)
Nx ^ nx.
Linear Differential Equations 83
As in the last proof, there is no loss of generality in supposing that
X = 0 in A1.35). It can be supposed that n0 > 0 and, in view of A1.35)
in Lemma 11.4, it can also be supposed that B has an eigenvalue which is
a positive integer.
The proof of Lemma 11.3 makes it clear that there exists a change of
variables A1.25) such that U(t) is a polynomial in / and that A1.1)
becomes A1.26) in which the eigenvalues of C are the same as those of B
except that any integral eigenvalue X = n > 0 of B has been replaced by
X = 0. Let nx(C) have the same meaning for C as nx does for B. It will be
shown that
A1.37)
no(C) ^ n0.
To this end, it suffices to re-examine the proof of Lemma 11.3 and show
that the analogue of A1.37) holds at each step. Thus, if the first step leads
from A1.1) to A1.29), it suffices to show that
A1.38)
^ n0
Let B be in a Jordan normal form, B = diag [/(I), . . ., J(g)] and let
X(l) > 0 be an integer. Make the change of variables A1.28) transforming
A1.1) into A1.29), where A1.30) holds. Since X(l) ^ 0 and B2 is in a
Jordan normal form, it is clear that n0 rows of B2 contain only zero ele-
elements. Hence the rank of C1 in A1.30) is at most d — n0, so that A1.38)
holds. Consequently A1.37) follows.
Since X = 1,2,... are not eigenvalues of C, the last part of Lemma 11.4
implies that A1.26) has no(C) linearly independent solutions tj(t) which are
regular at / = 0. Since [/(/) is a polynomial in /, A1.25) shows that A1.1)
has at least no(C) linearly independent solutions y(t) regular at / = 0.
In view of A1.37), the inequality A1.36) follows for X = 0. This completes
the proof of Theorem 11.4.
Exercise 11.5. (a) Let A(t) = (ajk(tj) be a d X d matrix of functions
analytic for |/| < a. Let a(;") = 0 or 1 and a < d if a = a(l) + • • • +
a(cT)- Then the system
has at least d — ex. linearly independent solutions analytic at / = 0.
(/>) Let ao(t), a^t) be analytic for |/| < a, then the differential equation
tu" + a&W + ao(t)u = 0
has at least one solution u(t) ^ 0 analytic at / = 0. For a generalization
of this exercise, see § 13.
84 Ordinary Differential Equations
12. Higher Order Equations
Consider a differential equation of dth order for a function u,
A2.1) u(d) + /ViM"-11 + • • • + Pi(t)u' + Po(t> = 0,
in which the coefficient functions are single-valued and analytic on a
punctured disc 0 < |/| < a. Instead of writing A2.1) as a first order
system in the standard way, transform it into a system for the vector
y = (y\ % h
A2.2)
Then
A2.3)
., /), where
ty? =(j- l)y> + y'+
tyd =(d- \)yd -
for; = \,...,d.
for; = 1, ...,</- 1,
k=l
Thus / = 0 is a simple singularity for this system \Ud-kpk(t) is analytic at
t = Qfork = 0,...,d-l; i.e., if p^i) has at most a pole of the first
order, pd-2@ has at most a pole of the second order, ... ,/?„(/) has at
most a pole of the dth order. In this case, let
A2.4) ak(t) = td-kpk(t) = bk + | pkntn for k = 0, . . ., d - 1,,
where 6,. and/?^ are constants and the series in A2.4) is convergent for
\t\<a. Then A2.1) is of the form
A2.5) uw + /-^-iM""*1 + ' • • +
y + rdflo(OM = 0,
where ao(/), ..., ad^(t) are analytic at / = 0. Correspondingly, A2.3) is
of the form A1.1), where B is the constant matrix:
A2.6) B =
/ 0
0
0
0
1
1
0
0
0
1
2
0
0
0
1
3
0
0
0
0
1
\
0 0 0 0 ...
\—b0 —b1 —b2 —b3 . . . (d — lj ^d_n
The coefficient matrix on the right of A2.3) reduces to the constant
Linear Differential Equations 85
matrix B ifpkn = 0 for k = 0,. . ., d — 1 and «= 1, 2, .... This is the
case when A2.1) is the differential equation
A2.7)
/-^h"*-11 + • • • + /-'"-d V + t-dbou = 0,
in which b0, .. ., bd_x are constants. This is called Euler's differential
equation. The solutions of A2.7) are easily determined from the fact that a
fundamental matrix for the corresponding system A2.3) is Y(t) = tB;
cf. the remark following A1.3). Thus the solutions of A2.7) are linear
combinations of functions of the form /A(log /)*.
The numbers X and permissible values of k are determined by the
Jordan normal form of B. The equation A2.7) obviously has a solution of
the form u = tk if and only if X is an eigenvalue of B. Substituting u = tx
into A2.6), it is seen that this is the case if and only if F{X) = 0, where
A2.8) F{X) =
}=0
- 1)... {X - j + 1) and bd = 1.
The equation F(X) = 0 is called the indicia! equation for A2.5).
Let X(l),.. ., X(g) be the distinct solutions of F(X) = 0 with the
respective multiplicities h(l),. .., h(g), where h{\) + • • • + h{g) = d.
Then a linearly independent set of solutions of A2.7) is /A(j)(log /)*, where
j=l,...,gandk = O,...,h(j)- 1.
Exercise 12.1. (a) Verify this last statement by the type of argument
in § 8(vi) following Exercise 8.1. (b) The remarks concerning A1.1) and
A1.3) show that the change of variables t = es reduces the system A2.3)
belonging to A2.7) to one with constant coefficients. Verify directly that
the substitution t = es reduces A2.7) to an equation with constant
coefficients and, hence, that (a) follows directly from § 8(vi).
Returning to the general equation A2.1) and its corresponding system
A2.3), the following theorem will be proved:
Theorem 12.1 (Fuchs). Let pk(t) be single-valued and analytic for
0 < |/| < a. Then t = 0 is a regular singular point for A2.3) if and only
ift = 0 is a simple singularity for A2.3) (i.e., if and only //A2.4) holds with
convergent series on the right).
It is clear that / = 0 is a regular singular point for the system A2.3)
if and only if the solutions of A2.1) are linear combinations of functions
of the form /A(log t)ka(t), where a(/) is analytic for |/| < a.
Proof. The "if" portion of the theorem is a consequence of Theorem
11.1. Thus it is sufficient to prove the "only if" portion.
It follows from Theorem 10.1 that A2.1) has at least one solution of the
form «,(?) = /A«i@' where ax(/) is analytic for 0 < |/| < a. If it is
assumed that / = 0 is a regular singular point for A2.3), then ax(/) has at
86 Ordinary Differential Equations
most a pole at t = 0. In fact, by changing A, it can be supposed that a^f)
is analytic at t = 0. In particular at@ 5* 0 for small |f | > 0.
The proof proceeds by induction on the order d. Consider first an
equation of order d = 1,
u + po(t)u = 0,
with a solution u^t) = t\{t), a^f) analytic at t = 0. It is clear that
po(t) = —UiiO/u^t) has at most a pole of order d = 1 at t = 0.
Assume J > 1 and that the theorem is correct for equations of order
d — 1. Let u = u-^t) be the type of solution described above and introduce
the new dependent variable v = ujux for small t > 0. Then A2.1) is
transformed into an equation of the form
A2.9) »<"> + qd.2(tyd-» + ¦¦¦+ qo(t)v' = 0,
which is an equation order d — 1 for v'. It is readily seen that
A2.10)
and since u = w^f) is a solution of A2.1),
A2.11)
0 = u
• • • +po(t),
where CJfc = j\jk\{j — k)\ are binomial coefficients.
The equation A2.9), as an equation of order d — \ for v', has the
solutions v = (m/«i)' for arbitrary solutions w of A2.1). Consequently,
t = 0 is a regular singular point for the system associated with A2.9).
Hence, by the induction hypotheses, td-x-kqk(t) is analytic at t = 0 for
k = 0, . . . , d — 2. Also, I//*'/"! has a pole of at most the order k at
t = 0. It follows from A2.10) and A2.11) that/^ has at most a pole of
order 1 at t = 0, /?d_2 has a pole of order at most 2, etc. This proves
the theorem.
Exercise 12.2. Let po(t),. . .,pd-i{i) be analytic for a < \t\ < 00.
The point t = 00 is called a simple singularity [or regular singular point]
for A2.1) if it is a simple singularity [or regular singular point] for A2.3);
cf. Exercise 11.2. (a) Necessary and sufficient that t = co be a simple
singularity for A2.1) is that td-k-1pk(t) -> 0 as \t\ -> co for k = 0, . . .,
d — 1. (Z>) Let /?0@, • • -,pd-i(t) be analytic for f 5* h,..., tn, co.
Necessary and sufficient that t = tlt..., tn, co be simple singularities
forA2.1) is 0 k O*@
Linear Differential Equations 87
for k = 1, . . ., d, where ak(t) is a polynomial of degree at most k(d — 1).
(Differential equations with this property are said to be of Fuchs' type.)
For a second order (d = 2) equation A2.5) having a regular singular
point at t = 0, it is comparatively simple to discuss the behavior of a set
of fundamental solutions. We attempt first to find a solution of the form
A2.12)
u =
k=0
by the method of undetermined coefficients. If the roots A1; A2 of the
indicial equation F(X) = 0 [cf. A2.4) and A2.8)] are such that At - A2
is not an integer, then we obtain two solutions A2.12) with X = A1; A2 in
this way; see Corollary 11.2. If At — A2 ^ 0 is an integer, we can still
obtain in this way a solution u{t) of the form A2.12) with X = At; see
Lemma 11.4. A second linearly independent solution v(t) can be obtained
from the fact that, by § 8(iv),
(»/«)' = u~\t) exp
-\iS-1a1
{s) ds.
Exercise 12.3. Discuss the nature of the solutions at the (finite)
singular points of the equations: (a) t2u" + tu + (t2 — /u2)u = 0 (Bessel);
(b) A - t2)u" - 2tu + n(n + l)u = 0 (Legendre); (c) A - t2)u" -
2tu + [«(« + 1) - m2{\ - f2)-1]" = 0 (associated Legendre).
13. A Nonsimple Singularity
In this section, we shall prove a theorem about the number of analytic
solutions of a particular type of linear homogeneous system for which
/ = 0 is a singularity, but not necessarily a simple singularity.
Theorem 13.1 (Lettenmeyer). Let A(t) = (ajk(t)) be a d x dmatrix of
functions analytic at t = 0. Let «(;) ^ 0 be an integer for j = 1, ..., d
and let a < d, where a = a(l) + • • • + a(d). Then the system
A3.1)
forj = l,
has at least d — a. linearly independent solutions analytic at t = 0.
This theorem generalizes Theorem 11.4 which corresponds to the case
a(y) = 0 or 1; cf. Exercise 11.5.
Proof. The solutions y{t) of A3.1) analytic at t = 0 will be determined
by the method of undetermined coefficients. Let the function ajk(t) have
the expansion
C1.2)
= 2 ajk,mt
88 Ordinary Differential Equations
Consider a solution y{t) = (y^t) y*(t)) of A3.1) with convergent
expansions
A3.3) y'O) = 2 yJtm-
Then the numbers yj satisfy
k
A3.4,-J
d m
-«(j)+i = 2 I,ajk,m-nynk
for J = 1 d and m = 0, 1 It is understood that y > = 0 if
m < 0. Conversely, if a set of numbers yns satisfies A3.4) and if A3.3) is
convergent for small |;| and j = 1 d, then y(t) = (y1^) yd(t))
is a solution of A3.1) analytic at t = 0.
Let N be a large fixed integer to be specified below. Divide the system
of equations A3.4jm) into two systems
A3.5) 2i: A3-4im) f°r;=1 d andO ^ m ^ N + </) - 2,
A3.6) 22-- A3.4,J for; = 1 d and m ^ N + <</) ~ *•
Since it is assumed that the series in A3.2) is convergent, there exist
positive numbers, c and p, such that
A3.7) |aifc.j ^4; for ^ fc = i d and ^ = °> i
P
Let 0 < 0 < 1 and define
A3.8) zim -
A3.9) clmJm ¦¦
The last set of relations is equivalent to
m
Cth.kn —
where h = m — a(j) + 1.
m — «0) + 1
Thus the system 22 of equations can be written as
«» +22 cih,knZkn = 0 for m ^ N + <x.(j) - 1,
where h = m — «(j) + 1, or in the form
*im + Z^T'c^knZkn = 0 for m>N,
or, finally, as
d m+a(j)-l d N-l
(i3.io) zim + 2 2 oMnzkn =-22 cim.,
Linear Differential Equations 89
It is understood that the inner sum on the left over N ^ n ^ m + a(/) — 1
is 0 if m + «(/) = ^r-
Let p, q be fixed integers satisfying 1 ^ p ^ d and 0 ^ q ^ N — 1.
Instead of A3.10) consider the set of equations
A3.11)
Zim +2, 2, Cjm,l:Aii = ~cim.vQ'
k=l n=N
for j = 1 J and m^N. It will be shown that if N is sufficiently
large, then A3.11) has a solution zjm = zj? satisfying
A3.12) laJJJ ^2 for j = 1 d and m ^ JV.
To this end, note that, by A3.7) and A3.9),
\cim.kn\ =
m
Thus, if c', c" are suitable constants (independent of p, q, N), then
_
2
I2 < _
l =
=iV JV A — 0 )
y \c i2 < ?_ y
Z, \cjm.kn\ = % 2,
Hence, if N is sufficiently large,
d oo
A3.13)
y y ic |2 < _ < i
Z/ Z, I'-jm.pal = »t2 ^ 1»
?2
N2
d d qo m+a(i)—1 oo t t
(i3.i4) 2 2 2 2 l<w»r = rfV 2 -2 ^;.
for all choices of/>, 9.
From now on, it is supposed that N is fixed so large that A3.13) and
A3.14) hold. Let ju > N denote a fixed integer and replace the infinite
system A3.11), where j = 1,..., d and m ^ N, by the finite system of
equations,
d v
A3.15) z?m +2t 2 <W«tf« = -c,».,«
for; = 1,..., J and N ^ m ^ ft, where j< in A3.15) is
A3.16) r =. </» »*, (*) = min [a, w + a(/) - 1].
The system A3.15) can be written in the form
A3.17) ? + C? = ?7,
where f is a vector of dimension e = d(jx — N + 1) with components
90 Ordinary Differential Equations
z?m for j = 1,. .., d and m = N, . .., ft; r\ is a vector of dimension e
with components —cjm>m for j =¦ 1,..., d and m = N,..., ft; and C
is an e X e matrix. Using Euclidean norms, it is clear from A3.13) and
A3.14) that
|| 9,11 ^ 1 and ||C|| ^ 1/2.
Hence A3.17) has a unique solution | satisfying ||||| ^ 2, since ||?7|| =
III + C||| ^ Hill - ||C||| ^ 4 |||||. Thus the finite system A3.15) has a
unique solution zfn satisfying
|a?J ^2 for j = 1,.. . , d and m = N,. .., p.
Then by the Cantor Selection Theorem I 2.1, there exists a sequence of
integers (N <) ,a(l) < fiB) <••• such that
zfm = lim zf» exists for j = 1,..., d and m^N.
ft-* OO
Note that v in A3.15) and A3.16) satisfies v(j, m, /j) -*• m + «(/) ~ ' as
/«-»• oo. Hence, by letting ,a = ft(h) -> oo in A3.15), it follows that zf?
is a solution of A3.11) for j = 1,. . ., d and m ^ N satisfying A3.12).
Consequently, if 2OT are any dN numbers for p = 1, . .., d and q =
0,. .., N — 1, then a solution of A3.10) for/ = 1, . . ., cfand m > N is
given by
A3.18,J */» = 2 ^E,*,.
j) = l 0=0
fory = 1, .. . , d and m^N. In other words, the equations of the system
^2 in A3.6) [i.e., the equations in A3.10) fory = 1, .. ., d and m ^ N ]
are satisfied if A3.18,m) hold fory = 1,... , d and m ^ N.
Thus the original system of equations A3.4jm) for the yj or, equi-
valently, for the zim are satisfied if ?i and A3.18) hold. If a* =
max [a(l),. . ., a.(d)], the system ^i involves the d(N + a* — 1) unknown
Zjm for y = 1,. . ., d and m = 0, ..., N + a* - 2; cf. A3.4,J for
y = 1, .. . , d and 0 ^m ^ N + a(j) — 2. Also the system 2i consists of
[N- 1 + a(l)] + • • • + [N - 1 + aid)] = d(N - 1) + a equations,
where a = a(l) + • • • + a.{d). Add to the system ^i> the (possibly
vacuous) set of equations
?3: A3.183m) for y = 1,. ..,d and N ^ m ^ N + a* - 2.
The system ^3 involves the same set of d(N + a* — 1) variables that occur
in 2i and consists of d{a.* — 1) equations. Thus, the combined systems,
2iand23,hasatleastrf(Ar+ a* - \)-[d(N- 1) + a + d{a* - 1)] =
d — a. linearly independent solutions.
Corresponding to any solution of ^i and 2a> the equations A3.183m)
for m > N + a* — 2 (together with the equations of 2a) g've a solution
Linear Differential Equations 91
of ^2- The resulting set of numbers yj = zimj(dp)m, for j = 1,. .., d
and m = 0, 1,. . . , is a solution of ^i and ^2- In view of A3.12) and
A3.18), there is a constant c0 such that \zjm\ ^ c0; hence \ym'\ ^ co/(dp)m
for y = 1,..., d and m ^ 0. Consequently, A3.3) is convergent for
|f| < dp. This proves Theorem 13.1.
Exercise 13.1 (Perron). Let a3@ be analytic for | f | < a, j = 0,.. ., d.
Let a be an integer, 0 ^ a < d. Show that
?Vd» + flrf-iW-1' + •"" + fli@"' + «o(O" = 0
has at least d — a. linearly independent solutions analytic at t = 0.
Exercise 13.2 (Lettenmeyer). (a) Let v4(f), X{i) be d X d matrices of
functions analytic at t = 0. Let det X(t) =? 0 and det X(t) have a zero of
order a, 0 ^ a < d, at t = 0. Then the system
A3.19)
has at least d — a. linearly independent solutions analytic at t = 0.
(b) Let Jf @. ^ (?) be analytic in a simply connected ^-domain E such that
det X(t) ^ 0 and det X(t) has exactly a zeros in E, counting multiplicities.
Let a < d. Then A3.19) has at least d — a. linearly independent solutions
y(t) analytic on E.
Exercise 13.3. Let X{t) be as in Exercise 13.2(a). Let f(t, y) =
f(t, y1,.. ., yd) be a ^-dimensional vector, each component of which is
an analytic function of (i.e., a convergent power series in) t, y1,..., yd.
Then
X(t)y'=f(t,y)
has a d — a parameter family of solutions y(t) analytic at t = 0. See
Bass [1].
Notes
section 1. Equation A.5) in Theorem 1.2 as well as the special case (8.4) are given
by Liouville [2] in 1828, although some authors refer to (8.4) as Liouville's formula and
A.5) as Jacobi's. Theorem 1.2 was also given by Jacobi [1, IV, p. 403] in 1845; for the
particular case when the system A.1) is replaced by one linear equation of the second
order, the corresponding formula (8.4) occurs in a paper A827) by Abel [1, I, p. 251].
The notion of a fundamental set of solutions is due to Lagrange (circa 1765); see
[2, I, p. 473]. The term "fundamental set of solutions" was introduced by Fuchs
[4, p. 117] in 1866.
section 2. The method of variation of constants (and Corollary 2.1) is essentially due
to Lagrange A774, 1775); see [2, IV, p. 9 and p. 159]. The result of Exercise 2.2 goes
back to Perron [11]. It has also been proved by Diliberto [1], [2]. The proof given in
Hints, using Exercise 2.1, is that of Reid [4].
section 3. The possibility of reducing the order when a solution is known goes back
A762-1765) to d'Alembert for a linear equation of order d. Corollary 3.1 when A.1)
92 Ordinary Differential Equations
is replaced by a single equation of order d [cf. (8.2) and (8.19)] is implicit in a result
A833) of Libri [1, pp. 185-194] (who also makes remarks about the possibility of
extending the result to systems). Corollary 3.1 for the case of an equation of rfth order
is given explicitly A874) by Frobenius [1].
section 5. For the analogous case of a linear rfth order equation, the general solution
[§ 8 (vi)] is due to Euler A743).
section 6. Theorem 6.1 for the case of a rfth order equation was given A883) by
Floquet [1].
section 7. The adjoint of a linear rfth order equation was given by Lagrange about
1765, see [2, I, p. 471]. The adjoint system G.1) was denned in 1837 by Jacobi; cf.
[1, IV, p. 403]. The term "adjoint" was introduced by Fuchs [1, p. 422] in 1873.
section 8. See comments on §§ 1-7. Exercise 8.3 contains results of Polya [1]; cf.
comments of Hartman [15].
section 9. The type of formalism discussed in connection with (9.1)—(9.6) was used
extensively by G. D. Birkhoff [3]. Exercise 9.1 is based on considerations used by
Cesari [1]; see also Levinson [3].
appendix. For a historical survey and references, see Schlesinger [2], Forsyth [1],
and the encyclopedia article of Hilb [1]; for later references, see A. Schmidt [1].
section 10. Problems arising out of A0.2) were initiated by Riemann in work dated
1857(cf. [l,pp. 379-390]) and independently by Fuchs in 1865(see [l,p. 124]). Theorem
10.1 is essentially due to M. Hamburger [1]; the proof in the text follows Coddington
and Levinson [2]. Although the paper by Wintner [5] contains many misstatements,
including the main theorem, it has a number of good ideas including the suggestion to
view A0.5) as a system A0.6) of d2 equations rather than as a matrix equation; for
applications, see § 11.
section 11. Theorem 11.1 is due to Sauvage [1]; cf. Hilb [1] for references to Horn
and Schlesinger. The proof in the text is that of Birkhoff [1]. The partial converse,
Theorem 11.2, was given by Sauvage and Koenigsberger; the first complete proof is due
to Horn [1]. The proof in the text is that of Schlesinger [1, pp. 141-162], who attributes
the arguments in the proof of Lemma 11.2 to Kronecker. For similar proofs of this
lemma, see Lettenmeyer [1] and Moser [2]; generalizations have been given by Hilbert,
J. Plemelj, and G. D. Birkhoff, see Birkhoff [2] for references. Corollary 11.2 and Lemma
11.3 are in Rasch [1, p. 113]. Their use in connection with A1.25>-A1.27) is that of
A. Schmidt [1 ]. Exercise 11.5 is a special case of Lettenmeyer's [1 ] result, Theorem 13.1.
section 12. Theorem 12.1 is due to Fuchs in 1868 [1, p. 212]. It was first proved in
special cases by Riemann [1, pp. 379-390]. The proof in the text for the "only if"
part is that of Thome [1]; see Hilb [1] for references to Frobenius.
section 13. Theorem 13.1, which is due to Lettenmeyer [1], generalizes the result of
Perron [1] in Exercise 13.1 dealing with one equation of dth order. The proof in the
text is a modification of Lettenmeyer's which, in turn, is based on Hilb's proof [2] of
Perron's theorem. For Exercise 13.2, see Lettenmeyer [1].
Chapter V
Dependence on Initial Conditions and Parameters
1. Preliminaries
Let f(t, y) be defined on an open (t, «/)-set E with the property that if
('o> 2/o) e E> then the initial value problem
A.1)
V'=f{t,V) and
has a unique solution y(t) = rj(t, t0, y0) which is then defined on a maximal
f-interval (cu_, cu+), where cu± depends on (t0, y0). In this chapter, the
problem of the smoothness (i.e., of the continuity or differentiability
properties) of r\(t, t0, y0) will be considered.
Often, a more general situation is encountered in which A.1) is replaced
by a family of initial value problems depending on a set of parameters
z = (z\ ..., z%
A.2)
V' =f(t, y, z) and y(t0) = y0,
where for each fixed z, A.2) has a unique solution y(t) = rj(t, t0, y0, z). In
most cases, the question of the dependence of solutions of A.1) on t and
initial conditions can be reduced to the question of the dependence on
t, z of solutions of a family of initial value problems A.2) for fixed initial
conditions y(t0) = y0; conversely, the question of the dependence of
solutions of A.2) on t, t0, y0, z can be reduced to the question of smooth-
smoothness of solutions of initial value problems in which extra parameters z do
not occur. The first reduction is accomplished by the change of variables
t,y ->¦ t — t0, y — y0 which changes A.1) to
A-3)
y' = fit -to,y- y0) and y@) = 0,
in which z = (t0, y0) = (t0, yj-, ..., yod) can be considered as a set of
parameters (and the initial condition y@) = 0 is fixed). The second
reduction is obtained by replacing A.2) by an initial value problem for a
(</ + eO-dimensional vector (y, z), in which no extra parameters occur:
A.4) y =f(t, y, z), z = 0 and y(t0) = y0, z(t0) = z0,
93
94 Ordinary Differential Equations
where (t0, y0, z0) denotes any of the possible choices for (t, y, z). For this
reason, some of the theorems which follow will be stated for A.2) but
proved for A.1).
Below x, y, r),fe Rd and z e Re, where d, e ^ 1.
2. Continuity
The assumption of uniqueness implies the continuity of the general
solution y = rj(t, t0, y0, z) of y' = f(t, y, z):
Theorem 2.1. Letf(t, y, z) be continuous on an open (t, y, z)-set E with
the property that for every (t0, y0, z) e E, the initial value problem A.2), with
z fixed, has a unique solution y{t) = r)(t, t0> y0> z). Let w_ < t < co+ be
the maximal interval of existence of y(t) = r)(t, t0, y0> z). Then w+ =
a>+(t0, y0, z) [or cu_ = co_(t0, y0, z)] is a lower [or upper] semicontinuous
function of (t0, y0, z) e E and rj(t, t0, y0, z) is continuous on the set co_ <
t < cu+, (t0, y0, z) e E.
It is understood that cu+[cu_] can assume the value +oo [— oo]. The
lower semicontinuity of cu+ at (tlt ylt zj means that if t° < (o+(tlt yu zj,
then co+(t0, y0, z) ^ t° for all (t0, y0, z) near (tlt yx, zj; i.e., cu+(?1; yu zj ^
lim inf w+(t0, y0, z0) as (t0, y0, z0) -> (tlt yu zj. The upper semicontinuity
of co_ is similarly denned.
It is easy to see that cu±(?0, y0, z) need not be continuous. For suppose
that (tlf yu 2j) e E and tx < t° < a)+(tlt ylt Zj). If E is replaced by the set
obtained from E by deleting the point (t, y, z) = (t°, rj(t°, tlt yu z{), 2j),
then (o+(tlt yu 2t) now takes the value t°, but co+ is not altered for all
points (t0, y0> z) near (tlt ylt zj.
Remark. Let f(t, y, z), r)(t, t0, y0, z) be as in Theorem 2.1. For fixed
(t, t0> z), the relation y = r)(t, t0> y0, z) can be considered as a map carrying
y0 into y. The assumption that the solution of A.2), for (t0, y0, z) e E, is
unique implies that this map is one-to-one. In fact the inverse map is
given by y0 = ri{t^t,y,z). A consequence of Theorem 2.1 is that the
map y0 —*• y is continuous.
Proof. Since A.2) can be replaced by A.4) and (y, z) by y, there is no
loss of generality in supposing that/does not depend on z. Thus, it will
be supposed that f(t, y) is defined on an open (t, «/)-set E and that A.1)
has a unique solution y(t) = rj(t, t0, y0) on a maximal interval of existence
co_ < t < cu+, where cu± = (o±(t0> y0). It will be shown that, in this form,
Theorem 2.1 is merely a corollary of Theorem II 3.2 for the case/n(/, y) =
f(t,y),n= 1,2,....
In order to verify that w+(t0> y0) is lower semicontinuous, choose a
sequence of points (flt y10), {t2, «/20),... in ? such that (tn, yn0) -»• (t0, y0) e
E and co+(tn, yn0) -»¦ c(^ oo) as n -*¦ oo, where c = lim inf cu+(?*, y*) as
Dependence on Initial Conditions and Parameters 95
(f*» y*)-*-Qo> Vo)- Since the solution of A.1) is unique, it follows from
Theorem II 3.2 that c ^ w+ = w+(t0> y0); i.e., w+(t0, y0) is lower semi-
continuous. The proof of the upper semi-continuity of w_(t0> y0) is the
same.
Note that, for the case where /„ =/, n = 1, 2,..., and the solution
of A.1) is unique, a selection of a subsequence in Theorem II 3.2 is un-
unnecessary. Thus it follows that r)(t, t0, y0) is a continuous function of
Oo»2/o) for every fixed t, co_(t0, Ho) < t < co+(t0,y0); in fact, this con-
continuity is uniform for t1 ^ t ^ t2, w_(t0, y0) < t1 < t2 < w+(t0, y0). In
other words, if e > 0, there exists a d€Q > 0, depending on (t0> y0, t1, t2),
such that
.t, to> yd) -
Uo - h\, \Vo ~
for t1 ^ t ^ t2. But since r)(t, t0> y0) is a continuous function of t for
fixed (t0, y0), there is a «5el > 0, depending on (t0, y0, t1, t2), such that
Hf, t0, y0) - r)(s, t0, yo)\< € if \t - s\< Scl, f1 ^ j, / ^ t\
Hence, if S€ = min (<5e0, <5el) and \t - s\, \t0 - h\, \y0 - y^ < S€, then
This completes the proof of Theorem 2.1.
3. Differentiability
If it is assumed that f(t, y, z) is of class C1, it follows that the general
solution y = rj(t, t0, y0, z) of A.2) is of class C1. In fact, even more is
contained in the following theorem.
Theorem 3.1 (Peano). Letf(t, y, z) be continuous on an open (t, y, z)-set
E and possess continuous first order partials df/dyk, df/dz' with respect to
the components of y and z: (i) Then the unique solution y = rj(t, t0, yo, z) of
A.2) is of class C1 on its open domain of definition a>_ < t < a>+, (t0, y0, z) e
E, where co± = w±(t0, y0, z). (ii) Furthermore, if J(t) = J(t, t0, y0, z) is
the Jacobian matrix (dfjdy) off(t, y, z) with respect toy at y = r)(t, t0, y0, z),
C.1) J(t) = J{t, to, y0, z) = M-1 at y = rfc, t0, y0, «),
\dyl
then x = dr](t, t0, yo> z)/3yok is the solution of the initial value problem,
C.2) x'=J(t)x, x(to) = ek,
where ek = (e^, . . ., ekd) with ek' = 0 if j j? k and ekk = 1; x =
dy(t, /„, y0, z)/dz> is the solution of
C.3) x'= J(t)x + gj(t), x(to) = O,
96 Ordinary Differential Equations
where g}{i) = g^t, t0, y0, z) is the vector df(t, y, z)jdz' at y = rj(t, t0, y0, z);
and dt](t, t0, y0, z)/dt0 is given by
C.4) ^ = -ijVu,^).
The uniqueness of the solution of A.2) is assured, e.g., by Theorem
II 1.1. Note that the assertions concerning dr)/dyok and C.2) or drj/dz'
and C.3) result on "formally" differentiating both equations in A.2),
i.e., both equations in
»?'('> t0, «/o> z) = f(t, n, z), »?('(» 'o- Vo> z) = Vo
with respect to yok or z1. Similarly, differentiating these equations formally
with respect to t0 shows that x = dr]ldt0 is also a solution of x' = J(t)x
satisfying the initial condition x(t0) = — r)\t0, t0, rH, z) = —f(t0, y0, z).
Writing f(t0, y0, z) = S/*(?o, y0, z)ek, it follows that C.4) is a formal
consequence of C.2) and the principle of superposition (Theorem IV 1.1)
for the linear system x' = J{t)x.
More generally, if y(t, s) is a 1-parameter family of solutions of y =
f(t, V, z) f°r fixed z, if y(t0, s0) = y0, and if y{t, s) is of class C1 in (t, s), then
the partial derivative x = dy(t, s)jds at s = s0 is also a solution of the
system x' = J(t)x. For this reason x' = J{t)x is called the equation of
variation of A.2) along the solution y = r)(t, t0, y0, z).
The assertion concerning x = dr)ldyok and C.2) shows that the Jacobian
matrix (drjldy0) is the fundamental matrix for x' = J(t)x which reduces
to the identity for t = t0. In particular, Theorem IV 1.2 implies
Corollary 3.1. Under the conditions of Theorem 3.1,
dy0 1 Ju i=i By1
for co_ < t < a>+, where the argument of the integrand is (s, r)(s, t0 y0, z), z).
Remark. By C.5), det (dr)ldy0) jt. 0. Thus, the continuous one-to-one
map y° -*• y = r)(t, t0, y0, z) for fixed (t, t0, z), considered in the Remark
after Theorem 2.1, is of class C1 and has an inverse of class C1 with respect
to (t, t0, y, z). This statement about the inverse is also clear from the
explicit formula y-*yo = r)(t0, t, y, z).
Exercise 3.1 (Liouville). Let f(y) be of class C1 on an open set E and
let y = r)(t, yo)bc the unique solution of the initial value problem y' = /(«/),
2/('o) = Vo f°r ('o. 2/o) e E- Show that the set of maps yo->y defined by
y = r)(t, y0) for fixed t are volume preserving if and only if div/(«/) =
S dfkldyk is 0.
Note that the assertion that x = dr]ldyok is a solution of C.2) implies
that the iterated derivative d(dr]ldyok)ldt exists and is J(t) dr)ldyok. The
C.5)
Dependence on Initial Conditions and Parameters 97
last expression is a continuous function of t, t0, y0, z; hence, Schwarz's
theorem implies that d(drjldt)ldyok exists and is d(dr]ldyok)ldt. A similar
remark applies to drjjdz'. Also, note that on the right side of C.4) the
variable t only occurs in drjldyok.
Corollary 3.2. Under the conditions of Theorem 3.1., the second mixed
derivatives d2r]ldy0k dt = ^rj/dt dyk, d^/dz1 dt = d^jdt dz\ d2r)ldt0 dt =
^rjjdt dt0 exist and are continuous.
In order to avoid an interruption to the proof of Theorem 3.1, the
following simple lemma will be proved first. This lemma is a convenient
substitute for the mean value theorem of differential calculus when dealing
with vectors, for it avoids some awkwardness in the fact that 6k depends on
k in y(b) - y(a) = (b - aW@J,..., y"@«)), where a < dk < b.
Lemma 3.1. Let f(t, y) be continuous on a product set (a, b) X K,
where K is an open convex y-set, and let f have continuous partials dfjdyk
with respect to the components of y. Then there exist continuous functions
fk(t, yu y2), k = 1, . . ., d, on the product set (a, b) X K X K such that
C.6)
and that if(t, yx, y2) e (a, b) X K x K, then
C.7) f(t, y2) - f{t, yi) = I fk(t, yi, y2)(yk - yk).
Infact,fk(t, yx, y2) is given by
0.8)
Proof. Put F(s) =f(t, sy2 + A — s)y1) for 0 ^ s ^ 1. The convexity
of K implies that F(s) is defined. Then dF/ds = 2 (y2k - yf) df(t, sy2 +
A - s)y1)jdyk. Hence F(\) - F@) is the right side of C.7) \ffk is defined
by C.8). Since F(\) =f(t,y2) and F@) =f{t,yx), the lemma follows.
Proof of Theorem 3.1. Since A.2) can be replaced by A.4) and (y, z)
by y, there is no loss of generality in supposing that/does not depend on
2 when proving the existence and continuity of the partial derivatives of r\.
Thus the initial value problem A.1) having the solution y = r)(t, t0, y0) on
(o_ < t < co+ is under consideration.
In order to simplify the domain on which the function r\ must be con-
considered, let a, b be arbitrary numbers satisfying co_ < a < b < co+,
@^=0) h(tu yx). Then, by Theorem 2.1, rj(t, t0, y0) is defined and con-
continuous for a ^ t ^ b and (t0, y0) near (tu yx). In the following only such
('.'o-.Vo) w'll be considered. Since the assertions of Theorem 3.1 are
"local," it clearly suffices to prove the assertions on the interior of such a
('. 'o- 2/o)-set.
98 Ordinary Differential Equations
(a) In order to prove the existence of dr)jdyok, let h be a scalar, ek the
vector in C.2) and, for small |A|,
C.9) yh(t) = rj(t, t0, y0 + hek).
This is defined on a ^ t ^ b and, by Theorem 2.1,
C.10) yh(t)-y0(t) asA->0
uniformly on [a,b]. By A.1), (yh(t) - yo(t))' =f(t,yh{t)) -f(t,y«(t)).
Applying Lemma 3.1 with y2 = yh(t), yx = yo(t),
C.11) [yh(t) - yo(t)]' = I Ut, yo(t), yh(t))\yh\t) - yk{t)l
Introduce the abbreviation
C.12)
xh =
yh(t) -
The existence of dr](t, t0, yo)l3yok is equivalent to the existence of lim xh(t)
as h -> 0.
By A.1) and C.9), yh(t0) = y0 + hek, and so^(f0) = ek. Thus, by C.11)
and C.12), x = xh(t) is the solution of the initial value problem
C.13)
x'=J(t;h)x, x(to) =
where J(t; h) is a d x d matrix in which the &th column is the vector
fk(t,y«@, Vn(t)). By C.6), the continuity of fk(t, ylf y2) and C.10), it
follows that J(t;h)-*J(t; 0) as h -> 0 uniformly on [a, b], where J(t; 0) =
J(t) is the matrix defined by C.1).
Consider C.13) to be a family of initial value problem depending on a
parameter h, where the right side J(t; h)x of the differential equation is con-
continuous on the open-set a < t < b, \h\ small, x arbitrary. Since the solu-
solutions of C.13) are unique, Theorem 2.1 implies that the general solution
is a continuous function of h [for fixed (t, t0)]. In particular, x(t) =
lim xh(t), h-+0, exists and is the solution of C.2) on a < t < b. Hence
3rj(t, t0, yo)ldyok exists.
In order to verify that this partial derivative is continuous with respect
to all of its arguments, rewrite C.2) as
C.14)
x = J(t, t0, yo)x, x(t0) = ek,
a family of initial value problems depending on parameters (t0, y0). Since
J(l> lo> Vo) is a continuous function of (t, y, t0, y0) and initial value prob-
problems associated with a linear differential equations have unique solutions,
Theorem 2.1 implies that the solution x = dr)(t, t0, yo)ldyok of C.14) is a
continuous function of its arguments.
Dependence on Initial Conditions and Parameters 99
(b) The existence and continuity of drj(t, t0, yo)ldto will now be con-
considered. Put
x(i)
The solution of the initial value problem y =f, y(t0) = rj(t0, t0 + h, y0)
is the same as the solution of y =f, y(t0 + h) = y0; i.e., rj(t, t0 + h, y0) =
V((> lo, y(to, to + h> yd))- Thus
hxh(t) = r)(t, t0, r)(t0, t0 + h, y0)) - rj(t, t0, y0).
Since r](t, t0, y0) has continuous partial derivatives with respect to the
components of y0 and r)(t0, t0 + h, y0) -> rj(t0, t0, y0) = y0 as h -> 0, it
follows that
blk(t0, to + h,y0) - yok]
as h -*¦ 0. By the mean value theorem of differential calculus and y0 =
r](t0 + h,to + h, y0), there is a 6 = dk such that
rjk(t0, t0 + h, y0) - yk = -hr)k'(t0 + dh, t0 + h, y0), 0 < 6 < 1.
Note that rjk'(t0 + Oh, t0 + h, y0) = fk(t0 + dh, rj(t0 + dh, t0 + h, y0)) is
fk(t0, y0) + o(l) as h -> 0. Thus, as h -> 0,
*.@ = -I I3*'! \ Vo) + o(l)][f\t0, y0) + od)].
L dyok J
This shows that dr)/dt0 = lim xh(t) exists as h -*¦ 0 and satisfies the ana-
analogue of the relation C.4). This relation implies that dr)ldt0 is a con-
continuous function of (t, t0, y0).
(c) Returning to A.2), so that/can depend on z and r\ = r)(t, t0, y0, z),
it follows that r) is of class C1. By applying the results just proved for A.1)
to A.4), it is readily verified that the assertions concerning C.2), C.3),
and C.4) hold. This verification will be left to the reader. *
The proof of Theorem 3.1 has the following consequence.
Corollary 3.3. Let f(t, y, z, z*) be a continuous function on an open
(t, y, z, z*)-set E, where z* is a vector of any dimension. Suppose that f has
continuous first order partial derivatives with respect to the components of
!/ and z. Then
C.15) y' =f(t,y,z,z*), y(t0) = y0
has a unique solution r\ = r)(t, t0, y0, z, z*) for fixed z, z* with (t0, y0, z, «*) e
/•.'; t] has first order partials with respect to t, t0, the components of y and
of z, and the second order partials d2rjjdt dt0, d2rjjdt dyok, d2rjjdt dz>;
finally, these partials off] are continuous with respect to (t, t0, y0, z, «*).
100 Ordinary Differential Equations
Proof. Since the partial derivatives of r\ involved in this statement are
calculated with z* fixed, their existence follows from Theorem 3.1. Their
continuity with respect to (t, t0, y0, z, «*) follows, as in the proof of
Theorem 3.1, by the use of analogues of C.1), C.2), Theorem 2.1, and an
analogue of C.4).
4. Higher Order Differentiability
The question of higher order differentiability of the general solution is
easily settled by the use of Theorem 3.1 and Corollary 3.3.
Theorem 4.1. Let f(t, y, z, «*) be a continuous function on an open
(t, y, z, z*)-set E such that f has continuous partial derivatives of all orders
not exceeding m, m ^ 1, with respect to the components ofy and z. Then
D-1)
y =f(t,y, z,
y(to) =
has a unique solution t] = rj(t, t0, y0, z, z*), for fixed z,z* with (t0, y0, z,z*)e
E, and rj has all continuous partial derivatives of the form
D.2)
df 34° d(y0T ¦ ¦ ¦ d(y0T
¦ ¦ ¦ d(zeY<
where i ^ 1, i0 ^ 1 and i0 + Z/?fc + ?a, ^ m.
Proof. The proof will be given first with i0 = 0 by induction on m.
The case m = 1 is correct by Corollary 3.3 for z* of any dimension.
Assume the validity of the theorem if m is replaced by m — 1(^ 1).
Consider the analogue of C.2),
D.3)
x' = J(t, t0, y0, 2, z*)x, x(t0) = ek,
where / = (df/dy) at y = r)(t, t0, y0, z, z*). By the assumption on / and
by the induction hypothesis, the right side, J(t, t0, y0, z, z*)y, of the
differential equation in D.3) has continuous partial derivatives of order
^m — 1 with respect to the components of y, y0, and z. Hence, by the
induction hypothesis, the solution x = drj(t, t0, y0, z, z*)jdyok of D.3) has
continuous partial derivatives of all orders ^ m — 1 with respect to the
components of y0 and each of these partials has a continuous partial
derivative with respect to t. Similarly, the analogue of C.3) shows that
dr)jdz> has continuous partial derivatives of all orders ^ m — 1 with
respect to the components of y0, and z, each of these partials has a con-
continuous partial derivative with respect to t. This completes the induction
and shows that r)(t, t0, y0, z, z*) has continuous partial derivatives of the
form D.2) with i0 = 0, / ^ 1, Sot*. + 20, ^ m.
Dependence on Initial Conditions and Parameters 101
The existence and continuity of derivatives of the form D.2) with
/0 = 1, / ^ 1, Sot*. + 2/?3 ^ m — 1 follows from the analogue of C.4).
This completes the proof.
Corollary 4.1. Let f(t, y, z) be of class Cm, m ^ 1, on an open (t, y, z)-
set. Then the solution y = r)(t, t0, y0, z) o/A.2) is of class Cm on its domain
of existence.
The proof of this useful corollary will be left as an exercise.
5. Exterior Derivatives
Several useful concepts will be introduced in this section. All concepts
are of a "local" nature.
By a (piece of) 2-dimensional surface S of class Cm, m ^ 1, in a Euclid-
Euclidean j/-space Rd is meant a set S of points y in Rd which can be put into
one-to-one correspondence with an open set D of points (m, v) in a Eu-
Euclidean plane by a function y = y(u, v) of class Cm on D such that the two
vectors dyjdu, dyjdv are linearly independent at every point of D. The
function y = y(u, v) is called an admissible parametrization of S.
If V = y(u, v) is any given function of class Cm, m > 1, on an open
(m, p)-set D such that dyjdu, dyjdv are linearly independent at a point,
hence near a point (m0, v^) of D, then the set of points y = y(u, v) for
(m, v) near (u0, v0) is a piece of surface. For by the implicit function
theorem, the map (m, v)->-y is one-to-one for (u, v) near (m0, v0).
Consider a piece of surface So of class C1 with an admissible parametriza-
parametrization y = y(u, v) defined on a simply connected, bounded open set Do and a
piecewise C1 Jordan curve C in Do bounding an open subset D of Do. Let
S, J be the «/-image of D, C, respectively. This situation will be described
briefly by saying "a piece of C1 surface S bounded by piecewise C1 Jordan
curve J."
A differential r-form on an open set ? is a formal expression
d d
co = J,. . .2 Ph...ij,y) dyh A • • • A dyir *
with real-valued coefficients defined on E, wherep( ...,(«/) = ±Pj .,.3 («/)
according as (j1,. . . ,jr) is an even or odd permutation of (iu ... , ir).
In particular, pilt..ir(y) = 0 if two of the indices iu . . . , ir are equal.
The form w is called continuous [or of class Cm or 0] if its coefficients are
continuous [or of class Cm or identically 0] on E. A sequence of differen-
lial r-forms on E is said to be uniformly bounded [or uniformly conver-
convergent] if the sequences of the corresponding coefficients are uniformly
bounded [or uniformly convergent]. Differential r-forms can be added
in the obvious way. Differential r- and i-forms can be multiplied to give
102 Ordinary Differential Equations
an (r + i)-form by the usual associative, distributive laws and anti-
commutative law dy' A dy' = —dy' A dy1; see § VI 2.
A continuous linear differential form A-form or Pfaffian)
E.1)
=l.P,{y)dyi
i
on E is said to possess a continuous exterior derivative dco if there exists a
continuous differential 2-form
d d
dm =2 J,Pu(y)
i=l 3=1
E.2)
on E such that Stokes' formula
E-3)
A dy\ pif = -pj{,
= \\ dco
holds for every piece of C1 surface S in E bounded by a C1 piecewise
Jordan curve / in E.
It is clear that if S is the image of D on the surface So: y = y(u, v) for
(m, v) e D, and / is the image of the Jordan curve C, then E.3) means that
f d r d d
E-4) 2 Pl(y(u, v)) dy'\u, v) = \ 2 1 P
dv,
«, »)) ^T
O{U,V)
with the usual convention as to orientation of C around D.
If the coefficients pfy) of E.1) are of class C1, then co has a continuous
exterior derivative with dco = 2 rf/?/y) A rfy3' or
There are cases, however, where E.1) has a continuous exterior derivative
when the coefficients of E.1) are only continuous. Consider, for example,
the case that there exists a real-valued function U(y) of class C1 such that
co = dU [i.e., Pj{y) = dUjdy'], then co has the exterior derivative dco = 0.
The fundamental lemma about the existence of continuous exterior
derivatives is the following:
Lemma 5.1. Let E.1) be a continuous linear differential l-form on an
open set E. Then E.1) has a continuous exterior derivative E.2) on E if
and only if, on every open subset E° with compact closure E° <= E, there exists
a sequence of l-forms co1, co2, . . . of class C1 such that con —»¦ co as n —*¦ oo
uniformly on E° and dco1, dco2, . . . is uniformly convergent on E° (in which
case, d(on —>¦ dco uniformly on E° as n —*¦ oo).
Proof. If a sequence oo1, co2, . .. of the specified type exists on E°
and if the case oo = oo" of E.3) is written in the form E.4), an obvious
Dependence on Initial Conditions and Parameters 103
term-by-term integration gives E.3). Thus, it is seen that the existence
of sequences co1, co2, . . . is sufficient for the existence of a continuous doo.
Conversely, if E.1) is a continuous 1-form on E with a continuous
exterior derivative E.2), approximate the coefficients of co, dco by the
method in §13: Let cp(t) be as in § I 3 and put, for e = 1/n,
=«
'" • • • ply - •
J—oo J—oo
/•OO /•OO
d\ ••• Pilv-
J—CO J —OO
ll2)^1...^,
1\\2)drj1...drjd.
Since the integrals are actually integrals over spheres ||?7|| ^ e, pln) and
plf are defined on the sets En consisting of points y whose (Euclidean)
distance from the boundary of E exceeds e = 1/n. In particular, they are
defined on ?° for large n and tend uniformly to /?„/?„-, respectively, on
E°, as n -»¦ qo.
Define the C00 forms con = 2 /?<•"> (y) dy>' and a" = SS/$> (y) dy* A dy1'
on ?° for large n. Let S be a piece of C1 surface in E° bounded by a C1
piecewise Jordan arc / in ?°. Then if e = 1/n is sufficiently small and
II?; || ^ e, the translation S(rj) of S by the vector — t) is in E and E.3) is
valid if S is replaced by S(rj). This can be written in a form analogous to
E-4),
f
Jc
- r,) dy\
u, v) = f f 2 I
JJd
u, ») -
d(u, v)
dv.
Let this relation be multiplied by ce~dcp(e~2 ||r;||2) and integrated over
||?7ll ^e with respect to drI... drf. An obvious change of the order of
integration shows that the result can be interpreted as the Stokes' relation
w" = a". Thus oon has the continuous exterior derivative dco11 = a."
in Eo. This completes the proof.
Remark. In deciding whether or not a continuous 1-form E.1) has the
continuous exterior derivative E.2), it suffices to verify Stokes' formula
E.3) for rectangles S on coordinate 2-planes y{ = const, for / j? j, k,
where 1 ^ j < k < d. This is a consequence of the following exercise.
Exercise 5.1. A continuous differential 1-form E.1) on an open set E
has a continuous exterior derivative if and only if there exists a continuous
differential 2-form E.2) such that for every pairy, k A ^j<k^d) and
fixed y', with / # j, i 5* k, the 1-form p^y) dy1 + pk(y) dyk has the con-
continuous exterior derivative pjk dy' A dyk + pkj dyk A dy1'; in fact, if and
only if Stokes' formula E.3) holds for all rectangle S on 2-planes y{ =
const, for / ^ j, k with S <= E.
104 Ordinary Differential Equations
Exercise 5.2. Let the continuous differential 1-form E.1) possess
a continuous exterior derivative and letp^y) have a continuous derivative
with respect to y' for a fixed y j? 1. Show that/?3(«/) possesses a continuous
partial derivative with respect to y1.
Exercise 5.3. Let p,{t), where j = 1,. . ., d, be continuous functions
on E such that p^y) has continuous partial derivatives with respect to the
components yk, k j^ j, of y. Show that E.1) has a continuous exterior
derivative.
For the sake of brevity, "vector" and "matrix" notation will be used
in connection with 1-forms and their exterior derivatives. For example,
an ordered set of e 1-forms cu1; . . . , we will be abbreviated co = (co^ . . . ,
co^; analogously if these forms have continuous exterior derivatives,
dco denotes the ordered set of 2-forms dco = (dco1, . . ., dcoe). Finally,
if A = (a^iy)) is an e X d matrix function on E, by co = A(y) dy
will be meant the ordered set of 1-forms co = {cox, . . ., coe), where coi =
d
2 «*,¦(*/) dy' for / = 1,.. ., e.
3=1
6. Another Differentiability Theorem
The main result (Theorem 3.1) on the differentiability of general solu-
solutions has the following generalization.
Theorem 6.1. Let f(t, y, z) be continuous on an open (t, y, z)-set E. A
necessary and sufficient condition that the initial value problem
F.1)
y'=f(t,y,z), y(to) =
have a unique solution y = r)(t, t0, y0, z) for all (t0, y0, z) e E which is of
class C1 with respect to (t, t0, y0, z) on its domain of definition is that every
point of E have an open neighborhood E° on which there exist a continuous
nonsingular d x d matrix A(t, y, z) and a continuous d X e matrix C(t, y, z)
such that the d differential \-forms
F.2)
co = A(dy -fdt)+ C dz
in the variables dt, dy1,..., dyd, dz1,.. . ,dze have continuous exterior
derivatives on E°.
In contrast to Theorem 3.1, the conditions of Theorem 6.1 are invariant
under C1 changes of the variables t, y, z.
It is understood that if A = (afi(t, y, «)) and C = (cik(t, y, «)), then F.2)
represents an ordered set of 1-forms, the rth one of which is
F.3)
3-1
; dy' -
dt+^,cikdzk,
k=l
Dependence on Initial Conditions and Parameters 105
/ = 1,. .. ,d. If this has a continuous exterior derivative, the latter is a
differential 2-form of the type
F.4) dco, =2 ioc^dy',
3=1 k=l
2 Pa dt A dy1 + I ya dt A dzk
1=1 fc=l
3 = 1 k=\
dzk A
3 = 1 fc=l
eiik
dzj A dzk,
where a.iik = —a.iki, ^(j, yik, dijk, eijk = — eiki are continuous function of
(t, y, z). In this case, define a d x d matrix F(t, y, z) = (f{i{t, y, «)) and
did y, e matrix N(t, y, z) = (n^t, y, «)) by
F.5)
F.6)
fa = Pu -
k=l
Theorem 6.1 will be proved in § 11 below. The proof of Theorem 6.1
will have the following consequence.
Corollary 6.1. Letf(t, y, z) be as in Theorem 6.1, let A(t, y, z), C(t, y, z)
exist on E° as specified, and consider only (t, y, z) e E°. Then x = dr)ldyok
is the solution of
F.7) [A(,t, rj, z)x]' = F(t, r,, z)x, x(t0) = ek
and x = dr)jdzi is the solution of
F.8) [A(t, rj, z)x + ct{t, r), «)]' = F(t, rj, z)x + nt{t, rj, z)', x(t0) = 0,
where ct{t, y, z), nt(t, y, z) are the ith columns of C(t, y, z), N(t, y, z), re-
respectively, for i = 1, . . . ,eandrj = rj(t, t0, y0, z).
Note that a solution x = x(t) of F.7) does not necessarily have a de-
derivative, but A(x, rj, z)x(t) has a derivative satisfying F.7). An obvious
change of variables reduces the linear equations in F.7) and F.8) to the
type considered in Chapter IV; cf. A1.1)-A1.3).
The statements concerning F.7) and F.8) can be written more con-
conveniently as matrix equations
F.9)
F.10)
Ut, rj, z)
= F{t, n,z)^L, |!L = / at t = t0,
dy0 oy0
= F(t,rj,z)%+ N(t,rj,z), ^ = 0 at t = t0,
dz dz
where drjldy0, dr/jdz denote Jacobian matrices.
106 Ordinary Differential Equations
Exercise 6.1. Let f(t,y,y') be a continuous ^/-dimensional vector
denned on an open (t, y, y')-set E. Let (ix,..., id) be any set of d integers,
0 < ij ^ d, such that no integer except 0 occurs more than once. Let
t = y° and suppose that fk(y°, y, y') has continuous partial derivatives
with respect to each of its arguments except possibly yik. Then the initial
value problem
v" = f(t, y, y'), y(t0) = y0, y'(t0) =y0',
where y'* ^ 0 if ik ^ 0, has a unique solution y = rj(t, t0, y0, «/„') and
¦>l(t, t0,2/o> 2/o')> ?}'(t, t0, y0, y0') are of class C1.
The following two exercises are applications of Theorem 6.1 and
Corollary 6.1 to differential geometry.
Exercise 6.2. (a) Let (gjk(x)), where x = (x1, . . ., x?), be a d X d
nonsingular symmetric matrix with real entries which are functions of
class C1 for small ||x|| and let (g'k(x)) be the inverse matrix. Consider the
initial value problem
F.11)
xr +
= 0, a<0) = x0 and x'@) = x0'
2 I
1=1 k=l
1
for the geodesies of ds2 = 22 gjk dx' dxk, where Tjk = T^k(x) are the
Christoffel symbols of the second kind denned by
2.
m=i
l_3ar dx' dx
Assume that "ds2 has a continuous Riemann curvature tensor" in the sense
that each of the d2 differential 1-forms co/ = Sj. Fjt dxk has a continuous
exterior derivative. Show that F.11) has a unique solution x = f(f, x0, x0')
for small \t\, \xo\ and arbitrary x0' and that f(f, x0, x0'), g'(t, x0, x0') are of
class C1 as functions of their 2d + 1 variables, (b) Let z = (z1,..., ze)
and x = (x1, . . ., xd), where e ^ d, and let z = z(x) be a function of class
C2 for small ||x|| with a Jacobian matrix C«3'/9x*) of rank J. Show that (a) is
applicable to ds2 = ||?fe||2 = US gft Jx3 Jx*, where glk is the scalar product
(dz/dx') ¦ {dzjdxk). (c) Show that *2 = [1 + 9(x2)^] [(dx1J + (dx2J],
where d = 2, has more than one geodesic through the point x0 = 0 in
the direction x0' = A,0).
Exercise 6.3. Let (hjk(y)), where «/ = (y1, «/2), be a 2 X 2 symmetric
matrix of real-valued functions of class C1 for small ||y|| such that det
(hjk) < 0. Let a denote the indefinite quadratic form
Then a has factorizations a = 2co1a>2, where a>i, co2 are linearly indepen-
independent differential 1-forms. (Note that a is not a differential 2-form and
Dependence on Initial Conditions and Parameters 107
a = 2a>xa>z is an ordinary, not exterior, product.) (a) There exists a unique
continuous differential 1-form co12 such that
dco1 = a>12 A a>2 and da>2 = a>1 A a>12.
(b) Assume that a has a continuous curvature K = K(y) in the sense that
the factors co^ co2 can be chosen so that (o12 has a continuous exterior
derivative, in which case, K is denned by
da>n = K co1 A a>2.
Show that there exist functions y(u) = y(ux, u2) of class C1 for small ||u||
such that «/@) = 0 and y = y(u) transforms a into the form
a = ITdu1 du2, where T = T(u) > 0
is of class C1 and has a continuous second mixed derivative such that
32(log 7Q
du'du*
+ KT=0.
(It can be shown that y = y(u) is of class C2; Hartman [16].) (c) Show
that (b) is applicable if a = 0 is the differential equation for the asymptotic
lines on a piece of surface of class C3 of negative curvature in Euclidean 3-
space. (d) Show that (b) is applicable if a = 0 is the differential equation
for the lines of curvature on a piece of surface of class C3 without umbilical
points in Euclidean 3-space.
7. S- and Z-Lipschitz Continuity
The proof of sufficiency in Theorem 6.1 falls into two parts: uniqueness
and differentiability. It turns out that the necessary and sufficient condi-
condition of Theorem 6.1 can be lightened considerably for "uniqueness" alone.
Consider the case in which no parameters z occur,
G.1) y'=f(t,y), y(to) = yo.
Correspondingly, A = A(t, y) and the analogue of F.2) is
G.2) co = A(dy-fdt).
Here, co = (co^ . . ., cod), where
G.3)
a>t=2,atldy'-
i=i
and when dco exists, it is of the form
G.4)
dm, = 2
ptt dt A dy\
108 Ordinary Differential Equations
where a.ij]c = — a.m, fty are continuous functions of (t, y). Correspondingly
where /«y = fti-- "
G.5)
F =
The notion of a differential 1-form co with a continuous exterior deriva-
derivative generalizes the notion of a C1 form co. Lemma 5.1 suggests the
following generalization of a 1-form co with uniformly Lipschitz continuous
coefficients:
A continuous linear differential form E.1) on a domain E will be said
to be S-Lipschitz continuous on E if there exists a sequence of 1-forms
co1, co2, .. . of class C1 on E such that con -*¦ co as n —>¦ oo uniformly on E
and dco1, dco2, ... are uniformly bounded on E.
Exercise 1.1. Show that if the coefficients of E.1) are uniformly
Lipschitz continuous on E, then E.1) is S-Lipschitz continuous on E.
Exercise 7.2. Consider the case of dimension d = 2. Let
a =p1(y)dy1
be continuous on a simply connected, open (y1, «/2)-set E with the property
that there exists a bounded, measurable function pn(y) on E such that for
any subset S of E bounded by a C1 piecewise Jordan curve / in E,
j Pi(y) dy1 + p,(y) dy2 = jj Pl2(y) dy1 dy\
Show that co is 5-Lipschitz continuous on every open subset Eo with
compact closure ?0 c ?. (It is understood that "measurable" means
measurable with respect to plane Lebesgue measure.) One might say that
co has a "bounded exterior derivative." This notion does not generalize
readily to arbitrary dimensions d, for a piece of B-dimensional) surface
S is of J-dimensional measure zero if d > 2.
The condition that each of the d forms of G.2) is S-Lipschitz continuous
can be generalized as follows: Let f(t,y) be continuous and A(t, y) be
continuous and nonsingular on E. The form G.2) is said to be L-Lipschitz
continuous on E if there exists a sequence of forms co1, co2,... of the type
G.6) co" = An(t, y) [dy -/„(/, y) dt]
of class C1 on E such that con —>¦ co uniformly on E as n -»¦ oo, and that
there exist constants c0, c satisfying
G.7)
col ^ An*Fn ^ cl for «= 1,2, ...
on is. Here jFn = Fn(t, y) is the matrix belonging to G.6) defined by the
analogue of formulae G.3)-G.5). The inequalities in G.7) have the follow-
following meaning: if B, C are two d X d matrices, B ^ C means that the
Dependence on Initial Conditions and Parameters 109
corresponding quadratic forms satisfy f ¦ B? < f ¦ Cf for all real d-
vectors f, where the dot indicates scalar multiplication. B ^ C is equiv-
equivalent to BH ^ CH, where BH = ?(# + B*) is the Hermitian part of B.
The form G.2) is called upper L-Lipschitz continuous on E if G.7) is
replaced by
G.8)
An*Fn ?
for « = 1,2,....
Consider these conditions when A,f are of class C1. In this case,
the exterior derivative of G.2) can be calculated formally from dco =
(dA) A(dy — fdi) — Adf hdt which gives
_ 1 ldaik dau\
G-9) a_*
Hence, G.5) shows that
G.10)
/«-l?+I^+I^-
at
If B = A*F = (bi}) and G = A*A, then
G.11)
2
where H = (/i,7) is skew-symmetric. This can be seen as follows: the
partial derivative of G = A*A with respect to Ms G' =*= A*A' + A*'A,
so that the Hermitian part of A*A' is |G'. This accounts for the replace-
replacement of the first term of G.10) by that in G.11). The same remark applies
to the third terms involving differentiation with respect to yk instead of t'.
For other applications of G.11), see § XIV 12.
Exercise 7.3. Let f(t, y) be continuous on E and satisfy [f(t, «/2) —
/('»2/i)]' (Vn — Vd ^ 0- Show that co = dy — f(t, y) dt is upper L-
Lipschitz continuous with An = I in G.6) and c = 0 in G.8).
8. Uniqueness Theorem
A generalization of the uniqueness theorem contained in Theorem 6.1
is the following:
Theorem 8.1. Let f(t, y) be continuous on an open (t, y)-set E and let
there exist a continuous, nonsingular matrix A(t, y) on E such that the
\-forms G.2) are S-Lipschitz continuous on E or, more generally, that G.2)
is L-Lipschitz continuous on E. Then G.1) has a unique solution y =
'/('»'o» Vo) for aH (t0, y0) e E. Furthermore, t](t, t0, y0) is uniformly Lip-
Lipschitz continuous on compact subsets of its domain of definition.
110 Ordinary Differential Equations
This theorem will be proved in § 10 below. It is possible to formulate a
one-sided uniqueness theorem analogous to Theorem III 6.2. This is
given by the following exercise.
Exercise 8.1. Let/, A be as in Theorem 8.1 except that G.2) is supposed
to be only upper L-Lipschitz continuous on E. Then G.1) has a unique
solution y = rj{t, t0, y0) to the right (t ^ t0) of t0 for all (t0, «/0) e E.
Furthermore, an inequality of the type
\n(t, h, «/i) - rfc, t2, ^l ^ const. \rj(t*, tx, yx) - r)(t*, t2, yj\
holds for t ^ t* ^ max (tlt t2) on compact subsets of the domain of
definition of rj(t, t0, «/„).
Exercise 8.2 (Another One-Sided Generalization of Corollary III6.1).
Let f(t, y) be continuous on R: 0 ^ t ^ a, \y\ ^ ft. On R, let there exist
a continuous, nonsingular yi(?,«/) such that G.2) is upper L-Lipschitz con-
continuous on^?: @ <) e ^ t ^ a, \y\ < ft with G.8) replaced by
F A < -
i.e.,
n A.n
on Re
for n = 1, 2,.... Then G.1) has a unique solution for t0 = 0, \yo\ < ft.
9. A Lemma
The proof of Theorem 8.1 will depend on the Uniqueness Theorem
III 7.1 and the following lemma.
Lemma 9.1. Let f(t, y) be of class C1 and let A(t, y) be a nonsingular
matrix of class C1 on an open set E such that
(9.1)
A*F ^ fi(t)A*A,
where fi(t) is a continuous function. Let y = rj(t) = r)(t, t0, «/0) be a solution
of G.1) and J(t,y) the Jacobian matrix (dfjdy). Then a solution x(t) of
the linear "equations of variation"
(9.2) x' = /(/, r,(t))x
satisfies, for t ^ t0,
(9.3) \\A(t, j?)a</)|| ^ Ufa, 2/oM'o)li exp \ fj,(s) ds.
Proof. A differentiation of A(t, r)(t))x(t), using (9.2) and G.10), shows
that (9.2) implies that"
(9.4)
(Aox)' = F<>z,
Dependence on Initial Conditions and Parameters 111
where the superscript 0 indicates that the argument is (t, y) = (t, rj) with
V = v(t) = ri(t, t0, y0). Thus
(9.5)
(A°x) ¦ (A°x)' = x ¦ (A°)*F°x,
since A°x ¦ F°x = x ¦ (A°)*F°x. Note that M°a;||2 = AH ¦ A°x = x ¦ (A0)*
A°x. Hence (9.1) and (9.5) imply that d \\AH\\*jdt ^ 2[i(t) {{AH^ and so
a quadrature gives (9.3); cf. Lemma IV 4.2.
Note that if
-cl ^ A*F <LcI,
m-1/ < A*A ^ ml,
c j> 0,
m > 1,
(9.6)
(9.7)
then —cmA*A ^ A*F^ cmA*A. In this case, (9.3) and the correspond-
corresponding inequality for t ^ t0 show that
(9.8) ||y40a;(f)|| ^ \\A(t0, yo)x(to)\\ exP cm \t — to\
where rj(t), hence x(t), is defined. In particular, since x = drj(t, t0, yo)ldyok
is a solution of (9.2) by Theorem 3.1, (9.6) and (9.7) imply that
(9.9)
dy0 II
Estimates for drj(t, t0, yo)/dto follow from the analogue of C.4),
Mo, 2/o) _ 4- d>?0, t0, y0) k
_ u J v'o> Vo)•
3s/o
(9.10) "'/v>> t0' yo; = - 2
10. Proof of Theorem 8.1
Consider first the caie that/, A are of class C1. Let (tu yx) 6 E and let
E° be a convex open neighborhood of (tlt yx) with compact closure E° c L7.
Then, by Peano's existence theorem (Corollary II 2.1), there exists an open
set Eo <= E° and numbers a, ft (> ^ > a) such that if (t0, y0) e ?0, then
y(t, t0, y0) exists for a ^ ? ^ ft and (t, rj) e L70. Note that Eo, a, ft depend
only on E° and a bound for |/| on E°.
Suppose, in addition, that (9.6) and (9.7) hold on E°. Then (9.9) and
(9.10) show that there exists a constant .ST(depending only on E°, abound
for I/I on E° and on m, c) such that
A0.1) \r,(t, t0, y0) - r,(t, t0, y°)| <K\y0- y°\
for a ^ / < ft, (/0,2/0) e ?0, (t, f)eE0;
A0.2) |,(/, t0, y0) - rj(t, t\ yo)\ <K\t0- t°\
for a-^f^b, (t0, y0) e ^o, (t°, y0) e ?0; and
A0.3) \V(t, to,yo) - V(t*, t0, yo)\ ?K\t- t*\
112 Ordinary Differential Equations
for a ^ t, t* ^ ft, (t0, «/„) 6 Eo. Finally, interchanging t and t0 in A0.1)
gives
A0.4) |y0 -y°\^K \rj(t, t0, y0) - r,{t, t0, y°)\,
provided that a^t<:b,(t, rj(t, t0, «/„)) e Eo> (t, rj(t, t0, y0)) e Eo. There is
a neighborhood Eoo c ?0 of (?1; yt) such that the last proviso [for A0.4)]
is satisfied if the interval [a, b] is sufficiently small and (t0, y0), (t0, y°) e ?00.
It must be emphasized that [a, b], Eo, Eoo and the constant K in A0.1)-
A0.4) depend only on ?°, a bound for | /1 in E°, and the validity of (9.6)-
and (9.7) on E°.
Return to the case that A,/are not necessarily of class C1; instead A,f
are only continuous, A is nonsingular and G.2) is L-Lipschitz continuous
on every open set E° with compact closure E° <= E. Then there exists a
sequence of d ordered 1-forms
co^ = An(t,y)dy-hn(t,y)dt,
on E° with C1 coefficients such that An -»¦ A, hn-+ Af uniformly on E°
as n -»¦ oo and G.7) holds on E°. Since A is nonsingular, An(t, y) is non-
singular on E° for large n, say, for all n, and coln) can be written as
A0.5) co^ = An(t, y) [dy -/„(/, y) dt],
where/„ = A~1hn ->-/uniformly on E°as n -»¦ oo. It can be supposed that
G.7) holds on is0 with c0 = — c ^ 0; also that there exists an m > 1 such
that
A0.6) w-1/ ? 4nMn ^ m/, m> 1, for «= 1, 2, ...
on?°.
For (t1} yx) e ?, let E° in the last paragraph be any convex open
neighborhood of (t1} yx) with compact closure E° <= E. Since //i,/2, ¦ ¦ ¦
are uniformly bounded for (t, y) e E°, there is an open neighborhood Eo of
(tu «/r) such that any solution y = «/(?) of G.1) or of
A0.7) y'=fn(t,y), y(to) = yo
for (t0, y0) e Eo exists and (?, «/@) e ?° for a ^ ? ^ ft, where a, ft (> ^ > a)
are independent of « and the solution y = y(t). Thus there exists a AT,
independent of n, such that if y = rjn(t, t0, y0) is the solution of A0.7),
then A0.1)-A0.4) hold r\ = r\n. In particular, the sequence rj^t, t0, «/„),
rj2(t, t0, y0), ... is uniformly bounded and equicontinuous for a 5! t ^ ft,
('o> 2/o) e ^o- Thus there exists a subsequence, which after renumbering
can be taken to be the full sequence, such that
A0.8)
rj(t, t0, y0) = lim rjn(t, t0, y0)
exists uniformly for a < t ^ ft, (/„, y0) e Eo.
Dependence on Initial Conditions and Parameters 113
By Theorem I 2.4, y = rj(t, t0, y0) is a solution of G.1) for a ^ t ^ ft.
Also, A0.1) and A0.4) hold under the conditions specified on t, /„, y0, y°.
The inequality A0.4) implies that if (t, t0, y0) is sufficiently near to (tlt tlt yj,
then no two distinct arcs y = r](t, t0, y0) pass through the same point
(t, y). Hence, by Theorem III 7.1, y = rj(t, tu y^) is the only solution of
G.1) with (t0, y0) = fe 2/0 on small intervals [t± - e, tj], [tlt tx + «].
Since (tlt y^) is an arbitrary point on E, Theorem 8.1 follows.
11. Proof of Theorem 6.1
Sufficiency. It is assumed that there exist continuous A, C such that
F.2) has a continuous exterior derivative and det A ^ 0 in a vicinity of a
point of E. It will first be shown that it is sufficient to consider the case
that/does not depend on z. To this end, write F.1) as A.4) and let
(y, z) be replaced by */*, so that correspondingly (/ 0) is replaced by/,,.
Let A^(t, y*) be the matrix
(A C\
0 I
and co* =
dy^ —/„, dt). Then co* = (co, dz), where co is given by F.2).
Hence co* has continuous exterior derivatives. Thus, if the asterisks are
omitted from «/*,/*, A*, it is seen that it is sufficient to consider the case
that/ = /(f, y) does not depend on z.
Let (*!, yO, E°, Eo, a, ft, A0.5), rjn(t, t0, y0) be as in the proof of Theorem
8.1. Then A0.8) holds uniformly for a ^ t ^ ft, (?„, «/„)'? ?0. Also, in
obvious notation, x = drjjdyok is the solution of
A1.1) [An(t, r,n)x\ = Fn{t,r,n)x, x(t0) = ek;
cf. the derivation of (9.4) from (9.2). Introducing the new variables
A1.2) x* = An(t,Vn)x
shows that
A1.3) x*' = Fn(t, t}n)A~\t, r,n)x*, x*(t0) = An(t0, yo)ek.
Since An(t,r)n)-+ A(t, rf), Fn(t, »?„)->- F(t,_V), A-\t,rjn) ->- A~\t,rj) as
/; -> oo uniformly for a ^ t ^ ft, (t0, y0) e Eo, Corollary IV 4.1 shows that,
for fixed (t0, */„) e ?0, lim An(t, rjn) drjjdyok exists uniformly for a ^ t ^ ft
as n —>¦ oo and is the solution of
A1.4)
x*' = F(t, rj)A-\t, r))x*, x*(t0) = A(t0, yo)ek.
Actually this limit is uniform for a ^ t ^ ft, (t0, y0) e Eo. This can be seen,
e.g., by Theorem 2.1, by constructing a family of linear differential
equations x*' = //(/, /„, */„, t)x*, where H(t, t0, y0, e) is a matrix continuous
114 Ordinary Differential Equations
in (t, t0, y0, e) which becomes Fn(t, rj^A'^t, r)n) for e = l/« and F(t, rj)
A~\t, rj) for e = 0.
If follows that
A1.5)
lim —^- exists uniformly
9y
for a ^ t ^ b, (to, y0) e Eo and is the solution of
A1.6) [A(t, rj)x\' = F(t, rj)x, x(t0) = ek.
Consequently, a standard term-by-term differentiation theorem implies
that drjldyok exists for a ^ t ^ b, (t0, y0) e Eo and is the limit A1.5). As
in the proof of Theorem 3.1, it is seen that drj/dto exists and is given by the
analogue (9.10) of C.4).
This proves that rj(t, to, y0) is of class C1 if (t, t0, y0) is sufficiently near to
(ti, ti, yx), where (tx, yx) is an arbitrary point of E. If now a and b are
chosen arbitrarily, subject only to the condition that rj(t, tx, yx) exists for
a ^ t ^ b, then a finite number of applications of formulae of the type
V(t, to, y0) = r)(t, t*, rj(t*, t0, y0)) shows that rj(t, t0, y0) is of class C1 on
its domain of existence.
All assertions of Corollary 6.1 except that concerning F.8) also have
been verified. The verification of this assertion will be left as an exercise.
Necessity. Assume that F.1) has a unique solution rj = rj(t, t0, 2/0> z)
which is of class C1. Let (tx, yx, zx) e E be fixed. Then there is a neigh-
neighborhood E° of (tx, yx, zx) such that rj(t, t0, y0, z) exists on an interval
containing t0, tx if (t0, y0, z) e E°. Also, since the Jacobian matrix dr)ldy0
is the identity matrix at (t, t0, yo> z) = (tx, tx, yx, zx), it can be supposed
that E° is so small that det (drj(tx, t0, y0, zo)ldyo) ^ 0 for (t0, y0, z0) e E°.
Consider the function rj(tx, t, y, z) of (t, y, z) e E° for fixed tx. Then
A1.7) dn(h, t, y, z) =
[dy - f(t, y, z) dt] + (|?) dz,
where C.4) has been used with (t0, y0) replaced by (t, y). Thus, if A(t, y, z)
= (.3rjldy0) at (t, t0, y0, z) = (tlt t, y, z) and C(t, y, z) = (dr)(h, t, y, z)jdz),
then co in F.2) becomes co = drj(tlt t, y, z) which has the continuous
exterior derivative dco = 0. Also, det A ^ 0. This completes the proof.
12. First Integrals
Consider a system of differential equations
A2.1) y'=f{t,y)
in which / is continuous on an open set E. A real-valued function u(t, y)
defined on an open subset Eo of E is called a first integral of A2.1) if it is
Dependence on Initial Conditions and Parameters 115
constant along solutions of A2.1). i.e., if y = y(t) is any solution of A2.1)
on a /-interval (a, b) such that (t, y(t)) e Eo for a < t < b, then u(t, y(t)) is
independent of t.
Lemma 12.1. Let u(t, y) be a function of class C1 on an open set Eo c E.
Then u(t,y) is a first integral of A2.1) if and only if it is a solution of
the linear partial differential equation
A2.2)
du S, du k,
dt *=i dy
0.
In fact, A2.2) is equivalent to du(t, y{t))jdt = 0 for all solutions y =
y(t) of A2.1) such that (/, y(t)) e Eo.
Theorem 12.1. Let u = rf(t, y), rj\t, y), . . ., rjd(t, y) be first integrals
o/A2.1) of class C1 on an open subset Eo c E such that the Jacobian matrix
(drj/dy) is nonsingular, where rj = (rj1, . . . ,rjd). Let (t0, y0) e Eo, rj0 =
r](t0, y0), and y = y(t, rj) the function inverse to rj = rj(t, y) for (t, rj) near
(t0, rjo). Then, for fixed rj,y = y(t, rj) is a solution o/A2.1). Furthermore,
if (to, Vo) e Eo and u(t, y) is of class C1 for (t, y) near (t0, yo)> then u(t, y) is a
first integral for A2.1) if and only if there exists a function U = U(rj) of
class O-for rj near rj0 such that u(t, y) = U(rj(t, y)) for (t, y) near (t0, y0).
Proof. Since u = ^(t, y) for i = 1 d is a first integral of A2.1),
it follows from A2.2) that
+ / = 0. Since y = y(t, rj) is inverse to rj = rj(t, y),
it is seen that y' = dyjdt = —(drjldyy^drj/dt), so that y = y(t, rj) is a
solution of A2.1) for fixed rj. This proves the first part of the theorem. ,
If U(rj) is of class C\ it follows readily from the criterion A2.2) that
u(t, y) = U(rj(t, y)) is a first integral. Conversely, let u(t, y) be a first
integral for (t,y) near (to,yo) and put U(rj,t) = u(t,y(t,rj)). Clearly,
u(t, y) = U(rj(t, y), t). Thus it sufficies to verify that U(rj, t) is independent
of t. But dUjdt = dujdt + Yi(dujdyk)yk' which is 0 by A2.1) and A2.2).
This proves the theorem.
Theorem 12.2. Letf(t, y) be continuous on an open set E. Then, for any
('0.2/o) e E, A2.1) has d first integrals rj = rj(t, y) of class C1 on a neighbor-
neighborhood of (t0, y0) satisfying det (drj/dy) ^ 0 if and only if the initial value
problem y' =f y(t0) = y0 has a unique solution y = rj(t, t0, y0) of class C1
(with respect to all of its variables.)
Proof. If y = rj(t, t0, y0) exists and is of class C\ put rj(t, y) = rj(t0, t, y)
for (/, y) near (/„, y0) and fixed t0. Then each component of rj(t, y) is a
first integral, for rj(t, y(t)) is the constant y(t0)- Also (drj(t, y)jdy) is the
unit matrix at t = /„, hence nonsingular for t near t0.
116 Ordinary Differential Equations
Conversely, if the components of r] = r](t, y) are first integrals of class
C1 on a neighborhood of a point (t0, «/„) of E, det (drjjdy) ^ 0 and y =
y(t, rj) is the inverse function, put rj(t, t0, y0) = y(t, r){t0, y0)). This is a
solution of the initial value problem y' =/, y(t0) = y0 and is of class C1.
For fixed t0, it follows that drj(t0, t, y) = (drj(t0, t, y)ldyo)[dy — f(t, y) dt];
cf. A1.7). Thus Theorem 6.1 implies that the solution y = rj(t, t0, y0) of
V = /> y({o) = Vois unique and of class C1.
Notes
section 3. Theorem 3.1 was first proved for d = 1 by the method of successive
approximations independently by Picard (see Darboux [1, p. 363]) and Bendixson [1].
(It had been proved earlier by Nicoletti [1] assuming an additional Lipschitz condition
on the partial derivatives of/.) Theorem 3.1, without the parameters z, was proved by
Peano [3] using a method similar to that in the text (except that, instead of using
Theorem 2.1, he employed an estimate of the type \xh(t)\ ^ exp c\t — t0 \ for C.12) where
c is related to bounds for \dfldy>\; cf. Lemma IV 4.1). The result was rediscovered by
von Escherich [2], using the method of successive approximations, and by Lindelof [2],
using a method similar to Peano's. In [1], Hadamard indicates a proof similar to that
in the text; Lemma 3.1 is given in Hadamard [3, pp. 351-352] with a different proof.
section 5. The definition of a continuous exterior derivative was given by E. Cartan
[1, pp. 65-71]. Lemma 5.1 is due to Gillis [2] (and, in a more general form, was used by
him to answer affirmatively the question of E. Cartan whether cor a w, has a continuous
exterior derivative if the r-form cor and i-form <o, have continuous exterior derivatives).
The proof of Lemma 5.1 in the text is that of H. Cartan [1, pp. 62-63].
section 6. Theorem 6.1 is in Hartman [17]; the proof in the text follows Hartman
[26]; cf. [14] for the case d = 1. For Exercise 6.2 and generalizations to extremals,
see Hartman [17]. For Exercise 6.3, see Hartman [14],
sections 7 and 8. See Hartman [26].
section 9. See Hartman [26]. Lemma 9.1 is essentially in Lewis [2] (cf. Opial [8]);
for a generalization see Lewis [3].
section 12. Theorem 12.1 goes back to Lagrange's work in 1779; cf. [2, IV pp.
624-634].
Chapter VI
Total and Partial Differential Equations
This chapter treats certain problems involving partial differential equations
which can be solved by the use of the theory of ordinary differential
equations. Parts I and II are independent of each other.
PART I. A THEOREM OF FROBENIUS
1. Total Differential Equations
Let H(y, z) be a continuous d X e matrix on a (d + e)-dimensional open
set E, say H = (ht]). Consider the set of total differential equations
A.1) dy - H(y, z)dz = 0 or dyl -J,h{j(y, z) dz1 = 0, i = 1, . . . , d,
3 = 1
and initial condition
A-2) yfc,) = y0'
for some («/„, z0) e E. Equation A.1) is an abbreviation for the set of
partial differential equations
(i.3) **¦:-
for i = 1, . . . , d and j = 1, . . . ,
e.
If e = 1, then A.3) is a set of ordinary differential equations; existence
and uniqueness theorems for corresponding initial value problems are
supplied by earlier chapters. If e > 1, then in general we cannot expect
A.1) to have solutions. For example, suppose that H{y, z) is of class
C1. If a C1 solution y = y(z) of A.1) exists, then A.3) makes it clear
that y(z) is of class C2 [since the right side of A.3) is of class C1]. But
then dy/dz' dzm = dty/dz™ dz1 and this leads to the condition
A.4)
dzn
dh
im
for / = 1,. .., d andy, m = 1, ..., e, which must hold along the solution
(?/. 2) = (?/(z), s). Thus, in this case, a necessary condition that A.1)-A.2)
117
118 Ordinary Differential Equations
have a solution y(z) = y{z, z0, «/„) for arbitrary points («/„, z0) of E, is that
the "integrability conditions" A.4) hold as an identity on E. When H is of
class C1, it will be seen that this necessary condition is also sufficient for the
existence of y = y(z, z0, y0) and, furthermore, in this case the solution of
A.1)—A.2) is unique, and y(z, z0, y0) is of class C1 with respect to all of its
arguments.
Instead of dealing with A.1)—A.2) directly, it is more convenient to
consider
A.5) m = A(y,z)[dy-H(y,z)dz],
where A(y, z) is a continuous, nonsingular d X d matrix, and to pose the
following problem: When does there exist a function
A-6) y = y(rj, z)
of class C1 on a (d + e)-dimensional neighborhood of a point (??„, z0)
such that
A.7) y(rH, z0) = y0, where (y0, z0) G E,
A.8)
det
0)
and that A.6) transforms A.5) into a differential form
A.9) oj = D(V,z)drj
in J?? = (drj1, . .., Jj/*) with coefficients depending, of course, on (rj, z).
(The insertion of A(y, z) in A.5) is for convenience only and does not
affect the problem.) When y = y(rj, z) exists, the system A.5) will be said
to be completely integrable at (*/„, z0).
If y = y(rj, z) is of class C1 and dy = (dy/drj) drj + (dy/dz) dz is inserted
into A.5), it is seen that the result is of the form A.9) if and only if
A.10)
in which case
A.11)
\dz
Ay,
-A = D(V, z).
In particular, A.8) implies that D(rj, z) is nonsingular.
The reduction of A.5) to a form A.9) is equivalent to satisfying A.10),
i.e., A.1) or A.3). Hence, the complete integrability of A.1) or A.5) is
equivalent to the existence of a family of solutions y = y(rj, z) of A.1)
depending on parameters rj = G?1, ..., if) and satisfying A.7), A.8).
The question of the existence of A.6) can be viewed from a slightly
different point of view: When does 00 in A.5) possess a local "integrating
Total and Partial Differential Equations 119
factor", i.e., when does there exist a nonsingular continuous matrix E(y, z)
such that E(y, z)oj is a total differential drj(y, z), E(y, z)co = drj. It is clear
that E(y, z) exists if and only if the problem just posed has a solution; in
which case, E(y, z) -- D~\rj(y, z),z) where rj(y, z) is the inverse function of
y = y{rj, z).
Finally, the question of the existence of A.6) can be dealt with in still
another way. Consider the problem of finding real-valued functions
u = u(y, z)ofd+e independent variables satisfying e simultaneous linear
partial differential equations
A.12)
I — = 0, j = 1, . . . , e,
By'
where H(y, z) is a continuous d x e matrix on an open set E. The system
A.12) is called the system adjoint to A.1).
The system A.12) is said to be complete on E if there exist d solutions
u = rj\y, z),. . . , rjd(y, z) on E such that the rank of the Jacobian matrix
(drjjdy, drjjdz) is maximal (i.e., d) at every point. Actually, A.12) shows
that this rank condition holds if and only if (dij(y, z)/dy) is nonsingular.
If d such solutions rj1,. . . ,rf exist, the system A.12) can be written as
Multiplication by (drj/dy) shows that this is equivalent to
A.14)
The condition det (drj/dy) 5^ 0 implies that rj = rj(y, z) has a local inverse
V = y(v> z) of class C1 and so, dy/dz = —{drjldy)-\drjldz). Thus A.14) is
equivalent to A.10), i.e., A.3). This argument can be reversed. Hence the
complete integrability of A.1) or A.5) at (*/„, z0) is equivalent to the
completeness of A.12) on a neighborhood of 0/o, z0). In particular, when
e > 1 and H(y, z) is of class C1, A.4) is therefore necessary and sufficient
for completeness of A.12) on a vicinity of every point («/„, z0) e E. The
condition A.4) can be written as
A.15) Xm[hit] = X,[him] for i=l,...,d and m,j = 1,.. ., e
by virtue of A.12)
The arguments of § V 12 show that if A.1)—A.2) has solutions for all
(.'/o- 2o) e E< then a function u(y, z) of class C1 on Eo <= E is a solution of
A.12) if and only if it is a "first integral" of A.1), i.e., u(y(z), z) = const,
for every solution y — y(z) of A.1) for which (y(z), z) 6 Eo.
A system A.1) or equivalent system A.10) for which the integrability
120 Ordinary Differential Equations
conditions A.15) are satisfied is called a Jacobi system. Theorem 3.2 is an
existence and uniqueness theorem for such systems. In Theorem 3.2, it
is not assumed that H(y, z) is of class C1, so that integrability conditions
take a form different from A.15), in fact, a more convenient algebraic
form [so that no calculations involve the rather formidable relations A.4)].
This theorem will be deduced from Theorem 3.1 which treats the question
of the complete integrability of A.1) or A.5).
Exercise 1.1. (a) Let blk(y), . . ., bdk(y) for A: = 1, 2 be 2d functions of
class C1. Define the partial differential operators Xk[u] = ?*jbik(y) dujdy1
for A: = 1, 2 and the operator X12[u] = H^X^b^] - X^by]) du/dy>. Show
that X12 is the commutator of Xlt X2 in the sense that if u(y) is of class C2,
then X12[u] = X^X^u)] - X^X^u)]. (b) Show that if u(y) is of class C1
and Xk[u] = 0 for k = 1, 2. Then X12[u] = 0. See Exercise 8.2F) for a
generalization.
Exercise 1.2. Let B(x) = (bu(x)), where i = 1, .. ., d + e and j =
1, . . ., e, be a (d + e) x e matrix of rank e, continuous on an open set
E in the (d + e)-dimensional x-space, x = (x1, . .., 3^+e). Define the
differential operators Y,[u] = ~L{ bu(x) dujdxi for j=l,...,e. The
system (*) Y,[u\ = 0,;" = 1, . . ., e, is called complete on E if there exist d
solutions u = ^(x), .. ., rjd(x) of class C1 such that the (d + e) X d
Jacobian matrix (d^/dx^ is of rank d. Using Theorem 3.1 show that if
B(x) is of class C1, then the system (*) is complete on a neighboorhood of
every point x0 e ? if and only if the commutator Ylk of Yt, Yk (cf. Exercise
1.1) is a linear combination of Ylt. . ., Ye [i.e., if and only if there exist
functions cjkm(x), for;, k, m = \,...,e, such that Yjk = ZmcjkmYm or,
equivalently, Yk[biS] - y,[6tt] = 2m cjkm(x)bim(x) for;, k = 1, . . ., e and
i = I,. . ., d + e].
2. Algebra of Exterior Forms
In order to be able to write integrability conditions in a convenient form,
some simple facts about exterior forms will be recalled.
Let 0 < r < d and Crd = d\ \r \ (d — r)! Consider a vector space W over
the real field of dimension Crd with basis elements eti ir, where
1 ^ i\ < i2 < ' ' ¦ < ir ^ d. Thus any vector co in the space W has a
unique representation
where cfi... ir are real numbers. Introduce the symbols e,-,...,-,. for
ji,. ¦ ¦, Jr = U ¦ ¦ ¦ ,d, where e5i... ir = 0 if two indices jt, ¦ ¦ ¦ ,jr are
equal and eSi... tr= ±^,... «r according as (y1;.. . ,jr) is an even or odd
permutation of (/1;.. ., ir), 1 ^ /\ < • ¦ • < ir ^ d. Then any vector co has
Total and Partial Differential Equations 121
a unique representation of the form
31=1 3r=l
subject to the conditions
B.1) ch---ir= ±c«i ¦¦•«,.
where (jlt. .. ,;r) is an even or odd permutation of (ilt . .., ir) and
1 ^ «! < ¦ ¦ ¦ < ir ^ d. In particular, c^ , .. 3-r = 0 if two of the indices
;1; ... ,;r are equal.
Change the notation for the "basis elements" e3i ¦ again, writing
ei1. . . tr = dy'1 A • • ¦ A dyir. Correspondingly, a vector is a differential
r-form
B.2) • oj=2,ch...udyhA-- ¦ Adyir
with constant coefficients subject to B.1).
As in the last chapter, multiplication of a differential r-form of and an
5-form of is defined as a differential (r + .s)-form obtained by the usual
associative and distributive laws and the anticommutative law dy1 A dy' =
— dy1 A dyf; so that of A of = (— 1)"ojs Aojr. This type of multiplication
will be referred to as "exterior" multiplication.
We can obtain a "change of basis" for the vector space W of differential
r-forms in the following way: Let T = (ttl) be a nonsingular d X d
matrix and let
B.3) dy* =2,1,^1', det(y^0,
and, more generally, let the "basis" elements be the exterior product
dy* A • ¦ • A dy1' = \2tHi drj'j A • • • A ^2 f,ri drj'J ¦
Then B.2) becomes a differential r-form of the type
B.4) oj=2 ?h ¦ ¦ ¦ u dV11 A ¦ ' ' A drI',
where the convention analogous to B.1) is observed.
In order to see that this is the usual type of change of basis for the space
W, it is necessary to prove the following:
Lemma 2.1. Let the "change of basis" B.3) transform B.2) into B.4).
Then B.2) is 0 (i.e., all ct 3- = 0) if and only if B A) is 0 (i.e., all yh . . . i
= 0).
This follows form the fact that "changes of basis" B.3) are associative,
i.e., if
dr,* = Zsti dV, where det (s(j) * 0,
i-i
122 Ordinary Differential Equations
transforms B.4) into
B.5) ca = 2,dll...]rd
then
3 = 1 \fc=l
transforms B.2) into B.5). Thus the choice (si}) = (t^)'1 gives Lemma 2.1.
If certain of the dyk are linear combinations of the others (e.g., if
dyh+1, . ¦ ., dy* are linear combinations of dy1, . . ., dyh with constant
coefficients), then B.2) becomes a differential r-form in dy1, . ¦ ., dyh.
This notion is involved in the following lemma:
Lemma 2.2. Let co1,...,cos be linearly independent differential l-forms
and co a differential r-form with constant coefficients. Then the exterior
product co1 A • • • A cos A co is 0 if and only if the relations co1 = ¦ ¦ ¦ = cos = 0
imply that co = 0.
Let the forms cox,. .. ,cos be given by
d
B.6)
ot = 2xij dy\
3=1
i = 1, . . ., s.
3
The assumption of linear independence means that ojx, . . ., a>s considered
as vectors are linearly independent; i.e., rank (sit) = s where i = 1, . . ., s
and j = 1, . . ., d. The relations co1 = ¦ ¦ ¦ = cos = 0 mean that s of the
dy' are expressed as linear combination of the other d — s dyk, in which
case co becomes an r-form in the latter. The statement of the lemma is
to the effect that this r-form is 0 (i.e., all coefficients are 0) if and only
if co1 A • • • A cos A co = 0.
The proof will show that the conclusion of Lemma 2.2 can be stated as
follows: co1 A • • • A cos A co = 0 if and only if there exist s differential
(r — l)-forms al5 . . ., as such that co = o^ A co-^ + • • • + a, A cos.
Proof. Join d — s rows to the d x s matrix (si}) to obtain a nonsingular
square matrix. Consider the change of basis B.3), where (tu) = (s^).
Then, with respect to the new basis, K>t = drf for 1 ^ / ^ s and, say, co
is given by B.4). The product co1A--AcosAco = 0 if and only if
every nonzero term of co contains a factor co{ = dr\l for 1 ^ ir ^ s, i.e.,
if and only if co = 0 when co1 = • • • = cos = 0. Thus the assertion is
correct with respect to the (drj1, . . ., J^)-basis and, by Lemma 2.1, with
respect to the (dy1,.. ., J«/d)-basis.
3. A Theorem of Frobenius
The following theorem of Frobenius is the main theorem concerning
A.5).
Total and Partial Differential Equations 123
Theorem 3.1. Let H(y, z) be a continuous d X e matrix on an open set
E. A necessary and sufficient condition for the complete integrability of
co0 = dy — (H(y, z) dz at («/„, z0) is that there exists a continuous, nonsingular
d X d matrix A(y, z) on a neighborhood of(y0, z0) such that if co = (cox, . . .,
cod) is defined by co = Aco0, then co has a continuous exterior derivative
satisfying
C.1) (co! A ¦ • ¦ A cod) A dcot = 0 for / = 1, . . . , d.
Conditions C.1) represent the integrability conditions. The expression
on the left is the exterior product of d differential l-forms cox, . . ., cod and
the differential 2-form dco{. Condition C.1) will be used in the equivalent
form (Lemma 2.2): co = 0 (i.e., co1 = • • • = cod = 0) implies that dco = 0
(i.e., dco1 = • • • = dcod = 0).
Exercise 3.1. Show that conditions C.1) reduce to A.4) if A(y, z) = I
and H(y, z) is of class C1.
Theorem 3.1 should be completed by the following:
Lemma 3.1. Let H(y, z) be a continuous d X e matrix on an open set E.
Let D. = D.(E) be the (possibly empty) set of continuous, nonsingular
d X d matrices A(y, z) on Esuch that co = A[dy — Hdz] has a continuous
exterior derivative. Then the integrability conditions C.1) hold for all
A efl or for no A e Q.
For example, if H(y, z) is of class C1, so that dy — Hdzh&s a continuous
exterior derivative, and if A.4) does not hold, then C.1) does not hold for
any choice of continuous nonsingular A.
Exercise 3.2 (A Simplified Version of Lemma 3.1). Let A(y, z) and
H(y, z) be continuous, det A ^ 0, and co = A[dy — H(y, z) dz] have a
continuous exterior derivative satisfying the integrability conditions C.1).
Let A0(y, z) be a C1, nonsingular d X d matrix. Show that A0(y, z)co has
a continuous exterior derivative given by d(A0co) = A0dco + (dA0) A co,
and hence that the form A0co satisfies integrability conditions analogous to
C.1).
Exercise 3.3. Let x, y, z be real variables; P(x,y,z), Q(x,y,z),
R(x,y,z) real-valued functions of class C1; and P* + Q2 + B? ^ 0.
Show that the integrability condition co A dco = 0 for the existence of local
integrating factors for co = Pdx + Q dy + Rdz is that P(Ry — Qz) +
Q(PZ - Rx) + R(Qy - Px) = 0.
Exercise 3.4. Show that if H(y, z) is continuous on E and there exists a
continuous nonsingular A(y,z) on E such that co = A[dy — H(y,z)dz]
has a continuous exterior derivative satisfying the integrability conditions
C.1), then every point («/„, z0) e E has a neighborhood Eo on which there
is defined a sequence of C1 l-forms co1, co2, . .. such that con —*¦ co and
</c>" --*¦ du> uniformly as n —»¦ oo and wn satisfies the integrability conditions.
124 Ordinary Differential Equations
Exercise 3.5. Let H(y, z) be continuous on E. Show that co = dy —
H(y, z) dz has a continuous exterior derivative dco satisfying the integra-
bility conditions C.1) if and only if H(y, z) has continuous partial deriva-
derivatives with respect to the components of?/ (cf. Exercise V 5.1) and, for
l^y < m < e and i = 1, . . . , d,
0.2, I *„ *.
a, *,„ _ & *„
for all rectangles S with boundaries /, S c E, in the 2-planes y = const,
and zk = const, for k ^j,m; cf. A.4).
Theorem 3.2. Le? H(y, z) be continuous on an open set E. A necessary
and sufficient condition for A.12) to be complete on a neighborhood of a point
(y0, z0) e E is the existence of a continuous nonsingular matrix A(y, z) on a
neighborhood of (yo,zo) as in Theorem 3.1. In particular, if e > 1 and
H(y, z) is of class C\ then A.4) [or A.15)] is necessary and sufficient for the
(local) completeness of A.12).
The concept of completeness leads at once to:
Corollary 3.1. Let H(y,z) be continuous on a neighborhood of a point
(y0, z0) and let A.12) be complete [i.e.,possess d solutions u = r)\y, z),. . . ,
rjd(y, z) such that rj = (rj1, . . . , rjd) satisfies det (drjjdy) ^ 0]. Put rH =
rj(y0, z0). Let u(y, z) be a real -valued function of class C1 on a neighborhood
qf(yo,zo). Then u(y,z) is a solution of A.12) if and only if there exists a
function U(rf) of class C1 for rj near rH such that u(y, z) = U(rj(y, z)) for
(y, z) near (y0, z0).
The proof of Theorem 3.1 will be given in §4 and that of Lemma 3.1
in §5. Theorem 3.2 follows from Theorem 3.1 and the considerations of
§ 1. The proof of Corollary 3.1 is similar to that of Theorem V 12.1 and
will be omitted.
4. Proof of Theorem 3.1
Necessity. Let there exist a function y = y(rj, z) of class C1 in a vicinity
of a point (rjo,zo) satisfying A.7), A.8) and transforming
D.1) co0 = dy- H(y,z)dz
into a form
D.2) co0 = D0(rj, z) drj.
Thus D0(rj, z) = (dy/drj) is nonsingular; cf. A.11). By condition A.8),
A.6) has an inverse rj = r\(y, z) of class C1 on a vicinity of (*/„, z0). Let
A(y, z) = Da-\r)(y, z), z); so that A is continuous and nonsingular,
furthermore, co = A(dy — H dz) = dr\ has the continuous exterior
derivative dco = d(drj) = 0. This proves the "necessity."
Total and Partial Differential Equations 125
The proof of "sufficiency" will depend on the following lemma which
exhibits the role of the integrability conditions C.1).
Lemma 4.1. Let f(t, y, z) be continuous on a neighborhood of a point
(t0,2/o» zo) and let there exist a continuous, nonsingular d X d matrix
A(t, y, z) and a continuous d X e matrix C(t, y, z) such that
D.3) co = A(dy — fdt — C dz) has a continuous exterior derivative
satisfying C.1). Let y = rj(t, t0, yu z) be the solution of
D.4) y'=f(t,y,*), y(to) = y1.
Then the matrix
(art©-<*•
is independent of t.
Proof of Lemma 4.1. By Theorem V 6.1, D.4) has a unique solution
y = rj(t, t0, ylt z) of class C1. In the notation of Corollary V 6.1 [cf.
(V 6.9)-(V 6.10)], Y = A(t, rj, z) (drj/dy^ is a fundamental matrix for the
linear system
D.6) Y' = F(t,r),z)A-\t,V,z)Y
and Y = A(t, rj, z)[(drj/dz) — C(t, rj, z)] is a solution of
D.7) r = F(t, rj, z)A-Hf, r,,z)Y + N(t, rj, z) + F(t, rj, z)C(t, rj, z).
Note some differences in notations here and in §V6; here yx plays the
role of y0, and — AC in D.3) that of C in (V 6.2).
For t0 fixed, the change of variables (t, y, z) -> (t, yx, z), where y =
V {t-. t0, y-i, z), transforms D.3) into
D.8)
co = A(t, rj, z){ (J^
Since the coefficients in D.8) possess continuous derivatives with respect
to t, the proof of Lemma V 5.1 shows that
dco=U —
dt A dz + ¦ ¦ ¦ ,
where the omitted terms involve dyx' A dy*, dyj A dzk, and dz' A dzk. In
view of the remarks concerning D.6) and D.7), this is
D.9) da = f(^-) dt A dy, + If ^\ + N\ dt A dz + ¦ ¦ ¦ ,
where the argument of A, C, F, N is (/, rj, z) and rj = rj(t, t0, «/i, z).
126 Ordinary Differential Equations
At a fixed (t, yx, z), choose dy1 to make to = 0, i.e.,
then D.9) has the form dto = {• • •} dt A dz + (terms in dz'A dzk), where
{• • •} is given by
<->-'[© -
Since D.10) makes w = 0, Lemma 2.2 and the integrability conditions
C.1) imply that dm = 0. In particular {¦¦¦} = 0, i.e., N + FC = 0. In
this case, D.7) reduces to D.6). Thus Y = A (drj/dy^ is a fundamental
solution of D.6) and Y = A[(dr)/dz) — C] is a matrix solution of the same
system D.6). Consequently there is a matrix C° independent of ? such that
cf. §IV 1. Since the matrix D.5) is C°, the lemma is proved.
Proof of "Sufficiency" in Theorem 3.1. By assumption, there is a
continuous, nonsingular A(y, z) satisfying the conditions of the theorem.
Change notation as follows: let t = z1; if e = 1, let z2 = 0 and if
e > 1, let z2 be the (e — l)-dimensional vector (z2, . . . , ze). Then A.1) can
be written as
w = A(y, t, z2)[dy -fdt - C2 dz2],
where/ is the first column of the matrix H and C2 is the matrix consisting
of the other columns of H. Let y = r\{t, z2, «/j) be the solution of y =
f(y, U z2), yiz,,1) = y1 for fixed z^. The change of variables (y, t, z2) -*
(yls ?, z2) transforms w into the form
— H2(yu z2) dz2],
This can be written as
where 772 does not depend on t = z1 by Lemma 4.1; cf. D.8) and D.5).
The form w, of course, satisfies the integrability conditions in the new
variables («/1; z).
If e = 1, so that z2 = 0, the theorem is proved. If e > 1, change
notation by letting z2 = t and z3 = 0 or z3 = (z3, . .., ze) according as
e = 2 or e > 2 and write
co =
-fxdt- C3dz3],
Total and Partial Differential Equations 127
where/j =f-l{y1, t, z3) is the first column of H2 and C3 is the matrix con-
consisting of the other columns. Let yx = rj^t, z3, y2) be the solution of
Vi =/i0/i> ?> Z3)> 2/i(zo2) = 2/2- Thus rj1 does not depend on z1. The change
of variables («/1; z1, ?, z3) —>¦ (y2, z1, ?, z3) transforms a> into a form of the
type
where, by Lemma 4.1, H2 = H3(y2, z3) does not depend on z2.
If e = 2, so that z3 = 0, the theorem is proved. It is clear that the
process can be continued to obtain a proof of the theorem for any e > 2.
5. Proof of Lemma 3.1
Since the integrability conditions are local conditions, it can be supposed
that E is the neighborhood of a point («/„, z0). Suppose that Q.(E) is not
empty and that the integrability conditions are satisfied for some element
of tl(E). Then, by Theorem 3.1, there exists a y = y(j\, z) satisfying A.7)-
A.8) and transforming
E.1) w0 = dy - H(y, z) dz
into
E.2) w0 =
Let A(y, z) be an arbitrary element of D.(E). Then
E.3) w = A{dy-Hdz)
is transformed into
E.4) « = D(rj, z) drj, where D = A(y, z) (— I.
XOTj'
Since the property of having a continuous exterior derivative is not lost
under a C1 change of variables y,z —>¦ r\, z [see the definition of continuous
exterior derivative in § V 5], A e Q.(E) implies that E.4) has a continuous
exterior derivative. It is clear from the proof of Lemma V 5.1 that dco
is of the form
Bik(V, z) drf A dz\
d d
E.5) dco =2
where Ajk = —Akj, Bjk are continuous J-dimensional vectors.
As D(rj, z) is nonsingular, to = 0 is equivalent to dr\ = 0, in which case
d(o = 0. Thus E.4) satisfies the integrability condition. But then E.3),
which results from E.4) by a C1 change of variables rj, z -+y, z, also
satisfies the integrability condition. This proves the lemma.
128 Ordinary Differential Equations
6. The System A.1)
The theorems of § 3 give necessary and sufficient conditions for the
existence of local solutions of initial value problems A.1)-A.2).
Theorem 6.1. Let H(y, z) be a continuous d X e matrix on an open set
E. A necessary and sufficient condition that A.1)-A.2) have a unique
solution y = y(z, z0, y0)for z near z0 and all (y0, z0) e E such that y(z, z0, y0)
is of class C1 is the complete integrability of(\.5) at every (y0, z0) e E, i.e.,
the existence of an A(y, z) for each («/„, z0) e E as in Theorem 3.1. In this
case, if the product of the two Euclidean spheres R: (\\y — yo\\ ^ b) X
(II2 - zo|| ^ a) is in E and if \r\- H(y, z)?| < M for all Euclidean unit
d-dimensional vectors r\, e-dimensional vectors ? and (y, z) e R, then
y(z) = 2/B> 2o> 2/o) exists on the sphere \\z — zo\\ ^ min (a, b/M).
It follows, as remarked in § 1, that if e > 1 and H(y, z) is of class C\
then A.4) is necessary and sufficient for the existence of local solutions
(l.l)-(l-2) with (yo,zo) arbitrary.
Particularly important cases of A.1) are those in which H(y, z) is linear
in y. Let H0(z), H^z),.. ., Hd(z) be continuous d X e matrices on an
open z-set D and let
F.1)
so that A.1)-A.2) becomes
F.2) dy - \ho(z) + J Hk(z)A dz = 0, y(z0) = y0.
Corollary 6.1. Let Hk(z) = (hkij(z)), where k = 0, . . ., d, be continuous
d X e matrices (i = 1,. . ., d and j = 1,. . ., e) on an open z-set D.
Necessary and sufficient for F.2) to have a solution y = y(z, za, y0) for all
z0 6 D, y0 arbitrary is that for 1 < j < m ^ d, and r = 0, 1, . .., d,
F.3) f hrlj dz' + hHm dzm=2 f (hkiihrkm - hkimhrk}) dz1 dzm
for every rectangle S with boundary J, S c D, on coordinate 2-planes
zh = const, for h ^ j, m. In this case, the solution y(z, z0, y0) of F.2) is
unique and of class C1 with respect to all of its variables. When H0(z), . . .,
Ha(z) are of class C1, conditions F.3) are equivalent to
This corollary is a consequence of Theorem 6.1 and the following
exercises.
Total and Partial Differential Equations 129
Exercise 6.1. (a) Let Ho, ..., Hd be of class C\ show that conditions
F.4) are the integrability conditions for co = dy — H(y, z) dz in the case
F.1). (b) Let Ho, . . . , Hd be continuous. Show that co = dy — H(y, z) dz
in case F.1) has a continuous exterior derivative satisfying the integrability
conditions if and only if the conditions on F.3) of Corollary 6.1 are
satisfied.
Exercise 6.2 (Continuation). Show that if F.1) is continuous, then there
exists a continuous nonsingular A(y,z) such that A(dy — H dz) has a
continuous exterior derivative satisfying C.1) if and only if the same is
true of dy — H dz.
The proof of Theorem 6.1 combined with the usual arguments of the
monodromy theorem will be used to show that, in the linear cases F.1), the
solutions exist in the large.
Corollary 6.2. Let H(y, z) be continuous for z e D and all y, where D is
a simply connected open set. Assume that the sufficient conditions of
Theorem 6.1 for the local solvability o/(l.l)-A.2) are satisfied. For every
compact subset Do of D, let there exist a constant K = K(D0) such that
\\H(y, z)\\ ^ K(\\y\\ + 1) for zeD0 and all y. Then the solution y(z) =
2/B, 20, y0) o/(l.l)-0-2) exists for all z e D.
It will be clear that the condition \\H(y, z)\\ ^ K(\\y\\ + 1) can be refined
along the lines of Theorem III 5.1.
Proof of Theorem 6.1. Necessity. Let A.1)-A.2) have a unique solu-
solution y = y(z, z0, y0) for z near z0 of class C1. The Jacobian matrix (dyldy0) is
the identity matrix at z = z0 and hence is nonsingular for z near z0. For
fixed z0, put y(z, rj) = y(z, z0, if). This is of class C1 near the point
O?o> zo) = B/o>2o); it satisfies A.7), A.8) and transforms D.1) into a form
D.2) since, for fixed rj, y(z) = y(z, rj) is a solution of A.1), i.e. of A.10).
Thus "necessity" in Theorem 6.1 follows from that in Theorem 3.1.
Uniqueness. Let y = y(z) be a solution of A.1)-A.2), say on the
Euclidean sphere \\z — zo\\ ^ a. For a fixed e-dimensional vector ? of
(Euclidean) length one, consider the values y^(t) = y(z0 + tt,) of y(z) on
the line segment z = z0 + ??, 0 ^ t ^ a. By A.1)—A.2),
F.5)
d]h
dt
= H(ylt z0
y1@) = y0.
If H(y, z) is smooth (e.g., is uniformly Lipschitz continuous in y), then the
initial value problem F.5) has a unique solution which is necessarily
?/i@ = y(z0 + tt,) for 0 ^ t ^ a. The same can be concluded under the
conditions imposed here; namely, that A, H are continuous, det A ^ 0,
and A.5) has a continuous exterior derivative. For then the forms
F6)
,, z0 + tQ[dyl -
dt]
130 Ordinary Differential Equations
in dyx, dt for fixed z0, ? have continuous exterior derivatives. (This follows
at once from the definition of "exterior derivative" in § V 5.) Hence
Theorem V 6.1 implies the uniqueness of the solution of F.5).
Existence. By Theorem 3.1 there exists a y(j], z) of class C1 satisfying
A.7), A.8) and transforming A.5) into A.9). For r\ = rj0, y(rH,z) is a
solution of A.1)-A.2); i.e., y(z, z0, y0) = y(rj0, z) exists. It only remains
to verify that y(z, z0, y0) is of class C1 in all of its variables. By A.8), A.6)
has a C1 inverse rj = rj(y, z) for (y, z) near (y0, z0). But y(rj(t/i, zx), z) for
fixed (yu Zj), hence fixed rj = rjiy^ zx), is a solution of A.1) reducing to
yx when z = zx. In other words, y(z, zu yx) = y{r](yx, zx), z) which shows
that y(z, 2l5 yx) is of class C1.
Domain of Existence. It is readily verified that the conditions on H in
the last part of Theorem 6.1 imply that the solution yx(t) = yx(t, Q of
F.5) exists for 0 ^ t ^ a, where a = min (a, bjM), for all unit vectors ?.
In fact, if \\yx\\' is a right or left derivative, F.5) and the conditions on H
imply that | ||yj'| ^ M; cf. the proof of Lemma III 4.2. The existence
and uniqueness of solutions of A.1)-A.2) for all (?„, y0) imply that yx(t, ?)
is a function «/(z) of z = z0 + ??andthat«/ = y(z)is asolutionof(l.l)-A.2).
This proves Theorem 6.1.
Exercise 6.3. Assume that H(y, 2) is of class C1 and that A.4) holds.
Prove the existence of a solution of A.1)—A.2) by the use of F.5).
Exercise 6.4. Assume that H(y, 2) is of class C1 and that A.4) holds.
Let20 = 0. Prove the existence of a solution of A.1)-A.2) in the following
manner: Let h}(y, 2) be the ^-dimensional vector which is theyth column
of H{y, 2). Define y = y^z1) as the solution ofdyjdz1 = hx(y, 21, 0,. .., 0),
2/@) = y0. If y^z1,..., z'-1) has been defined, let 2//21,..., 2') be the
solution of dyjdz'' = hfy, z\ . . ., z'\ 0,..., 0), y@) = y^z1,..., z'-1).
Show that y = yjfe1,..., ze) is the desired solution of A.1)-A.2).
Proof of Corollary 6.2. It is clear from Theorem III 5.1 and the
conditions of Corollary 6.2 that, for a fixed ?, the solution yx = yx(t) of
F.5) exists on any ^-interval / containing t = 0 on which 2 = z0 + tl, is in
D. By the proof of unique ness of solutions y(z) = y(z,z0,y0)o( A.1)-A.2),
a pair of solutions y(z, z01, y01), y(z, z02, «/02) of A.1) on 2-spheres about z01,
202, where zOi = zo + t?, yOi = y^t,), and tx, t2 e J, coincide on any common
domain of definition. Consequently, the solution y = y(z, z0, «/0) can be
defined on an open subset of D containing the line segment z = z0 + ??,
teJ.
These arguments show that the same is true if the line segment on
2 = 20 + tt, is replaced by any polygonal path P in D which begins at
20 and has no self-intersections. If two such polygonal paths Px, P2 which
begin at z0 and end at zx are considered, the two solutions y{z, z0, y0)
defined on neighborhoods of Pu P2 agree at 2 = zx. First, this is clear if
Total and Partial Differential Equations 131
the paths Px, P2 are sufficiently near to each other. Second, it then follows
without this nearness condition on Pu P2 by virtue of the simply connected-
connectedness of D. Hence the solution y(z) = y(z, z0, y0) can be defined (as a
single-valued function of 2) on D, as was to be proved.
PART II. CAUCHY'S METHOD OF CHARACTERISTICS
7. A Nonlinear Partial Differential Equation
Consider a partial differential equation
G.1)
for a real-valued function u = u(y) of d independent real variables, where
F(u, y,p) is a real-valued function of 1 + d + d variables on an open-set
is2ti+i- A solution of G.1) is a function u = u(y) of class C1 on an open
y-set Ed such that (u(y), y, uy(y)) e is2ti+1 for y e Ea and G.1) becomes an
identity in y. Here uv(y) = (du/dy1, . . . , du/dy*) is the gradient of u.
In general, solutions of initial value problems are sought, i.e., solutions
of G.1) which take given values on a piece of a hypersurface 5. To be more
explicit, let 5 be a piece of CMiypersurface in «/-space, i.e., let 5 be a set of
points
G.2) S:y= ?(y), where y = (/,. .., /-*),
?(y) is of class C1 in a vicinity of y = y0 and rank (dfydy) = d — 1. Let
<p be a given function on 5 or, equivalently, let 95 = <p(y) be a given func-
function of y for y near y0. Then the "initial condition" is the requirement
that the solution u = u{y) reduces to 95 on 5, i.e.,
G.3) uU(y)) = <p(y).
The existence theorems to be obtained are local in the sense that only
solutions u = u(y) defined for y near y0 = ?(y0) will be obtained. The
method to be used is that of Cauchy which reduces the problem to the
theory of ordinary differential equations and is called Cauchy's method
of characteristics. There is no analogue of this method for systems of
first order partial differential equations.
The following abbreviations Fu = dFjdu, Fy = (dF/dy1, ..., dFfiy*)
and Fv = (dFjdp1, . . ., dF/dp*) will be employed. A dot denotes the usual
scalar product of J-dimensional vectors.
In order to motivate the method to be employed, consider the following
heuristic arguments in the special case of G.1) which is a linear partial
differential equation of the form
G.4)
132 Ordinary Differential Equations
i.e., F{u,y,p) does not depend on u and is of the form F(u,y,p) =
?>fk(y)pk = Fv(y) -p. If u = u{y) is a solution of G.4) and y = y(t) is a
solution of the system of ordinary differential equations
G.5) y'=f=Fv,
then G.4) shows that u(y(t)) is a constant. Solutions y = y(t) of G.5) are
called characteristics of the partial differential equations G.4).
Figure 1.
Suppose that no characteristic is tangent to 5. This condition can be
expressed by
G.6)
det
dy
. Fv{y{y))
if 5 is given by G.2). In G.6), [(9?/3y), Fv] is a d x d matrix, the first
d — 1 column of which constitute the d x (d — 1) Jacobian matrix
C?(y)/3y) and the last column is the vector Fp.
In this case, S is said to be noncharacteristic and the characteristics
with initial points on 5 fill up a small piece Ed of «/-space. The value
«(«/) of a solution u at a point y e Ed must be the same as the given value
of <p(y) at the initial point ?(y) on 5 of the characteristic through y; see
Figure 1. Conversely, it is to be expected that a function u(y) defined on
Ed in this fashion is a solution of G.3)-G.4). Under suitable smoothness
conditions on Fthis turns out to be the case; cf. § V 12 for the relationship
between solutions of G.4) and first integrals of G.5).
Instead of a linear equation G.4), consider a somewhat more complicated
equation, say, a quasi-linear equation, i.e., an equation in which the highest
order (=first) partial derivatives occur linearly,
G.7) F = 2 f\u, y) |2 + U(u, y) = 0.
*=i dyh
If a solution u = u(y) is known and we consider a solution y(t) of G.5),
Total and Partial Differential Equations 133
where the right side is /(«, y) = /(«(«/), y), then the equation F = 0
implies that du{y{t))jdt = — U{u{y{t)), y(t)). This leads to the set of
ordinary (autonomous) differential equations
G.8) y'=f, u' = -U
in which the right sides are functions of u and y, but not of the independent
variable t. It will turn out that problems for the quasi-linear equation
F = 0 can be reduced to problems for this system of ordinary differential
equation.
Returning to the general nonlinear case G.1), characteristics will be
defined. These are not generally level curves of a solution as in the linear
case. Assume that F(w, y, p) is of class C1 on some open (w, y, p)-set
and assume that u = u(y) in G.1) is of class C2. We can reduce the
"nonlinearity" of G.1) by differentiating G.1) with respect to a fixed
component ym of y to obtain a second order partial differential equation
for u which is quasi-linear, i.e., linear in the second order partials of u.
This equation can be formally written as a first order, quasi-linear equation
for pm = dujdym,
Thus, in analogy to the above, we are led to the ordinary differential
equations
dt dp'
to which we can add dujdt = ^{dujdy^y'' or
7 — 1 J ^— —
j — i, . . . , a, —— — —
dt
dt »fi dp1
The differential equations for y' and u do not depend on m. Letting
m = 1,..., d gives a set of autonomous ordinary differential equations
for m, y, p which can be written as
G.9) y' = Fp, p'=-Fv-PFu, u'=p-Fv,
where the argument of Fu, Fy, Fv is (w, y, p) and the independent variable
t does not occur. A solution y = y(t), p = p(t), u = u(t) of G.9) is called
a characteristic strip and the projection u(t), y(t) of a solution into the
(w, «/)-space is called a characteristic. A condition of the type
G.10) Fv(u, y, p) 9* 0
prevents a characteristic from reducing to a point. The "derivation" of
G.9) will be stated as a formal result in Lemma 8.2.
134 Ordinary Differential Equations
A solution of the initial value problem G.1), G.3) cannot exist unless
there exists a function p = p(y) on 5 which is the gradient of u{y) at y =
Z,{y) and satisfies
G.11) F{<p{y),i{r\p) = 0
G.12)
a/
for i=l,...,d-L
The last condition results upon differentiation of G.3) with respect to y\
In particular, there is a vector p0 = p(y0) such that
G.13) F(u0, y0, po) = 0
(?14) Po.^ = ^ for i = l,...,d-l.
Assume that the "initial data is noncharacteristic at y = y0," i.e., that
G.15)
det
y0,
0,
where the first d - 1 columns of the matrix in G.15) are the vectors
dZ,(yo)ldy\ i = 1,. ¦ ¦, d - 1, and the last is Fp(u0, yo,po). Then, by the
implicit function Theorem I 2.5, G.13), G.14), G.15) and y = i(y) e C\
F e C\ <p e C2 imply that G.11 )-G.12) has a unique solution p = p{y) of
class C1 in a vicinity of y = y0 which satisfies p(y0) = p0-
Note that in the nonlinear case, we cannot speak of being non-
characteristic" but only of "the initial data being noncharacteristic."
The initial data consists of 5: y = Uy), the function <p(y), the vector p0,
and, when G.15) holds, the implicitly determined p(y). By continuity,
G.15) implies that
det
for y near y0. When G.16) holds, the initial data is called noncharacteristic.
Exercise 7.1. Suppose that F in G.1) does not depend on u. What is
the form of the system G.9) for the characteristic strips? Note that the
determination of u in G.9) reduces to a quadrature.
The initial value problem G.1), G.3) is often considered in another form,
to be obtained now. This form of the problem will be used in § 11. Suppose
that 5 in G.2) is of class C1. Without loss of generality, it can be supposed
that y0 = 0 and that 5 in G.2) is given in the form / = ipiy1, ¦¦¦, y*-1),
where y is of class C2 for (y\ . . ., y"-1) near 0 and y@) = 0. If
Total and Partial Differential Equations 135
are introduced as new coordinates, again called y, then in the new coordi-
coordinates 5 is a piece of the hyperplane yd = 0 near y = 0. The partial differen-
differential equation G.1) is transformed into another of the same type, although if,
e.g., the original F is of class C2, the new F has continuous first and second
derivatives except possibly for those of the type d^Fjdy1 dyk. The condition
G.15) in the new coordinates system becomes dF(u0, yo,po)ldpd 9* 0. Thus,
if G.13) holds, the equation F(u, y, p) = 0 can be solved for// in terms of
m, y,p\ ... ,pd~1, say/?" = —H{u, y,px,... ,pd-1\ and G.1) is equivalent to
pd + H(u, y,p\ . . . ,/-!) = 0 for (w, y,p) near (w0, yo,po).
Thus, if the notation is changed by replacing d — \ by d and y by
(y, t), then the initial value problem takes the form
G.17)
G.18)
ut + H(u, t, y, uy) = 0,
"@, y) = <p(y),
where u = u(t, y) is the unknown function, 95B/) is the given initial function,
ut = dujdt, and uy = (du/dy1,.. ., dujdyd).
Exercise 7.2. (a) Write H as H{u, t, y, q), where y = (y1,. . ., yd) and
q = (q1,.. ., qd). Find the differential equations for the characteristic
strips for the partial differential equation G.17), using t as the independent
variable, (b) Simplify the result of part (a), assuming that H(t, y, q) is
independent of u. [Note that in this case, when H is a Hamiltonian
function, G.17) is the Hamilton-Jacobi partial differential equation and
the nontrivial parts of the equations for the characteristic strips are the
equation of motion in the Hamiltonian form.]
8. Characteristics
The relationships between solutions of G.1) and characteristic strips or
characteristics will now be determined.
Lemma 8.1. Let F(u, y, p) be of class C1. Then F(u, y, p) is a first
integral of the system G.9); i.e., F is constant along any solution of G.9).
Proof. It suffices to verify that if y{t), p(t), u(t) is a solution of G.9),
then the derivative of F(u(t), y(t),p(t)) is 0. This is equivalent to
,- (-Fv - PFU) + Fu{p
= 0.
Lemma 8.2. Let F(u, y,p) be of class C1 on an open set E2a+1 and let
u = u{y) be a solution o/G.1) of class C2 on an open set Ed. For y0 e Ed,
there exists a characteristic strip y(t),p(t), u(t) for small \t\, such that
u(t) = u(y(t)),p(t) = u?df(t)) and y@) = y0.
In particular, in the (u, y)-space, the arc (u, y) = (u(t), y(t)) lies on the
hypersurface u = u(y) and (—1, p(t)) is a normal vector to this hyper-
surface at the point (u, y) = (u(t), y(t)). Thus if u = u(y) is a solution of
136 Ordinary Differential Equations
G.1) of class C2, the hypersurface u = u(y) can be considered to be made
up of characteristics.
Corollary 8.1. If the solutions of initial value problems associated with
G.9) are unique (e.g., if Fe C2) and u = u^y), u2(y) are two solutions of
G.1) of class C2 which "touch" at y = y0 [i.e., ux(y^ = u2(y0), ulv(y0) =
M2&B/o)]> then they "touch" along a characteristic arc y = y(t).
Proof of Lemma 8.2. [This is a repetition of the "derivation" of G.9).]
Consider a solution y = y(t) of the initial value problem
(8.1)
y' = Fv(u(y), y, uy(y)), y@) = y0.
Differentiating G.1) with respect to ym gives
(8.2)
Bu_
dF
+
dF d2u
3=1 dp1 dyj dyT
= 0
Putp(t) = uv(y(t)). Then (8.1) implies that (8.2) can be written as the wth
component of Fup + Fy + p' = 0, where the argument of Fu, Fy is
(u(y(t)), y(t), p(t)). Also, if 11 = u(y(t)\ then u' = uy(y(t)) • y' is p • F„ by
(8.1). Thus y = y(t),p = uv(y(t)), u = u(y(t)) is a solution of G.9). This
proves the lemma.
Remark. It remains unknown whether Lemma 8.2 is valid if it is only
assumed that u(y) e C1 and d > 2. In this direction, there is a partial
result for d > 2 and a complete result for d = 2:
Exercise 8.1. Let F(u, y, p) be of class C1, u(y) of class C1 in a neighbor-
neighborhood of y0 and a solution of G.1). (a) Let m be fixed, 1 ^ m ^ d. Show
that there exists a solution y(t) of the initial value problem
y' = Fv(u(y), y, uy(y)), y@) = y0
such thatpm(t) = [du(y)ldym]v=v{t) has a continuous derivative with respect
to t satisfyingpm' = —dF\dym — Fupm, where the argument of dFjdym, Fu
is [u(y(t)), y(t), uv(y(t))]. (b) In particular, if the solution of the initial value
problem y' = Fv(u(y), y, uy(y)), y@) = y0 is unique [so that y(t) in part (a)
does not depend on m], then the conclusion of Lemma 8.2 is valid. (This is
applicable, e.g., if F(u,y,p) depends linearly on p.) (c) If Fv =? 0 at
(u, y,p) = (u(y0), y0, uv(y0)) and d = 2, then the conclusion of Lemma 8.2
is valid (under the assumptions F eC1, ue C1).
Exercise 8.2. Let F(u, y,p) and G(u, y,p) be of class C1 on an open
(u, y, /?)-domain. Define the function H(u, y, p) by
H=FV-(GV+ PGU) - Gv ¦ (Fy + pFJ.
(If F, G are linear in/? and independent of u, //corresponds to the "com-
"commutator" of/7, G defined in Exercise 1.1.) Let u(y) be a solution of class C1
of both F = 0 and G = 0 on some y-domain D. (a) Show that if, in
Total and Partial Differential Equations 137
addition, u(y) is of class C2, then u(y) is a solution of H = 0. (b) Show that
if u{y) e C1 and if through each point (u, y,p) = (u(y0), y0, uy(y0)), there
passes a characteristic strip for F = 0 as in Lemma 8.2 [e.g., if Exercise
8.1F) is applicable], then u(y) is a solution of H = 0.
9. Existence and Uniqueness Theorem
The main theorem on G.1) is the following.
Theorem 9.1. Let F(u,y,p) be of class C2 on an open domain E2d+1.
Let (uo,yo,po) e E2d+1. Let G.2) be a piece of hypersurface of class C2
defined for y near y0 and ?(y0) = y0. Let <p(y) be a function of class C2for y
near y0 and <p(y0) = u0. Finally, let G.13), G.14), and G.15) hold. Then,
on a neighborhood Ed of y = y0, there exists a unique solution u = u(y) of
class C2 of the initial value problem G.1)-G.3).
Note that G.15) implies that Fv(u0, yo,po) ^ 0 and that rank (dt,(y)jdy)
is d — 1. The condition Fp(u0, yo,po) 9* 0 implies the existence of hyper-
surfaces 5 satisfying G.15). For example, if dF/dp" # 0 at (uo,yo,po),
then the hyperplane yd = yod is an admissible 5.
Proof. By the argument in §7, there is a unique function p = p(y) of
class C1 for y near y0 satisfying G.11)-G.12) andp(y0) = p0. Let y = Y(t, y),
p = P(t, y), u = U(t, y) be the solution of G.9) satisfying the initial
condition
(9.1) 7@, y) =
P@, y) = p(y), U@, y) = <p(y).
By Theorem V 3.1, this solution is unique and Y(t, y), P(t, y), U(t, y),
Y'(t, y), P'(t, y), U'(t, y) are of class C1 for small \t\ and y near y0. Since F
is a first integral for G.9),
(9.2)
F(U(t, y), Y(t, y), P(t, y)) = 0.
In fact, the function in (9.2) does not depend on t by Lemma 8.1 and, at
t = 0, (9.2) reduces to G.11).
By the first part of G.9) and by (9.1), at (t, y) = @, y0), the Jacobian
dY(t, y)ld(t, y) at t = 0, y = y0 is
d(y\...,f) _.,„
Thus assumption G.15) shows that this Jacobian determinant is not 0.
Hence, there exists a unique map
(9-3) t = t(y), y = y(y)
for y near y0 inverse to
(9.4) y = Y(t, y).
138 Ordinary Differential Equations
The map (9.3) is of class C1. Put
(9.5) u(y) = U(t(y), y{y))
for y near y0. Then (9.2) becomes
F(u(y), y, P[t(y), y(y)}) = °-
Thus the existence assertion will be proved if it is shown that
(9.6) uy(y) = P{t(y), y(y)),
i.e., du(y) = P{t{y), y(y)) ¦ dy. Under the change of variables (9.4),
y _>. (tt y\ with nonvanishing Jacobian, this is equivalent to
dU(t, y) = iW, Y) ¦
ay
+ P(t, y) • Y'(t, y) dt
or to
(9.8) U'(t, y) - P(t, y) • Y'(t, y) = 0.
The relation (9.8) follows from G.9). Thus only (9.7) remains to be
verified.
For a fixed y, let A/0 denote the expression on the left of (9.7). Thus it
suffices to show that A/0 = 0. Note that, by G.12) and (9.1),
(9.9) A/0) = 0.
In what follows, let F = F(u, y,p) and u = U(t, y), y = Y(t, y), p =
P(t, y). Differentiating the left side of (9.7) with respect to t gives
, , du' , dy dy'
The change of order of differentiation is permitted since y , p, u are of
class C1. Using G.9), the last relation becomes
If (9.2) is differentiated with respect to y\ it is seen that the sum of the first
two terms on the right is — Fu dujdy'. Hence,
Total and Partial Differential Equations 139
Since A/0 satisfies a linear homogeneous differential equation and the
initial condition (9.9), it follows that A/0 = 0.
Hence (9.5) is a solution of G.1)—G.3). Also u(y) is of class C2 since its
gradient (9.6) is of class C1. Finally, when F e C2, so that solutions of
G.9) are uniquely determined by initial conditions, the uniqueness of C2
solutions of G.1)—G.3) follows from Lemma 8.2, the remarks following it,
and the existence proof just completed. (For another uniqueness proof,
see the next section.)
It should be mentioned that when the initial data is not "noncharacter-
istic," in general there will not exist a solution.
In a sense, the existence Theorem 9.1 is unsatisfactory for it produces
solutions of class C2 when it is natural to ask for solutions only of class C1.
It is reasonable to inquire as to whether or not the differentiability con-
conditions can be lightened in Theorem 9.1 and still obtain solutions of class
C1. To some extent, this question can be answered in the negative.
Exercise 9.1. Let x, y be real variables and f(x) a continuous nowhere
differentiate function of a;. Show that ux — uy +f(x + y) = 0, «@, y) =
0 has no C1 solution. Thus continuity of F is not sufficient to assure the
existence of solutions.
Exercise 9.2. Even FeC1 and analytic initial data is insufficient to
assure existence of solutions. Let x, y, q be real variables. Let/(<7) be a
real-valued function of class C1 for small \q\ such that dfjdq is not Lip-
Lipschitz continuous at q = 0. Show that, on the one hand, the procedure
in the proof of Theorem 9.1 does not lead to a solution of ux =f(uv),
"@, y) = |«/2. (The difficulty arises from the fact that the analogue of the
map (9.4) has no inverse.) On the other hand, Exercise 8.1(c) implies that
if a solution exists, then it is obtainable by such a procedure. Hence there
is no C1 solution.
Exercise 9.3. The last exercise shows that, in a sense, the following is
the "best" theorem: Theorem 9.1 remains correct iff, ?(y), <p, uiy) e C2"
is replaced by "F, t,(y), cp, u(y) are of class C1 with uniformly Lipschitz
continuous partial derivatives." See Wazewski [3]. This can be proved
by a suitable modification of the proof of Theorem 9.1 using, e.g., the fact
that uniformly Lipschitz continuous functions possess total differentials
almost everywhere. For a different proof, see Digel [2].
10. Haar's Lemma and Uniqueness
Let G.1), G.3) be replaced by G.17) and G.18), i.e., by
A0.1) ut + H(u,t, y, uv) = 0,
A0.2) "@,2/) =
140 Ordinary Differential Equations
It follows from Theorem 9.1 that if H(u, t, y, q) is of class C2 on an open
set E2+2d containing the point (u, t, y, q) = (99@), 0, 0, ^@)) and cp(y) is of
class C2 for y near y = 0, then A0.1), A0.2) has a unique solution u =
u(t,y) of class C2 for small \t\, \y\. The situation as to uniqueness for
solutions of A0.1)-A0.2) is very simple.
-L(a-t)
Figure 2. The case d = dim y is 1.
Theorem 10.1. Let H(u, t, y, q) be defined on an open set is2+M containing
the point (u, t, y, q) = 0 and satisfy a uniform Lipschitz condition with
respect to (u, q). Let <p(y) be a function of class C1 satisfying 95@) = 0,
^@) = 0. Then A0.1)-A0.2) has at most one solution of class C1 on a
neighborhood Edofy = 0.
This follows by applying the following lemma (with C = N = 0) to the
difference v = u2(t, y) — u^t, y) of two solutions u^t, y), u2(t, y).
Lemma 10.1. Let v = v{t, y) be a real-valued function of class C1 on a
set R: 0 g t ^ a«a), \y{\ ^ L(a - t)for i = 1,. . . ,d and satisfy
A0.3)
A0.4)
\v@,y)\<:C,
dv_
dy«
+ M \v\ + N,
where L, M > 0 and C, N ^ 0 are constants. Then, on R,
A0.5)
See Figure 2.
\v(t, y)\ ^ Ce
m
N
eMt -
M
Total and Partial Differential Equations 141
Proof. Let C", N' be arbitrary constants satisfying
A0.6) C<C, N<N'
and put
u(t, y) = C'em + N'
M '
so that
A0.8) ut = L 2 —z + Mu + N' = Mu + N'.
*=i dyh
It will be shown that
A0.9) u(t, y) - v(t, y) > 0
on R, so that letting C ^-C and TV' -> N gives
v(t, y) ^ Cem + N '"'
A0.10)
M
Replacing v by — v in this argument gives A0.5).
It is clear from A0.3), A0.6), and A0.7) that u — v > 0 for small t > 0.
If A0.9) does not hold on R, there is a point (t0, y0) of R such that 0 <
t0 < a, A0.9) holds on that portion of R where 0 ^ t < ?0, and equality
holds at (?0,2/0).
For any of the 2d choices of ±, the points of the line segments
A0.11) (t, y) = (?, ±L(t0 -t) + yo\ ..., ±L(t0 - t) + yod)
are in R for 0 < t < t0, for |±L(?0 - t) + y*\ ^ L{a - t) since |yo*| ^
L(a — ?0); see Figure 2. The difference w — v at the point A0.11) is
positive for 0 ^ t < ?0 and 0 at f = f0. Consequently, the derivative of
u — v along the line A0.11) is nonpositive at t = t0. This gives
A0.12)
<0
at
From A0.8), w, = M« + W, so that ut = Mv + N' = M \v\ + N' at
(t, y) = (t0, y0). This fact and uy = 0 give
i;, ^ M \v\
± 1^
at
If ± is chosen so that ±dvjdyk = \dvjdyk\ at (t0, y0), the resulting inequality
contradicts A0.4) since W > N. This implies A0.9) on R and proves the
lemma.
Exercise 10.1. (a) Let B be the (?, y)-set 5 ={(?,«/): 0 ^ ?< a,
ck + Lkt ^ yk ^ dk - Lkt for k = 1, . . . , d}, where Lfe ^ 0, ck < <4, and
142 Ordinary Differential Equations
2L,fi ^ dk — ck. Let u(t, y) be a real-valued function of class C1 on B and
let m(s) = max u(s, y) taken over the set Bs = {(t, y):(t, y)eB,t = s}.
Then m(t), 0 ^ t < a, has a right derivative DRm{t) and there exists a point
d
(t, y0) e Bt such that m(t) = u(t, y0) and DRm{i) =
evaluated at (t, «/„). (b) Let co(t, u) be continuous for 0 < t < a, u ^ 0
and such that the only solution of u' = w(t, u) defined for 0 < t ^ e (<a)
and satisfying u(t) ->¦ 0 and u{t)jt ->¦ 0, as t ->¦ +0, is u{t) = 0. Let
//(w, t, y, q) be continuous for (w, ?,«/, ?) near (w, t, y, q) = 0 and
1/7A/!, ?, y, <7i) — /7(m2, ?, y, qz)\ = 2 Z.,. l^* — ^s*| + «(?, \^ — u2\). Let
9)(y) be of class C1 for small \y\ and satisfy ?)@) = 0, ^@) = 0. Then
A0.1), A0.2) has at most one solution on the set B.
Exercise 10.2. (a) Let B denote a bounded (t, «/)-set defined by in-
inequalities: 0 < t < a, bit, y) ^ 0 for / = 1, . . ., m, where b}(t, y) is a
real-valued function of class C1. It is assumed that every boundary point
of B lies on either t = 0, t = a, or k of the m hypersurfaces bf{t, y) = 0,
1 ^ k ^ m; also, if fc of the hypersurfaces *,(?,«/) = 0, say/ =j\,. . . ,jk,
d
have a point (t,y) in common, then the k differential 1-forms 2C6,/
dy*)dy\ where j =j\,... ,jk, are linearly independent at (t,y). Let
/7(«, ?, y, q) be defined on a A + 1 + d + fiT)-dimensional domain E with
a projection on the (t, «/)-space containing B. Let u{t, y), v(t, y) be real-
valued functions of class C1 on B such that (m, ?,«/, !<„) e E, (v, t, y, vy) e E.
Suppose that ut > H{u, t, y, uy), vt < H(v, t, y, vy) on B and that m@, y) >
v@, y). Finally, suppose that, at every boundary point (t, y) of B common
to k hypersurfaces b^, y) = 0 for / =j\, . . . ,jk, we have the inequality
u,t,y,\u -I^
for all non-negative numbers A1; . . ., lk such that (w, ?,«/, [u — S A46} .]„) e
B. Then w(?,«/) > v(t, y) on 5. See Nagumo [3]. (b) Let 5 be the same as
the (t, «/)-set in Exercise 10.1(a). Let H(u, t, y, q) be defined on a A + 1 +
j -I- rf)-dimensional set E with a projection on the (t, «/)-space containing
B and satisfying |//(w, t, y, qj - H(u, t, y, q2)\ < S Lk \qf - q2k\. Let
u(t, y), v(t, y) be C1 functions on 5 such that (w, ?,«/, «„) e E, (v, t, y, vy) e
E; that ut > H(u, t, y, uv) and vt ^ H(v, t, y, vy) on E; and that m@, «/) >
«@, y). Then m(?, «/) > v(t, y) on 5. (c) Deduce Lemma 10.1 from part (b).
Notes
section 1. Connections between the systems A.1) and A.12)were considered by Boole
in 1862; see E. A. Weber [1] for historical remarks and early references. Discussions
of integrability conditions and existence theorems for Jacobi systems go back A862)
Total and Partial Differential Equations 143
to Jacobi [1, V, p. 39] and to Clebsch [2] who introduced the concept of "complete
systems" A.12). (See also the reference to A. Mayer [1] in connection with Exercise 6.3.)
The result in Exercise 1.1F) is due to E. Schmidt. For references to Schmidt, Perron,
and Gillis and for generalizations of part F) see Ostrowski [1]. Further generalizations,
based on Plis [2], are given by Hartman [13]; see Exercise 8.2.
section 2. See E. Cartan [1, pp. 49-64].
section 3. Under analyticity assumptions, Theorem 3.1 is due to Frobenius [2].
The statement of the theorem in the text, avoiding differentiability assumptions, is due
to Hartman [17]. The proof in the text is adapted from E. Cartan [1, pp. 99-100]. For
a related but somewhat less general theorem on Jacobi systems, see Gillis [1].
section 6. The arguments in the "sufficiency" proof and the existence proof for
A.1)—A.2) suggested in Exercise 6.3 go back to A. Mayer [1]; cf. Caratheodory [1,
pp. 26-30]. The proof outlined in Exercise 6.4 was used by Weyl [2, pp. 64-68]. Still
another proof, using successive approximations, is given by Nikliborc [1]; cf. also
Gillis [1].
sections 7-9. The initial value problem considered in § 7 is the simplest example of
the type called "Cauchy's problem" in the theory of partial differential equations.
The differential equations G.9) for the characteristic strips, Theorem 9.1, and its proof
are due to Cauchy (about 1819) under conditions of analyticity; see, e.g., [1, pp.
423-470]. Actually, a few years earlier, Pfaff [1] had considered the problem of finding
solutions of G.1) and also introduced, in a cumbersome manner, the system G.9) for
the characteristic strips in order to reduce the problem to the theory of ordinary
differential equations. A treatment of nonanalytic equations G.1) awaited, of course,
a knowledge of Theorem V 3.1. In fact, both Picard and Bendixson's work in 1896,
mentioned in connection with Theorem V 3.1, were written from the point of view of
solving a nonanalytic linear partial differential equation. A theorem similar to Theorem
9.1 was given by Gross [1]. For fuller treatments of this problem, see Caratheodory [1]
and Kamke [5]. For a theorem involving slightly less differentiability conditions, see
Exercise 9.3 and the reference to Wazewski [3] and Digel [2]. Regions of existence for
the solutions were investigated by Kamke and also by Wazewski [2]. For Exercise 8.1,
see Plis [2]. The example in Exercise 9.1 was given by Perron [1].
section 10. Lemma 10.1 was given by Haar [1] for the purpose of proving the
uniqueness Theorem 10.1. (This paper contains a wrong proof for Lemma 8.2 under
the assumption that u? C1.) For Exercise 10.1, see Wazewski [1] (and Turski [1] for
a generalization which is contained in Exercise 10.2). For Exercise 10.2, see Nagumo [3].
Chapter VII
The Poincare-Bendixson Theory
The main part (§§ 2-9) of this chapter deals with the geometry of solutions
of differential equations on a plane (d = 2). The restriction to a plane
appears essential since the arguments will make repeated use of the Jordan
curve theorem. In § 10, the results obtained are applied to certain non-
nonlinear second order differential equations.
Recent extensions of the Poincare-Bendixson theory from planes to
2-dimensional manifolds are presented in the Appendix in § 12. The last
section (§ 14) concerns the behavior of solutions of differential equations
oh a torus.
1. Autonomous Systems
A system of differential equations in which the independent variable t
does not occur explicitly,
(i.i) y =f{y\
is called autonomous. A trivial but important property of such systems is
the fact that if?/ = y{t), a < t < /S, is a solution of A.1), then y = y(t + t0)
is also a solution for a— ?0<?</5 — t0 for any constant t0. An orbit
will mean a set of points y on a solution y = y(t) of A.1) without reference
to a parametrization.
Any system y' = f(t, y) can be considered autonomous if the dependent
variable y is replaced by the (d + l)-vector (t, y) and the system y' =/is
replaced by t' = 1, y' =f(t,y), where the prime denotes differentiation
with respect to a new independent variable. For most purposes, however,
this remark is not useful.
A point y0 is called a stationary or singular point of A.1) if/C/0) = 0
and a regular point if/C/0) / 0. The stationary points y0 are characterized
by the fact that the constant y(t) = y0 isasolution of A.1). When solutions
of A.1) are uniquely determined by initial conditions, f(y0) = 0 and
y(t0) = y0 for some t0 imply y{t) = y0. This need not be the case in
general.
144
The Poincare-Bendixson Theorey 145
If A.1) has a solution C+:y = y(t) defined on a half-line t ^ t0, its set
li(C+) of co-limit points is the (possibly empty) set of points y0 for which
there exists a sequence t0 < ?x < . . . such that tn -> oo and y(tn) -> y0 as
n -> oo. Correspondingly, if C~:y = y(t) is denned for t ^ t0, we define
the set A(C~) of a-limit points and if C:y = y(t) is denned for — oo < t <
oo, its set of limit points is denned to be A(C) U Q(C).
Remark 1. Q(C+) is contained in the closure of the set of points
C+:y = y(t),t^ t0.
Theorem 1.1. Assume that f(y) is continuous on an open y-set E and
that C+:y = y+(t) is a solution of(\.\)for t ^ 0. Then Q(C+) is closed. If
C+ has a compact closure in E, then Q(C+) is connected.
Proof. The verification that Q(C+) is closed is trivial. In order to prove
the last part, note that by Remark 1, fi(C+) is a compact set. Suppose
that Q(C+) is not connected, then it has a decomposition into the union of
two closed (hence, compact) sets Cu C2 such that dist (Cu C^) = d > 0.
It is clear that there exists a sequence 0 < t1 < t2 < ¦ ¦ ¦ of ^-values
satisfying dist (y+(t2n+1), CO -> 0, dist (y+(t2n), C2) -> 0 as n -> oo. Hence,
for large n, there is a point t = tn* such that tn < tn* < tn+1, dist (y+(tn*\
Ct) ^ E/4 for / = 1,2. The sequence «/+(?/), y+(t2*), ¦ • ¦ has a cluster
point y0, since C+ has compact closure. Clearly, y0 c Q(C+) and dist
G/o> Q) = <V4 for / = 1,2. This contradiction proves the assertion.
Theorem 1.2. Letf, C+ be as in Theorem 1.1 andy0 eE O Q(C+). Then
A.2)
y'=f(y\
has at least one solution y = yo(t) on a maximal interval («_, a>+) such that
yo(t) e Q(C+) for w_ < t < co+. In particular, when C+ has a compact
closure in E, then Co: y = yo(t) exists on (— oo, oo) and Co U A(C0) U
Q(C0)
An orbit C0:y = yo(t), w_ < t < co+, which is contained in some
Q(C+), C+ <? Co, is called an (cu-) limit orbit. If, in addition, y = yo(t) is
periodic, yo(t + p) = yo(t) for all t and some/7 > 0, the orbit C0:y = yo(t)
is called an (co-) limit cycle. (The condition C+ <? Co assures that not
every periodic C0:y = yo(t) is a limit cycle; cf. the case of a family of
closed orbits.)
Proof. Let t0 < t1 < .. . and tn -> oo, yn-+y0 as n -> oo, where
yn = y+dn)- Then yJj) = y+({ + Ois a solution of
A.3)
y' =/C/)> 2/@) = yn-
It follows therefore from Theorem II 3.2, where fn(t, y) =f(y) for n =
1,2,..., that A.2) has a solution yo(t) on a maximal interval (co_, w+)
and that there exists a sequence of positive integers n(l) < nB) < . ..
146 Ordinary Differential Equations
such that
A.4)
yo(t)=limyn(k)(t) =\im y+(t + tn(k))
holds uniformly on compact intervals of co_ < t < co+. It is clear that
yo(t) e Q(C+) for co_ < f < co+. This proves the first part of the theorem.
The second part concerning existence on (— oo, oo) follows at once from
Theorem II 3.1 which implies that the right maximal interval [0, co+) for
y = yo(t) is either [0, oo) or yo(t) tends to dE as t -* a>+ < oo.
The last part concerning A(C0) and Q(C0) follows from A.4) and the
fact that Q(C+) is closed.
Remark 2. If solutions of all initial value problems associated with A.1)
are unique, then "selection" in the proof of the theorem is unnecessary;
thus yn = y+(tn) -* y0 as n -* oo implies that
A.5) yo(t)=hmy+(t + tn)
holds uniformly on every closed, bounded interval in (a>_, a>+).
Corollary 1.1. If Q(C+) consists of a single point y0 e E, then y0 is a
stationary point and y+(t) -*¦ y0 as t -*¦ oo.
2. Umlaufsatz
In the plane (d = 2), where the Jordan curve theorem is available, the
notions of the last section can be carried much further to give the Poin-
care-Bendixson theory. This will be done in §§ 4-6. In order to avoid an
interruption of the proofs, the idea of the index of a plane stationary point
will first be discussed.
Recall that a Jordan curve J is defined as a topological image of a circle;
in other words, J is a «/-set of points y = y(t), a ^ t ^b, where y(t) is
continuous, y(a) = y(b), and y(s) 5* y(t) for a < s < t < b. The Jordan
curve theorem will be stated here for reference. For a proof, the reader is
referred, e.g., to Newman [1, p. 115].
Jordan Curve Theorem. If J is a plane Jordan curve, then, its comple-
complement in the plane is the union of two disjoint connected open sets, Ex and E2,
each having J as its boundary, dE1 = dE2 = J.
One of the sets Et or E2 is bounded and is called the interior of J;
furthermore the interior of J is simply connected.
Consider a continuous arc J:y = y(t), a ^ t ^ b, in the y = (y1, y2)
plane. Let rj = rj(t) 5* 0, a ^ t ^ b, be a continuous 2-dimensional
vector attached to the point y(t), i.e., rj 5* 0 is a vector field on J. Consider
an angle (p = (p(t) from the positive ^-direction A,0) to rj(t), so that
cos <p = »77ll»7ll» sin <p = rj2l\\rj\\, where ||»?||2 = to1J + (r)*)*. These
formulae determine cp(t) up to an integral multiple of 2v but if <p(t) is fixed
The Poincare-Bendixson Theory 147
at some point, say t = a, then <p(t) is uniquely determined as a continuous
function. By (p(t) below is always meant such a continuous determination.
Define^) by
2w;,(J) = <p(b) -
For example, if rj(t) is continuously differentiate,
If J = J1 + J2 in the sense that J:y = y(t), a ^ t ^ b, and a < c < b,
Ji'-y = y(t), a ^ i ^ c and J2:?/ = «/(f), c ^ t ^ ft, then
1\(t, I)
Figure 1.
Actually, if t)(t) is given,y^/) has nothing to do with J but, in applications,
rj(t) will be a "vector beginning at the point y = y(t)" of J.
The main interest below will be in the case that J is a Jordan curve, in
which case it will always be assumed that J is positively oriented and rj 5^ 0
is a continuous vector on J [so that y(a) = y(b) and rj(a) = rj(b)]. [Only
Jordan curves J:y = y(t) which are piecewise of class C1 will occur, so
that the positive orientation means that the normal vector (—dy2/dt,
dyx\di) 5* 0, defined except at corners of J, points into the interior of J.]
It is clear thaty(/^) is an integer. It is called the index ofr\ with respect to J.
Theorem 2.1 (Umlaufsatz). Let J : y = y(t), 0 5j t ^ 1, be a positively
oriented Jordan curve of class C1 and rj(t) = dyjdt (^ 0) the tangent vector
field on J. Thenjp)= 1.
Proof. On the triangle A : 0 ^ s ^ t ^ 1, define t](s, t) as follows:
y(s, t) = [y(t) - y(s)]l(t - s) if s * t or (s, t) * @, 1), r,(t, t) = y'(t), and
148 Ordinary Differential Equations
^@,1)= — rj@,0). It is clear that »;(?, discontinuous and rj(s,t) 5* 0 on A.
Note that rj(O, t) and r\(t, 1) are oppositely oriented vectors; see Figure 1.
Suppose that the point y = 2/@) on J is chosen so that the tangent line
through 2/@) is parallel to the j/1-axis and no part of J lies below this
tangent line. Since A is simply connected, it is possible to define (uniquely)
a continuous function cp(s, t) such that 95@, 0) = 0 and cp(s, t) is an angle
from the positive ^-direction to tj(s, t). Then lirj^J) = y{\, 1) — <p(°> 0),
as can be seen by considering <p(t, t).
2ir
Figure 2.
The position of J implies that 0 :_ 95@, t) :_ v and that 95@, 1) is an odd
multiple of n, hence 95@, 1) = it. Similarly, a consideration of cp{s, 1) —
95@, 1) = cp(s, 1) — 77 for 0 :_ s :_ 1 shows that 95A, 1) — n = n. Con-
Consequently, 95A, 1) = 2tt. Since 2tt/^(J) = <p(l, 1) — 95@, 0) = 95A, 1), the
theorem is proved.
The "rounding off" of corners in the case of a J which is piecewise of
class C1 gives the following:
Corollary 2.1. Let J :y = y(t), 0 _? f _s 1, be a positively oriented
Jordan curve which is piecewise of class C1 with corners at the t-values
@ <) tx < • • • < tn « 1) and tj(t) = dy/dt for t 5* tk. Let Jk:y = y(t),
h-i = * = h* for k = I,. . . ,n + I with t0 = 0 and tn+1 = 1. Then
n + l n
2t 2 JifJk) + 2 (9k ~ ¦"¦) = 27T, where <pk is the exterior angle 0 _ <pk ^
*i *i
2v at y(tk).
Note that <pk — v is an angle, — tt ^ <pk — n _ tt, from t](tk — 0) to
t](tk + O); see Figure 2.
The Poincare-Bendixson Theory 149
The essential idea in the proof of Theorem 2.1 is contained in:
Lemma 2.1. Let J : y = y(t), a ^ t ^ b, be a Jordan curve, f(f) and
r}{t) two vector fields on J which can be deformed into one another without
vanishing. Thenji(J)=jt){J).
The possibility of a deformation without vanishing means the existence
of a continuous vector r\ = r\(t, s) for a ^ t ^ b, 0 ^ s ^ 1, such that
tj(t, 0) = f(f), n(t, 1) = n(t), r}(a, s) = r](b, s), and that tj(t, s) * 0. For
example, t](t, s) = A — s)f(f) + s»;(f) is such a deformation if f(f),»;(?)
are not in opposite directions for any t.
Proof. Let j(s) be the index of tj(t, s) for a fixed s. It is clear that7E) is
a continuous function of s. Since y(s) is an integer, it is constant. In
particular, y@) = j A).
3. Index of a Stationary Point
In what follows, f(y) =f(y\ y2) is continuous on an open plane set E.
As before, a point where/= 0 is called a stationary point and a point
where/5^ 0 is a regular point.
Let J.y = y(t), a ^ t ^ b, be an arc in E on which f(y) 5* 0. Define
jf(J) to bejn(J), where »?(f) =f{y{t)). For example, if/(«/), j/(f) are of class
C1, theny}(J) is given by the line integral
C.1)
ll/ll2 '
where /=(/1,/2).
When J is a positively oriented Jordan curve in E on which/5* 0, the
integery/J) is called the index of f with respect to J.
Lemma 3.1. Let Jo and Jx be two Jordan curves in E which can be
deformed into one another in E without passing through a stationary point.
Thenj,(J0)=j,(J1).
The assumption here means the existence of a continuous y(t, s),
a^t^b, 0^s^\, such that (i) for a fixed s, J(s):y = y(t, s) is a
Jordan curve in E; (ii) 7@) = Jo, J(\) = Jx; and (ui) f(y(t, s)) 5* 0. The
proof is the same as that of Lemma 2.1.
Corollary 3.1. Let J be a positively oriented Jordan curve in E such that
the interior of J is in E and thatf(y) 5^ 0 on and inside J. Thenjf(J) = 0.
Proof. Since the interior of a Jordan curve is simply connected, J can be
deformed (in its interior) to a small circle J1 around a point y0 of its interior.
Since/(?/0) 5* 0, it is clear that if the circle J1 is sufficiently small, the change
of the angle between f(y) and the j^-direction around Jt is small. Since
y/Ji) is an integer, jf(Ji) = 0. By Lemma 3.1, j^J) = 0.
Let y0 e E. Lemma 3.1 shows that the integer jf{J) is independent of the
Jordan curve J in the class of curves J in ? with interiors in E containing
150 Ordinary Differential Equations
no stationary point except possibly y0. This integer jf(J) is called the index
jAVo) °f Vo with respect to/. By Corollary 3.1, jf(y0) = 0 if y0 is a regular
point. For this reason, only the indices of isolated stationary points y0 are
considered.
Corollary 3.2. Let J be a positively oriented Jordan curve in E on which
f{y) j? 0 and let the interior of J be in E and contain only a finite number oj
stationary points yls. .., yn. Thenj,(J) =jf{y1) + ¦¦• + jf(yn).
Figure 3.
For J can be deformed into a path consisting of circles around each
stationary point and "cuts" between the circles traced in both directions;
see Figure 3.
Exercise 3.1. Show that according as ad — be > 0 or ad — be < 0,
the index of the origin with respect to /„(«/) = (ay1 + by2, cy1 + dy2) is
+ 1 or -1.
Exercise 3.2 (Continuation). Let fo(y) be as in the last exercise and
fi(y) a continuous function defined for small \\y\\ such th&tf^yJlWyW —*¦ 0
as y-+0. Show that if/(«/) = fo(y) + /iG/), then y = 0 is an isolated
stationary point and the index y/0) = ±1 according as ad — be ^ 0.
Exercise 3.3. Let f(y) be of class C1 on an open set E with a Jacobian
determinant det (df/dy) = 3(/1,/!!)/3(y1, y2) different from 0 wherever
/ = 0. Let J be a positively oriented Jordan curve in E with interior I'm E
and f(y) ^ 0 on J. Show that there are at most a finite number of station-
stationary points yx, . . ., yk in / and that _//¦/) = n+ — n_, where n+ or n_ is the
number of these points at which det (dfjdy) > 0 or det (dfjdy) < 0.
Theorem 3.1. Letf{y) be continuous on an open set E and let y = yp(t)
be a solution ofy' =f(y) of period p, yv(t + p) = yp(t)for — oo < t < go.
Let y = yv(t), 0 ^ t ^ p, be a Jordan curve with an interior I contained in
E andf(yp(t)) j? 0. Then I contains a stationary point.
The Poincare-Bendixson Theory 151
Proof. Let J be the Jordan curve, y = yv(t), 0 ^ t ^p, with a positive
orientation. By Theorem 2.1, jf(J) =1^0. Thus the theorem follows
from Corollary 3.1.
A return to the study of stationary points and their indices will be made
in § 6 below.
4. The Poincare-Bendixson Theorem
The discussion of the differential equation y' =f(y) in § 1 will now be
continued for the plane case (d = 2) with the aid of the Jordan curve
theorem. The main result is the following theorem of Poincare-Bendixson.
U(C+)=CP
Figure 4.
Theorem 4.1. Letf(y) =f(y1, y2) be continuous on an open plane set E
and let C+:y = y+(t) be a solution of
D.1)
y' =f(y)
for t ^ 0 with a compact closure in E. In addition, suppose that «/+(fi) ^
y+(td for 0 = h < h < °° and that Q(C+) contains no stationary points.
Then O,(C+) is the set of points y on aperiodic solution Cv:y = yv(t) o/D.1).
Furthermore, ifp > 0 is the smallest period of yv(t), then yjj^) ?^ yv(t%)for
0 ^ ?! < t2 < p; i.e., J.y = yv(t), 0 ^ t ^ p, is a Jordan curve.
In the case that initial value problems associated with D.1) have unique
solutions, either «/+(fi) 5* y+(t2) for 0 ^ tt < f2 < oo or y+(t) is periodic
[i.e., y+(t + p) = y+(t) for all t for some fixed positive number p]. In the
latter case (excluded in Theorem 4.1), O.(C+) coincides with the set of
points on C+.
The proof of Theorem 4.1 will show that C+ is a spiral which tends to
the closed curve Q.(C+):y = yv(t) either from the exterior or from the
interior; see Figure 4. It will have the following consequence:
152 Ordinary Differential Equations
Corollary 4.1. Assume the conditions of Theorem 4.1 and let p > 0 be a
period of y = yv{t). Then there exists a sequence @ ^) tx < t2 < . . . such
that
D.2) y+(t + tn) -> yv{i) as « ^ oo
uniformly for 0 ^ t ^ p and
\r-->) ln+i — ln P as> n—*¦ ou.
Proof of Theorem 4.1. A closed, bounded line segment L in E is called
transversal to D.1) if/(«/) 5* 0 for y e L and the direction of/(«/) at points
2/ e L are not parallel to L. All crossings of L by a solution 2/ = 2/@ of
2/' = / are in the same direction with increasing t.
y+2(t2)
Figure 5.
The proof will be divided into steps (a)-(e).
(a) Let y0 e E, f(y0) 5* 0, L a transversal through y0. Then Peano's
existence theorem implies that there is a small neighborhood Eo of y0
and an e > 0 such that any solution y = j/°(f) of the initial value problem
y' =/, 2/@) = j/° for «/° e Eo exists for |f | ^ e and crosses L exactly once
for \tI ^ e. In fact, if E > 0 is arbitrary, Eo and e can be chosen so that
y°(t) exists and differs from y° + (/"(j/0) by at most d \t\ for \t\ ^ e. Thus
if Eo is sufficiently small, y = j/°(f) crosses L at least once, but can cross L
at most once for |f| ^ e since crossings of L are in the same direction.
In particular, it follows that if y = y°(t) is a solution of y' = f on a
closed bounded interval, then y = y°(t) has at most a finite number of
crossings of L.
(b) Let L be a transversal which, without loss of generality, can be
supposed to be on the j/2-axis, where y = (y1, y2). Suppose that y = y+(t)
crosses L at f-values tt < f2 < . . . , then y+\tn) is strictly monotone in n.
In order to see this, suppose without loss of generality that crossings of
L occur with increasing?/1 (i.e., y1 changes from negative to positive values
at crossings). Consider the case that */+2(f,) < y+2{t2); see Figure 5.
The Poincare-Bendixson Theory 153
The set consisting of the arc y = y+(t), tx ^ t ^ t2, and the line segment
2/+2(fi) = y2 = y+Kh) on the j/2-axis forms a Jordan curve /. For all
t > t2,y = y+(t) is in the exterior of/ or in the interior of/, by the assump-
assumption on y+(t) and the fact that crossings of L occur only in one direction.
This makes it clear that y+\t2) < y+\t3) and the argument can be repeated.
(c) It will now be verified that if L is a transversal, Q(C+) contains at
most one point on L. For if y0 e L Pi Q(C+), part (a) implies that y =
y+(t) crosses L infinitely many times [in fact, whenever y+(t) comes near
y0]. With increasing t, the intersections of y = y+(t) and L tend monoto-
monotonously along L to y0, by part (b). Thus L n Q(C+) cannot contain any
other point y = y0.
(d) Since C+ is bounded, Q(C+) is not empty. Let y0 e Q(C+). By
Theorem 1.2,y' = f y@) = j/ohasa solution C0:y = j/0@> — 00 < f < 00,
contained in Q(C+); thus Q(C0) <= Q(C+).
Q(C0) is not empty. Let y° e Q(C0), so that y° is a regular point since
Q(C+) contains no stationary points. Thus there is a transversal L°
through y° and y = yo(t) has infinitely many crossings of L° near y°, but y°
and every such crossing is a point of Q(C+). By (c), these points coincide.
In particular, there exist points tt < t2 such that y° = J/O(fi) = 2/0(^2)- It
follows that D.1) has a periodic solution y = yv{t) of period p = t2 — tx
such that 2/j,(f) = j/0(f) for 11 ^ f ^ f2. Since j/0(f) is not constant on any
f-interval, it can be supposed that yv(t°) 5* yv{Q for 0 ^ t0 < t° < p.
(e) It must be shown that Q(C+) coincides with its subset Cv:y = yv(t),
— 00 < t < 00. If not, ?i(C+) — Cp is not empty. Then, Cv contains a
point y1 which is a cluster point of ?i(C+) — Cv, since Q(C+) is connected
by Theorem 1.1. Let Lx be a transversal through yv Any small sphere
about y1 contains points y2 e Q(C+) — Cv. For any such y2, y' =/has A
solution y = y2(t), — 00 < t < 00, such that j/2@) = j/2 and y2(t) is
contained in Q(C+), by Theorem 1.2. If y2 is sufficiently close to y1} then
j/2(f) crosses the transversal Lx. The crossing is necessarily at the point yx
by part (c).
Since j/2 ? C,,, this is impossible when solutions of initial value problems
belonging to D.1) are unique. That it is impossible in the general case can
be seen as follows: Let y2(tp) e Cv, while y2(t) ? Cp for f between 0 and f„.
Since y2(t„) is a regular point, there is a transversal Lj, through y2(tj,). Then
a small translation of Lv in a suitable direction is a transversal Lj,0 which
meets Cv and 2/ = y2(t) in two distinct points; see Figure 6. This contra-
contradicts part (c) and proves the theorem.
Remark. For subsequent use, note that the argument in part (e) shows
that whenever C+:y = y+(t), t ^ 0, has the property that y+(tx) 5* y+(t2)
for f, 5^ f2 and «/0 e Q(C+) n ? is a regular point, then there is a neighbor-
neighborhood ?¦„ of y0 such that the solution of y' = /", ?/@) = y0 in Q(C+) n Eo is
154 Ordinary Differential Equations
unique. If, in addition, Q(C+) is connected and there is a periodic orbit
Cv:y = yv{t) in Q(C+) consisting only of regular points, then Q(C+) = Cv.
Proof of Corollary 4.1. Let y° = yjO) and let L° be a transversal
through y°. Let the successive crossings of L° by y = y+(t) occur at @ ^)
fi < h < ¦ ¦ ¦ ¦ Then y+(tn) tends monotonically along LP to y°. Since
y = ^(f) is the unique solution of y' = /, 2/@) = 2/° in Q(C+), an analogue
of Remark 2 following Theorem 1.2 shows that D.2) holds uniformly for
bounded f-intervals, in particular for 0 ^ t ^ p.
Figure 6.
Note that y+{tn +p)-+ yv(p) = yo,n~+ °°- Thus if e > 0 and n is
large, y+(t) crosses L° in the interval [tn+ p — e, tn + p + e]. Hence
'«+i ^ '„ + /» + e- A1so ||y+0« + 0 - ^(Oll is small for large n,
0<e^f^/j — e</j, which implies that there is a d > 0 such that
Wy+iK + 0 — 2/oll = ^ f°r 0 < e = f ^Z7 ~ e- In particular, there is no
crossing of L° for e ^ t < /> — e. Hence fn+1 ^ *„ + p — e for large n.
This proves the corollary.
Theorem 4.2. Lef/, C+ 6e as in Theorem 4.1 exce/jf that Q(C+) contains
a finite number n of stationary points o/D.1). If n = 0, Theorem 4.1
applies. Ifn = 1 and Q(C+) is a point, Corollary 1.1 applies. If I ^ n < oo
an^fQ(C+) w nof a point, then Q(C+) consists of stationary points yx, y2, ¦ ¦ ¦,
yn and a finite or infinite sequence of orbits C0:y = yo(t), — oo ^ oc_ < f <
a+ ^ oo, which do not pass through a stationary point buty^a.^) = lim y(t),
as t-+ a±, ex/sf an^f are among the set yx,.. . ,yn.
It is possible that 2/0(oc+) = 2/o(a-)- It is not claimed that (oc_, oc+) is the
maximal interval of existence of y = yo(t). But when initial conditions
uniquely determine solutions of D.1), so that the only solution of D.1)
through a stationary point yk is y(t) = yk, then it follows that oc_ = — oo
and oc+ = oo; cf. Lemma II 3.1.
Proof. Consider the case that n ^ 1 and Q(C+) is not a point. Since
Q(C+) is connected by Theorem 1.1, it contains regular points y0. For any
The Poincard-Bendixson Theory 155
such point y0, there exist solutions C* :y = y#(t), — oo < t < oo, of
y' =f j/@) = 2/0 in Q(C+). Let C, denote any such solution.
Consider only C+ + :y = 2/*@ for t ^ 0. The treatment of t ^ 0 is
similar. There are two cases: (i) there is a first positive t = oc+, where
2/*(«+) is a stationary point (hence one of the points yu . , ., yn) or (ii)
2/*(f) is not a stationary point for any finite t ^ 0.
Consider case (ii). If Q(Q+) contains a regular point 2/°, then, by part
(d) in the proof of Theorem 4.1, C*+ contains a periodic solution path
Cp'-y = 2/jj@- But then Cj, contains a stationary point yt; otherwise, by
part (e) and the Remark at the end of the proof of Theorem 4.1, &(C+) =
Cv. This is impossible since 2/*@ ^ 2/i for t ^ 0. Hence in case (ii),
Q(C*+) can contain only stationary points and, since it is connected, only
one stationary point 2/*.
Thus, in case (ii), Q(C*+) is a stationary point 2/* and 2/*(O~*2/* as
t -*¦ oo by Corollary 1.1. Thus any regular point y0 e ?i(C+) is on an arc
Co'-y = 2/o(O> <*¦- < t < oc+, in Q(C+) of the type specified.
It remains to show that the set of such arcs Co is at most denumerable.
Note that if 2/0 e Q(C+) is a regular point, the solution C*:y = 2/*@
above is unique on a sufficiently small interval \t\ < e; cf. part (e) and the
Remark following the proof of Theorem 4.1. Thus, no two of the arcs
Co can meet.
Since 2/+(fi) 5* 2/4.(^2) for t2 > fl5 it can be supposed that y+(t) is not one
of the stationary points 2/1, • • ¦, yn for t ^ 0. Otherwise, 2/+@. f ^ 0, is
replaced by y+(t), t ^ t0, for a suitable f0 > 0. Also if y0 e Q(C+) is a
regular point, then y+(t) 5* 2/0 for t ^ 0 for the sequence of intersections of
2/ = 2/+(f) with a transversal through y0 tends to y0 in a strictly monotone
sense; cf. part (b) of the proof of Theorem 4.1. Thus C+ and Co have no
points in common.
Suppose, if possible, that there exists a nondenumerable set of Co which
can then be assumed to join the same, not necessarily distinct, pair of
stationary points. Any one or two of these joined together forms a Jordan
curve/. Since there is at most a denumerable set of/with pairwise disjoint
interiors, there exist three such distinct /, say Jt, /2, /3, such that /3 is
contained in the closure of the interior /„ of /„ for n = 1, 2. This is
impossible, for since C+ does not meet /„, C+ is either between Jx and /2 or
/2 and /3.
In the next theorem, the assumption /+(fi) 5* 2/4.O2) for t1 56 t2" is
omitted.
Theorem 4.3. Let f(y) =f(y1,2/2) be continuous on an open plane set E
and C+:y = 2/+@ a solution of D.1) for t ^ 0 with compact closure in E.
Then Q(C+) contains a closed {periodic) orbit Cv:y = yv{t) o/D.1) which
can reduce to a stationary point yP(t) = y0.
156 Ordinary Differential Equations
Proof. Suppose that Q(C+) does not contain a stationary point. Let
2/ogQ(C+) and C0+:y = yo(t), 0 ^ t < oo, be a solution supplied by
Theorem 1.2, so that Co+ «= Q(C+). Since Q(C+) is closed, Q(C0+) «=
Q(C^). If 2/0(A) 5* 2/0(fa) for 0 ^ ft < f2 < oo, then Q(C0+) is a closed
orbit y = yv{t) by Theorem 4.1. If 2/0(A) = yo(t2) for certain fl5 f2 with
0 ^ ft < t2 < oo, then the orbit Co+ contains the periodic solution path
y = yv(t) of period p = t2 — tx which coincides with yo(t) for ft ^ t ^ f 2.
In either case O.(C+) contains a closed orbit y = «/j,(f)-
Theorem 4.4. Letf{y) be continuous on an open, simply connected plane
set E where f(y) 5* 0 and y = y(t), a solution of D.1) on its maximal
interval of existence (<w_, a>+). Then y = y(t) does not remain in any
compact subset Eo of E as t —*• a>+ [or t —>¦ a>_].
Proof. If, e.g., C+ : y = y(t), t0 ^ t < a>+, is in a compact set Eo in ?
for some t0, then <w+= 00 by Theorem II 3.1 and O.(C+) contains a
periodic solution y = yv(t) by the last theorem. Since/9^ 0 on ?, yv(t)
does not reduce to a constant on any f-interval. Let t1 be the first t > f0,
where j/jX^) = yp(t0). Then «/ = yv{t), to^t ^ ^, is a Jordan curve /.
Thus D.1) has a periodic solution y = yo(t) of period tx — t0 such that
yo(t) = «/j,(f) for f0 ^ f ^ ft. Since E is simply connected, the interior of
J:y = «/0(f), t0 ^ t ^ tx, is contained in E. By Theorem 3.1, this interior
contains a stationary point. This contradicts //O and proves the
theorem.
5. Stability of Periodic Solutions
Return to the situation in Theorem 4.1.
Theorem 5.1. Letf C+, Cv be as in Theorem 4.1 Then there exists an
e > 0 such that ify0 is within a distance e o/Q(C+) = Cv:y = yv(t) and on
the same side {interior or exterior) of Cv as C1", then
E.1)
y'=f{y), 2/@) = 2/0
has a solution C0:y = yo(t) for t ^ 0 such that yQ{t^) 5* yo{t2) for tx ^ f2
and?l(C0+) = Cv.
Proof For sake of definiteness, consider the case that C+ is exterior to
CB. Let y° = yp@), LP a transversal through y°, the successive crossings
of L° by y+(t) occur at @ ^) tx < t2 < ... . Thus y+(tn) tends monoton-
monotonously along L° to y".
Let /„ be the Jordan curve consisting of the arc y = y+(t), tn ^ t ^
fn+1, and the open segment /„ of L° joining y+(tn), j/+(fn+1). Let Dn be the
interior of /„ and En = Dn — Dn+1. Note that En is a simply connected
open set in E since Jn+1, except for the point y+(tn+i), is interior to /„;
see Figure 7.
It is clear that if
The Poincare-Bendixson Theory 157
> 0 is sufficiently small, then \jEn contains all points
on the same side as C+. It is also
clear from Corollary 4.1 that the union \}En, n ^ N, for large
Nis within a distance e of Cv. Since f(y) ^ 0 for y e Cv, it can be supposed
that UEn contains no stationary points; otherwise, tu t2, . . . is replaced
b
y0 ? C+ U Cv within a distance e of C
c+
Figure 7. Shaded area is Dn+1.
In the proof, it is sufficient to consider only those y0 $ C+ U Cv, so that
y0 e En for some n. Let 2/ = yo(t) be a solution of E.1) for t near 0. By
Theorem 4.4, a continuation of this solution for increasing t meets the
boundary dEn of E at a finite f-value. Let f0 be the first t0 > 0 at which
j/0(f0) e d?n, where dEn <= C+ U /„ U 7n+1. It can be supposed that
^(f1) 5^ yo(t2) for 0 ^ fx < f2 ^ f0. Because of the direction of crossings
of solutions on /„, yo(to) ? In so that yo(to) eC+U In+1. If j/0(f0) e C+,
say 2/0(f0) = ^(f0). then yo(t) can be defined for t ^ f0 to be j/0(f) =
,y+(f + t° — t0). If j/0(f0) e 4+!, then j/0(f) exists for t(> t0) near f0 and is
-n+l-
Continuing this procedure, we obtain yo(t) defined for t ^ 0 so that
either yo(t) = y+(t + a) for some a and large t or the solution y = yo(t)
with increasing t successively passes through En, En+1,.... This completes
the proof.
Let Cv:y = yjj) be a periodic solution of y' =/of period/? > 0 such
that J:y = yv{t), 0 ^ t ^ />, is a Jordan curve. C,, is called orbitally stable
from the exterior as /-*+<» if, for every e > 0, there exists a E = de > 0
158 Ordinary Differential Equations
with property that if y0 is exterior to but within a distance d of /, then all
solutions C0+:y = yo(t) of E.1) exist and remain within a distance e of /
for t ^ 0. Cv is called asymptotically orbitally stable from the exterior as
t -* oo, if there exists an d > 0 such that if y0 is exterior to but within a
distance d of/, then all solutions C0+:y = yo(t) of E.1) exist for t _ 0 and
Q(C0+) = /. Similar definitions hold with "exterior" replaced by "in-
"interior" and/or "t -* oo" by "t -* — oo".
Theorem 5.2. Let f{y) be continuous on an open plane set E with the
property that initial values determine unique solutions of y' =f Let
Cv:y = yv{t) be a periodic solution of y' = f with a least positive period p.
(i) Then Cv is asymptotically orbitally stable from the exterior as t-* oo if
and only if the orbit Cv is Q(C0+) for some solution C0+:y = yo(t), t ^ 0,
on the exterior of Cv. (ii) Then Cv is orbitally stable from the exterior as
t-* co if and only if either the orbit Cv is ?i(C0+) for some solution Co+
exterior to Cv or, for every e > 0, there is a periodic solution of A.1)
exterior to and within a distance e of Cv.
Proof. In (i), "only if" is trivial and "if" follows from Theorem 5.1.
In (ii), "if" is clear from Theorem 5.1. In order to prove the converse,
suppose that Cv is orbitally stable from the exterior. Since solutions of
initial value problems E.1) are unique, f(y) ^ 0 on C, and, hence, in some
eo-vicinity of Cv. Let C0+:y = yo(t), t ^ 0, be a solution of y' = f
exterior to but within a distance e < e0 of Cv. Then Co+ has a compact
closure in E and Q(C0+) contains no stationary points, so that Q(C0+) is a
periodic solution path of y' =/by Theorem 4.1. Thus either Q(C0+) = Cv
or Q(C0+) is periodic solution path exterior to and within a distance e of Cv.
6. Rotation Points
In §§ 6-9, the behavior of solutions of
F.1) y'=f(y), 2/@) = 2/0
in the vicinity of an isolated point will be considered. Let f(y) be defined
for small \\y\\, say, on an open set containing \\y\\ _ b, and let
F.2) /@) = 0 and f(jf)*O if 2/5*0.
Note that if F.1) has a unique solution for all y0, then Theorems 4.1 and
4.2 imply that if 0 < \\yo\\ < b and C0+:y = yo(t) is the solution of F.1) on
the maximal interval [0, co+), then only the following (not mutually
exclusive) cases can occur: (i) there is a least t0, 0 < t0 < co+, such that
||j/0(f0)|| = b; (ii) a>+ = oo and the solution path Co+ is a Jordan curve
with y = 0 in its interior; (iii) a>+ = oo and Co+ is a spiral approaching
such a closed orbit; (iv) a>+ = oo and yo(t)-+O as f-* oo; and (v)
<w+ = oo, Co+ is a spiral around y = 0 and Q(C0+) consists of a finite or
The Poincare-Bendixson Theory 159
infinite sequence of orbits y(t), — oo < t < oo, such that y(t) -* 0 as
t -* ±oo.
By a spiral y = yo(t), 0 ^ t < w+, around y = 0 is meant an arc
yo(t) j? 0 such that a continuous determination of arc tan yo\t)jyQ\t)
tends to either oo or — oo as t -* a>+.
If every neighborhood of y = 0 contains closed orbits surrounding
2/ = 0, the stationary point y = 0 is called a rotation point.
When y = 0 is a rotation point and solutions of arbitrary initial value
problems F.1) are unique, the set of solutions ofy' =/in a neighborhood
of y = 0 can be described as follows: there is a neighborhood Eo of y = 0
such that the solution C0:y = yo(t) of F.1) for every y0 e Eo is either a
closed orbit surrounding y = 0 or is spiral such that Q(C0), A(C0) are
closed orbits surrounding y = 0.
This is illustrated by the following: Consider the differential equations
F.3) 2/1' = 2/M>0 - y2v{r), y*' = ylv(r) + y*u(r),
where u, v are continuous, real-valued functions of r = \\y\\ for small
r ^ 0. In polar coordinates, these equations become
F.4) r' = ru(r), d' = v(r).
Example 1. Suppose that u(r) = 0, v(r) = jS 5* 0. Then F.3) is
F.5) j/i' = -W, y« = 0yi
and F.4) is r' = 0, 0' = 1, so that all orbits (y =? 0) are circles.
Example 2. Suppose that u{r) = r sin A/r), v(r) = 1. Then F.4) is
r' = r2 sin A/r), 0' = 1. Thus, besides the trivial solution y = 0, there are
closed orbits r = 1/aztt, n = 1, 2,.... Between the orbits, r = l/mr and
r = l/(n + 1)tt, (—1)V > 0 and so the corresponding orbits are spirals
which tend to the circles r = l/mr, r = \j{n + \)n as t -> 00, f -* — 00 or
f -* 00, f -*- — 00, depending on the parity of n.
Example 3. In the last example, u{r) can be redefined to be 0, say,
between r = l/rnr and r = l/(n + 1)tt for a finite or infinite sequence of
n-values 1, 2, .... Correspondingly, the spiral orbits between r = l/mr
and r = \j{n + X)n are replaced by circular orbits.
Exercise 6.1. Let C be a closed set on 0 _: r _^ 1. Show that there
exist functions u(r), v{r) which are uniformly Lipschitz continuous on
0 <: r ^ 2, uV) + t'2(r) ^ 0 for r 5* 0, and the solution of F.3) with
initial condition y@) = y0 is a closed orbit if 0 < \\yo\\ e C and is a spiral
if||yoll*c,o< H2/0II ^1.
A rotation point y = 0 such that all orbits, except y = 0, in a vicinity of
// = 0 are closed curves is called a center. The simplest illustration of a
center is the linear system F.5) in Example 1.
160 Ordinary Differential Equations
7. Foci, Nodes, and Saddle Points
Assume that the only solution of y' =f(y), 2/@) = 0 is y = 0 [so that
no solution y(t) ^ 0 can tend to 0 as t tends to a finite value].
The simplest nonrotation points are called "attractors." The isolated
stationary point y = 0 is called an attractor for t = oo [or t = — oo] if all
solutions 2/ = 2/oW of F.1) for small \\yo\\ exist for t ^ 0 [or t ^ 0] and
yo(t) -* 0 as t -»¦ oo [or t ->¦ — oo]. If, in addition, all orbits yo(t) ^ 0 are
G.1) with a<0,/3>0
G.2) with X2<
17.3) with X<0
G.2) with Xi<0<X2
Figure 8.
The Poincare-Bendixson Theory 161
spirals, then the attractor y = 0 is called a focus. If all orbits, yo(t) ^ 0
have a tangent at y = 0; i.e., if a continuous determination of d(t) =
arc tan 2/o2@/2/o1(O tends to a limit 0O, — oo < 60 < oo, then the attractor
2/ = 0 is called a no<ie. A node is called a proper node if for every 0O mod
lit, there is a unique solution y = 2/0@ such that d(t) -*¦ d0; otherwise it is
called an improper node.
Illustration of these cases of attractors are given by real linear equations;
see Figure 8. The system
G.1) 2/1' = a2/1 — i^2/2) 2/2' = fit/1 ~)~ a2/2
is an attractor for t = oo [or t = — oo] if a < 0 [or a > 0]. It is & focus
if a, jS 5* 0 and a proper node if a 5* 0, /? = 0. In the case,
G.2) 2/1' = ^i2/\ 2/2' = A2J/2
2/ = 0 is an improper node if AjA2 > 0, Aj 5^ A2. The case
G.3)
2/1' = Ay1, 2/2' = 2/1 + A2/2,
where A 5^ 0 is also an improper node.
There are nonrotation points which are not attractors and attractors
which are neither foci nor nodes. The simplest example of a stationary
point which is not an attractor is a saddle point. This is a stationary point
y = 0 with the property that only a finite number of orbits tend to 0 as
f-*+ooorf-*— 00. This is illustrated by G.2) where Al5 A2 are real and
A1; A2 < 0.
Exercise 1.1. Verify the statements just made about Gi 1)—G.3).
Exercise 7.2. Consider the linear system y' = Ay, where A is a real
constant 2x2 matrix with det A 5^ 0, so that y = 0 is the only stationary
point. Let A1; A2 be the characteristic values for A. Show that y = 0 is an
attractor for t = 00 [or t = — 00] if and only if Re Xk < 0 [or > 0] for
k = 1,2; y = 0 is a center if and only if Re At = Re A2 = 0; 2/ = 0 is a
focus if and only if A1; A2 are complex conjugates, but not real or purely
imaginary; y = 0 is a proper node if At = A2 and the elementary divisors
are simple; y = 0 is an improper node if A1; A2 > 0 but either At 5* A2 or
At = A2 and the elementary divisor is multiple.
8. Sectors
The general nonrotation point will now be considered. It will be
convenient to have the following terminology. A solution y = y(t) ^ 0
of y' =/defined on an interval [0, co) (or an interval (-co, 0]) for 0 <
m ^ 00 is called a positive [or negative] null solution if 2/@ —*¦ 0 as t -*¦ a>
[or -co]. When the solution of y' = f 2/@) = 0 is unique, then necessarily
w = 00.
162 Ordinary Differential Equations
Lemma 8.1. Letf{y) be continuous for small ||«/||,/@) = 0, andf{y) ^
Ofor y 9^ 0. Suppose that y = 0 is not a rotation point. Then there exists
at least one null solution.
Proof. Suppose that e > 0 is so small that there is no closed orbit in
\\y\\ ^ e surrounding y = 0 and suppose, if possible, that there is no null
solution in \\y\\ ^ e.
Then there is no solution C0:y = yo(t) 5* 0 of y' = f defined and
satisfying \\yo(t)\\ ^ e for t ^ 0. For otherwise yo(tj) 5* yo{Q when
t± 5* f2 (since there are no closed orbits in \y\ ^ e) and Theorem 4.2
implies that there exists at least one positive null curve. Similarly, no
solution y = yo(t) 5* 0 if«/' =/exists and satisfies \\yo(t)\\ ^ e for t ^ 0.
Thus, if \\yo\\ < e and yo(t) is a solution of F.1), there is a bounded
interval — s ^ t ^ 0 such that yo(O) = y0, \\yo(t)\\ < e for — s < t ^ 0
and \\yo(—s)\\ = e. Correspondingly, the solution y = yo(t — s) is defined
for 0 ^ t ^ s, \\yo(t - 5)|| < € for 0 < t ^ s, \\yo(t - s)\\ = € for t = 0
and yo(t — s) = y0 for t = s.
By considering a sequence of points y0 = yx, yz,. . . tending to the
origin, we obtain a sequence of solutions y = yn(t), 0 ^ t ^ sn, such that
ll»»@)|| = «, ||2/n(f)ll < e for 0 < t ^ sn, \\yn(sn)\\ -0 as «— 00. After
a selection of a subsequence and renumbering, it can be supposed that
y0 = lim «/n@) and to = lim $„, n -* 00, exist, where ||j/0|| = e and 0 ^
co ^ 00. Also, if co > 0, it can be supposed that yo(t) = lim j/n(f),« ~* °°,
exists uniformly on every closed bounded interval of [0, co) and is a
solution of y' = /.
It is clear that a> > 0. For otherwise, for large n, y = «/„(?) is in a small
vicinity of y0 for 0 ^ t ^ sn; thus «/n(.O ->¦ 0, n ->¦ oo, is impossible. If
0 < ft) < 00, then Peano's existence theorem shows that, for large n,
yn(t) can be defined on an interval containing 0 ^ t ^ co, and «/0(f) =
lim yn(t) exists uniformly for 0 ^ f ^ co; thus, j/oC^) = um 2/«(O = 0-
Finally, if co = 00, then yo(t) is defined and ||2/0@ll = e f°r f = 0- This is
a contradiction and proves the lemma.
Hypothesis. In what follows, assume that solutions of arbitrary initial
value problems y' = f(y), y@) = y0 are unique.
Let C be a positively oriented Jordan curve surrounding y = 0. A
solution y = y(t) of y' = f(y) is called a positive or negative base solution
for C if y(t) is defined for either t ^ 0 or t ^ 0,2/@) e C, «/(f) is interior to
C for f ?? 0, and «/(f) is a null solution.
Let y = 2/i(f), 2/a@ be base solutions for C. The open subset S of the
interior of C with boundary consisting of y = 0, the arcs y = y^t), yz(t)
and the (oriented closed) subarc C12 from ^@) to «/2@) will be called the
sector of C [determined by the ordered pair y = y^t), y = «/a(f)]- It is not
excluded that y^O) = y2@) so that Cia can be C or reduce to a point.
The Poincare-Bendixson Theory 163
Consider the case that there exists a solution y = yo(t), — 00 < t < 00,
°f V' = f which is interior or on C for all t and yo(t + tj) = y^t) for
t ^ 0, yo(t + f2) = 2/2(f) for t ^ 0 for some tu f2(^ ft); see Figure 9.
The point 2/ = 0 and the arc y = yo(t), - 00 < t < 00, form a Jordan
Figure 9. Elliptic sectors.
curve J with interior /. If S contains /, then it is called an elliptic sector.
When ft = f2 [so that j/^0) = j/2@) = yo(tj], and C12 reduces to the point
yo(t]), then S is elliptic and coincides with /. When tx ^ f2, S can contain
points not in /. By considering the possibilities (i)-(v) mentioned after
F.2), it is seen that if y = y(t) is a solution of F.1) with j/@) e /, then y(t)
exists for — 00 < t < 00 and y(t) ->¦ 0 as t ~* ± 00.
Figure 10. (a) Hyperbolic sector. F) Parabolic sector.
A sector S with the properties that it is not an elliptic sector arid that
S u C12 contains no base solution is called a hyperbolic sector; see
Figure 10(a). Part (a) of the proof of Theorem 9.1 below implies that one
of the boundary arcs y = y^{t), y2(t) of a hyperbolic sector is a positive
and the other a negative base solution.
A sector S with the properties that both boundary arcs y±(t), y2(t) are
positive [or negative] base solutions and that the closure of S contains no
164 Ordinary Differential Equations
negative [or positive] base solution is called a positive [or negative]
parabolic sector; see Figure 10F).
Note that any type of sector S (elliptic, hyperbolic, or parabolic) can
contain solutions y = y{t), — <x> < t < <x>, such that y(t) -* 0 as t -»± oo.
Such an arc and the point y = 0 constitute a Jordan curve. There can even
be an infinite sequence of such orbits in S with the interiors of the corre-
corresponding Jordan curves pairwise disjoint. Denote by Se, the elliptic portion
ofS, the sets of points of S on such orbits and the point y = 0. Then Se is
closed. In the case of hyperbolic and parabolic sectors S, Se and C12 are
disjoint.
Hyperbolic sectors S do and parabolic sectors S can contain open
solution arcs y = y(t) having both endpoints on C12. The closure of the
set of points y on such arcs will be called Sh, the hyperbolic part of S. It
will be clear from parts (a) and (b) of the proof of Theorem 9.1 that
y = 0 e Sh or y = 0 <? Sh according as S is hyperbolic or parabolic.
Lemma 8.2. Let f(y) be continuous on a simply connected open set E
containing y = 0 such that /@) = 0, f(y) ?=¦ 0 if y 5* 0, and that the
solutions of initial value problems y' =/(«/), y@) = y0 are unique. Let C be
a Jordan curve in E surrounding y = 0. Then there is at most a finite
number of elliptic and hyperbolic sectors in C.
Proof. If the lemma is false, then there is a point yo?C and a sequence
of points 2/A,@), 2/B)(O),... of C tending monotonously to y0 along C
such that 2/Bn)(O), y{2n+i)@) is the initial point of a positive, negative base
solution, yi2n)(t), y{2n+i)(t), respectively. Then y = yin)(t) is in a sector
with boundary arcs y = 2/(m_d@, 2/u+i>@ for h ^ 1. Clearly, y{0)(t) =
lim y{2n)(t), ya)(t) = lim y{2n+1)(t), n ->¦ 00, exist uniformly on bounded
intervals of t ^ 0, t ^ 0, respectively, and are solutions of/ =/. But as
point sets, the two arcs y = yi0\t), ya)(t) are identical. This is impossible
as can be seen by considering y{0)(t) for small t ^ 0 and ya)(t) for small
— f ^ 0. Thus the lemma is correct.
Lemma 8.3. Letf(y), C be as in Lemma 8.2. If the closures of all of the
hyperbolic and elliptic sectors are deleted from the interior ofC, then the
residual set is either empty, the interior of C, or the union of a finite number
of pairwise disjoint parabolic sectors.
Proof. It is sufficient to consider the case when there exist hyperbolic
and/or elliptic sectors and that the residual set is the union of a finite
number of disjoint sectors. It has to be shown that these sectors are
parabolic.
Let S be a sector not containing any hyperbolic or elliptic sectors.
Suppose first that of the two boundary base solutions y = y^t), y2(t) of S,
one is positive and the other negative. It will be shown that this leads to a
contradiction.
The Poincare-Bendixson Theory 165
For the sake of definiteness, let y±(t) be a positive, y2{t) a negative base
solution; see Figure 11. Moving on the boundary arc C12 from yx@) to
2/2(O), there is a last point y±* [possibly ^@)] such that the solution
V = 2/i*@ of y' =/, 2/@) = «/!* exists and is in S for t ^ 0 (and hence is a
positive null solution). Then yf 5* j/2@), since S contains no elliptic
sectors. Moving on C12 from j/2@) toward y^, there is a last point y2*
such that the solution y = y2*(t) of y' =f y@) = y2* exists and is in S
for t < 0. Then y2* ?=¦ 2/1* since S contains no elliptic sector.
,, ci
Figure 11.
Let Cf2 be the subarc of C12 joining y±* and y2*. The solution y =
2/i*@ [or y2*(t)] with t increasing [or decreasing] from 0 has a last point 2/n
[or y22] on C, where j/u [or y22] can coincide with yx* [or y2*] and is on the
arc (or point) of C from j/^0) to yf [or y2* to y2]. Let Cu [or C22] be the
subarc on C from y^ to yx* [or y2* to y22] and C12 = Clt U Cf2 U C22.
Let y = yn(t) [or y22(t)] be the solution of y' = f, y(Q) = yn [or y22] for
/ ^ 0 [or t ^ 0]. Then there is a sector S* with boundary consisting of
?/ = 0, the base solutions y = j/u(f), y22(t), and the arc C12.
Since S* is subset of S, it is not an elliptic sector. No solution y = yo(t)
of F.1) with y0 an interior point of C12 is in S* U C12 for f ^ 0 or t ^ 0. x
This is clear if y0 e Cf2 by the definition of the endpoints of yf, y2* of C^.
It is also clear if y0 <= cu U C22, for the solution arc beginning at such a
point ?/„ either leaves S or is a part of the solution arcs y = j^*^), 2/2*(O
which for large \t\ are not in S* (but on the boundary of S*).
Thus S* is a hyperbolic sector in S. This contradiction shows that the
boundary base solutions y = y^t), y2(t) cannot be of opposite type (i.e.,
positive and negative).
166 Ordinary Differential Equations
Consider the case that both boundary base solutions y = y^t), y^t) of
S are positive [or negative]. Then if S is not parabolic, it contains a
negative [or positive] base solution y = ys(t). But then yx(t\ ys(t) and
2/2(O» 2/3@ define sectors of the type just discussed. This is impossible,
hence S is parabolic and the lemma is proved.
In order to avoid a consideration of special cases, the following con-
convention will be adopted: If y = 0 is a nonrotation point, then, by Lemma
8.1, there exists at least one base solution y = yY{t) if C is in a sufficiently
small neighborhood of y = 0. If there are no hyperbolic or elliptic sectors,
the arcs y^t) and y2(t) = yx{t) define a parabolic sector with C12 = C.
Thus, for small C around a nonrotation point, there is always a decom-
decomposition of the interior of C into a finite number of elliptic, hyperbolic,
and parabolic sectors.
Exercise S.I. Let/, Cbe as in Lemma 8.2. Suppose that the closure of
the interior I of C contains no periodic solutions. Show that there is a
point y0 g C such that the solution C0:y = j/0@ of F-1) exists and is in /
for either t ^ 0 or t ^ 0. Thus yo(t) is either a null solution or is a spiral
around y = 0 such that O(C0) or A(C0) contains a solution y = y(t),
— 00 < t < 00, which is both a negative and a positive null solution.
9. The General Stationary Point
The object of this section is to prove the following theorem:
Theorem 9.1. Let f(y) be continuous on a simply connected open set E
containing y = 0 such that /@) = 0 and f(y) 5* 0 for y # 0 and that
solutions of initial value problems F.1) are unique. Let C be a Jordan curve
in a sufficiently small neighborhood of y = 0 surrounding y = 0, ne the
number of elliptic and nh the number of hyperbolic sectors in C. Then
the index jf@) of the point y = 0 is given by
(9.1) 2jf@) = 2 + ne-nh.
It is clear that for a rotation point y = 0, ne = nh = 0 andy/O) = 1, so
that (9.1) holds and such points need not be considered. Thus it can be
supposed that there is a decomposition of the interior of C into elliptic,
hyperbolic and parabolic sectors.
Proof. C will be replaced by a piecewise C1 Jordan curve C", around
y = 0, made up of solution arcs and orthogonal trajectories, as shown in
Figure 12, where E, H, P represent elliptic, hyperbolic, and parabolic
sectors. If r\ is the tangent vector to C" and / is a subarc of C, with
endpoints which are not corners, then, in this proof l-nj^J) will represent
the contribution of/to ivj^C1) = 2n, i.e., the variation of the turning of
r\ along / taking into account discontinuities of r\ in accordance with
Corollary 2.1. Furthermore ^@) =jt(C).
The Poincare-Bendixson Theory 167
If L is an arc joining two points A and B, the notation [AB], (AB), etc.
will be used to denote the corresponding closed arc, open arc, etc. For
any sector with boundary base solutions y = y^t), y2(t), let At = 2/i(±e0)
for a fixed e0 > 0, where ± is chosen according as yt(t) is a positive or
negative base solution.
(a) Hyperbolic Sectors. Let 5 be a hyperbolic sector determined by
2/ = 2/i@> 2/2@- In order to fix ideas, suppose that y = y^t) is a positive
base solution. Thus Ax = y1(+e0), where e0 > 0. It is clear that At $ Se,
the elliptic part of S. Since Se is closed, points y of S near Ax are not in Se.
Figure 12.
Consider the differential equation
(9.2) y'=g(y), where g = (-p(y),f\y)),
for the orthogonal trajectories of solutions of y' =/. Let Lt = [^t-5J,
/ = 1, 2, be a solution arc of (9.2) with initial point At such that {Afi^ is
in S — Se; see Figure 13. It will be shown that if Lx = [A^] is suffi-
sufficiently short and yoe{AlBl], then the solution of y = yo{t) of y' = f
2/(%) = 2/o exists and yo(t) e S on an interval e0 ^ t ^ t*, yo(t*) is on and
is the only point of y = yo(t) on L2 = (A2B2], and yo(t*) -> A2 as y0 ->- Ax.
Let T > 0, € > 0. Then, by Theorem V 2.1, there exists a d = d(e, T) >
0 such that \\y0 — A^l <d implies that yo(t) exists and satisfies \\yo(t) —
.Vi@ll < e for eo ^ t^ T. In particular, yo(t) e S for €0 ^ t ^ Tif T and
1/e are sufficiently large. Let t1 = *%„) be the least t > €0 such that
?/0(/1) e C12, so that t1 > T. The fact that S is hyperbolic and that-?/0 ? 5e
implies the existence of t1. It is clear that t\y0) -»• 00 as y0 -»• ^x.
Choose y01) y02, . . . such that yOn e (^A], /^ = limyOn, y° = lim^o^1)
exists on C12 as «—»• 00, where yOn(t) is the solution belonging to
?/o = Van and f1 = ^(yon)- Thus, the solution y_n(t) = yOn(t + t1) exists
and is in S u d2 for —t1 ^ t ^ 0 and 2/-n@) ->¦ y°, n -+ 00. By Theorem
168 Ordinary Differential Equations
V2.1, y-x(t) = \\my_n(t) exists uniformly on every bounded interval
-T = t < 0, is the solution of y' =/, y@) = y°@) ? C12, and y_M@ ? 5
for f = 0. Since 5 is a hyperbolic sector, y_M@ is on the boundary of 5
for large -t; i.e., y_x(t - t0) = y2(t) for f < 0 and y-x(-t0) is the last
point of y_M@ ? C12 when / decreases from 0.
Figure 13.
It follows that according as t0 > 0 or f0 = 0, yo(fx - t0) or y^f1) tends
to y2@) as y0 -»¦ y4x. In particular, if y0 is sufficiently near to Alf there
exists a least f* > e0, t* = t*(y0), such that yo(t*) ? L2 = 04252]. Also
Let Li = Mi^J be so small that t*(y0) exists for all y0 ? Lx. Choose B2
to be yo(t*) for yQ = Bx; see Figure 13. Let C4 be an interior point of
[AfBf] and /the arc CXBXB2C2 consisting a piece of the orthogonal trajec-
trajectory Llt the solution arc y = yo(t) joining Bx and 2?2, and the piece [C22?2]
of L2. It is easy to see that
(9.3) 277/X/) = 2ltjn(J) - 77,
for the angle from r\ to/is \n on [Q5J. It changes to 0 on going through
2?! and is 0 on (BXB^. It jumps to — n/2 on passing through B2 and is
-tt/2 on E2C2]. This gives 2n[Jf(J) - jn{J)] 77/2 - 77/2 = -77, i.e.,
(9.3). If yx(t) is a negative and y2(t) a positive base solution, the formula
(9.3) is still valid.
F) Parabolic Sectors. Let S be a parabolic sector. In order to fix
ideas, let S be a positive parabolic sector, so that At is a point yX€o) where
€0> 0.
The Poincar6-Bendixson Theory 169
Note that y = 0 $ Sh, the hyperbolic part of S. For otherwise, the
arguments of part (a) show that there exists a negative base solution in
S U C12. Also, if y j? 0 is a point of 5e, then y ^ Shby the definitions of
5e, 5ft. Hence 5e n Sh is empty, so that dist (Se, Sh) > 0. It is also clear
that Se O C12 is empty. Furthermore, y?t) $ Se for / = 1,2 and / ^ 0,
for otherwise y^0 6 5e f°r 1 = 1 or 2 and all t ^ 0 and, in particular,
y/0) e C12 O 5e, which is impossible. Similarly, yt(t) $ Sh for i =1,2 and
/ > 0, otherwise yt(t) ? 5ft for / = 1 or 2 and all t ^ 0 and, in particular,
the limit point y = 0 e Sh.
Figure 14.
If Sh is not empty, there are points yoe S for which a solution y =
yo(O of F.1) on some interval a_ < t ^ a+ is such that yo(O e 5 for a_ <
/ < a+ and yo(«±) ? Q2. The arc y = yo(t), ct_ ^ t < a+ and the corre-
corresponding subarc of C12 joining yo(a.±) form a Jordan curve. There is a
finite or infinite sequence of such maximal Jordan curves Jlf J2, . . . in the
sense that the interiors of J1} J2, . . . are pairwise disjoint and the union of
Ji, J2, . . . and their interiors contains S n Sh; see Figure 14. The set of
points on C12 not on any/„ together with the points on the closures of the
arcs /„ O S form a Jordan arc C[2 in S U C12 joining 2^@), y2@). Let 5'
be the interior of the Jordan curve consisting of C[2, y = 0, andjhe arcs
.'/ = Vi(t) for i = 1,2 and / ^ 0.
If Sh is empty, let 5 = 5' and C12 = Ci2. Then, whether or not Sh is
empty, 5,, c 5'; and so 5e O Q'g is empty. Hence there is a polygonal
path P\ y = p{s), 0 = 5= 1, which joins A1 = p@), A2=p{\) and
p(s) e 5' - 5, for 0 < s < 1.
170 Ordinary Differential Equations
If 0 < t < 1, the solution y = y,{t) of y' = /, 2/@) = p(s) is such that
ys{t) e S for t ^ 0 and y?-ts) e C12 for some f, > 0. Through each point
p(s)eP, draw an open orthogonal trajectory arc L; i.e., a solution arc of
(9.2) through y = p(s) such that the closure of L is in S' — Se. In the
casesp@) = At andp(l) = A2, let the corresponding L be half-closed and
have Ai as an endpoint instead of an interior point.
The orthogonal trajectories L can be taken so short that they lie in
S' - Se. It follows that the solution y = yo(t) of y' = f, 2/@) = yo<= L
exists and is in S for t> 0 and meets C for some t < 0.
The set of orthogonal trajectory arcs L can be considered to form an
"open covering" of the s-set 0 ^ 5 ^ 1 in the sense that s is "contained"
in an L if an arc y = ys(t), 0 ^ / ^ fs or -t* ^ / ^ 0 is in S and contains
a point of L. If an s0 is "in L," then s near s0 is "in L." Thus, by the
theorem of Heine-Borel, there is a finite set Lt = [A1} B1), L2 = [A2, B2),
L3 = (A3, B3),. .. of these arcs such that every solution y = ys{t) meets at
least one of the L1; L2,.. . at some f (| 0); in which case y?t) e S for
Let A, = 2/(d, 2/B), • ¦ •, 2/(n> = ^2 be the endpoints of the arcs L1;
?».... The solution y(Jt)(/) of 2/' =/B/), 2/@) = 2/(*> exists and is in S for
t ;> 0. Also, since yik) e S' - Se, there is a least t = tk > 0 such that
Pfc s 2/(S:)(-0 ? C. Thus y(fc)(/ - tk), / ^ 0, is a positive base solution.
After a suitable change of enumeration, it can be supposed that Px =
2/i@), P2, • • •, Pn = 2/2(O) are ordered on C and that Pt # P, for / # y;
see Figure 15.
Each solution pair y = ylk){t - tk), y = y{k+1)(t - tk+1) defines a sector
Sk which is a subsector of 5. Let s0 be the largest rvalue, 0 g s ^ 1, such
The Poincare-Bendixson Theory 171
that p(s) meets the arc y = yik)(t — tk) and s1 the least rvalue sx > 50 such
thatp^) is on the arc y = y{k+1)(t — tk+1). Now y = ys(t), for s fixed in
s0 < 5 < «!, is in 5fc and meets at least one of the selected L^L^.i...
Since the arcs Llt L2, . . . have no endpoints in Sk, there is at least one of
the arcs L1; L2, . . . which meets every solution arc y = ys(t) for s0 <
s < sv Thus, there are closed orthogonal trajectory arcs L1,.. ¦, LN~X
each of which is a subarc of one of the closed arcsLi, L2, . . . , such that V
joins 2/ = 2/(*>(' - h) and 2/ = 2/(*+i>0 - W- It can be supposed that
I}, L* begin and terminate, respectively, at Alt A2; say L1 = [AXB^\,
I* =[B2A2].
Let d, C2 be interior points of L1, LN-1 and /the arc joining C[, C2
consisting successively of subarcs of L1, y = y^)(t — t2), L2, y =
2/C>(' - Q, ¦ ¦ ¦, 2/ = 2/w-dO - *jv-i)> i-*- It will be verified that
(9.4) 27T/X.7) = 2^,G).
Note that the tangent vector r\ to / at a point y e Lk is in the direction
±gB/), where g(y) occurs in (9.2) and ± is independent of A:. (That the ±
is independent of A: can be seen as follows: Suppose that g(y) points into
5X at Au then it clearly points into S2 at Bv By continuity it points
into 52 along the solution arc y = y{2)(t) and, hence, at the end-point of
L2 on y = 2/B)(O- Similarly, it points into S3 at the endpoint of L2 on
V = 2/3@- This argument can be continued and shows that ± does not
depend on A:.)
Thus the sign of the sine of the angle from r\ to/is =F 1 and, consequently,
the angle is of the form 2mr =F J"", where n is an integer and =F is independ-
independent of k. On L1 = [Q5J, the angle from rj to/is \tt (so that on L2,. ..,
I-*, it is of the form 2«7t + \ri)- On the part of/consisting of the solution
arc y = y{2)(t — t2), the angle from r\ to/becomes 0 or n. Hence on I?, it is
|7r. Continuing this argument, it is seen that the angle from r\ to/is \tt on
every V. Thus 27r[y/(/) -/,(/)] = |tt - \* = 0; i.e., (9.4) holds.
It is readily verified that if S is a negative parabolic sector, a similar
construction also leads to (9.4).
(c) Elliptic Sectors. Let 5 be an elliptic sector with boundary solutions
?/ = 2/i@- 2/2@ which are subarcs of some y = yo(t), — 00 < t < 00, in S.
Suppose that the constructions just described have been made on all of the
hyperbolic and parabolic sectors. Since S is adjacent to such sectors, there
is a solution arc [/^J on the arc y = yo(t) [containing the points 2/i@),
.11/2@) in its interior] and two orthogonal trajectory arcs [C2A2), (A^j] in
the interiors of the adjacent sectors, respectively; see Figures 16 and 17.
If/ is the arc [C2CJ consisting of the orthogonal trajectory [C2A2), the
solution arc [^2/l1], and {A^C^, then
(9.5)
2njf(J) =
172 Ordinary Differential Equations
For, on [CZAJ in Figure 16 [or in Figure 17], the angle from r\ to / is
] ll (A^J it b
it becomes
[or 3tt/2 -
g
-\n [or \n]; on (AZA{), it is 0 [or n]; and finally, on
\it [or 3tt/2]. Thus 2n[jf(J) - jn{J)] is fr - (~K
177 = 77]; so that (9.5) holds.
(d) Completion of the Proof. In the constructions in parts (a) and (b),
the same number €0 > 0 has been used for all of the hyperbolic and
parabolic sectors. Thus, if a base solution y = y(t) is on the boundary of
Figure 16.
Figure 17.
two such adjacent regions, there is an orthogonal trajectory arc [C2Cj]
cutting y = y(t) at a point A. For the arc / = [QCJ, it is clear that
(9.6)
= jn(J).
Thus if the relations (9.3), (9.4), (9.5), and (9.6) are added for all arcs /,
we obtain 2njf@) = 2n — nnh + nne, where the 2t7 on the right is 2t7
?;,(/) by the (Umlaufsatz) Theorem 2.1 and its Corollary 2.1. This
proves (9.2).
Remark. For the purpose of the next exercise, note that the assumption
in Theorem 9.1 that "the solution of y' = f(y), y{0) = y0 is unique" can
be relaxed to the assumption that "the solution of y' = f(y), y@) = y0 is
unique when y0 # 0." (This involves an obvious modification of the
definitions of "base solution," "elliptic sector," "hyperbolic sector," and
"parabolic sector.") For y' = f(y) can be replaced by y' = h(y), where
%) = \\y\\J\y)- The two indicesy/O),;^) are obviously equal. It is clear
that an arc y = y(t), where y(t) # 0, is a solution of y' = f(y) if and only
if it becomes a solution of dyjds = h(y) after the change of parameters
t-*s where ds = dt/\\y(t)\\. Thus "ne, nh" are the same for both systems
y' =f(y), y' = %). Finally, since y = h implies that ||y'|l ^ IMI for
small ||y||, it follows that y = 0 is the only solution oft/ = h{y),y@) = 0.
The Poincare-Bendixson Theory 173
Exercise 9.1. Let U(y) be a real-valued function of class C1 for \\y\\ <b
such that t/@) = 0 and the gradient g(y) = (dU/dy1, dU/dy2) vanishes
only for y = 0. Thus if f(y) = {dU/dy\ -dU/dy1), then U is constant on
solutions of y' = f. Show that either (i) there is an e > 0 such that U(y) ^
0 in 0 < \\y\\ < e or (ii) the set of arcs in 0 < \\y\\ < b which join y = 0
and \\y\\ = e and on which U = 0 consists of a finite (even) number 2« of
arcs, In > 0; furthermore, j/0) = 1 — n. [Note: The initial value
problems y' = f(y), 2/@) = y0 # 0 have unique solutions. (Why?)
Case (i) occurs only if y = 0 is a rotation point, in which case it is a center.
This happens only if U has a strict local maximum or minimum at y = 0.
Case (ii) occurs if y = 0 is a nonrotation point, in which case there are no
elliptic or parabolic sectors for any Jordan curve C surrounding y = 0.]
Theorem 9.2. Let f(y) be continuous on a simply connected open set E
such that solutions of initial value problems F.1) are unique. Let C be a
positively oriented Jordan curve of class C1 in E with the property that
f(y) #0o«C and thatf(y) is tangent to C at only a finite number of points
y1} ..., yn of C. Let ne, nh be the number of these points yt where the
solution arc y = y(t)ofy' =fy@) = y i for small\t\ is internally, externally
tangent to C at yt (so that ne+ nh ^ n). Then 2jf(C) = 2 + ne — nh.
The solution arc y(t) of y' = f y@) = yi e C is said to be internally
[or externally] tangent to C at y{ if there exists an e > 0 such that y(t) is
interior [or exterior] to C for 0 < |/| ^ e.
Proof. Let r\ denote the positively oriented tangent vector on C. If
ne = nh = 0, then it is clear that the angle from r\ to / along C does not
pass through a value 0 mod 77. Hence the two integers jt(C),jn(C) are
equal. Since jn(C) = 1 by Theorem 2.1, it follows that 2jf(C) = 2 if
ne =nh = 0.
Hence it can be supposed that not both n" and nh are zero. A point y0
of C will be called an elliptic [or hyperbolic] point if the solution arc
through y0 is internally [or externally] tangent to C at y0. Thus ne [or nh]
is the number of elliptic [or hyperbolic] points.
Let / = [AB] be a subarc of C such that A, B are elliptic or hyperbolic
points but no interior point of J is elliptic or hyperbolic. Then at a point /
//„ interior to /, the solution arc through y0 crosses C in a direction (from
interior to exterior or from exterior to interior of C) independent of yo.~~
Thus by considering the angle from r\ to/, it is seen that 2njf(J) — 2njn(J)
is —77, 0, or 77.
If one of the points A, B is elliptic and the other hyperbolic, then f(y)
and the tangent vector r\ have the same orientation at both A, B or have
the opposite orientation at both A, B; see Figure 18a. In this case,
2nj,(J) - 2njtl(J) = 0.
If both points A, B are elliptic or both are hyperbolic, then/(t/) and the
174 Ordinary Differential Equations
tangent vector r\ have the same direction at one of the points A, B and have
the opposite orientation at the other point. In this case, 2nj/J) —2-irjn{J) =
±7r. It will be left to the reader to verify that 2vjf(J) — 2njn(J) is n or — n
according as both A, B are elliptic or both are hyperbolic; cf. Figure 186.
Thus, if n'(J) or n\J) denotes the number @, 1, or 2) of the endpoints of
J which are elliptic or hyperbolic, then
= 2njn{J) + MnV) ~
(bl
Figure 18.
holds in all cases. Summing this relation for all subarcs / gives the desired
result since
I;„(¦/) = 1, * 1 n\J) = n\
n\J) = n\
j j
10. A Second Order Equation
An application of the theorem of Poincare-Bendixson will be given in
this section in Corollary 10.1. It concerns the second order equation
A0.1)
u" + h(u, u')u + g(u) = 0
for a real-valued function u or the equivalent autonomous first order
system
A0.2)
u' = v, v = —h(u, v)v — g(u).
If, in A0.1), g(u)u > 0, then the termg(w) is a "restoring force" (as for the
harmonic oscillator u" + u = 0). If h > 0, then the "frictional" term hu
tends to decrease the speed \u'\ (as in the equation u" + hu = 0 with
h > 0 a constant). The next theorem can be interpreted as saying that if
the restoring force and frictional term are not too small, then the solutions
of A0.1) are bounded.
The Poincare-Bendixson Theory 175
Theorem 10.1. Let g(u), h(u, v) be real-valued, continuous functions for
all u, v with the properties: (i) solutions of A0.2) are uniquely determined
by initial conditions; (ii) there exists a number a > 0 such that
g(u)u > 0 for \u\ ^ a,
fu
G(u) = g(r) dr —»• oo as |m|—>-oo;
Jo
A0.3)
A0.4)
(iii) there exists a number m > 0 such that
A0.5) h(u, v) ^ — m, for |w| ^ a, — oo < v < oo;
(iv) ifho(u) = inf h(u, v)for — oo < v < oo, then
A0.6) ho(u) > 0 for \u\ ^ a,
A0.7) H(u) = I ho(r) dr-+co as u -* oo.
Then there exists a Jordan curve C bounding a domain E, containing the
origin (u, v) = @, 0), such that no point of C is an egress point for E and
that ifu(t), v(t) is a solution o/A0.2) starting, say, at t = 0, then u(t), v(t)
exists for t ^ 0 and (u(t), v(t)) e Efor large t.
A sufficient condition for (iv) is that there exists a number M > 0 such
that h(u, v) ^ M > 0 for |w| ^ a, — oo < v < oo. Actually, the proof
below does not use the full force of (iv) but only that h(u, v) > 0 for
\u\ ^ a, that h°(u) > 0 for u ^ a if h°(u) = inf h(u, v) for v ^ -v0, and
that h\u) ^0 for u ^ a and h\r) dr -* oo as u -* oo if /;!(«) =
Ja
inf A(m, u) for v ^ u0 > 0 for a fixed tv
Proof. Along a solution arc w@, v{f) of A0.2),
A0.8)
aw
Below m', c' refer to the derivatives in A0.2) and dvjdu refers to A0.8).
Let \g(u)\ ^ b for \u\ ^ a. Then, by A0.5) and A0.8),
A0.9) — ^m+ — ^
rf ||
if \u\ ? a, \v\ ^ —.
m
Let q>(u) be a positive, continuous increasing function of u ^ a such that
A0.10) <p(u) >
ho(u)
and
<p(u) -+oo as u —*¦ oo.
176 Ordinary Differential Equations
v < -<p(u) < - -^ implies that v < 0, h(u, v) ^ ho(u) >
Then
- — ; so that
A0.11)
— <0 if « ^ a, v<-<p(u)
du
Consider the arcs
A0.12) Ca: %v2+G(u) = a for m| ^ |a
for a large constant a > 0. These arcs are symmetric with respect to the
w-axis. A portion of these arcs and a segment of the line u = a [or u = —a]
Figure 19.
form a Jordan curve bounding a domain E+(a) [or ?_(a)]; see Figure 19.
Along a solution u(t), v(t) of A0.2), the quantity rp(t) = \v\t) + G(u(t))
has a derivative tp' = v(—hv — g) + gv = —h(u, v)v2 < 0 if \u(t)\ ^ a,
v(t) # 0. Thus ip(t) is decreasing and so the arc u = u(t), v = v(t) enters
.E±(a) with increasing t as soon as it meets thecurved boundary Ca of E±{x)
and remains in E±{a) as long as |m(?)| ^ a.
For convenience, the construction of C will refer to Figure 20. A letter
on this diagram denotes either a point or one of its coordinates; e.g., v2
denotes either the point (a, v2) or the ordinate v2.
Choose a > a so large that
A0.13)
^- and
m
4a
> 2m.
Choose a so that Ca passes through the point A = {a, —cp{a)); i.e.,
a = \<p\o) + G{a). In particular, Ca passes through the point (m, v) =
(a, 9?(<t)). Let t > 0 denote the point where Ca meets the w-axis.
Let u°(t), v°(t) denote the solution of A0.2) determined by the initial
condition (mo(O), ^o(°)) = (T> °)- As t decreases from 0, u°(t) decreases and
The Poincare-Bendixson Theory 177
v°(t) increases until m°(/) takes the value a at some point t = t2 < 0. This
is clear from v' = — hv — g ^ — g < 0, hence u' = v > 0, as long as
u ^ a. Let C° denote the arc (u, v) = («°@, v°(t)). In view of the remarks
above, the part of C° for /2 < / ^ 0 has only the point t in common with
Ca. In particular, if v°(t) = vt when u"(t) = a and if v0 = <p(o), then
Figure 20.
Vl > v0. Also, by A0.8), dv/du ^ -h(u, v) < -ho(u) on C° for a < u ^
ff. Hence, if v2 = u°(/2), then
A0.14)
Ja
Ao(r) dr=Vl + H{a)
H{a).
Put y = u0 + \H{o) <v2 — ?H(o). Thus the slopes of the line seg-
segments y to v2 and — y to — u0 are at least \H(a)\2a and thus exceed 2m,
by A0.13). Define /S by /S = |y2 + G(a), so the arcs C^ pass through the
points dby.
Let C denote the Jordan curve consisting of the arc C° from t to u2, the
line segment from f2 to y, the arc C^ from y to — y, the line segment from
— y to — v0, the horizontal line segment from — v0 to A, and Ca from A to
t. Let ? denote the interior of C.
It will first be verified that the points of C, except for the points on C°,
are strict ingress points of E. To this end, it is sufficient to verify that the
178 Ordinary Differential Equations
solution arcs of A0.2) reaching the lines segments on C cross C as indicated
in Figure 20. This is clear along the horizontal segment from — v0 to A,
for u' = v < 0 and dvjdu < 0 by A0.11) and the monotony of q>(u) for
u ^ a. The slope of the segment from — y to — v0 is H(a)l4a > 2m while,
along this segment, u' = v < 0, dvjdu ^ 2m by A0.9) since v0 = c/>(ar) >
6/m by A0.13). Similarly, the slope of the segment from y to v2 exceeds
2m and on this segment v ^ y > v0 ^ i/w so that m' = c > 0 and
dvjdu ^ 2w.
It remains to show that every solution u(t), v(t) of A0.2) starting at
t = 0 exists for t ^ 0 and that (u(t), v(t)) e E for large t. For this purpose,
rename a to a0 and the Jordan curve C to C(ff0). Then, for each a > ff0,
the above construction leads to a Jordan curve C(a) and its interior ?(<x).
The sets E{a) are increasing with a and the union U = UC(<r) for a > ct0
is the exterior of C(cr0). Let u(t), v{t) be a solution of A0.2) starting at a
point of U for t = 0. Then, as long as m@, ^@ remains in U, there is a
unique a = a{t) such that (u(t), v(t)) e C(a(t)) and a(t) is a nonincreasing
function of t. This implies that (u(t), v(t)) exists for t ^ 0; cf. Corollary
II 3.2.
Suppose, if possible, that u(t), v(t) does not enter ?(<x0) eventually, so
that a(t) ^ ff0 for all t > 0. Let crx = lim a(t) as f -»- oo. Thus (w(O, c(O)
is arbitrarily near to C(ffx) for large /-values. It is clear in this case,
however, that (u(t), v(t)) e E(a{) for arbitrarily large t. But then a(t) < ctx
for large t. This is a contradiction and proves the theorem.
Corollary 10.1 In addition to the assumptions of Theorem 10.1, suppose
that g(u) ^ 0 for u 5^ 0 [so that g(u)u > 0 for u # 0 and the origin is the
only stationary point for A0.2)] and suppose that the origin is not an co-limit
point for every solution of A0.2). Then A0.2) has a periodic solution
(uo(t), v 0@) ^ 0 which is asymptotically orbitally stable from the exterior as
t —»¦ oo and the interior of the Jordan curve u = uo(t), v = voiO contains the
origin.
Proof. If some solution (u(t), v(t)) does not have the origin as an
co-limit point, then the theorem of Poincare-Bendixson (Theorem 4.1)
implies that its set of co-limit points is a periodic solution (mo(O, vo(t)),
since (u(t), v(t)) e ? for large t.
The Jordan curve u = uo(t), v = vo(t) surrounds the origin by Theorem
3.1. It also follows that if there are two periodic solutions, then one of the
corresponding Jordan curves is contained in the interior of the other.
Since all periodic orbits are contained in the compact set ? u C, it follows
that there exists a unique periodic solution mo(/), vo(t) such that the Jordan
curve C0:u = uo(t),v = vo(t) contains all other periodic orbits in its
interior.
The periodic solution uo(t), vo(t) is asymptotically orbitally stable from
The Poincare-Bendixson Theory 179
the exterior as t —»• oo. In order to see this, consider a solution u(t), v(t)
starting for t = 0 at a point exterior to Co. Since the origin is not an
colimit point for u(t), v(t), its set of colimit points is a periodic orbit
(Theorem 4.1) which is necessarily Co. Thus the assertion follows from
Theorem 5.2.
Exercise 10.1. (a) Let g(u),h(u, v) be continuous for all u, v and let
e(t) be continuous for all t. Suppose that solutions of
A0.15) u" + h(u, u')u' + g(u) = e(t)
are uniquely determined by initial conditions. Let there exist positive
constants a, e0, m, M such that (i) h(u, v) ^ M for \u\ ^ a, \v\ ^ a;
(ii) h(u, v) ^ — m for all u, v; (iii) \e(t)\ ^ e0 for all t; and (iv) ug(u) > 0
for large |w| and lim inf \g(u)\ > ma + e0 as |m| —»¦ oo. Then there exists a
Jordan curve C in the (u, u)-plane such that if u(t) is a solution of A0.15)
starting at t = t0, then u(t) exists for t ^ t0, (u, v) = (u(t), u'(t)) is in the
interior ? of C for large t and the (u(t), u'(t)) cannot leave ? with increasing
t. (b) If, in addition, e(t) is periodic of period/? > 0, then A0.15) has a
solution of period p. See Opial [7].
Exercise 10.2. Show that Theorem 10.1 remains correct if condition
(iii) is strengthened to h(u, v) ^ m > 0 for |m| ^ a, — oo < v < oo, and
(iv) is relaxed to h(u, v) ^ 0 for |m| ^ a.
A particular case of A0.1) is the equation
A0.16) u" + h(u)u' + g(u) = 0.
If g(u) = u, A0.16) is called Lienard's equation. An elegant simple
argument shows, under certain conditions, the existence of a periodic
solution which is unique (up to translations of the independent variable t).
The equation A0.16) will be treated as the first order system
A0.17)
where
A0.18)
«' = v - H(u), v = -g(u),
(M) = I
Jo
h{r) dr and G(;
Jo
g(r) dr.
Theorem 10.2.. Let h(u), g(u) be continuous for all u with the properties:
(i) that solutions of A0.17) are uniquely determined by initial conditions;
(ii) that h(u) = h(-u) is even, H(u) <0for0<u<a, H(u) > 0 and is
increasing for u > a, H(u) —»¦ oo as u —*¦ oo; finally, (hi) that g{u) =
—g(—u) is odd and ug(u) > 0 for u # 0. Then A0.17) has exactly one
periodic solution (uo(t), vo(t)) ^ 0, up to translations of t, and this solution
is asymptotically, orbitally stable {from the exterior and interior) as t —»• oo.
Of course, (u(t), v(t)) is a [periodic] solution of A0.17) if and only if u(t)
is a [periodic] solution of A0.16).
180 Ordinary Differential Equations
Proof. The functions on the right of A0.17) are odd functions of
(u, v); so that if u(i),v(t) is a solution, so is — u(t), —v(t). The tangent to a
solution arc u(t), v{t) is horizontal at a point (u, v) if and only if u = 0 and
is vertical if and only if v = H(u).
The arguments to follow refer to Figure 21. Along a solution starting
at vx > 0, u increases and v decreases until v = H(u), say, at the point y.
Then u decreases, v continues to decrease and the solution arc remains
/v = H(ul
Figure 21.
below the curve v = H(u) until the solution meets the r-axis, say, at v = v2.
Otherwise the solution would have a horizontal tangent at a point where
u # 0 or v would tend to — oo as u tends to a finite value, which is im-
impossible by
A0.19)
du
v- H(u)
By symmetry, it is clear that a continuation of the solution is closed if
and only if v2 = —vv It is clear that we can start at any point y, i.e.,
(y, H(y)), and determine the points vx, v2 by moving along the correspond-
corresponding solution arc with decreasing, increasing t. Denote J(t>22 — vx2) by
q>(y). Thus if ip(u, v) = \v2 + G(u), then
<p(y) = dxp,
Jviyvz
where the integral denotes a line integral along the solution arc from v1 to
y to v2.
If 0 < y ^ a, then cp{y) > 0, for along a solution arc, dip/dt = vv' +
g(u)u' = —g(u)H(u) > 0. If y > a, let a and /S denote the points where the
The Poincare-Bendixson Theory 181
solution arc meets the line u = a; see Figure 21. Write
<p(y) =
\ where <Pi = \ +\ ^ =
Along the solution arcs contributing to (plt dtpjdu = v(dvjdu) + g(u) =
-g(u)H(u)l(v - H(u)), by A0.19). Thus dip/du > 0, du > 0 on Vlx and
dipjdu < 0, du < 0 on fiv2, so that cp^y) > 0. As y increases, the arc v^a.
is raised, so that g(u) \H(u)\/(v — H(u)) decreases; and fiv2 is lowered, so
thatg(u) \H(u)\j\v — H(u)\ decreases. Hence y-fy) decreases as y increases.
On the solution arc ot.yE, dv < 0 and dtpjdv = v + g(u) dujdv = H(u) >
0. Thus
Jzy
H(u) dv < 0 for y > a.
If a.yP has a parametric representation u = u(v) = u(v, y), it is clear that
u(v, y), hence H(u(v, yj) is an increasing function of y. Thus dv < 0
implies that cp2(y) decreases as y increases. If u ^ a + d > a, then
H(u) ^ e > 0 for some constant e. This assures that y2(y) -+-oo as
y —*¦ oo.
Thus, cp(y) > 0 for 0 < y ^ a, cp(y) = (p-^iy) + <p2{y) is a decreasing
function of y > a, and cp(y) —*¦ — oo as y ->¦ oo [since cp2{y) ->¦ — oo and
q?!^) is decreasing]. Hence, there exists a unique y0 such that (p(y0) = 0.
The corresponding solution of A0.17) is a nontrivial periodic solution,
which is unique up to translations of the parameter t.
The fact that cp(y) < 0 for y > y0 [cp(y) > 0 for 0 < y < y0] makes it
clear that this periodic solution is asymptotically orbitally stable from the
exterior [interior] as t -*¦ oo.
Exercise 10.3. (a) Verify that Theorem 10.2 applies to van der Pol's
equation
A0.20)
u" + fi(u2 - l)w' + u = 0,
where fi > 0 is a constant, (b) If A0.17) is the system corresponding to
A0.20), what is the nature of the stationary point (u, v) = @, 0)?
Exercise 10.4. Show that if the condition "H(u) > 0 and is increasing
for u ^ a" is omitted in Theorem 10.2, then the conclusions of Theorem
10.2 are valid with "exactly one" replaced by "at least one" and "exterior
and interior" replaced by either "exterior" or "interior."
Exercise 10.5. Let h, g satisfy the conditions of Theorem 10.2; in
addition, let g(u) be increasing for u > 0; and let fj. > 0 be a constant.
Let T(ji) denote the limit cycle in the (m, u)-plane belonging to
A0.21)
u" + fih(u)u' + g(u) = 0;
182 Ordinary Differential Equations
v = H(ul
Figure 22.
i.e., the unique nontrivial periodic solution arc of
A0.22) u' = v- f*H(u), v' = -g(u).
Show that, as yu -» oo, Y(jj) tends to the closed curve consisting of two
horizontal line segments and two arcs on c= H(u) as in Figure 22.
Compare Lefschetz [1, pp. 342-346].
APPENDIX: POINCARfi-BENDIXSON THEORY ON
2-MANIFOLDS
Although the proof of the Poincare-Bendixson Theorem 4.1 for plane
autonomous systems depended on the validity of the Jordan curve
theorem, it turns out that analogous results hold under suitable smoothness
assumptions on arbitrary 2-dimensional differentiable manifolds. These
analogous results will be the main object of this appendix.
11. Preliminaries
The objects of study of this appendix will be flows on 2-manifolds.
Definition. 2-manifold of class Ck. Let M be a connected, Hausdorff
topological space for which (i) there is given an open covering M = U Ua,
where a. e A and A is some index set, and for each a, a continuous, one-to-
one map ga of Ua onto an open (plane) square such that (ii) if Ua n Up is
not empty, theng/g-1) is a map of ga(Ua O Up) onto gp(Ua O Up) which
is of class Ck (so that, if k ^ 1, this map has a nonvanishing Jacobian).
Then {M; Ua;ga} is called a 2-manifold of class Ck.
If m denotes a point of M, then ya = ga(m) is a function of m e Ua, the
values of which are binary (real) vectors ya = (y^1, yj). As m varies over
Ua, y^ varies over a square; e.g., \y?\, \y?\ < 1. Ua is called a coordinate
neighborhood of any point m e Ga and y^ the local coordinates of m =
g~\y^). Condition (ii) concerns the map yp = gp(g*\yj). (The paren-
parenthetical part concerning the Jacobian d{y^, y*)ld{yj;2/a2) is redundant
The Poincard-Bendixson Theory 183
since the inverse maps gpig^1) and ga(gp~x) are assumed of class Ck,
k^ 1.)
It is important to emphasize that the 2-manifold {M; Ua; gj consists of
the space M, the given covering U Ua, and the given set of homeomorphisms
ga. Usually, Ua and ga are fixed and {M; Ua; gj will be abbreviated to M.
Definition. Let M be a 2-manifold of class Ck, k^.0. Let U, g be an
open set of M and y = g(m) a homeomorphism of U onto an open square.
Then U is called an admissible coordinate neighborhood on M and y =
g(m) (admissible) local coordinates of m e U provided that {M; Ua and
U; ga and g} satisfies conditions (i) and (ii) of the last definition.
Definition. Let M be a 2-manifold of class Ck, k^.0. By a flow
A1-1)
m
* =
= Vm or ml =/u(t,m)
of class Ck is meant a function fi{t, m) defined for — oo < t < oo, m e M
with values in M such that (i) for a fixed t, Tl: M -> M is a homeomorphism
of M onto M; (ii) 71' is a group of maps
(i i .2) rr = r+\ i.e., M/, h{s, m)) = M' + *, ™),
in particular, ;u@, m) = m; (iii) yu(/, w) is a continuous function of (t, m);
finally, (iv) if k ^ 1, then fi{t, m) is of class Ck as a function of (t, m).
The last two conditions have the following meaning: Consider any
given (/„, m0), let Ua be a coordinate neighborhood containing m0, 2/a =
gjm) local coordinates on (/a, and let fi(t0, m0) e Up. Then (iii) means
that fi{t, m) e (Tg for (t, m) near (/„, m0) and (iv) means that
(ii-3) yf = gM*> s~x\yx)\)
is of class Ck as a function of (t, ya) on the open (t, 2/J-set on which the
right side is defined.
Lemma 11.1. Let k ^ 1. Then, for a fixed t, the Jacobian of the map
Sap'-y* -* y? given by A1.3) does not vanish {wherever S'^ is defined).
Proof. For, by (ii), 5^ has the inverse
yx =
-t, gj\yfi)]),
which by assumption is also of class C*. This implies the lemma.
A "flow" is a generalization of the concept of "general solution of
autonomous differential equations on M"; cf., e.g., § 14 or § IX 2.
For any given point m0, the subset C(m0) = {m0( = fi{t, m0), — oo <
t < oo} of M is called the orbit through m0. Since each point moe M
uniquely determines its orbit, it is clear from the group property (ii) that
two orbits are either identical or have no points in common.
When the orbit C(m0) reduces to the point m0 [i.e., fi{t, m0) = m0 for
— oo < t < oo], then m0 is called a stationary point.
184 Ordinary Differential Equations
Lemma 11.2. Let k^.\ and m0 e Ua. Then m0 is a stationary point if
and only ify = gj^(t, w0)) has the derivative 0 at t = 0.
Exercise 11.1. Verify this lemma.
A subset Mx of M is called an invariant set if TtM1 = Mx for every t,
— oo < t < oo. Thus Mx is an invariant set if and only if TlMx <= Mx
for — oo < / < oo or, equivalently, if and only if C(mx) <= Mx whenever
mx 6 Mx. In particular, if Mj is a closed invariant set and mx e Afl5 then
C^), the closure of the orbit through mx, is contained in Mx.
A subset N of Mis called a minimal set if (i) TV is a closed invariant set
and (ii) TV contains no proper subset which is closed and invariant.
For example, if m0 is a stationary point, then the set consisting of m0 is a
minimal set. More generally, if fi(t, m0) is periodic [i.e., if there exists a
number/? > 0 such that fi(t + p, m0) = fi{t, w0) for — oo < t < oo], then
C(m0) is a minimal set.
Exercise 11.2. Let M be a 2-manifold and Tl a flow on M. (a) If
Mx, M2 are invariant sets, then Mx U M2, Mx n M2 and Mx — M2 are
invariant sets, (b) If Mx is an invariant set and dMx, Mx° = Mx — dMx
are its boundary and set of interior points, then dMx and Mx° are in-
variantsets. (c) IfTVis a minimal set, then either TV = M or TV is a nowhere
dense set.
Let moe M and let C+(m0) denote the semi-orbit C+(m0) = {/t(t, m0) for
t ^ 0} starting at m0. A point meMis called an co-limit point of C+(w0)
if there exists a sequence of /-values 0 < tx < t2 < . . . such that /„ —»• oo
and fi(tn, m0) -*¦ m as n -*¦ oo. The set of co-limit points of C+(m0) will be
denoted by O.(m0). The analogues of Theorems 1.1 and 1.2 are valid.
Exercise 11.3. Let M be a 2-manifold and Tl a flow on M of class C°.
Let w0 e M. (a) Let C'(m_o) = {w = /«(*, m0) for t ^ j} = C+(ju(j, m0)).
Then O(w0) = C1^) O C2(m0) O ... and O(w0) is closed, (b) If wx e
O(m0), then Cfaj) c Q(w0); i.e., O(w0) is an invariant set. (c) If, in
addition, C+(w0) has a compact closure, then O(w0) is connected.
Exercise 11.4. Let M, 71' be as in Exercise 11.3, TV a minimal set, and
m0 e TV. (a) Then Q(m0) <= C(w0) = TV. (b) If, in addition, O(m0) is not
empty (e.g., if TV is compact), then O(w0) = TV.
Let M, Tl be of class Ck, k ^ 1. Let C/a be a coordinate neighborhood
on M and y = ga(m) local coordinates of m e Ua; thus ga: Ga —»• 5 is a
homeomorphism of Ua onto a square 5, say, S-.ly^, \y2\ < 1. A closed
[or open] arc Y:y = yo(s) for |,s| ^ a [or |,s| < a] of class Ck in 5 or its
image f :wo(i) = gal(yo(s)) in C/a is called a transversal of the flow 71' if no
orbit is tangent to f, i.e., if the differentiable arc y = gjj*[t, mn(s)]),
which is defined for small \t\ and starts at yo(s) when t = 0, is not tangent
to F for any s on |,s| < a [or |,s| < a].
Lemma 11.3. Let k ^ 1, moe Uaa nonstationary point of the flow Tl
The Poincare-Bendixson Theory 185
and y = ga(m) local coordinates on Ua. Then there exists a Ck transversal
arc Y:y = yo(s) or f :mo(s) = g^Hy0{s))for \s\ ^ a0 satisfying mo(O) = m0.
Furthermore, for any such transversal and sufficiently small a > 0, the set
U = {m = f4t, mo(s)), where \t\ < a, \s\ < a} and y = g(m) = (s, t) for
m eU are an admissible coordinate neighborhood and local coordinates on
U, respectively.
Exercise 11.5. Verify this lemma.
12. Analogue of the Poincare-Bendixson Theorem
The main object of this section is to prove:
Theorem 12.1 (A. J. Schwartz). Let M be a 2-manifold of class C2 and
Tl a flow on M of class C2. Let N be a nonempty compact minimal set.
Then N is either (i) a stationary point m0; or (ii) a periodic orbit (which is
homeomorphic to a circle); or (iii) N = M.
In case (iii), M = TV is compact and has a flow Tl on it without stationary
points. It follows from the Euler-Poincare formula (relating genus and the
sum of indices of singular points of a vector field on M) that the genus of
M is 1; thus M is homeomorphic to a torus or a Klein bottle. Actually,
a flow on a Klein bottle without stationary points necessarily has a
periodic orbit. Thus, in case (iii), M is a torus. For a discussion of these
remarks, see H. Kneser [2].
The assumptions concerning the C2-property of Tl cannot be reduced
to C1 even if M is of class C°°; see Exercise 14.3.
Proof of Theorem 12.1. In view of Exercise 11.2(c), TV = M or TV is a
nowhere dense set. Suppose that the theorem is false. Thus TV # M is a
compact, nowhere dense, minimal set in M which does not contain a
stationary point or a periodic orbit. It will be shown that this is impossible.
Let m0 g TV and let U and y = g(m), where y = (s, t), be an admissible
neighborhood and local coordinates as furnished by Lemma 11.3. Thus,
ifm = g~\y) = g-\s, 0, where \s\, \t\ < a, thenm0 = g~\0, 0), f :mjs) =
g~\s, 0) is a transversal arc of class C2 and, for a fixed s, g~*{s, t) =
fj.(t, mo(s)) is part of the orbit C(mo(s)).
A point m and its local coordinates (s, t) will be identified; e.g., m0
will be referred to as the point @, 0). Let F represent the open line
segment |,s| < a, t = 0, which is the transversal arc T:y = g(mo(s)) =
(s, 0). The point (s, 0) of F will be referred to simply as s.
Since TV contains no interior points, it is clear that K = TV O F contains
no ^-interval, so that K is a nonempty, nowhere dense set, closed relative to
r. Let
A2.1)
W=(-a,a)-K,
186 Ordinary Differential Equations
so that W is an open s-set and has a decomposition W = U(a;, /S;) into
pairwise disjoint, open s-intervals (a1; j3j), (a2, /S2), ....
Let U be the set of ^-values on |,y| < a such that the semi-orbit C+(m0(s))
starting for t = 0 at the point mo(s) e f meets F for some t > 0. It is
clear that U is an open ,y-set on |si < a. For s e U, let t(s) be the least
positive /-value [in fact, t(s) ^ a] such that (t(t, mo(s)) e F and let f(s)
denote the s-coordinate of fjt(t(s), mo(s)); i.e.,
A2.2)
(f(s),O)=gQi[t(s),mo(s)]) for s e U.
We can obtain t(s) and /(a1) in another way. Let soe U and define the
real-valued functions a, t of (s, t) by
(a, r) = ?(/*[/, mo(s)]) =
g-1^, 0)])
for small Is1 — jo|, \t — t(so)\. Thus, on the orbit starting at mo(s), (a, t)
are coordinates of the point corresponding to the time t. In particular
(a, t) = (f(s), 0) if (s, t) = (s, t(s)). Then a, t are of class C2 with a non-
vanishing Jacobian 9(cr, r)jd{s, t) by Lemma 11.1. Consider the inverse
functions s = so(a, t), t = to(a, t) for small \a —f(so)\, \r\. Then t(s) =
to(s, 0) and/fa) = a(s, t(s)) for small |j — so\. In particular, t(s), as well as,
A2.3) f(s) is of class C2.
Note that cr^, t), t(s, t) satisfy
(a(s, t
, r(s, t
= (a(s, t(s)), t)
for small |j — so\, \t\. Since 9(cr, r)jd{s, t) ^ 0, it follows that the Jacobian
of (a(s, t + t(s)), t), t) with respect to (s, t), which is the product of the
Jacobians 9(a, t)IB(s, t) and d(s, t + t(s))ld(s, t), does not vanish. Hence
the partial derivative of a(s, t + t(s)) with respect to s is not 0. Since
f(s) = a(s, t + t(s)) for small |/|, it follows that
A2.4) Df(s) ^ 0 for s e U if D = djds.
Since N is a compact minimal set, m1e N implies that C{m^) = N; cf.
Exercise 11.4. This shows that if s1 e K = N n F, then jx e C/, i.e., that
K c C/. Let K be an open subset of U such that
A2.5) ^cFcfc[/.
In particular, A2.3) and A2.4) imply that there exist constants R and C
such that
A2.6)
A2.7)
0 < l/R < \Df(s)\ ^R
< C for
for s e V and /? > 1,
jeK and C> 1.
The Poincare-Bendixson Theory 187
It is clear that
A2.8) fu^(-a,a)
is a one-to-one map and has an inverse/^). Below/*($), k = 0, ± 1,...,
denote the iterates of/and/; e.g., f°(s) = s and if s e U and/(,y) e U,
then /2(s) =f(f(s)). In particular, if k > 0, then /*(*) is defined on
?/ r\f-\U) n • • • (^/-^-"(C/); a similar remark applies to /c < 0.
In addition to the properties A2.6), A2.7), the function f(s) has the
properties that
A2.9) f(K) = K
since N is invariant; that
A2.10) /*($) = s for some jeX implies that k = 0,
since ./V contains no periodic orbits; and finally, that
A2.11) K is a nonempty minimal set with respect to /
in the sense that K contains no proper subsets A"o such that A"o is closed
with respect to K and that/(A) = A"o [or equivalently,/±1(A0) <= A"o].
The remainder of the proof consists in proving the assertion:
Lemma 12.1. If K is a closed set and V, U are open sets of \s\ < a
satisfying A2.5), then there cannot exist a function f(s) defined on U
satisfying A2.3)-A2.11).
Proof, (a) Assume that this lemma is false and that f(s) exists. Put
A2.12) e = dist (A", (-a, a) - V), so that 0 < e < a.
Note that if (a;, p,) is an interval of W = (—a, a) — K, then
A2.13) Pi - a, < e implies that [a,, ft] <= V <= U
since a;-, ps e K.
(b) It is easy to see that
A2.14) f(K1) = K1 if *! = {alf ft, a,,/Jfc . . .}
is the set of endpoints a;-, j3;.. For j0 e A" — A^ if and only if s0 is a limit
point of both K O (—a, ,y0) and K n (i'o, a). Thus A2.14) follows from
A2.4) and A2.9).
(c) It will be shown that there exists an integer / such that a = ai;
ft = ftt satisfy
A2.15)
f\[a, p]) c V for k = 0, 1,....
Let Q be the finite set of endpoints a,, /3;- of intervals (o^, j3;) of W such
that Pj — oij ^ e. In view of A2.10), there is an integer n such that
188 Ordinary Differential Equations
/*(ai) $ Q for k ^ n. Then, by A2.14),/"(«i) = «* or/"(ai) = /3,.for some
i and l/^/SO -/*(aj)| < e for « ^ A:. Thus A2.15) follows from A2.13).
(d) Let k ^ 0 be an integer and [p, q] a closed ^-interval such that
f\[p, ?]) e K for 0 ^ A: ^ n. Then
A2.16)
Dfk+\s)
Dfk+\r)
< exp CR 21/J(P) -/'(«)I
J=0
for 0 ^ k ^ n and/) ^ r, 51 ^ ^. In order to verify this, note that fk+1(s) =
/(/*($))• Thus if Df°fk(s) denotes the derivative of/evaluated at/*($),
then we have
A2.17) Dfk+\s) = [Dfofk(s)]Df(s), hence
Consequently,
=IT0/°/J'(s).
TT
log
Dfk+\s)
^2 | log I Dfof(s)\ - log |D/o/'(r)l
Z)/*+1(r)
and, by the mean value theorem of differential calculus,
log
,)I • !/'(*) - /'
. Dfk+\r) |
for suitable 0, between f'(r) and f'(s). Thus A2.16) follows from A2.6),
A2.7).
(e) Let a, /3 be as in A2.15) in (c). Then
A2.18) 5 = ?|D/*(a)| satisfies 1 5i <5 <i 2a(/3 - a)-^2'^.
In order to see this, note first that Df(s) = 1, so d ^ 1. There is a 0k c
(a, j3) such that
l/*(«) ~f\P)\ = \Df\ek)\ (/J - a).
Since the intervals with endpoints/*(a),/'i:(j3) are contained in (—a, a) and
are pairwise disjoint for k ^ 0 by A2.10), it follows that
In addition, by A2.16),
|D/*(a)|^ \Df\Bk)\ eiaCR for k = 0, 1,. . .
This proves A2.18).
(/) Let d denote the number
A2.19) d = €/3CRd(l + a); in particular 0 < d < e.
The Poincare-Bendixson Theory 189
It will be shown that
A2.20A:) /*([<x - d, a + d]) <= K,
A2.21/c) |/*(j) -/A"(a)| < e for |j - a| ^ rf,
A2.22A:) \Df(s)\ ^ 3 |D/*(a)| for \s - a\ ? d
hold for A: = 0, 1, In view of/*(a) e /iT and the definition A2.12) of
e, it is seen that A2.20A:) is a consequence of A2.21A:); cf. A2.13).
The relations A2.21&), A2.22A:) are trivial for k = 0. Assume that
A2.21A;), A2.22/c) hold for k = 0, ...,«. Then, by A2.16) in (d),
\Dfn+\s)\ <
(a)| exp
J=0
By the mean value theorem and A2.22/), fory = 0, ...,«,
^ 3 |s - a|?
where 0; is between a and s. Thus A2.18) implies that
\Dfn+\s)\ ^ \Dfn+\oL)\ icmi .
But since 3C/?J6 < e/a ^ 1 and e < 3, the inequality A2.22 « + 1)
holds. This inequality shows that
\fn+\s) -/"+1(«)l ^ k - a| • |Z)/"+1@)l ^ 3J |D/"+1(o^)| ^ 3J6,
where |a — d\ < |j — a|. Since C > 1 and R > 1 imply that 3dd < e,
the inequality A2.21 n + 1) holds. This proves A2.20)fc)-A2.22)fc) for
k = 0, 1, ....
(g) In view of A2.18) and A2.22A;),
A2.23) lim Dfn{s) = 0 uniformly for |a - s\ ^ d.
n—*-oo
By the minimality property of K, the closure of the sequence of points
/J+"(a) for n = 0, 1, . . . (and j fixed) is the set K. Hence there exists a
large value of & (>0) such that
(a) - a| ^ | and
for |j - a| ^ rf.
Thus |/*(a ± J) — a | < d, and so fk(s) — s has opposite signs at s = a ± J.
Hence there is an s-value ,y0 such that fk(s0) = j0 and l^,, — a| < d. In
addition, fk(s0) = s0 for « = 1, 2, . . . and, by A2.23), /B*(a) -* ^0 as
« -*¦ oo. Since AT is closed and <xe K implies that/"*(a) e A", we have s0 e A".
This contradicts A2.10) and proves Lemma 12.1 and Theorem 12.1.
190 Ordinary Differential Equations
Lemma 12.2. Let M be a 2-manifold and P a flow on M of class C°.
Let M1 be a nonempty, compact, invariant subset. Then Mx contains at
least one nonempty, minimal set.
Since the intersection of invariant sets is an invariant set, this lemma is
an immediate consequence of Zorn's lemma. (For the statement and
proof of the latter, see Kelley [1, p. 33]). Theorem 12.1 and Lemma 12.1
give an analogue of the Poincare-Bendixson theorem:
Theorem 12.2. Let M be an orientable 2-manifold of class C2, P a
flow on M of class Ca, and m0 e M. Suppose that O.(m0) ^ M and that
Q(m0) is a nonempty compact set which contains no stationary points. Then
Q(m0) is a Jordan curve and C+(m0) spirals toward Q(m0).
When N is a Jordan curve on M containing no stationary points, then
C+(m0) is said to spiral toward N if, for every m e N, there is a transversal
arc F = F(m) through m such that successive intersections of C+(m0) =
{m0* = fj,(t, m0), t ^ 0} and F tend monotonically to m.
The concept of the "orientability" of M is used here to mean the
following: Let N be a Jordan curve on M of class C1 containing no sta-
stationary points. Then there exists an open set K containing N and a homeo-
homeomorphism of the cylinder {(s, y1, y2):\s\ < 1, (j/1J + (j/2J = 1} onto V
such that the circle s = 0 is mapped onto N and the line segments (y1, y2) =
const, are mapped onto transversal arcs.
Proof. By Lemma 12.2, Q(m°) contains a nonempty, compact minimal
set N and, by Theorem 12.1, N is a Jordan curve (containing no stationary
points). Let V be a neighborhood of N described above. By considering
the image of the flow on the cylindrical image of V, it is clear that the
arguments in parts (b) and (c) of the proof of the Poincare-Bendixson
Theorem 4.1 are valid. Hence C+(m0) spirals toward N. This implies, in
particular, that N = Q(m0) and completes the proof of the theorem.
13. Flow on a Closed Curve
In view of the remarks following Theorem 12.1, it is seen that the torus
is an exceptional 2-manifold M in that it can admit flows for which M is a
minimal set. It seems therefore of interest to examine flows on a torus.
This section is a preparation for such an examination and concerns a
"flow" on a Jordan curve F or, equivalently, a topological map of F onto
itself.
Let F be a Jordan curve and S:F->-r an orientation preserving
homeomorphism of F onto itself. The discrete group of homeomorphisms
Sn,n = 0, ±1,..., of T onto itself will be called a flow on F. Asin§ 11,
we can define the orbit C(y):{Sny:n = 0, ±1,. . .} through a point y e F,
semi-orbit C+(y) = {Sny:n = 0, 1, . ..}, the set Q(y) of cu-limit points of
C+(y), invariant sets, minimal sets, etc.
The Poincare-Bendixson Theory 191
The points of the curve F can be considered to be parametrized, say
y = y(y% where 0 < y < 1, and the points y e F corresponding to
y = 0 and y = 1 are identical. Or, more conveniently, F can be considered
as a line, — co < y < co, on which we identify any two points yl5 y2 for
which y1 — y% is an integer. Then y1 = Sy can be represented as real-
valued function yx = f(y) satisfying
@/B/) is continuous and strictly increasing,
00/B/ + 1) =/B/) + 1, so that f(y) - y has the period 1.
The fact that/is nondecreasing merely reflects the fact that S is orientation
preserving. The condition that / is strictly increasing and satisfies (ii) is
a consequence of the fact that S is a homeomorphism. Conversely, any
function f(y) satisfying (i) and (ii) induces an orientation preserving
homeomorphism S of F onto itself.
Let f°(y) = y and /"%) be the function inverse to f(y). Let f\y) =
/(/B/)), /3B/) =/(/2B/)), • ¦ • and similarly/-%) = f~\f-\y)), f~\y) =
f~\f-*<!))), • • ¦, so that fn(y) corresponds to Sn for n = 0, ±1,... and
each/n(j/) satisfies (i) and (ii).
Lemma 13.1. Let f(y) be a continuous function for — oo < y < oo
satisfying (i) and (ii), then there exists a number a such that
A3.1)
/%)/«-¦« as \n\-+co for all y;
in fact, if dn, en are defined by
n<x - dn = min (/%) - y), nx + en = max (/%). - y)
for — oo < y < oo, so that
- 2/ - «a ^ en,
then
A3.2) dn^0, 6n^0 and 6n + en<\.
The number a is called the rotation number of the map S or of the flow
S" on F.
Proof, (a) The continuous function
A3-3) Vm(y) =fm(y) - y.
has the period 1, by the analogue of (ii), and satisfies
A3.4) \m\ OS™ - 0J< 1 if p*[or ftm] = max [or min] <pjy)\\m\.
For suppose that A3.4) does not hold. Then (pm{y^) — cpm(y2) = 1 for
some points ylt y2. Since cpm is periodic, it can be supposed that y2 <
y1<y,+ \. Then, by A3.3), f™^) -fm(y2) = 1 + yx - y2 > 1; so
that /n(y1) > 1 +fm(y2) =/m(l + y2)- But this contradicts the in-
increasing character of/m.
192 Ordinary Differential Equations
(b) Let k ^ 1 be an integer. Then A3.3) shows that
Hence
for y = 1, 2,... ;
in particular, for k = 1,
0i ^ /%) ~ /"(y) ^ P for i = 1, 2,....
Then if n > 0 and n = kd + r, where 1 ^ r < /c, a sum of these relations
fory = 1, ..., d and i = kd + 1,.. ., kd + r gives
<*& + rPi ^ /%) - 2/ ^ </fc0* + r/31.
By the definition of j3n, /3", and 9?nG/), this implies that
Consequently
A3.5)
^liminf/3n^limsup/3n ^ /J*.
But since /3* — /3fc -*¦ 0, /c -> oo, by A3.4), it follows that these upper and
lower limits are equal, say, to a. Thus <pn{y)jn -> a as « -*¦ oo uniformly
in y. This gives the part of A3.1) concerning n -*¦ + oo.
By A3.4) and A3.5), kfik ^ cpk(y) < kfik and fc/3fc < k<x < kfik. Hence if
dk = kix- kfik and efc = fc/3* - /ca, then A3.2) holds for n = 0, 1,
Note that/-"(«/) -y = z- fn(z) if z =f~n(y) and so A3.2) holds with
<5_n = en and e_n = dn for « = 1, 2,.... This completes the proof.
Lemma 13.2. The rotation number a of S is a topological invariant, i.e.,
is independent of the parametrization ofY.
Proof. Let S determine f(y). A change of parameters z = g(y) on F,
preserving orientation, is given by a function g(y) satisfying the analogue
of (i) and (ii). If R:z = g(y), then, in the z-parametrization, S becomes
RSR-1^! = h(z), where h(z) = g(J{g~\z))) satisfies (i) and (ii). Since
(RSRr1)" = /?5"/?-1:zn = hn(z) = g(J\g-\z))\ it follows that h\z) =
g(g-\z) + nx + rj where -dn ^rn^en byA3.2). Let^(z) = g(z) - z,
so that yi(z) has the period 1. Then, as n -*¦ oo,
hn(z) g-\z) +n* + rn + y>{g-\z) + nx + rn)
n n
This proves the lemma.
Theorem 13.1. Let Sn be a flow on a Jordan curve T. Then the rotation
number a of S is rational if and only if Sk has a fixed point y0, Sky0 = y0,
for some integer k > 0.
Proof. Let S belong to f(y). Suppose that Sky0 = y0 for some k > 0,
i.e., that there is a number y0 and an integer r such that f\y0) = y0 + r.
The Poincare-Bendixson Theory 193
Hence
/"*(!*>) = 2/o + nr and
nk k
is rational.
Conversely, suppose that a = rjk is rational, where k > 0. Then/%) —
y — r attains non-negative and non-positive values by A3.2). Hence there
is a y0 such that/%,,) — Vo ~ r = 0; i.e., Sky0 = y0 for the point y0 e F
corresponding to y0. This proves the lemma.
Lemma 13.3. Let S have an irrational rotation number a. Let yoeT be
fixed, yn = Sny0, j and k fixed integers, and Fo either of the closed arcs on
F bounded by yjt yk. Then there exists an integer n > 0 such that F c F",
where
U • •
F" = Fo U
S-n\i-k\T()
Hence, ifyeT, there is an i = i(y), 0 ^ i ^ n, such that S*y e Fo.
Proof. For the sake of definiteness, let Fo be the oriented subarc of
F from yk to yi and let F(ri) be the union of arcs
A3,6) F(ni = Fo U S-u-k)T0 u
us
Thus F" = F(n» or F" = 5"(J-*»r(n» according as; > k or; < k and so,
F c rn if and only if F c F("». Thus it suffices to show that if n > 0 is
large, then F <= F(n). The mth and (m + l)st arc on the right of A3.6)
abut at a common Midpoints-**1"-*')/*- HenceifF + F(n'for« = 1,2,.. .,
the endpoints S~nU~k)yk tend monotonously to a point y° on F as n —»¦ oo.
But then y° = lim S-nU~k)yk = lim S-{n-lW~k)yk = SJ-*)/0 whereas, by
Theorem 13.1, S'~k has no fixed point. This contradiction shows that
F c r("' for large n.
Thus, if y e F, then y e S^I^F,, or Sn^y e Fo for some m = m(y),
0 ^ m ^ «. This proves the lemma.
If Sky0 = y0 for some k > 0, the orbit C(y0) consists of the points
y0, Sy0,. .., S*)/o. If no Sk, k ^ 0, has a fixed point, the situation is as
follows:
Theorem 13.2. Let S" be a flow on a Jordan curve F having an irrational
rotation number a. For any y e F, let Cl(y) be the set of co-limit points of
the semi-orbit C+(y). Then No = Q.(y) is independent of y and hence is a
unique minimal set of Sn. Furthermore No = T or No is a perfect nowhere
dense set on V.
When No = F, the flow S" is said to be ergodic.
Remark. Since No is a unique minimal set of the flow Sn, as well as
S"n, it is clear that S" is ergodic if and only if S~n is ergodic. Thus the set
jV0 is the set of limit points of any semi-orbit C+{y) = {Sny, n = 0, 1, ...}
or of any semi-orbit C~(y) = {S~ny, n = 0, 1,.. .} or of any orbit
= {S"y,n = 0, ±1,.-. .}.
194 Ordinary Differential Equations
Proof. Consider the semi-orbit C+(y0) and its set O(y0) of co-limit
points. Let y° e O(y0) and y any point of F. Since there are points
yp yk e C+(y0) arbitrarily near to y° and the smaller of the arcs Fo bounded
by them contains a point Sy for some / ^ 0, it follows that y° e Q(y);
i.e., O(y0) <= O(y). Interchanging the roles of y and y0 shows that
No = Q(y) is independent of y.
The set No is closed and invariant. If y e No, then y is a limit point of
C+(y) <= No, hence a limit point of No. Thus A^ is perfect. Oearly A^
is minimal. Hence A^ = F or No is nowhere dense; cf. Exercise 11.2(c).
This proves Theorem 13.2.
Lemma 13.4. Let S have an irrational rotation number a. For a given
y0, the function g(yn + k) = rnx + k, where yn = fn(y0) and n, k = 0, ±1,
.. . is an increasing function on the sequence of numbers {yn + k}.
Proof. It has to be shown that
A3.7) /%„) + k <fm(y0) + j implies that na + k < ma + j.
Applying/-"* to the first inequality in A3.7) gives
k < 2/o + j or /"-mB/0) -yo<j-k.
Since fn~m(y) — y cannot be an integer, max (fn~m(y) — y) <j — k and
so A3.2) implies that (n — m)a <y — k. This proves A3.7) and the
lemma.
For the next theorem, the following simple lemma will be needed.
Lemma 13.5 (Kronecker). Let a be an irrational number and S" the
flow on a Jordan curve F such that the corresponding f(y) is f(y) = y + a.
Then Sn is ergodic (i.e., No = F).
In view of the Remark following Theorem 13.2, this is equivalent to
the statement that the set of points {y = mx + k, where k, n = 0, ± 1, . . .}
is dense on — oo < y < oo. In other words, if [y] denotes the largest
integer not exceeding y and yn = ncn — [net] is the fractional part of net,
then the lemma is equivalent to the statement that the set {y = yn, where
n = 0, ±1, . . .} is dense on 0 ^ y ^ 1.
Exercise 13.1. Prove Lemma 13.5.
Theorem 13.3. Let S" be a flow on a Jordan curve F having an irrational
rotation number a. Then Sn is ergodic (i.e., No = F) if and only if S is
topologically equivalent to a "rotation," i.e., if and only if there exists an
orientation preserving change of parameters R:z = g(y) such that
A3.8)
-^! = 2 + a,
Proof. Let there exist an R satisfying A3.8). The minimal set of the
map RSR'1 is the set A^ of limit points of a, 2a, ... on F by Theorem
The Poincare-Bendixson Theory 195
13.2. Since a is irrational, this minimal set -/Vo is F. Hence the minimal
set for 5 is F.
Conversely, let No = F be the minimal set for 5. Let y0 = 0 and yn =
/"@) in Lemma 13.4. Then the increasing function z = g(y) is defined on
the sequence {yn + k, n, k = 0, ± 1,. . .} which is dense on — oo < y < co.
It is continuous since {«a + k, for n, k = 0, ±1, .. .} is dense on — oo <
z < oo and, hence has a unique continuous extension to an increasing
function for — oo < y < oo. Furthermore, g(y + 1) = g(y) + 1. Thus
R.z = g(y) defines a change of parameters on F.
The relation g(f@) + k) = n<x + k implies that if y =/"@) + k,
then g(f(y)) = g(fn+1@) + k) = (n + l)a + k = g(y) + a. Since the
set of the points {/"@) + k} is dense on — oo < y < oo, it follows that
g(f(y)) = g(y) + « for all y. That is, RSR^.z, = g(f(g-\z)) = z + a.
This proves the theorem.
Exercise 13.2. Let F be a circle and No any perfect, nowhere dense set
on F. There exists an orientation preserving homeomorphism 5 of F
onto itself having (an irrational rotation number a and having) A^ as its
unique minimal set. See Denjoy [1].
Lemma 12.1 occurring in the proof of Theorem 12.1 has the following
consequence.
Theorem 13.4. Let S" be a flow on F having an irrational rotation
number a. Let S belong tof(y) and suppose thatf(y),f-\y) are of class C2.
Then S is ergodic (i.e., No = F).
This assertion can be improved slightly as follows:
Exercise 13.3. (a) Let the condition that ilf(y),f-\y) are of class C2"
be relaxed to "/B/),/~1B/) are of class C1 and dfjdy is of bounded variation
for 0^2/^ 1-" Then S is ergodic. Denjoy; see van Kampen [1].
(b) This assertion is false if the condition that "df/dy is of bounded
variation for 0 ^ y ^ 1" is omitted. See Denjoy [1].
14. Flow on a Torus
A torus M will be viewed as a square 0<x^l, 0^2/^1 in the
(x, 2/)-plane in which the points (x, 0), (x, 1) or @, y), A, y) on opposite
sides of the square are identified or, even more conveniently, as the entire
(x, 2/)-plane in which pairs of points (x1; y^, (xa, y2) are considered
identical if and only if x1 — x2, y1 — y% are integers.
Consider a continuous flow
A4.1) Tt:xi = ^(t,x,y), y* = rj(t, x, y)
on the torus M, so that |, r\ are continuous for —oo < t, x, y < oo and
T' is a group of homeomorphisms of the (x, 2/)-plane onto itself. Further-
Furthermore, since (x, y), (x + 1, y), (x, y + 1) are considered identical points of
196 Ordinary Differential Equations
the torus M
?(t, x+\,y) = f0, x, y) + 1 and fa x + \,y)= fax,y),
A4.2)
?0, x, y + 1) = 10, x, y) and fax, y + 1) = r){t, x, y) + 1.
[The first line means, e.g., that if the point (x, y) is translated to (x + 1, y),
then its orbit is merely translated a distance 1 in the x-direction.] The
group property T*+t = "FT* is equivalent to
A4.3) ?(s + t, x, y) = ?(j, ?(t, x, y), fa x, y)) for ? = f or ? = V.
Suppose that f, t] have continuous derivatives with respect to t. Then
Fi(?, V) = tfttt, x, y)ldt]t=0, F2(x, y) = [dfa x, y)/dt]t=0 satisfy
A4.4) Fix, y) = Fix + 1, y) = F/z, </ + 1) for ; = 1, 2
and x = |(r, x0, y0), y = rj{t, x0, y0) is a solution of the initial value
problem
A4.5) x' = F.ix, y), y' = Fix, y),
A4.6) z@) = x0, y@) = 2/o,
by A4.3).
If, e.g., | and r\ are of class C1, then it follows from Theorem III 7.1
that A4.5), A4.6) has the unique solution x = |(r, x0, y0), y = rj(t, x0, y0).
Lemma 14.1. Let M be a torus and A4.1) a flow on M of class Ck,
k ^ 1 [or a continuous flow such that f, r\ have continuous partials with
respect to t]. Suppose that T* has no stationary point [or that Fj2 + F22 ^ 0].
Then there exists a Jordan curve F of class Ck [or of class C1] on M which is
transversal to the flow. Furthermore, such a transversal curve F cannot be
contracted to a point on M, so that F does not bound a 2-cell (i.e., does not
bound a subset of M homeomorphic to a disc).
For example, suppose that Fix, y) does not vanish and let F(x, y) =
-L, so that A4.5) has the same solution paths as
A4.7)
ax
i.e., x' = 1, y' = F(x, y).
Here every circle x = const, on M is a transversal curve F.
Proof. Consider the differential equations for the orthogonal tra-
trajectories
A4.8) x' = Fix, y), y' = -Fix, y).
Let xo(t), yo(t) be a solution of A4.8), so that (xo(t), yo(t)) & const, since
The Poincare-Bendixson Theory 197
Fj2 + F22 ^ 0. Also (xo@,2/0@) is of class Ck [or of class C1] since
(Fl F^ is of class C* [or class C0].
Suppose first that x = xo(t), y = yo(t) is a closed curve F on M, i.e.,
let there exist a least number/) > 0 and integers r0, s0 such that xo(t + p) =
xo(j) + ro» vlf + P) = 2/o(O + Jo- It is clear that F is a transversal curve.
If xo(t), yo(t) is not a closed curve on M, then the half-trajectory xo(t),
ya(t) for t ^ 0, viewed as a path on M, has at least one co-limit point, say,
(xi> 2/i)- The point (x1; j/j) is contained in arbitrarily small curvilinear
rectangles R.ABCD on M, in which the arcs AB, CD are solution arcs of
A4.5) and BC, AD are solution arcs of A4.8). The point xo(t), yo(t) is
inside R for some large t = t0 and leaves at some point Px on CD [or AB]
at a first time tx > r0 and then meets AB [or CD] at a point P2 for some
first ?2-> *i- It is clear that if R is small enough that there exists an arc
iV^i in R which together with the arc x = xo(t), y = yo(t) for t1 ^ t ^ t2
constitutes a transversal curve F of class Ck.
Suppose, if possible, that a transversal curve P on M can be contracted
to a point. Then it has an image F in the (x, j/)-plane which is a Jordan
curve of class C1 bounding an open set Q.0. Clearly the points of F are all
egress or all ingress points of Q° for A4.5). Then the index of F relative to
A4.5) is +1 or — 1; cf. §2. Hence O.0 contains at least one stationary
point by Corollary 3.1. This is a contradiction and completes the proof of
Lemma 14.1.
In what follows, it is supposed that
(Hj) M is a torus, T* is a flow on M of class Ck,k ^ 1, without stationary
points.
Let F be a transversal (Jordan) curve on M. After a suitable Ck homeo-
morphism of the plane, it can be supposed that
(H2) The circle F: x = 0 is a transversal curve. In particular, Fin, y) ^
0 if « = 0, ± 1,.... Without loss of generality it can be supposed that
A4.9)
Fln,y)>0 for n = 0, ±1,.. . ;
for otherwise t is replaced by — t.
Let m0 be a point of F on M and suppose that the semi-orbit C+(m0)
through m0 meets F again for some tx > 0. Let mx be the first such point.
In other words, if m0 = @, y0), then there is a unique t1 > 0 such that
f Ci, 0, j/o) = 1 [cf. A4.9)], so that m1 = A, fa, 0, y0)). Put f(y0) =
r}(ti, 0, y0). The set U of points y0 where U is defined is open and of period
1 (in the sense that y0 e U if and only if y0 + 1 e U). It will be supposed
that
(H3) Every orbit meets F and yx = f(y) is defined for ally, — oo < y < oo.
The hypothesis (H3) holds, e.g., if Fix, y) does not vanish [so that A4.5)
is "equivalent" to A4.6)] or if there are no closed orbits.
198 Ordinary Differential Equations
Exercise 14.1. Verify the assertions of the last paragraph.
The function f(y) has the following properties: (i) f(y) is strictly in-
increasing; (ii) f(y + 1) =f(y) + 1, so that f(y) — y has the period 1;
(iii) f(y) is of class Ck. The property (i) follows from the fact that two
orbits are either identical or have no points in common. The property
(ii) follows from the last line of A4.2); for if x = 0 and t = tx, then
f(A> 0> y) = 1 and f(y) = v((i> 0,2/)- The proof of property (iii) is
similar to (but simpler than) the proof of A2.3) in § 12.
The results of § 13 can now be transcribed to results about certain
flows on a torus. Let the number a of Lemma 13.1 for the function/(y)
be called the rotation number of the flow TK Thus Lemma 13.2 implies
the assertion:
Theorem 14.1. Let T* be a flow of class C1 on a torus M satisfying (Hx),
(H2), and (H3). Then there exists a periodic {closed) orbit if and only if the
rotation number <x of T* is rational.
Exercise 14.2. Let T* be as in Theorem 14.1 and let the rotation number
a be rational. Show that every semi-orbit C+(m0) on M is either a
Jordan curve or that Q(w0) is a Jordan curve and C+(m0) spirals toward
CKm0).
Theorems 13.2 and 13.4 give
Theorem 14.2. Let T* be a flow of class C2 on a torus M satisfying (Hx),
(H2), and (H3) and let its rotation number a be irrational. Then M is a
minimal set {and every semi-orbit C+(m0) is dense on M).
When M is a minimal set, the flow T* is called ergodic.
Remark. This theorem is false if the condition "T* is of class C2" is
relaxed to "T* is of class C1." The situation becomes even worse if it is
only assumed that "T* is continuous, f and r\ have continuous partials
with respect to t, and F^ + F^ ^ 0," although the other results above
have analogues in this case. This is indicated by the next exercise which is
an extension of the results of Exercise 13.3.
Exercise 14.3 Let points (x, y), (x + 1, y), and (x, y + 1) be identified
so that the plane becomes a torus M and the line x = 0 a circle T. (a)
Let ./Vo be an arbitrary, perfect, nowhere dense set on T. There exists a
continuous function F(x, y) = F(x + 1,2/) = F(x, y + 1) such that initial
value problems belonging to A4.7) have unique solutions and such that
if r*:x* = t + x0, y* = r){t, x0, y0) is the flow induced by A4.7) on M and
5 is the corresponding homeomorphism on F, then (the rotation number
a is irrational and) No is the unique, nonempty, minimal set of S. (b)
There exist functions F(x, y) = F(x + 1, y) = F(x, y + 1) of class C1 such
that the corresponding flow T* of class C1 has an irrational rotation
number a but T* is not ergodic (i.e., M is not a minimal set). See Denjoy
[1].
The Poincare-Bendixson Theory 199
The simplest flow on M is given by
A4.10) x* = t + x, yt = ait + y
arising from the differential equations
x' =1, y' = a,
where a is a constant (in fact, the rotation number of the flow). Consider
a flow of the form
A4.11) x* = t + x, y* = rj{t, x, y)
arising, e.g.; from a system of differential equations of the form x' = 1,
y' = F{x,y). The function f{y) belonging to A4.11) is
A4.12) f(y) = r)(\,0,y).
In examining the orbits of A4.11), it is sufficient to consider r](t, 0, y),
i.e., the orbits beginning for t = 0 at a point @, y) of the j/-axis. For the
orbit starting at (x, y) meets the x-axis when t = — x, and so
A4.13) rj(t, x, y) = n(f + x, 0, rj(-x, x, y)),
since T* = Tt+XT~X.
It will be verified that
A4.14) rj{t, 0, y) — <xt — y is bounded
for — co < t, y < 00. To this end, note that f(y) = r){n, 0, y), so that
rj{n, 0,y) — mx — y is bounded
by Lemma 13.1. If t = n + 0lt where 0 ^ d1 < 1, then rj{t, 0, y) =
rj(n, 0, »X0i> °> V)) satisfies
rj{t, 0, y) — mx — ^(flj, 0, y) is bounded, where mx = <xt — 0ja.
Also, rj(t, 0, y + 1) = 1 + rj(t, 0, y) shows that if y = J + 02, where
0 =: 62 < 1, then
»7@i, °> y) — j = y(Qi> °> #2) is bounded, where j = y — 02.
The last two relations give A4.14). The assertion A4.14) can be greatly
improved in the ergodic case. This is the essence of the following result.
Theorem 14.3 (Bohl). Let T* be a continuous flow on the torus M of the
form A4.11) such that T* is ergodic and has the (irrational) rotation number
a. Then there exist continuous functions xp(y), G(t, z) such that
A4.15) yfa + 1) = y>(y) and G(t, z) = G(t +1,2) = G(t, z + 1)
and that
A4.16) r)(t, 0, //) = <xt + y + xp(y) + G(t, cut + y + f(y)).
200 Ordinary Differential Equations
Note that for a fixed y [or t], rj(t, 0, y) — cut — y is an almost periodic
function of t [or periodic function of y\.
Proof. Let z = g(y) be the function supplied by Theorem 13.3 and let
y>(y), ipo(y) be the functions of period 1 defined by
A4.17) giy) = y + y>(y), g~\z) = z + y>0(z).
Make the change of variables R:(x, z) = (x,g(y)). Then T* becomes
A4.18) RTtR-i.x* = t + x,z*= ?(/, x, z),
where
A4.19) i{t,x,z) = g{n[t,x,g-\z)]).
By the analogue of A4.2)
A4.20) ?(/, x + 1, z) = ?(/, *, z) and ?(/, s, z + 1) = 1 + ?(/, x, z)
and, by the group property of RPR'1 [cf. A4.3)],
A4.21) ?(/ + 1, x, z) = i(t, x+l, C(l, x, z)) = ?(/, a;, ?A, z, z)).
Introduce the abbreviations
A4.22) rj(t, y) = rj(t, 0, y) and ?(/, z) = ?(/, 0, z).
Note that, since f(y) = r){\, y), Theorem 13.3 gives ?A, z) = z + a.
Hence A4.20)-A4.22) imply that
?(/, z + 1) = 1 + ?(/,«) and ?(/ + 1, z) = Ut, ?A, z)) = ?(/, z + a).
Thus the continuous function
V>i(t, z) = i(t, z - xt) - z
has the period 1 with respect to t [or z] for fixed z [or /]. Write the last
relation as
A4.23) ?(r, z) = a/ + z + ^(f, a/ + z).
In view of A4.19), V(t, y) = g-i(?[/, g(</)]), so that, by A4.17),
Consequently A4.23) shows that
A4.24) r,{t, y) = oit + g(y) + G(t,
if G(t, z) is defined by
Git, «) = Viit, «) + Vo(z +
It is clear that G satisfies the last part of A4.15). Finally, A4.24) and the
first part of A4.17) give A4.16). This proves the theorem.
The Poincare-Bendixson Theory 201
Notes
Most of §§ 1-9 of this chapter is contained in (or related to) the four-part memoir of
Poincare [3]; see also Bendixson [2], L. E. J. Brouwer [1] considers similar problems
without analyticity assumptions and even without the assumption that there is only
one solution through a given point.
section 1. The terminology "a- and co-limit points" is due to G. D. Birkhoff [3];
"limit cycle" to Poincare [3].
section 2. The Umlaufsatz (Theorem 2.1) goes back to Riemann [1, pp. 106-107].
It was first given a formal statement and proof by G. N. Watson [1]. The proof in the
text is that of H. Hopf [1]; cf. also van Kampen [4].
section 3. The definition of "index" was given by Poincare [3, I].
section 4. See Poincare [3] (in particular, part II) and Bendixson [2].
section 5. See Poincare [3].
sections 6-7. See Poincare [3].
sections 8-9. For Lemma 8.1, cf. Bendixson [2, p. 26]. In the analytic case, Theorems
9.1 and 9.2 are due to Poincare [3] (in particular, part I) and to Bendixson [2]. The
exposition in the text of Theorem 9.1 is an adaptation and simplification of Brouwer's
treatment [1, II] (which is complicated by the fact that initial conditions do not determine
a unique solution). For Exercise 9.1, see Schilt [1].
section 10. Problems of the type considered here go back to van der Pol [1] and to
Lienard [1]. Theorem 10.1 is a variant of a result of Levinson and Smith [1]. An
analogous result concerning A0.15) was given by Levinson [1], improved by Langenhop
[1], and further improved by Opial [7] as in Exercise 10.1. Theorem 10.2 is a general-
generalization of a result of Lienard [1]. For Exercise 10.5, see, e.g., Lefschetz [1, pp. 342-346]
and references to Flanders and Stoker [1], LaSalle [2], and Stoker [1]. For fuller
treatments of related problems and for references to the work of Cartwright, Littlewood,
Duff, Massera, Reuter, Sansone, Schimizu, etc.; see Andronow and Chaikin [1],
Bogolyubov and Mitropol'ski [1], Bogolyubov and Krylov [1], Lefschetz [1], Minorsky
[1], Conti and Sansone [1], and Stoker [1].
appendix. This type of investigation was initiated by Poincare [2], [3, III] for the
case of differential equations on a torus; see notes on §§ 13-14.
section 11. The ideas in this section are due to Poincare.
section 12. The results and methods of this section are those of A. J. Schwartz [1];
for a generalization, see Sacksteder [1]. The analogue of Theorem 12.1 was known
earlier for the torus; Denjoy [1]. See notes on §§ 13-14.
sections 13-14. Except for Theorem 13.4 and Exercises 13.1, 13.2 in § 13 and
Theorems 14.2, 14.3, and Exercise 14.3 in § 14, the results and methods of these sections
are essentially in Poincare [3, III]. The presentation in the text follows the completed
arguments and simplifications introduced by Denjoy [1] and van Kampen [1]; cf.
also Siegel [1]. (After Poincare, proofs for the main part of Lemma 13.1 have been
given by E. E. Levi, Bohl, H. Kneser, Nielsen, and Denjoy; see Bohl [1] and van
Kampen [1] for references.) Theorem 13.4 and its consequence, Theorem 14.2, were
conjectured by Poincare [3, III] for the analytic case. They were first proved by Denjoy
[I] in a slightly stronger form (cf. Exercise 13.3). A very simple proof of Denjoy's
result has been given by van Kampen [1]. A similar proof which avoids, however, the
use of the rotation number has been given by Siegel [1]. Denjoy [1] has given examples
(cf. Exercises 13.2 and 14.3) showing that the results are false if the smoothness
assumptions are lightened. Theorem 14.3 is a result of Bohl [1].
Chapter VIII
Plane Stationary Points
This chapter continues the discussion of the behavior of solutions of plane
autonomous systems. The basic existence theorems will be proved in the
first section for autonomous systems of arbitrary dimension.
1. Existence Theorems
In this section, there will be considered an autonomous system
A.1)
z> =f(z)
for a real d-dimensional vector z = (z1,. . ., zd). By a half-trajectory of
A.1) will be meant a solution arc in the z-space: C+: z = z{t), 0 ^ t <
°o+ (= °°)> or C~: z = z{t), 0 ^ t > co_ (> — oo), defined on a right or
left maximal interval of existence. Correspondingly, F will be called of
type C+ or C~. The point z@) will be called the endpoint of F.
Theorem 1.1. Let f(z) be continuous on an open z-set Q. Let Qo be an
open subset of Q such that the part of its boundary in Q {i.e., 9Q0 n Q)
is the union of two disjoint sets L U R, where R is compact, the points of L
are egress points, and 9Q0 C\ Q. = L KJ R is not compact. Then Qo U L
or Qo U R contains at least one half-trajectory F of A.1) with endpoint
in L or R, respectively.
For the definition of egress point, cf. § III 8. It is not assumed in this
theorem that L exhausts the set of egress points of Qo. Since the boundary
9Q0 is closed relative to Q, the condition that L U R is not compact implies
that either L U R is unbounded or has a limit point zB which is not in Q..
It will be clear from the proof that either there exists a half-trajectory F
of type C~ in Qo U L with endpoint on L or there exists a half-trajectory (of
type C+ or C~) in Qo U R with endpoint on R. Sufficient conditions for
the latter case are given by the next theorem. The conclusion of this
theorem does not specify, however, whether F is of type C+ or C~.
Theorem 1.2. In Theorem 1.1, assume that solutions of {I A) are uniquely
determined by initial conditions, that R is not empty, and that L U R is
202
Plane Stationary Points 203
connected, then there exists a half-trajectory in Qo U R with its endpoint
on R.
The condition that "L U R is connected" can be relaxed to "Lo U R is
connected, where Lo c ]_ and Lo U R is either unbounded or has a limit
point not in L U R (i.e., not in Q)."
Remark. In Theorem 1.2, the condition that solutions of A.1) are
uniquely determined by initial conditions can be omitted if there exist
Figure 1.
smooth functions fn{z), n = 1, 2, .. ., on Q which approximate f{z)
uniformly on compact subsets of Q and if Theorem 1.2 is applicable to
z' =fn{z). This will be illustrated in Corollary 1.1 below.
Proof of Theorem 1.1. For brevity, we write a>_{z0), a>+{z0), t{z0), . ..,
although solutions are not determined by the initial point z0, so that a>_,
a>+, t, . . . depend on z0 and the selected solution. Let z{t, z0) be any
solution of A.1) determined by z@) = z0 and let its maximal interval of
existence be a>_{z0) < t < oj+{z0). If z0 e L, then z{t, z0) e Qo for small
— t > 0. If there exists an z0 e L having a half-trajectory z{t, z0) in Qo U L
on its left maximal interval a>_{z0) < t ^ 0, then the conclusion of
Theorem 1.1 holds.
Suppose therefore that, for every z0 e L and solution z{t, z0), there
exists a t = t{z0), a>_{z0) < t{z0) < 0, such that z{t, z0) e Q.o for t{z) <
t < 0, but z{t, z0) ? Qo. Then z(t, z0) e R; in particular, /? is not empty.
Since R is compact and L U /? is not compact, there exists a sequence
of points Z], z,, ... of L such that if rm = t{zw), then, as m -»¦ oo, z° =
lim z{rm, zm) e /? exists and either ||zm|| -»¦ oo or zn = lim zm exists and
z^ ^fi; sec Figure 1.
204 Ordinary Differential Equations
Put zjt) = z(t, z(rm, zj), so that zm@) = z(rm, zj e R, zm(-rj =
zm e L, zm(t) e Qo for 0 < t < -rm. Since zm@) -* z° as m -* oo,
Theorem II 3.2 implies that if the sequence zls z2,. . . is suitably chosen, it
can be supposed that there exists a solution z°(t) of A.1) satisfying z@) = z°,
having a maximal interval of existence (a>0, a>°), and such that
A.2)
uniformly as m -* co
on any /-interval [/*, /*] in (co0, co0). In particular, co_(zj <'*<'*<
co+(zj for large m.
L-L
Figure 2. ««(/) 6 fi0 ^ * for 0 ^ / < w+(z<>).
Suppose, if possible, that z°(t) $ Qo U R for 0 ^ / < co0. Then z°(/) ?
O0 n Q = Qo U L U i? for 0 ^ / < co° and, hence there is a t = tx,
0 < ^ < co° such that z°(O $ Do- Let tx < t2 < co°. It follows that, for
large m, zjt) is denned for 0 < / < t2 and that A.2) holds for 0 ^ / ^ ta.
But then zm(^) ^ O0 for large m. Consequently, 0 < —rm < ^ for large
m. In this case, z°(—rm) — zm(—rm) -*0, m-+co, by A.2). Since
zm(—rm) = zm and either ||zj| ~* co or zm -*¦ zx as t -* co, it follows that
either ||z°(— rm)|| -> co or zo(—rm) -*«„ ^Qasm -* co. This is impossible
as — rm < t1 < co°. Hence Theorem 1.1 is proved.
Proof of Theorem 1.2. In view of the proof of Theorem 1.1, it suffices
to consider the case that there exist points z0 e L such that (the unique)
z(t, z0) is in Qo U L U i?, hence in Qo U R, for co_(z0) < / < 0. Let Lx
be the (nonempty) subset of L of such points. Clearly L1 is closed relative
to L, since L consists of egress points; cf. Theorem II 3.2.
If there is a point z° e R which is a limit point of Lx, then C~ : z(t, z°),
co_(z°) < t ^ 0, is in Qo U i?. Suppose therefore that Lx has no limit
point in R. Thus Lx is closed relative to L KJ R.
Consequently L — Lx has a limit point jo in Lx. Otherwise L \J R
Plane Stationary Points 205
has a decomposition into nonempty, disjoint sets (L — Lx) U R and Lx,
which are closed relative to L U i?. But L U i? is connected.
Thus the constructions of the last proof can be repeated with the
modifications that zm e L — Lx and zm-*zcc e Lx. If z°(t) e Qo U i? for
0 ^ t < co+(z0), the proof is complete; cf. Figure 2. If not, it becomes
necessary to examine the conclusions at the end of the proof to the effect
that —rm < tx < co° and z°(—Tm) -+zK as m -*¦ oo.
Hence there exists a t = too( ^ 0) which is a (finite) limit point of the
sequence t1s t2, . . . . Then z\—tx) = zB. It is clear that ia < 0 for
Figure 3. z°(t) e Qo U R for co_C0) < (/) <; 0.
z°@) e i?, zn e Li (and L, R are disjoint). Since jo e Z,! and z(/, zj is
uniquely determined by zn, it follows that z(t, zx) e Qo U R for cojz^,) <
/ < 0; see Figure 3. [This is the critical point where the assumption that
solutions of A.1) are uniquely determined by initial conditions is used.]
In particular, the half-trajectory z°(t — tx) = z(t, zj, co_(zx) < t ^
i9, is in Qo U R. Since the endpoint of this trajectory is z(tx, zj =
z° e R, the theorem is proved; i.e., z°(t) efl0Ui! for co_(z°) < t ^ 0.
Corollary 1.1. Le/ dim z ^ 2, /(z) be continuous on an open set Q
containing the closure of the spherical sector
A.3)
Qo = (z: 0
< ||z||<
-^--z*\\<r1<2
\\z\\ II
where d, r\ are positive numbers and ||z*|| = 1, and let
A.4) /@) = 0.
Let L, R be the lateral and spherical parts of the boundary o/Q0,
A.5)
A.6)
= (z:
0 < ||z||
< d, M--
II \\z\\
= \z: \\z\
206 Ordinary Differential Equations
respectively. Suppose that every point of L is an egress point for Qo. Then
A.1) has at least one half-trajectory F in Qo U R U {0} with endpoint in R
{and defined on a half-line t ^ 0 or t ^ 0); see Figure 4.
Proof. Suppose first that the solutions of A.1) are uniquely determined
by initial conditions, then Theorem 1.2 is applicable if the open set Q of
Corollary 1.1 is replaced by the open set obtained by deleting 2 = 0 from
Q, so that 9Q0 O Q = L U R. Then there exists a half-trajectory, e.g.,
C+ : z{t), 0 ^ t < w+ (< oo), in Qo U R with 2@) e R. If w+ < oo, then
(a)
Figure 4. (a) T of type C+. F) T of type C~.
z{t) -* 0 as t -*¦ w+; cf. Lemma II 3.1. But, in this case, the definition of
z{t) can be extended to 0 _ t < oo by defining z(t) = 0 for f ^ a>+.
[Actually, this situation cannot arise when the solutions of A.1) are
uniquely determined by initial conditions, since z(t0) = 0 for somef0implies
that z(t) = 0. It can arise, however, in the general case to be considered
now.]
It has to be shown that Corollary 1.1 is valid if it is not assumed that the
solutions of A.1) are uniquely determined by initial conditions. Since
2 e L is an egress point, if follows that the trajectory derivative of u(z) =
z z - z
*||2
— rf is non-negative at z e L; i.e.,
A.7)
B • 2*)B •/) - ||2||2(Z
^ 0
forO < ||2|| = d, \\zl\\z\\ - 2*|| = 7]-, cf.§1118. Thus,iff€(z) =f(z) - ez*,
then
A.8)
(z ¦ z*){z-fe) - \\z\\\z* -fe) ^ e(\\z\\* - B • z*f),
which exceeds a positive constant ecn > 0 when
A.9)
for large n.
1 ^ \\z\\ ^ d,
n
2
Plane Stationary Points 207
Let/1B),/2B), ... be a sequence of smooth functions such that/mB) ->¦
f(z) as m -* oo uniformly on an open set Q1 => Cl0. Let the integer n > 0
be so large that l/« < d. Then if/mB) is replaced by fm(z) — z*/n, if
necessary, it can be supposed that there is an m = m(n) such that
A.10)
B-2*)B-/J-||2||2B*-/J =
2n
when A.9) holds, and that m{ri) -* oo as n -*¦ oo.
Let Qn be the open set obtained by deleting the sphere ||z|| < 1/n from
a1. Let
' < \\Z\\ <
— 2*
and let Ln, R be the lateral and spherical parts of the boundary of QOn in
Q.n. By A.10), 2 e Ln is an egress point of QOn with respect to the differential
equation
A.11) *'=/»(*)•
Thus, by Theorem 1.2, A.11) has a half-trajectory Vm in QOn U R with
endpoint on R. For the sake of definiteness, let Tm = Cm+ : z = zm{t),
0^t<rm(^ co), where zm@) e R.
Here 0 _ t < rm is the right maximal interval of existence if A.11) is
considered only on Qn. Let the right maximal interval of existence of
zjt) be 0 ^ t < com (<oo) if A.11) is considered on Q1., Thus tm < wm
and rm < wm implies that ||zm(r J|| = 1/n.
By choosing a subsequence, if necessary, it can be supposed that z0 =
lim 2m@) e R exists as m = m(n) -»¦ oo. By Theorem II 3.2, it can also be
supposed that A.1) has a solution zo(t) satisfying 20@) = z0 and that
zm(t) -* zo(t) as m = m{n) -* oo uniformly on every compact interval of the
right maximal interval [0, w0) of existence of zo(t) relative to Q1.
Suppose that the half-trajectory F : zo(t), 0 ^ t < w0, is not in Qo U
R U {0}. Thus 20(^) ^ Do for some tlt 0 < h < w0. If e > 0, then 0 <
Tm < h + e and rm < o>m for large m. Consequently,
and ||2m(Tm)|| = 1/n -+0 asm(n) -»¦ co. Thus if tb is a limit point of
t15 t2, . . ., then zo{tx) = 0. Here, it is possible to change the definition
of zo(t) as follows: if t0 > 0 is the least f-value where zo(t) = 0, put
zo(t) = 0 for t ^ t0. Now T : zo(t) is defined for 0 ^ t < oo and 2m(f) ->
20@, w(n) -»¦ oo, uniformly for 0 ^ f ^ f0 < oo. Repeating the argument
just concluded, it follows that zo(t) c Qo u 5 u {0} for t ^ 0. This
completes the proof.
208 Ordinary Differential Equations
Corollary 1.2. Let dim z = 2. In addition to the assumptions of Corol-
Corollary 1.1, suppose that
A.12) /(z) ^ 0 for 0 5* z e O0.
Then the half-trajectory Y : z(t) in Corollary 1.1 is defined for t^.0 or
t ^ 0 and satisfies z{t) -»¦ 0 as t -»¦ co or t -»¦ — 00.
Proof. If this corollary is false, then F : z@ for t ^ 0 or t ^ 0 remains
at a positive distance from z = 0; cf. Lemma II 3.1. In view of A.12)
this is impossible by the general theory of plane autonomous systems;
cf. Theorem VII 4.4.
Figure 5.
Corollary 1.3. Let the assumptions of Corollary 1.2 hold, (i) If in
addition, z •/(«) < Ofor z e R, then any half-trajectory r«Q,U5U {0}
with endpoint on R is defined for t ^ 0. (ii) If 2 • f{z) >0forzeR and
2@ to any solution of A.1) wM z@) rnteno/- to 7? [i.e., ||z@)|| = 6, ||z@)/
||2@)|| - 2*|| < v], then 2@ can 6e defined for t <| 0, z@ eO0U«u{0}
am/ z@ -»¦ 0 as t -»¦ — co.
fxOTwe 1.1. Prove Corollary 1.3.
A different situation is considered in the next result.
Corollary 1.4. Let dim z = 2, so that L in A.5) is the union of two
disjoint open line segments Lu L2. Let the conditions of Corollary 1.2 hold
but, instead of assuming that every point z of L is an egress point for Qo>
assume that every z e Lx is an egress point for Qo and that every z e L2 is an
ingress point for Qo. Then there exists no, or at least two, half-trajectories
in Qo with endpoint on L U R.
Proof. Assume that there exists a half-trajectory T0 in Q° with end-
point z° e L U R. Suppose that T0 is of type C+ (otherwise, replace t by
— t and interchange y and L2). Then z° e L2 U R. _
Suppose that z°eL2. Then every point z0 of the segment of L2 from
0 to z° is the endpoint of a half-trajectory T0 of type C+ in Qo U L2;
cf. Figure 5(a).
Plane Stationary Points 209
Suppose that z° is an interior point of R and that no point z1 of L2 is an
endpoint of a half-trajectory in Qo U L2. Then a solution arc of A.1)
through z1 does not meet T0 for increasing t, but does meet R; cf. Figure
5(b). The arguments used in the proof of Theorem 1.1 show that there
exists a half-trajectory To of type C~ in Qo U R with endpoint z0 e R.
This proves Corollary 1.4.
Exercise 1.2. In Corollary 1.4, assume that z-f{z) 5^0 for z e R.
Show that there exists no, or infinitely many, half-trajectories in Qo with
endpoints on L U R.
For arbitrary d = dim z, Corollary 1.3 has the following analogue.
Corollary 1.5. Let d^.2. Let the assumptions of Corollary 1.1 hold and,
in addition, z ¦ f{z) ^0forzeQ.0 uii. Then any half-trajectory T:z@ in
O0 U R u {0} with endpoint in R is defined on a half-line t ^ 0 or t ^ 0
and tends to z = 0 as t -»¦ co or t -»¦ — 00 according as z -f(z) < 0 or
z-f(z)>0.
This is clear because ||z|| is decreasing or increasing with t according as
z -/(z)< 0 or z-f(z) >0.
2. Characteristic Directions
In this section, dim z = 2. Write z = (x, y) and A.1) as
B.1) x' = X{x,y), y'=Y{x,y).
It will be supposed that X, Y are continuous for small \x\, \y\ and that
B-2) X@, 0) = r@, 0).
Introducing polar coordinates x = r cos 6, y = r sin 8 transforms B.1)
into
B.3)
r = X{r cos 6, r sin 6) cos 6 + Y(r cos 6, r sin 6) sin 6,
rd' = Y(r cos 6, r sin 6) cos 6 — X(r cos 6, r sin 6) sin 6.
A direction 6 = d0 at the origin is called characteristic for B.1) if
there exists a sequence (rl5 00, (r2, d2), . . . such that 0 < rn ->¦ 0 and
0n -* 0O as r -* co; (Xn, Yn) = (Jf, Y) at (ar, 2/) = (rn cos ©„, rB sin 0n) is
not @, 0), and
B.4)
Yn cos fl0 — Xn sin fl0
0
as n
co.
The condition B.4) means that the angle (mod tt) between the vectors
(Xn, Yn) and (cos 60, sin 0O) tends to 0 as n -»• co.
Lemma 2.1. Let X(x,y), Y(x,y) be continuous for small \x\, \y\ and
210 Ordinary Differential Equations
X2 + Y2 ^ 0 according as x2 + y2 ^ 0. Let B.1) possess a solution
(x(t), y{t))for 0<t<w(<oo) such that
B.5)
0 < x2(t) + y\t) -> 0 as t
CO.
Let r{t) = (x2(t) + y\t))l/i > 0 and 6(t) a continuous determination of
arc tan y(t)/x(t). Let 0 = 0O be a noncharacteristic direction. Then either
6'(t) > 0 or d'@ < Ofor all t near w for which 6(t) = 0O mod 2n.
Figure 6.
Proof. Clearly 6'(t) ?* 0 for all t near co for which d(t) = 0O mod 2n.
Otherwise there exists a sequence t1 < t2 < ... such that tn ->¦ a>, d(tn) =
0O (mod 2tt) and B\tn) = 0. But then B.4) holds with (rn, 0n) = (r(tj, 0O)
because the expression in B.4) is zero by B.3). This is impossible since
0 = 0O is noncharacteristic.
Suppose if possible that the lemma is false. Then there exists ^ < t2 <
. . . such that tn -> w, KO > r(t2) > ..., (-\)n6\tn) > 0 and 6(tn) = 0O
(mod 2tt); see Figure 6. Let ©(/-, 6) denote the right side of the second
equation in B.3), so that (— l)n@(r(tn), 0O) > 0. By the continuity of
©(/-, 0O) with respect to r > 0, it follows that there exists an rn such that
r(Q >rn> r(tn+1) and ©(/•„, 0O) = 0. Since X2 + Y2 ^ 0 if a* + y2 * 0,
it follows that B.4) holds with (/•„, 6n) = (/•„, 80); i.e., 0O is characteristic.
This is a contradiction and proves the lemma.
Theorem 2.1. Let X(x, y), Y(x, y) and (x(t), y(t)) be as in Lemma 2.1.
Suppose that every 6-interval, a. < 6 < /?, contains a noncharacteristic
direction. Then either
B.6)
60 = lim
exists (and is finite)
Plane Stationary Points 211
or (x(t), y(t)) is a spiral; i.e.,
B.7) |0(OI - oo
as t
¦ co.
In the case B.6), 8 = 60 is a characteristic direction.
Proof. Suppose, if possible, that lim 6(t), as t -»¦ co, does not exist
either as a finite or infinite value. Then there exist numbers a, /? such that,
as t —»• w,
(- oo) < lim inf 0(f) < a < /3 < lim sup 6(t) (< oo).
By assumption, there is a noncharacteristic 0O satisfying a < 0O < /?.
Since 0(?) is continuous, there exist f-values arbitrarily close to co where
6(t) = a. < 60 and f-values close to a> where d(t) = ft > d0. It follows
that there are f-values arbitrarily close to a> where d(t) = 60, 6'{t) ^ 0 and
other f-values where 6(t) = 60, d'(t) ^ 0. This is impossible by the last
lemma. Hence lim 6(t), t -»• w, exists as a finite or infinite value. Since
the last assertion of Theorem 2.1 is clear from the definition of "character-
"characteristic direction," the proof is complete.
Exercise 2.1. Let X, Y be continuous for small \x\, \y\ and satisfy
X = Y = 0 at @, 0). Let ip{r) be a positive continuous function for small
r > 0 such that y(+0) = 0. Suppose that the limits
.„ ,. X(r cos 6, r sin 6)
pF) = hm —i f- -'
r-o i/\r)
,. Y(r cos 6, r sin 6)
= hm -^ f- '-
r-o ip(r)
exist uniformly for 0 near 60 and that/?2@) + q\6) ^ 0. Show that 6 = 6o
is a characteristic direction if and only if q{0) cos 6 — p(d) sin 6 = 0 at
0 = 0O-
Theorem 2.2. Lef ^(a;, ?/), y(x, y) be continuous on the triangle T:
0 <. x ^ a, \y\ ^ rjx and such that 1^0. Let co(x, u) be a non-negative,
continuous function for 0 <x ^ a, 0 ^ u ^ 2r\x with the properties that
w(x, 0) = 0 and that the only solution u{x) of
B.8)
for small x > 0 satisfying
B.9) u(x),
is u = 0. Lef the function
B.10)
du
— = w(x, u)
dx
«(?)
U(x, y) =
as x
Y(x, y)
X(x, y)
+0
212 Ordinary Differential Equations
satisfy
B.11) U(x, yt) - U(x, 2/1) <| co(x, y2 -
for y1 ^ yt
and (x, 2/j), (x, y^ e T. Then, up to replacement of the parameter t by t +
const., B.1) has at most one solution (x(t),y(t)) which for large t [or —t]
satisfies
B.12)
and
x(t)
0
as t [or — t] -*¦ oo.
Exercise 2.2. Prove Theorem 2.2. Note that B.1) is equivalent to
dy/dx = U(x, y).
Exercise 2.3. Show that the conclusion of Theorem 2.2 is valid if
B.11) is replaced by U(x, y2) — U(x, yt) < (y2 — yj/x for —r\x < y1 <
2/a ^ r\x and 0 < x ^ a. In particular, this is the case if X > 0 and
X, Yhave continuous partial derivatives Xv, Yv with respect to y satisfying
XYy - Xy Y < X2lx on T.
Exercise 2.4. Replace T in Theorem 2.2 by R0:0 < x < a, \y\ < b;
also, let co(x, u) be continuous on 0 < x ^ a, \u\ ^ 2b. Show that an
analogue of Theorem 2.2 is valid if "m(x)/x -> 0" and "y@M0 ->" °" are
deleted from B.9) and B.12), respectively.
3. Perturbed Linear Systems
The results of §§ 1 and 2 will be applied to a 2-dimensional system
obtained by perturbing a linear system
C.1)
z' = Ez,
in which z = (x, y) is a real 2-dimensional vector and ? is a constant
matrix with real entries. Unless otherwise specified, it will be assumed
that
C.2)
det E 5^ 0.
Let Xu X2 be the eigenvalues of E; so that A1( X2 are real or are complex
conjugates since E is real. If X = Ax or X = A2 is a simple eigenvalue or if
X = X1 = Xz and E has a double elementary divisor, then, up to factors
±1, there is only one (real) unit eigenvector z* satisfying Ez* = Xz*.
Recall from Exercises VII 7.1 and 7.2 that if Xu X2 = ±ifi are purely
imaginary (with j8 ^ 0), then z = 0 is a center; that if Xu Xz = a ± */? are
complex conjugates but not real or imaginary (a, j8 real, ^ 0), then 2 = 0
is a focus (at f = ±oo according as a. $ 0); that if Xlt X2 are real and
Plane Stationary Points 213
det E = XtX2 > 0, then z = 0 is an attractor, in fact, a node for t = ± oo
according as Xlt X2 ^ 0 (and a proper node only if X1 = X2 < 0 and ? has
simple elementary divisors); finally, if Xu X2 are real and X^2 = det E < 0,
then 2 = 0 is a saddle point.
After a real linear change of variables, it can be supposed, when con-
convenient, that E is in one of the normal forms
C.3-1)
E =
0 -p
P 0
C.3-2)
-j8'
= a ±
C.3-4)
? =
n
A = X1 = X2 real
C.3-3)
E =
Xlt X2 real;
C.3-5)
(simple elementary divisors)
The system to be considered in this section is of the form
C.4) z' = Ez + F(z),
where F(z) is continuous for small ||2|| and satisfies
F(z)
X = X1 = X2, ?#0
(double elementary divisor)
C.5)
- 0
as
Theorem 3.1. Assume C.2) and that the continuous F(z) satisfies C.5).
Let z = 0 be an attractor for C.1) at t = 00, 50 that otfc = Re Afc < O/o/-
*= 1,2.
(i) In this case, z = Ois an attractor for C.4) at t = 00. Afore generally,
if<*k < —c < Ofor k = 1,2, f/ien f/iere exists a constant M = M(c) such
that if ||20|| 5^ 0 w sufficiently small, every solution o/C.4) satisfying the
initial condition 2@) = 20 exists for t _ 0 and satisfies
C.6)
C.7)
IK0II =
t~l log I
for f = 0;
as t -»• 00,
where a. = ax or a =
(ii) T/"*! < a2 < 0,
f/ien
= a2 are rea/, and if C.7)
C.8)
lim
214 Ordinary Differential Equations
exists and is an eigenvector of E belonging to a (=XX or Aj). Inparticular, if
E is in the normal form C.3-3), then
C.9)
or
x(t)
y(t)
0
as t
oo
according as a. = ?n or a. = Xv
(iii) Let ax < aa < 0. Ifz° is either of the two real unit eigenvectors of E
belonging to X = al5 then C.4) /las a? least one solution z(t) satisfying C.8)
and C.1) with a = ax. V/"z° « ez7/ie/- o/f/ie two real unit eigenvectors of E
belonging to X = u.iandif\\z0\\ j^ Oan^||zo/||zo|| — z°\\ are sufficiently small,
then any solution of C.4) determined by z@) = z0 exists for t^O and
satisfies C.8) and C.7) wzY/i a = aa.
Proof of (i). After a real linear change of variables, it can be supposed
that E is in one of the real normal forms C.3). In C.3-5), it can also be
supposed that e > 0 is so small that Re X — \e < — c. It is then readily
verified that if r = \\z\\ and r 5^ 0 is small, then /•' 5! — cr. This implies
that along a solution z(t), r(t) ^ /-@)e~cf for small t > 0. Consequently
if r@) > 0 is sufficiently small, then z(t) exist for f ^ 0 and satisfies C.6)
with M = 1.
When E is not in a normal form and ? is a nonsingular matrix such that
L~XEL is in the form just used, then C.6) holds with M = 1 if z is replaced
by r\ = Lz. In this case, C.6) holds if M = \\L\\ ¦ || JL-—x||.
Wheno^ = ota < 0 and E is in a normal form C.3-2), C.3-4), or C.3-5),
it is easy to see that 0 ^ z(i) -»¦ 0 as t -»¦ 00 implies C.7). This completes
the proof of (i) for this case. [The cases ax 5^ «a will be considered in the
proofs of (ii) and (iii).]
Proof of (ii) and (iii). Assume that E is in the normal form C.3-3), so
that the unit eigenvectors of E are (±1,0) for X = X1 and @, ±1) for
X = Aa. On introducing polar coordinates z = (r cos 0, r sin 0), C.4)
takes the form, as r —> 0,
C.10)
/•' = [Xx cos8 0 + Aa sin8 0 + o(\)]r,
r6' = (Aa - Ax) sin 0 cos 6 +
It follows that the only characteristic directions (mod In) are 0 = 0,
tt/2, 77, 3tt/2; cf. Exercise 2.1. Hence, by Theorem 2.1, if 0 ^ z@ -»¦ 0 as
t -*¦ 00, then z = z(f) is a spiral or C.9) holds.
Let Qo be a wedge Q0:0 < \\z\\ < 6, ||z/||z|| - z°|| < jj, where z° =
@, ±1) and 6, r\ are small. It is seen that if a solution z(t) starts in Qo or
enters in Qo, it remains in Qo, for the boundary points are strict ingress
points for Qo; cf. Corollary 1.3 (ii) with t replaced by —t. Thus such
solutions satisfy z(t) ->¦ 0 as t -»• 00 and the second part of C.9), i.e., C.8).
Plane Stationary Points 215
The first equation of C.10) implies C.7) with a = Aa. In particular, no
solution z = z(t) ^ 0 tending to 0 as t ->¦ 00 is a spiral.
If, in the definition of the wedge ft0, z° is taken to be (±1,0), the
existence of solutions z(t) j? 0 satisfying C.8) follows from Corollary
1.3(i). As above, such solutions satisfy C.7) with a = Ax. This proves
Theorem 3.1.
Theorem 3.2. Let the eigenvalues of E be a. ± ifi, where a. < 0
and j8 ^ 0 are real; let C.5) hold; let z(t) be a solution of C.4) such
that 0 < ||z@ll < <5<, for all t and 6(t) a continuous determination of
arc tan y(t)jx(t). Let 0 < e < |/8|. Then there exists a d€> 0 such that if
d0 < <5?, then
C.11)
16(t)- fit\<et for large t.
In particular, d(t) -*¦ ± 00 as t -*¦ 00 according as
a. < 0,
0. 7/", m addition,
C.12) d'(t)^p, hence r
as f
00.
z/
5, if z = 0 is a center for C.1), z7 m a center or focus for C.4)
z = 0 « a focus for C.1), f/ien zY w a focus for C.4).
Proof. It is readily verified that neither the assumptions nor conclusions
are affected if z is subjected to a real linear transformation. Hence, it can
be supposed that E is in the normal form C.3-2), with a ^ 0. Let F(z) =
(F\ F8) and write C.4) as
C.13) x' = ex - py + F\x, y), y' = fix + cy + F\x, y).
Introducing polar coordinates gives
C.14) /•' = ar + R(r, 6), e' = p+S(r,6),
where
C.15) rR = F1 cos 6 + FHind, rS = F8 cos 6 - F1 sin 6;
so that R(r, 6), S(r, 6) -»¦ 0 as r -»¦ +0. Thus, there exists a <5? > 0 such
that \S(r, 0)| < e if 0 < r < <5?. Hence |0' - 0| < e if 0 < ||z@ll ^ <5?,
and C.11) holds. If a < 0, then, by Theorem 3.1, r(t) -»¦ 0 as f -»¦ 00, and
so S(r@, 0@)-*- 0 as t -»¦ 00. In this case, C.12) follows. This proves
Theorem 3.2.
As can be expected, the property that z = 0 is a center (i.e., that a
solution starting at any point z0 ^ 0 at t = 0 returns to exactly the point
z0 at some positive t) is very sensitive to perturbations. This is illustrated
by the following exercise which shows that no condition of smallness on
F(z) ^ 0 at z = 0 can assure that if z = 0 is a center for C.1), then it is a
center for C.4).
216 Ordinary Differential Equations
Exercise 3.1. Let h(r) be a continuous function for 0 _ r _ 1 such
that h(r) ->¦ 0 as r ->¦ 0. Consider a system C.4) of the form
C.16) x' fa + xh(r), y' = fa + yh{r),
where r = (x2 + y2fA, the function F(z) = (xh(r), yh{r)) is continuous for
||z|| = 1, and ||/{z)||/||z|| = \h(r)\^0 as r^O. If h(r) = 0, i.e., F= 0,
then 2 = 0 is a center. Show that if h(r) < 0 for 0 < r ^ 1, then 2 = 0
is a focus (at f = oo) for C.16).
It might also be guessed that the other cases, Ax = A2, determined by
equalities (rather than by inequalities) are sensitive to perturbations.
This turns out to be the case. For example, the next two exercises show
that if 2 = 0 is a node for C.1) with X1 = X2< 0, then, even if C.5) holds,
2 = 0 can be a focus for C.4), whether or not E has simple or double
elementary divisors. However, as will be shown in Theorems 3.5 and 3.6,
suitable conditions of smallness on Fat z = 0, more stringent than C.5),
preserve the character of this type of stationary point.
Exercise 3.2. Let E = diag [X, X], X < 0. Show that there exist
continuous functions F(z) for ||z|| < d satisfying C.5) and such that (a)
2 = 0 is a focus for C.4); and (b) the equation C.4) has a solution z{t) -»¦ 0
as t -»• oo satisfying any one of the seven possibilities compatible with
— oo _ lim inf Q(t) = lim sup 8(t) = oo.
(-> oo (->oo
Exercise 3.3. Let E be as in C.3-5) with X < 0 and e = 1, so that E
has a double elementary divisor and is in a Jordan normal form. Show
that there exist continuous F(z) for ||z|| _ d satisfying C.5) such that (a)
all, (b) some but not all, (c) no solutions z{t) of C.4) which tend to 0 as
t—>- 00 are spirals (i.e., \6(t)\ -*¦ 00 as t-*¦ 00). [Case (b) cannot occur if
the solutions of C.4) are uniquely determined by initial conditions]. See
Theorem 3.3.
Theorem 3.3. Let E be as in C.3-5) with X < 0 and e = 1. Let F(z) be
continuous for small \\z\\ and satisfy C.5) and let z(t) 5^ 0 be a solution of
C.4) for large t satisfying z(t) ->¦ 0 as t ->- 00. Then either z = z(t) =
(x(t), y{t)) is a spiral (i.e., \8(t)\ -»¦ 00 as t -»¦ 00) or 6(t) -»¦ 0 (mod n) as
t-> 00.
Exercise 3.4. Prove Theorem 3.3.
Conditions which assure that all or that no solutions in Theorem 3.3
are spirals will be considered subsequently; cf. Exercise 4.5.
Theorem 3.4. Let E = diag (A1; X2), where Xx < min @, X2); let F(z)
be continuous and satisfy C.5). Jfz° = A, 0) or (-1, 0), then C.4) has at
least one solution z(t), t _ 0, satisfying C.8) and C.7) with a = Ax;
furthermore, if X2 > 0 and z(t) is a solution of C.4) for large t such that
2@ -»¦ 0 as t -»¦ 00, then C.7) holds with a = Xt andy(t)jx(t) -»¦ 0 as t -> 00.
Plane Stationary Points 217
Exercise 3.5. Deduce Theorem 3.4 from Corollary 1.3 (i).
Exercise 3.6. Let E = diag (Xu X2), Ax < min @, X2), and let ^B) be
continuous and satisfy C.5) and
C.17)
- FB2)||
0
as zx, 22
-0
(with Zi 5^ z2). Then, up to reparametrizations (i.e., replacements of t by
t + const.), C.4) has unique pair of solutions z±(t) for large t such that
0 =? 2±@->-0 as t -> 00. These solutions satisfy z±@/||z±@ll -"(±1, 0)
as t ->¦ 00, and, hence C.7) with a = Xx.
Exercise 3.7. Let E = diag (?.lt XJ with Ax < min @, A2) and let F(z)
be continuous for small ||z|| and satisfy C.5). Use Theorem 2.2 and/or
Exercise 2.3 to find conditions, more general than C.17), to assure that
C.1) has at most one solution (up to changes of the parameter) satisfying
2@ - 0 and 2@/1|2@1| - A, 0) as t - 00.
This completes the discussion of C.4) under the assumption C.5).
Except in the case of a center, assumptions slightly stronger then C.5)
suffice to preserve the character of the stationary point 2 = 0 in passing
from the linear system C.1) to the perturbed system C.4). Results of
this type are consequences of general theorems in Chapter X (in
particular, in § X 16). Some will be stated here for the sake of
completeness. The deduction of these theorems from results in Chapter X
will be given as Exercises 3.7-3.11; cf. also Theorem X 13.1 and its
corollaries in § X 16.
The first condition to be imposed on F(z) will involve the function
C.18)
<po(r) = max ||FB)|| for
= r
[so that cpo(r) is a continuous, nondecreasing function for small r _ 0
and <j90@) = 0] and the condition
C.19)
f
J+o
dr < 00.
This last condition is satisfied if, e.g.,
C.20)
0
as
for some e > 0, since C.20) implies that <po(r)lr1+€ -»¦ Oas/- -> 0. Conditions
of the type C.18), C.19), or C.20) are invariant under linear changes of the
variables, z—>Lz where L is a constant matrix, so that, in the theorems to
follow, the assumption that the matrix ? is in a normal form is no loss of
generality. This does not apply, e.g., if C.18) is replaced by <po(r) =
max ||FB)|| for ||z|| = r and it is not assumed that cpo(r) is monotone.
218 Ordinary Differential Equations
Theorem 3.5. Let F(z) be continuous for small \\z\\ and satisfy C.18)-
C.19) and let z(t) be a solution of C.4) satisfying 0 ^ z(t) ->¦ 0 as t -»• oo.
(i) Let E be as in C.3-2), where a. < 0, 0 # 0. Then there exist constants
c>0,d0 such that
C.21) z(t) = ce"(cos [fit + 60 + o(l)], sin [fit + 60 + o(l)])
as t —»• oo; conversely ifc > 0, 0O are gwen constants, there exists a solution
2@ o/C.4) satisfying C.21).
(ii) Lef ? = diag [ku A2] w/fA Ax < A2 < 0. Then there exists a constant
c^jt. 0 or a constant c2 ^ 0 jmc/i that either
C.22)
or
C.23)
2@ =
as
as t
co
co:
conversely if ct ^ 0 am/ c2 ^ 0, f/iere exwf solutions z{t) of C.4) satisfying
C.22) an^ C.23), respectively.
(iii) Lef ? = diag [A^ A2] wz7/i Ax < 0 < A2. TTien f/iere existe a constant
q^O jmc/i f/iaf C.22) holds; conversely, ifcx ^ 0, there is a solution z{t) of
C.4) satisfying C.22).
(iv) Let E = diag [X, A] vwfA A < 0. 77ien there exist constants cu c2,
not both 0, smc/i
C.24)
2@ = eAi(ci + o(
as t
co;
conversely, if cx, c2 are g/yen constants, not both 0, f/ien C.4) /ias a solution
satisfying C.24).
Exercise 3.8. Denote by Theorem 3.5* the analogue of Theorem 3.5 in
which the hypothesis C.18)—C.19) is replaced by the slightly heavier
condition:
\\F(z)\
C.25)
C.26)
cp(r) = sup •
for 0 < ||z|| ^ r,
r lq>{r) dr < co.
J+o
(a) Deduce parts (i), (ii) of Theorem 3.5* from Theorem X 1.1 (i.e., from
variants of Corollary X 1.2). (b) Deduce parts (iii), (iv) of Theorem 3.5*
from Lemma X 4.3 (i.e., from Corollary X 4.2).
Exercise 3.9. Using an analogue of the Remark 2 following Lemma
X 4.3 and the result of Exercise 3.8, prove Theorem 3.5.
Theorem 3.6. Let the condition C.18), C.19) of Theorem 3.5 be replaced
by the assumption that there exists a non-negative, non-decreasing, continuous
Plane Stationary Points 219
function <p{r)for small r ^ 0 such that
C.27) ||FB0 - F{z2)\\ ^ cp(r) \\Zl -
for
^ r
and that C.26) holds. Then the constants c, 0O in C.21), the constant cx in
C.22) in both (ii) and (iii), and the constants c1; c2 in C.24) uniquely deter-
determine the solution z{t). {In particular, in case (iv), z = 0 is a proper node.)
Exercise 3.10. (a) Deduce the assertions concerning C.21) and C.24)
from Theorem X 1.1 (i.e., from variants of Corollary X 1.2). (b) Deduce
the assertion concerning C.22) in both parts (ii), (iii) from Exercise 3.6.
In the case of a multiple elementary divisor for E, the condition C.19)
of Theorem 3.5 has to be strengthened to
C.28)
J+o
r 2 |log r\ (po(r) dr < co.
This condition is also satisfied if C.20) holds.
Theorem 3.7. Let F(z) be continuous for small \\z\\ and satisfy C.18),
C.28). Let E be as in C.3-5) with X < 0 and e = 1. Let z(t) be a solution
of C.4) with small ||z@)|| ^ 0. Then z(t) exists for t > 0 and either there
exists a constant cx ^ 0 such that
as
oo
C.29) 2@ = qe^l + o(l), t + o@)
or there exists a constant c2 # 0 such that
C.30) 2@ = c2eA((o(l/0, 1 + o(l)) as t-+oo;
conversely, if cx^ 0 and c2 ^ 0 are given, then there exist solutions z(t)
satisfying C.29) and C.30), respectively.
Exercise 3.11. Deduce Theorem 3.7 from Corollary X 4.1. In order to
obtain the assertions concerning C.29) [or C.30)], make the change-of
dependent variables z = (x, y) ->- (u, v) denned by x = euu, y = eMt{u + v)
[or x = euujt, y = eu{u + v)] and the change of independent variable
t = e$. This deduction is more straightforward if the condition C.18),
C.28) is strengthened to C.25),
C.31)
r l |log r\ y{r) dr < oo.
Otherwise, the necessary arguments involve the analogue of Remark 2
following Lemma X 4.2; see also Corollary X 16.3 and Exercise X 16.2.
Theorem 3.8. Let the condition C.18), C.28) of Theorem 3.7 be replaced
by the assumption that there exists a non-negative, non-decreasing con-
continuous function cp(r) for small r ^ 0 satisfying C.27), C.31). Then the
constant c2 in C.30) uniquely determines the solution z(t).
Exercise 3.12. Deduce Theorem 3.8 from the changes of variables in
Exercise 3.11 and from Theorem X 8.2.
220 Ordinary Differential Equations
4. More General Stationary Point
A discussion similar to that of the last section will be given for a plane
autonomous system of the form
D.1) x' = P(x, y) + p(x, y), y' = Q(x, y) + q{x, y),
where P, Q are homogeneous polynomials of degree m ^ 1 and
D.2) p\x, y) + q\x, y) = o(r2m) as r2 = x2 + y2 -> 0,
D.3) (P + pf + (Q + qJ^0 according as x2 + y2 ^ 0.
In terms of polar coordinates, x = r cos 0 and y = r sin 0, define
D.4) RF) = r-m(P cos 0 + Q sin 0),
D.5) SF) = r-m(Q cos6-Psin 0);
thus R, S are homogeneous polynomials of sin 0, cos 0 of degree m + 1.
In terms of polar coordinates, D.1) can be written as
D.6) /•' = rm[R{6) + P(r, 0)], r6' = rm[SF) + a{r, 0)],
where
D.7) p(r, 0) = r~m(p cos 0 + q sin 0), a{r, 0) = r~m(q cos 0 - p sin 0)
tend to 0 as r -*¦ 0 uniformly in 0.
If SF) ^ 0 and R(d) ^ 0, then a necessary condition for 0 = 0O to
be characteristic is that S(d0) = 0 and a sufficient condition is that
SF0) = 0, RF0) ^ 0; cf., e.g., Exercise 2.1. If SF) ^ 0, it has only a
finite number of zeros (mod 2tt). Theorem 2.1 implies the following:
Theorem 4.1. Assume D.2), D.3) and S(d) ^ 0. If (x(t), y{t)) is a
solution of D.1) for large t > 0 [or — t > 0] satisfying
D.8) 0 < z2@ + y\t) -> 0 as f -+ oo [or f -> - oo],
then a continuous determination of d(t) = arc tan y(t)/x(t) satisfies either
D.9) 60 = lim 6(t) exists (and is finite)
and S@O) = 0 or
D.10) |0(f)|-»-oo as f-»-oo [orf->-— oo].
The question to be considered first is the following: If S@O) = 0, do
there exist solutions of D.1) satisfying D.8), D.9)? After a rotation of
the (x, 2/)-plane, it can be supposed that 0O = 0. Suppose 6 = 0 is a zero
of degree k > 0 for S@),
I
D.11) S@) = co0fc +
and c0 * 0, k > 0.
Plane Stationary Points 221
In order to state the next result, introduce the sector
D.12) Qo(<5, rj) = {(x, y) : 0 < r < 6, |0| < -r\ < \tt}.
Theorem 4.2. Assume D.2), D.3), D.11) and that k is an odd integer.
Then if 6, r\ > 0 are sufficiently small, D.1) has a half-trajectory F in
^o(A v) with endpoint on r = 6. For any such half-trajectory, D.8) and
D.9) hold with 0O = 0. If, in addition, R@) ^ 0, then F is defined for large
t > 0 or — t > 0 according as R@) $ 0.
Exercise 4.1. (a) Using Theorem 2.2 (and Exercise 2.3), obtain
sufficient conditions to assure that F in Theorem 4.2 is unique (up to the
replacement of t by t + const.). F) In addition to the conditions of
Theorem 4.2, assume that R@) ^ 0. Using Exercise 2.4, applied to D.6)
rather than D.1), deduce sufficient conditions for the uniqueness of F.
For example, show that if
= 5@) + a(r, 0)
RF) + P{r, 0)
satisfies
/•, 02) -
> 0 is continuous and satisfies
for -€ < 0x ^ 02 < €
y(>) dr/r <
*^+0
and small r > 0, where
oo, then F is unique.
Proof. If 0 < r ^ d and 0 = ±rj, where 6, 77 > 0 are sufficiently
small, then, by D.11), co0' ^ 0. Thus, if c0 > 0, then the lateral boundaries
0 = ±r] of Qo are strict egress points and Corollary 1.1 is; applicable. If
c0 < 0, this corollary becomes applicable if t is replaced by — t. Also,
D.3) implies that Corollary 1.2 can be used. This gives the existence of
F : (x(t), y{t)) satisfying D.8). Also, D.9) follows from Theorem 4.1
since r\ > 0 can be taken so small that 0 = 0 is the only characteristic
direction in |0| ^ r\. This gives the first part of Theorem 4.2. The second
part of Theorem 4.2 is much simpler since Corollary 1.3 (or even Cor-
Corollary 1.5) can be used in place of Corollary 1.1.
In order to obtain refinements of Theorem 4.2 and to deal with the case
that k > 0 is even, suppose that
D.13) R(8) = do6s + o(|0|O, 0^0 and
0, j ^ 0.
Theorem 4.3. Assume D.2), D.3), D.11) with k > 0 an even integer
and R@) ^ 0 [i.e., D.13) with j = 0]. Then, ifd,t]>0 are sufficiently
small, D.1) has no or infinitely many half-trajectories F in ?lo(d, rj). For
any such half-trajectory, D.8) and D.9) hold with 0O = 0.
The first part of this assertion follows from Exercise 1.2; the second
part from Theorem 4.1. See Theorem 4.5 and Exercises 4.6, 4.7 for
criteria for the alternatives in Theorem 4.3.
222 Ordinary Differential Equations
Wheny = 0 [so that R@) = d0 5* 0], k is odd, and
D.14) codo > 0,
then D.1) has infinitely many half-trajectories F satisfying D.8) and D.9)
with 0O = 0; cf. Corollary 1.3(ii). In the next theorem, there will be no
assumption on the parity ofy, k; instead, it will be supposed that
D.15) k>j+ 1 and Cod0 > 0.
[It can be mentioned that if bothy, k are even, then condition D.14) in-
involves no loss of generality. For the substitution 0 —»- — 0 (i.e., y -»• —y)
changes the sign of c0 if k is even (cf. D.6)) but not that of d0 if/ is even.]
Whenever D.14) holds, it can also be supposed that
D.16)
c0 < 0 and d0 < 0,
otherwise t is replaced by — t.
Exercise 4.2. Show that there are examples of P, Q with codo < 0,
k > 0 even,y > 0 odd, k > j + 1 such that no condition of smallness on
p, q assures that D.1) has a half-trajectory (x(t), y{t)) satisfying D.8).
Theorem 4.4. Assume D.2), D.3), D.11), D.13), D.15), D.16). Then
there exists a positive e0 = eo(co, do,j, k) such that if
D.17)
{x, y)\, \q{x, y)\ <
holds for small r > 0 [e.g., if p, q = o(rm+€) as r —*¦ + 0 for some e > 0],
then D.1) possesses infinitely many half-trajectories defined for t^0
satisfying D.8) and D.9) with 60 = 0.
In order to prove this, introduce the following notation: Let d, r\ > 0;
Cj and d1 are positive constants satisfying
D.18) -S@) <; Cl0fc, -
d1Bi for O^d^r).
Let fiir), yJ(r) be positive continuous functions for 0 < r ^ d such that
the functions D.7) satisfy
D.19)
and that
D.20)
[-a(r, 0)] for O^B^r,,
ip?r) ^ max [p(r, 0)] for 0 ^ 0 < -r\,
Vi(r), Viir) -»- 0 as r + 0.
It follows from D.6) that if r\ > 0 is small and 6 = 6{r\) > 0 is sufficiently
small, then
D.21)
6' < 0 for 0 < /• ^ 6, 0 = 77.
Plane Stationary Points 223
Let Qx be the set
D.22) Qi = {(*, y):0<r?6,0<d?ri,dJ'^ y,(r)},
where ^2 is any fixed constant, 0 < d2 < ^t; see Figure 7. Then D.6)
implies that, on Q1;
D.23) -/•' ^ d3rmd> > 0, where d3 = d1-d2> 0,
D.24) -r0' < [c^* + Wi(r)Vm-
Thus, along a solution of D.1) in Q1; /•' < 0 and it is permissible to
Figure 7.
introduce r as as independent variable, so that
D.25)
dd 5@) + a{r, 0)
r dr ~ /?@) + P(r, 0)'
D.26)
In addition, by D.21) and D.23),
D.27)
— > 0 for 0 < r ^ d, 0 = ri.
dr
Theorem 4.4 will be deduced from the following:
Lemma 4.1. If there exists a continuously dijferentiable function
8 = 0o(r), 0 < r ^ d, satisfying the differential inequality
D.28)
dr
and di60i ^ fi(r), then D.1) has infinitely many half-trajectories defined
for t ^ 0 satisfying D.8) and D.9) with 0O = 0.
224 Ordinary Differential Equations
Proof of the Lemma. If the new variable
D.29)
is introduced, D.28) becomes
v = 0)+1
D.30)
where
D.31) A = -t- > 1,
dv ^
r— ^
dr
c* =
ClU + 1)
j +
The relation D.30) implies that 0o(r) ->¦ 0 as r ->¦ 0. For 0o(r) ^ 0 is
increasing and if 0O, hence u, has a positive limit as r -»¦ +0, then, by
D.30), r </i>/tf> ^ const. > 0 for small /• > 0. Since this leads to the con-
contradiction v(r) ^ v{rj) — const, log (??//•)->-—oo as /-->-+0, it follows
that0o(/-)->-Oas/-->-+O.
Let r0 > 0 be so small that /•„ < 5, 0o(ro) < »?• Let 0 = 0(r) be a
solution of D.25) satisfying an initial condition rj > 6(r0) > do(ro); see
Figure 7. By D.27), 0(r) < 77 on any interval [rl5 /•„), /^ > 0, on which
0(r) exists. Since D.26) holds as long as (x, y) = (/• cos 0(r), r sin 0(r)) is in
Qi and since 0o(r) satisfies D.28), Theorem III 4.1 implies that 0o(r) ^
0(r) < r? on any interval [rl5 r0), /"i > 0, on which 0(r) exists and the
corresponding point (x, 2/) g Qx. Since d2d0}(r) ^ ^aW. the solution 0(r)
can be denned on @, /-0] and the corresponding point (x, y) e Qx. This
implies the lemma.
Proof of Theorem 4.4. In view of D.17) and D.19), it is possible to
choose
^) *M (log 1
For a constant e > 0 to be specified, put
so that
(log
<j +1)
(log 1/
The inequality D.28) or D.30) is equivalent to
I
(k-j-1)- d3
Since A > 1, it is clear that if e0 > 0 is sufficiently small, it is possible to
choose an e > 0 satisfying this inequality. Finally, the condition
Plane Stationary Points 225
<WW ^ V>z(r) becomes, by D.29),
(log llry^-i-1' - (log 1
But if d2 > 0 and e > 0, this holds for small r since k>j+ 1 > / Thus
Theorem 4.4 follows from Lemma 4.1.
When j = 0 in Theorem 4.4, the result can be sharpened somewhat.
Theorem 4.5. Assume D.2), D.3), D.11) with c0 < 0, R@) = do<O,
and k > 0 even. Put
D.32) e* = (-dof11"'"
r r l-*Y?. i\ii/(t-D
(i) Let 0 < e0 < e*, d > 0, r\ > 0. Suppose that a(r, 0) w D.7) satisfies
D.33) -o(r, 0) ^ __^L__ for 0<r^d, 0 < 0 ^ V.
Then D.1) /ios infinitely many half-trajectories defined for t ^ 0 satisfying
D.8) am/ D.9) wtt 0O = 0. (ii) Lef e* < e°, d > 0, r\ > 0. SM/7/wje that
— for 0 < r ^ 6, |0| ^ n.
D.34) -o(r, 0) ^ -—
(logl
TAen no half-trajectory satisfies D.8) ant/ D.9) w/fn 0O = 0.
The proof of (i) is similar to that of Theorem 4.4. Let 0 < — c0 < cu
0 < dx < -do, vi(r) be defined as in D.19) and
It is clear from D.25) that if d, r\ > 0 are sufficiently small, then (x, y) e Q2
implies that
D.35)
dr
and D.27). The required analogue of Lemma 4.1 is the following:
Lemma 4.2. Let c = cjd^ y{r) = v)i('")K- #" ^e^ exwf5 a con-
continuously differentiable function do(r) > 0, 0 < r ^ r\ satisfying
D.36)
dr
then the conclusion o/(i) in Theorem 4.5
Exercise A3. (a) Prove Lemma 4.2. (b) Deduce Theorem 4.5(i) from
it.
For the proof of Theorem 4.5 (ii), let 0 < cx < — c0, — d0 < dlf and
0 < ip\r) ^ min [-a(r, 0)] for |0| ^ r\.
226 Ordinary Differential Equations
Then if d, r\ are sufficiently small,
D.37) dxr ^ ^ cj1* + V\r).
Lemma 4.3. Let c = cjd±, ip(r) = y>\r)ldi- # for every smal1 *? > °
and r0 > 0, the solution of
D.38)
r-= cdk
dr
satisfies 0(rx) = —rj at some r±, 0 < rx < r0, then the conclusion of Theorem
4.5(ii) holds.
Exercise 4.4. (a) Prove Lemma 4.3. (b) Deduce Theorem 4.5(ii) from
it.
Exercise 4.5. Apply Theorem 4.5 to the case m = 1, P(x,y) = he,
Q(x, y) = x + Xy, and I < 0 [so that D.1) is the system considered in
Theorem 3.3].
Exercise 4.6. The proof of Theorem 4.5(i) makes it clear that if
f(r) ^ 0 is any continuous function for 0 < r ^ r\, y@) = 0, and if
D.39)
r ^ = cdk
dr
c > 0, has a solution 0o(r) > 0 for 0 < r ^ v\, then we can obtain
analogues of Theorem 4.5(i) by replacing D.33) by — a(r, 0) ^ eyi(r) for a
suitable e > 0. This exercise deals with conditions on y ^ 0 to assure
that D.39) has positive solutions for 0 < r ^ rj. Introduce the new
independent variable t defined by r = e~tle, so that dr/r = -dt\c and
r-=0 corresponds to / = oo. Writing <p(t) = ip((rtle)lc and X = k
transforms D.39) into
D.40)
V ¦¦ Bx - (p(t), where 0' = — ,
dt
X > 1, and (p(t) ^ 0 is continuous for large /. The problem is to find
conditions on continuous (p(t) ^ 0 to assure that D.40) has a positive
solution for large t; cf. § XI 7 for the case X = 2. For brevity, a function
<p(t) ^ 0 continuous for large / for which D.40) has a positive solution
for large / will be called of class Nx.
/•oo
(a) Show that if <p(t) is of class N^, then <p(t) dt ^ oo. (b) Show
that <p(t) e Nx if and only if there exists a continuously differentiable
positive function 6 = do(t) for large / such that
D.41) B'+Bx?-<p;
hence if ^>0(/)GiSTA and 0 < <p(t) ^ <po(j), then <p(t)eNx. (c) If p =
- 1), e* = max (« - «A)/(A - iy«^> for u > 0, and 0 ^ ?</) ^
I
Plane Stationary Points 227
/"CO
e*lt", then cp(t) g Nx. (d) If y(t) dt < oo and
/•oo /t -i \{A—1)/{A — 2)
D.42) cp{s) ds ^ —3 , where e* = :
/•OO
is max [u - (X - 1)ma] for u > 0, then <p(t)eNx. (e) If tlKX-l)<p(t)
dt < 00, then (p(/) g ^a. ^
Exercise A.I. Formulate analogues of Theorem 4.5 (i) using parts (d)
and (e) of the last exercise.
Notes
section 1. Theorems 1.1 and 1.2 may be new and are suggested by the result of
Hartman and Wintner [1] refining a paper of Perron [3]. The results of this section
have the advantage of permitting the treatment, e.g., of some cases of D.6) when
7?@), S(ff) have a common zero.
section 2. The main result (Theorem 2.1) of this section goes back to Bendixson [2]
under conditions of analyticity. The treatment in the text follows that of Nemytskii
and Stepanov [1]; cf. Hartman and Wintner [11] and Kowalski [1]. For results
related to Theorem 2.2, see Hoheisel [1], Hartman and Wintner [1], Hartman [1], and
Keil [1].
section 3. For early references on the subject of this section, see the encyclopedia
articles of Painleve [1] and Liebmann [1]; see also Dulac [1], [2]. Investigations on the
questions considered here were begun by Briot and Bouquet [1] for equations of the
form x dy/dx = ax + by + . . . which, because of § 2, contain most of the cases of
C.4) when the eigenvalues of E are real. Poincare [1 ] initiated the discussion of solutions
of C.4), under conditions of analyticity, when C.5) holds. Perron [3], [5] was the first
to systematically investigate the questions of § 3 for nonanalytic differential equations.
He obtained existence and uniqueness theorems of the type Theorems 3.5-3.8 but with
much heavier conditions. Weyl [4] obtained existence and uniqueness theorems for the
cases of real eigenvalues under conditions similar to those of Theorems 3.6, 3.8; cf.
also Hoheisel [1]. Wintner was the first to omit a Lipschitz condition of the type
occurring in Theorems 3.6 and 3.8 in considering questions of existence. Theorems
3.5, 3.7 are due to Wintner [6], [11] (and are based on his papers [3], [8]). This type of
result has been generalized for nonautonomous systems of arbitrary dimension by
Hartman and Wintner [19], see § X 13. Examples of the type occurring in Exercise 3.1
and 3.2(a) were given by Perron [5,1] and were modified by Hartman and Wintner [11]
to obtain all the assertions of Exercises 3.2 and 3.3.
section 4. Many of the papers mentioned in connection with § 3 are relevant here
in some cases of m = 1. Most papers in the literature on the problems of this section
invoke the hypothesis S@O) = 0, _RF0) 5* 0 (i.e., deal with the cases k > Q,j = 0); see,
e.g., Frommer [1], Forster [1], Lonn [2], Grobman and Vinograd [1], and Nemytskii
and Stepanov [1]. The particular uniqueness criterion involving ip(f) in Exercise
4.1(A) is due to Lonn [1]; for other criteria, see Vinograd and Grobman [1]. Theorem
4.4 may be new. Theorem 4.5 is a result of Lonn [2]; for the case k = 2 corresponding
to D.1) of the type x' = he + p, y' = x + Xy + q as in Exercise 4.4, the number e* in
D.32) is —rfoJ/4co and part (i), but a less sharp form of part (ii) was given earlier by
l.onn [1].
I
Chapter IX
Invariant Manifolds and Linearizations
This chapter concerns the behavior of solutions of an autonomous system
(or arbitrary dimension) in the vicinity of a simple type of stationary
point or of a periodic solution. Many results to be obtained will be
extended to nonautonomous systems in the next chapter by very different
methods; cf., e.g., §1X6 and §§ X 8, 11. The lemmas of this chapter,
however, dealing with local maps from one Euclidean space to another
have intrinsic interest, give some insight which is not furnished by other
methods, and are applicable to the study of both stationary points and
periodic solutions.
1. Invariant Manifolds
For every (real) t, let V: f —>¦ f ( be a continuous mapping of a neighbor-
neighborhood Dt of f = 0 in a Euclidean f-space into a neighborhood of f = 0
in the same space, with T'@) = 0. A set S is called invariant with re-
respect to the family of maps {T*} if T\Dt n S) <= S for all t. A set S is
called locally invariant with respect to {T*} if there exists an e > 0 such
that ?eS implies that T'°? e S for all t0 for which ||r'||| < e on the
^-interval joining 0 and t0.
The problem of the behavior of solutions of a smooth autonomous
system near a stationary point can, in some cases, be viewed as the com-
comparison of the solutions of a linear system with constant coefficients
A.1) ?' = E?
and solutions of a perturbed system
A.2) ?' = E? + FQ).
Unless the contrary is stated, it will be supposed that F(i) is of class C1
for small ||||| and that
A.3)
F(i) = o(|| f ||) as ?-0;
228
Invariant Manifolds and Linearizations 229
or, equivalently,
A.4) F@) = 0, djF(O) = O
where dsF is the Jacobian matrix of F with respect to f.
Let it = r](t, f0) be the solution of A.2) satisfying the initial condition
rj(O, |0) = |0. For a fixed t, consider |, = rj(t, |0) as a map 2"':?„->-1,
of a neighborhood Dt of ? = 0 in the f-space into a neighborhood of
I = 0 in the same space. The map Tl is defined on the set Dt of points f 0
for which the solution rj(t, |0) is defined on a ^-interval containing 0 and
/. [The maps T' are the germ of a group; cf. B.2).]
A set S in the f-space which is [locally] invariant with respect to the
family of maps T* will be called [locally] invariant with respect to A.2).
S is invariant [or locally invariant] with respect to A.2) if and only if it
has the property that f0 e S implies that r\(t, f0) e Sfor all t on the maximal
interval of existence of the solution r\(t, f0) [or for some e > 0, rj(t0, f0) e S
whenever \\rj(t, |0)|| < e on the ^-interval joining 0 and t0].
If S is an invariant set, then the intersection of S and a sphere ||||| < e
is locally invariant. Conversely, if S is a locally invariant set, then So =
\JT\S n Dt) is an invariant set. Thus the investigation of invariant sets
can be reduced to the study of locally invariant sets and vice versa. This is
convenient by virtue of the following remark: If F(i) is altered outside of a
small sphere, ||||| < e, and an invariant set So is determined for the
new differential equation, then the intersection of So and the sphere
|| III < e is a locally invariant set for the original differential equation A.2).
Locally invariant sets are convenient for another reason. The condi-
conditions imposed on F are of "local" nature and it is not reasonable to expect
that invariant sets, involving notions "in the large," should be simple sets.
For example, suppose that dim 1 = 2 and that A.2) has a pair of solutions
I = li@, I2(O for - co < t < co, such that |x@, I2(O -> 0 as t -+ ± co
as in Figure 1. Then the set So consisting of I = 0 and the points | =
!/0» — °o < ? < co and y= 1,2, is an invariant set So. So is a curve
with a self-intersection. But each of the sets S±= {? = (I1, 0), \P\ < e}
or S2 = {? = @, |2), 1121 < e}, for a sufficiently small e > 0, is a locally
invariant set and is a C^-arc.
After a linear change of variables, with a constant matrix N,
A.5) ? = N?, detiWO,
the equation A.2) becomes
A.6) ?' = N^ENC + N-WiNQ.
Suppose that N is chosen so that
A.7) N-*EN = diag [P, Q],
230 Ordinary Differential Equations
where P is a d X d and Q an e x e matrix with eigenvalues pv . . . ,pd
and q±,. . . , qe, respectively, where d > 0, e ^ 0 and
A.8) Repj < a < 0 and Re ?fc ^/5 > a.
Write ? = (y, z), where y is a ^-dimensional vector, z an e-dimensional
vector and (Ar~1.E'Ar)? = (Py, 2Z)- Thus, in ^-coordinates, the lineal
equation A.1) becomes
A.9) y' = Py, z'=Qz.
The solution (y(t), z(t)) of A.9) with an initial point (y@), 2@)) = B/@), 0)
satisfies z(t) = 0 and \\y(t)\\ ^ const. tjeat < const. e(ot+e>( for some integei
Ut)
Figure 1.
j and for arbitrary e > 0 and large /; cf. § IV 5. In addition, if (y(t),«(/))
is a solution of A.9) for which \\(y(t), z(t))\\ ^ e("-?>' for some e > 0 and
large / ^ 0, then z(t) = 0. Thus the ^-dimensional flat 2 = 0 in the
? = (y, «)-space is invariant with respect to A.9) and is made up of all
solutions (y(t),z(t)) satisfying \\(y(t),z(t))\\ < e("~e)( for some e > 0 and
large /.
The first question concerning A.2) to be considered is whether or not
an analogous situation holds for A.2). More precisely, for the system
A.2) or A.6) written as
A.10) y' = Py + F^y, z), z' = Qy + F2(y, z),
where Fu F2 are of class C1 for small ||y||, ||z||,
A.11)
as (y, z) -> 0,
is there a ^/-dimensional locally invariant manifold S of the form S:z =
g(y) defined for small ||y|| which is made up of all solutions («/(/), z(t)) of
Invariant Manifolds and Linearizations 231
A.10) in a neighborhood of (y, z) = 0 for large / satisfying \\(y(t), z(t))\\ =
e(/5-e)< for some e >. o. It will be shown in § 6 that the answer is in the
affirmative.
2. The Maps T'
(i) Consider the unique solution f = rj(t, ?0) of the initial value problem
B.1) ?' = E?
Since the solution rj(t, 0) = 0 for f0 = 0 exists for all /, rj(t, f 0) exists on an
arbitrarily large interval \t\ _ /0 if |||0|| is sufficiently small; cf. Theorem
V2.1.
For a fixed /, consider |( = rj(t, 10) as a map T':?o^>- ?t from the f-
space into itself. The set of maps T' behaves like an Abelian group in the
sense that if ||fj is so small that f( = rj(t, f0) is defined on a /-interval
containing t = 0, tlt t2, and /x + t2, then
B.2) J(i+B = J'lJ'a
for the given f0, i.e., ri(tx + t2, f0) = rj(tu r](t2, f0)) since the solution of
B.1) is unique.
(ii) Consider a change of variables R: ? = Z0(i) which together with its
inverse R~x:i = X0(Q is °f class C1- Then B.1) becomes of the form
B.3) ?' = (N
where G(?) = o(||?||) as ?->-0 and N is the Jacobian matrix N =
(9.Xo/9O{-o = 8cxo(°)- In general, G(?) is not of class C1. Solutions
It = ?(*, ?0) of B.3) are, of course, unique since solutions of B.1) are
unique and the map R is one-to-one. The map ?0 —>- ?(, is given by
ylA) Kl K . L,t — L,(t, C,oj-
This can be seen by considering the action of RT'R-1; thus /?-1?0 is a
point |0 = ^(?0), T'R-1^ is the solution f( = rj(t, f0) of B.1) for fixed
|0 and RiT'R-1^) is therefore the solution ?( = ?(/, ?0) of B.3) for fixed
?o-
(iii) By Theorem V 3.1, rj(t, ?0) is of class C1 and its Jacobian H(t, f0) =
9f r] with respect to f0 satisfies the linear initial value problem
B.5) H'(t, So) = [E + dfF(v)]H0, f0), MO, f 0) = /•
In particular, if 10 = 0,
B.6) #'(*, 0) = ?//(/, 0), H@,0) = /;
thus
B.7) H(t, 0) = eEt.
232 Ordinary Differential Equations
Therefore, the expansion for rj(t, f 0), for a fixed /, in terms of linear terms
in f0 and higher order terms is of the form
B.8)
where
B.9)
r](t, f 0) = eE% + S(/, f 0),
S(t, 0) = 0 and dhZ(t, 0) = 0.
3. Modification of /"(?¦)
In order to avoid technical difficulties (as, e.g., the fact that the domain
Dt of the map T': f0 -> f ( depends on /), it will be convenient to replace
F(f) in B.1) by a function which is defined for all f, is identical with F(f)
for small |||||, say, for ||||| ^ is, and vanishes for ||||| ^ s > 0. If the
new function is called F(f) again, then the solution f = r](t, f0) of B.1)
is defined for all f. Thus, for every /, the domain of r(:f0->f( is the
entire fo-space and the set of maps T' is indeed a group.
Lemma 3.1. Let F(f) be a vector function of class C1 for small |||||
satisfying F@) = 0, 3fF@) = 0. Let d > 0 be arbitrary. Then there
exists a number s = sF) > 0 (which tends to 0 with 0) and a function G (f)
of class C1 defined for all f satisfying G(|) = FQ)for ||||| ^ ?j, G(|) = 0
/or Hill ^ j, and \\dfi\\ ^ 0/or a// f.
In this lemma, F and f need not be of the same dimension. Here and
in the remainder of this chapter, the norm \\A\\ of a rectangular matrix A
is the norm of A as a linear operator from one Euclidean space into
another, i.e., the least constant c such that \\Ay\\ < c \\y\\ for all y.
Proof. Let s>0 be so small that ||3fF(f)|| ^0/8, in particular
||f (|)|| < 0 m||/8, for m|| ^ s. Let <p(t) be a smooth real-valued function
of * for * ^ 0 such that <p(t) = 1 for t ^ (I*J, 0 < cp(O < 1 for (%sJ <
t < s2, (p(t) = 0 for t > s2 and 0 ^ -d<p/dt ^ 2/j2 for all t ^ 0. Put
G(|) = F(f)cp(m||2) or G(|) = 0 according as ||||| ^ or ^ j. Then
3fG = 0 for m|| > s. For ||?|| < s, dfi = $&&?*)<? + 2(F^0 d(p/dt and
so ||3{G|| < @/8) + 2@ m||2/8)B/j2) < 0. This proves the lemma.
Thus, in dealing with solutions of B.1) only in a small neighborhood
of ? = 0, Lemma 3.1 shows that there is no loss of generality in supposing
that F(f) is of class C1 for all f,
C.1) ||3jF(f)|| ^ 0
for all |, and
C.2) F($) = 0 for mil > s,
where j = s(d).
It will now be verified that there exist s0 = so(s, 0) > 0, 0O = do(s, 0)
such that s0, 0O -> 0 as s, 0 -> 0 and that if the solution f = >j(/, |0) of
Invariant Manifolds and Linearizations 233
B.1) is written as B.8), then
C.3) S(/, f0) = 0 for 0 ^ / ^ 1, ||?0|| ^ ^o
C.4) ||3jS(/, fo)|| ^ 0O for 0 ^ / < 1, arbitrary f0.
In order to see this, note that C.1) implies that ||F(f)|| ^ 0 m||, thus a
solution of B.1) satisfies ||f'|| ^ c0 ||||| for c0 = ||F|| + 0. Hence the
solution | = |@ of B.1) satisfies m(/)|| ^ mo|| exp (-c0/); cf. Lemma
IV 4.1. Thus, if llfoll^Jo. where so = sexpco, then \\?(t)\\ ^ s for
0 ^ / ^ 1. In this case, B.1) reduces to f = Ff, 1@) = |0 for 0 ^ / ^ 1
and the solution f@ is eE%; i.e., in B.8), S(/, |0) = 0 for 0 ^ / < 1 and
IIfoil ^*o-
The relation E(t, |0) = »?(/, |0) - eEt?0 implies that 3{()H(f, |0) =
3j,S(/, f0) = eEt[K(t, f0) - /], where K(f, f0) = e"?(H(/, f0).
The matrix K(t, |0) has the derivative X' = e~E\H' - EH) or, by B.5),
K'(t, f0) = e-?( 3fF(»,)e?(ii:(f, ?0), X@, f0) = /.
Since \\e~Et d(F(rf)eEt\\ has the bound c-fi for 0 < t ^ 1, where cx =
(eIIEHJ( it f0ii0ws from Lemma IV 4.1 that \\K(t,g)\\ has a bound of
the form exp cx0 for 0 ^ / ^ 1. Hence \\K'\\ ^ (cx0) exp cx0, and so
\\K(t, |0) - /1| ^ (cx0) exp cx0 for 0 ^ / < 1. Consequently,
||3j0S(/, f0)ll ^ em M) exp Cl0 for 0 ^ / ^ 1;
i.e., C.4) holds with 0O = el|E|l(Ci0) exp cx0.
4. Normalizations
After a linear change of variables, I = JV?, B.1) can be written as
D.1) y' =Py + F^y, z), z' = Qz + F2(y, z) and y@) = y0, z@) = z0,
where N^EN = diag [P, g]. It is supposed that the eigenvalues pp qk
of P, Q satisfy
D.2) Re/7,-^a<0 and Reft. ^ 0 > a.
The eigenvalues of the nonsingular matrices
D.3) A = ep, C = eQ
are ep<, eq«, respectively, where 0 < |ep'| < ea < 1-, |e9i| ^ e" > ea. Thus
if e > 0 is arbitrary, there exist real nonsingular matrices Nlf N2 such that
= exp (JVf^JVi) has a norm ^ ea+e and that N^C'1^ =
p
exp (N21C~1N2) has a norm ^ e"'3"'. This follows by considering the
"real" analogue of the Jordan normal forms in which the usual 1 on the
subdiagonal is replaced by an arbitrarily small e; cf. § IV 9.
234 Ordinary Differential Equations
Since diag [A, C] = exp diag [P, Q], it can be supposed that N is re-
replaced by the product N diag [Nlt N2], so that
D.4) \\A\\^ea+e, HC-
It will be supposed that e > 0 is so small that
D.5) a=\\A\\, l/c=||C-i||
satisfy
D.6) a < c, a<\.
It will also be supposed that Flt F2 are of class C1 and
D.7) F» dvFf, dtFt vanish at (y, z) = @, 0) for i=l,2,
D.8) 113,^,11,113,^11^0 for all (y,z) and i=l,2,
D.9) Fly, z) = 0 for \\y\\* + INI2 ^ s2 > 0, i = 1,2.
Correspondingly, the general solution of D.1) defines, for fixed t, a map
V from (y0, z0) to the point («/, z) = («/;, z,) such that T* is of the form
D.10) T(: y, = ePty0 + Y(t, y,,z,), zt = eQ% + Z(t, y0, z0),
where
D.11) Y, Z and the Jacobian matrices dVotZo Y, Z vanish at (yo,zo) = 0
for all t,
D.12) ||3Voy||,||32oF||,||3vZ||, ||3,Z||^0O for all yo,zo
and 0 < t ^ 1,
D.13)
y=0, Z=0
if
+
and 0 ^ t ^ 1. In D.12)-D.13), 60, s0 depend on F, s) in such a way that
0o> •*(>->0 as ^> J-»-0. Finally, the set of maps T( form a group:
5. Invariant Manifolds of a Map
One of the basic results to be proved concerns one map T: (y0, z0) ->
(yl9 Zj) rather than a group of maps TK In the application of this result,
T= T1.
Lemma 5.1. Let A be a d x d matrix, C an e x e nonsingular matrix
such that D.5), D.6) hold. Let T: (y0, z0) -»¦ (ylf zx) be a map of the form
E.1) T: 2/x = Ay0 + Y(y0, z0), zx = Cz0 + Z(y0, z0),
7, Z are of class Cxfor small ||yj, ||zo|| and satisfy D.11).
Invariant Manifolds and Linearizations 235
exists an e-dimensional vector function z = g(y) of class C1 for small \\y\\
such that
E.2) g@) = 0, 3^@) = 0
and that the maps
E.3) R:u = y, v = z — g(y) and R'1: y = u, z = v + g(u)
transform T into the form
E.4) RTR-1: ux = Au0 + U{u0, v0), vx = Cv0 + V(u0, v0),
where
E.5) U, V and their Jacobian matrices vanish at (u0, u0) = 0
and
E.6) V(u0, 0) = 0.
Condition E.6) means that the set of points (u0, v0) near the origin on
the flat v0 = 0 is invariant under the map E.4); i.e., the manifold z = g{y)
is locally invariant for E.1). In applications of Lemma 5.1, the following
two remarks will often be used. Remark 2 will be used in §§ 8-9.
Remark 1. In view of D.11) and Lemma 2.1, it can be supposed that
Y, Z are of class C1 for all (y0, z0) and satisfy D.12)-D.13), where 60, s0 are
arbitrarily small positive numbers. Let 0O satisfy
E.7) 0 < 0O <
4
2
It will be shown, in this case, that g(y) can be defined for all y [so that
RTRr1 is defined for all (w0, v0)] and E.6) holds for all u0. Furthermore,
there is a constant a = a(d0) such that
E.8)
and ct->-0 as 6o
|| dvg(y)|| ^ a < 1
0.
Remark 2. If, in addition, it is assumed that c > 1, then g(y) -*¦ 0 as
112/11 -* =o.
Lemma 5.1 will now be proved by the method of successive approxi-
approximations. Another proof will be given in Exercises 5.3 and 5.4 at the end
of this section.
Proof of Lemma 5.1 and Remark 1. Assume for a moment that R [i.e.,
g(y)] is known. Then E.1), E.3) show that
91
-1- wi = Au<>
Z(«o, »0 + ?(«,»))
- g(Au0 + Y[u0, v0 + g(u0)]).
236 Ordinary Differential Equations
Hence, by E.4)
E.10) V(u, v) = Cg(u) + Z(u, v + g(u)) - g(Au + Y[u, v + g(u)]),
and E.6) holds if and only if
E.11) g(u) = C~l{g(Au + Y[u, g(u)]) - Z(u, g(u))}.
Hence, it must be shown that the functional equation E.11) for g(u) has
a solution of class C1 satisfying E.2)—E.8).
The equation E.11) will be solved by successive approximations. Let
E.12) go(u) = 0
and, if gn_i(tt) has been defined, put
E.13) gn(u) = C-i{gn_x(Au + Y[u, g„_&)]) - Z(u, gn^(u))}.
Below let gn^ = gn_lu), gn_x = g^Au + Y°), where Y° = Y(u, g^iu)),
Z° = Z(u, gn-i(u)). It is clear that g0, gu ... are defined and of class
C1 for all u. In addition, if dgn is the Jacobian matrix of gn, then
E.14) dgn = c-^idgl^iA + dvY° + (dj^idgn-j]
- [dvz° +
where, e.g., dyY° = dvY(y,z) at (y, z) = (u, g^u)).
Define the number a by
00
E.15)
a =
so that 0 < a < 1
c - a - 30O
by E.7). It will be shown by induction that
E.16) ||3^n(«)|| <a for all u.
It is clear that E.16) holds for n = 0. Assume that E.16) holds when n is
replaced by n — 1. Then, by E.14) and a < 1,
llfo.ll ^ c~l{a[a + do + doa] + [d0 + 60o]} ^ c~l[a(a + 30O) + fl0].
Since c~1[a(a + 30O) + 0O] = a, E.16) follows and the induction is
complete.
It will now be verified that dg0, dgx,... are equicontinuous. For any
function / = /(«) or f = f(y, z), let A/ = /(« + Aw) -f(u) or A/ =
/B/ + Ay,z + Az)-f(y,z). Put
E.17) Ax(«) = sup \\Ady>z Y, Z\\ for ||Ay||, ||Az|| ^ «,
where dVt% Y, Z means any of the four Jacobian matrices dvY, dzY, duZ,
dzZ of Y(y, z), Z{y, z). It will be shown by induction that
E.18) UAdsJ ^ h(d) for ||A«|| ^ 6 < 1,
I
Invariant Manifolds and Linearizations 237
where
E.19)
c - a - 460
It is clear that E.18) holds for n = 0. Assume its validity if n is replaced
by n — 1. Note that, by E.16),
hence
E.20)
\\A[Au
where the last two inequalities follow from D.12) and E.7). Using the
analogue of A[fiiu)f2(u)] =f1(u + Aw) A/2 + (A/J/^ii) and a < 1, it
follows from E.14) that if ||Am|| ^ 3 < 1, then
E.21)
20O]
4d0)
where the right side is h{8) by E.19).
Next, it will be shown that the sequence g0, glt.. . converges uniformly
on every bounded w-set. This is true if there exist constants M, r such that
0 < r < 1 and for n = 1, 2,...,
E.22) \\gn(u) - gn^(u)\\ <: M ||u|| r".
This inequality holds for n = 1, if M and r are chosen subject to Mr = a.
Assume the validity of E.22) when n is replaced by n — 1. By E.13),
c llg»(«)-g»-i(«)ll is at most
\\gn_x(Au + Y[u,gn_J) - gn_2(Au + Y[u,gn_2])\\
+ l|Z(«,?n_1)-Z(!i,sn_8)||.
The first term is majorized by
||?n_1U«+ Y[u,gn_1])-gn_2(Au+ Y[u,g*-M
+ ll^»-^« + Y [u, gn_J) - gn_2(Au + Y[u, gn_2])\\.
Hence c \\gn(u) - ^n_i(«)|| is not greater than
M \\Au + Y(u, gn_M rn~l + od0M \\u\\ rn~l + 60M \\u\\ rn~\
which is at most Mrn~x \\u\\ (a + 40O). Thus, if r = (a + 40o)/c and
M = ajr, then E.22) holds and r < 1 by E.7).
Consequently, g{u) = lim gn{u) exists uniformly on every bounded
w-set. In view of E.13), this limit function g(u) satisfies the functional
equation E.11). Finally, since the sequence dgo,dgY, ... is uniformly
bounded and equicontinuous, there is a subsequence which is uniformly
238 Ordinary Differential Equations
convergent on every bounded w-set. It follows that g(u) is of class C1.
This completes the proof.
Proof of Remark 2. Let M = max ||g(«)|| for ||w|| < s0. By D.13) and
E.11),g(w) = C^giAu) if \\u\\ ^ soandg(u) = C-ng(Anu) if WA^uW ^ s0.
Thus if \\An-lu\\ ^ s0 but ||y4n«|| < s0, then ||#(m)|| < Mcn. This implies
Remark 2 since c > 1 and, for large ||w||, there exixts a unique integer
n = n(u) satisfying \\A
Hi\\ ^
\\Anu\\ and n(u) ->¦ oo as
oo.
Exercise 5.1. (a) This part of the exercise concerns variants of Lemma
5.1 under various smoothness assumptions on Y, Z. Instead of the assump-
assumption that Y, Z is of class C1 and satisfies D.11)—D.12), suppose that
Y, Z satisfies one of the following hypotheses: (i) Y = 0, Z = 0 at
(y, z) = 0 and Y, Z is uniformly Lipschitz continuous with an arbitrarily
small Lipschitz constant for small ||y||, ||z|| [i.e., if e > 0 is arbitrary, then
||Ay|| + ||AZ|| ^ e(\\Ay\\ + ||Az||) for sufficiently small (||y||, \\y + by\\,
\\z\\, ||z + Az||)]; (ii) Y, Z is of class Cm, 1 ^m^ oo, and satisfies D.11);
(iii) Y, Z satisfies (ii) with 1 < m < oo and its partial derivatives of order
m have a degree of continuity majorized by a (constant times a) monotone,
non-negative function hm(d) ->¦ 0, <5->- + 0; (iv) Y, Z are analytic and
satisfy D.11). Then the analogue of Lemma 5.1 holds with a g{y)
having the corresponding property (i), (ii), (iii) or (iv), instead of being of
class C1.
(b) Verify that if Flt F2 in D.1) have the analogues of property (i), (ii),
(iii) or (iv), then Y= Y(t,yo,zo), Z = Z(t,yo,zo) in D.10) have the
corresponding property (i), (ii), (iii) or (iv) with respect (y0, z0) uniformly
forO<; t < 1.
Exercise 5.2. Show that the restriction a < 1 in Lemma 5.1 is not
needed if Y, Z are uniformly Holder continuous. [Note that the condition
a + 20O < 1 was used in the proof only in connection with E.20), E.21).
This difficulty can be avoided if hx{8); hence h(S), can be chosen to be
constant multiple of <5e for a sufficiently small e > 0.]
Corollary 5.1. Let T,g(y), 0O be as itiLemma 5.1 and Remark 1 following
it. For a given (y0, z0), put (yu zx) = T(y0, z0), (y2, z2) = T(ylt zx),....
Then, on the one hand, z0 = g(y0) implies that \\(yn, zn)\\ = O((a + do)n)
as « ->- oo (in fact, if D.2), D.3) hold and y0 ^ 0, then yn ^ 0 for all n,
HzJI/ll2/JI -*¦ ° °nd lim sup n~x log \{yn, zn)|| ^ a. as n ^- co); and, on the
other hand, z0 ^ g(y0) implies that (c - 2do)n = O(\\(yn, zn)\\) as n -> oo.
Remark 3. If c > 1 (so that a < 1 < c) then the manifold z = g(y)
in a neighborhood of (y, z) = @, 0) can be described as the set of points
(y0, z0) such that (yn,zn) = Tn(y0, z0) satisfy \\(yn, zn)\\ -+0 exponentially
as n—*- oo and/or ||(yn, zn)|| ->¦ 0 as «->¦ oo and/or (yn, zn) remains in a
neighborhood of @,0) for n = 0, 1, 2 In the case a < 1 < c, the
manifold z = g(y) is called the stable manifold of E.1) as n^- oo; the
Invariant Manifolds and Linearizations 239
corresponding manifold for n ->¦ — oo is called the unstable manifold of
E.1).
Proof of Corollary 5.1. Note that z0 = g(y0) is equivalent to v0 = 0.
In this case, d0 = dx = • • • = 0 by E.4), E.6). Correspondingly,
un = Aun_x + ?/(«„_!, 0), so that ||«n[| < (a + 0O) ||mm_1|| and ||mJ| ^
(a + 60)n || Mo || -+ 0 as n->- oo. Thus, if e > 0 is arbitrary, there is an
N = N€ such that ||«J| < (a + e) ||mot_1.|| for n ^ JV" and ||Mn+iV|| <
(a + c)" ll%ll for n ^ 0. Since yn = Mn, zn = g(un) = o(||Mn||) as n ^ oo,
it is seen that \\(yn, zn)\\ ^ A + a) \\un\\ and the first assertion follows. Also
lim sup n-1 log ||(yn, zj|| ^ log a. Since a linear transformation of the
2/-variables can bring log a arbitrarily near to a, it follows that lim sup n~x
By virtue of E.10), we have the relation
dvV(u, v) = dzZ(u, v +g(u)) — dg(Au + Y[u, v + g(u)]) dzY(u, v + g(u)\
so that \\dvV\\ ^ 0O + a60 ^ 20O, and by E.6), || V(u, v)\\ ^ 20o||d||. Thus,
vn = Cvn_x + V(un_x,vn_d implies that ||Dn|| ^ (c - 20O) ||t;M_1.|| or ||dJ| ^
(c - 2flo)n || foil - Also \\(yn,zn)\\ ^ ||(iin,on)|| - \\g(un)\\ ^ A - a) \\(un,vn)\\
which implies the last assertion.
Theorem 5.1. In the map T: f 0 —*• f 1(
E.23) J: fx = rf0 + S(f0),
fe/ H(|o) ^e of class Cxfor small ||fj anJ ja/;^y H@) = 0, 9f S@) = 0;
F a constant, nonsingular matrix having d, e0, e eigenvalues of absolute
value less than 1, equal to 1, greater than 1, respectively, where d, e0, e > 0.
Then there exists a map Rofa neighborhood of f0 = 0 onto a neighborhood
of the origin in the Euclidean (u0, v0, wo)-space such that R is of class C1
with a nonvanishing Jacobian, and RTR~X is of the form
E.24)
u1 = Auo+ U(u0, d0, w0),
RTR-1: Wl = Bw0 + W(u0, v0, w0),
Vx = Ct'o + V(UO, Do, W0),
A, B, C is a square d x d,e0 x e0, e x e matrix with eigenvalues of
absolute value less than 1, equal to 1, greater than 1, respectively; U, V,
W and their partial derivatives vanish at the origin; and
E.25)
E.26)
V = 0, W = 0 if d0 = 0, w0 = 0,
U = 0, W = 0 if u0 = 0, w0 = 0.
The condition E.25) [or E.26)] means that the plane d0 = 0, w0 = 0 of
dimension d [or u0 = 0, w0 = 0 of dimension e] is a locally invariant
manifold. When F has no eigenvalues of absolute value 1, so that dim
So = d + e, then the variables w0, Wj are absent in E.24).
vt =
V(t, u0, v0),
240 Ordinary Differential Equations
Proof. (Details will be left to the reader.) By Lemma 5.1, there is a
map Ro: f„ ~~*" (wo> vo> wo) °f class C1 with nonvanishing Jacobian such
that if RoTRfT1 is given by the right side of E.24), then E.25) holds. If
Lemma 5.1 is applied to (RoTR^1)-1 = R0T^-1R0-\ a new map Rt results
and the desired map R in Theorem 5.1 is given by R = RxR0.
The results of this section are applicable to differential equations by
virtue of the arguments used to obtain the following corollary of Lemma
5.1 and Corollary 5.1.
Corollary 5.2. Let D.10) be a group of maps T* of class C1 for all
{y0, z0) satisfying D.11) and D.12) for 0 ^ / ^ 1, where P, Q are constant
matrices such that D.2), D.3), D.5), and D.6) hold, and 60 satisfies E.7).
Let g(y) be the function furnished by Lemma 5.1 and the Remark 1 following
it when T = T1. Then RVRr1 is of the form
E.27) RT'R-1: ut = ePtu0 + U(t, u0, v0),
where
E.28) V(t, u0,0) = 0 for all /, u0.
Furthermore, ifyo^O and z0 = g(y0), then zt = g(yt)for all t, yt ^ Ofor
all t, \\zt\\I\\yt\\-*-0 and lira sup I'1 log \\yt\\ ^ a. as t ^ co; ifzo^g(yo),
then (c - 2doy = O(||B,(, z()||) as t -+ oo.
If c > 1 > a, a remark similar to that following Corollary 5.1 is
applicable here.
Proof. It will first be verified that if n ^ / ^ n + 1, then there exist
positive constants clt c2 such that
E.29) Cl\\{yn, Zb)|| ^ \\(yt, z()|| ^ c2\\(yn, Oil-
In order to see this, note that T* = T'-nTn. Thus D.10) and D.11) for
0 ^ / — n ^ 1 give
E.30) \\yt - ep(t-n)yj ^ 60(\\yJ + \\zj)
and a similar inequality for zt. These inequalities imply E.29).
Let z0 = g(y0), then the behavior of (yn, zn) for large n is described by
Corollary 5.1. The last part of E.29) gives lim sup r1 log \\(yt, zt)\\ ^ a
as t —>¦ oo. Suppose, if possible, that zt ?? g(yt) for some /, say / = t0,
then (c - 2do)n = O(\\(yn+h, zn+t<)\\) as n ^ co by Corollary 5.1. But this
is a contradiction. Hence zt = g(yt) for all /; i.e., vt = 0 for all / so that
E.28) holds.
Note that if yt = Ofor some /, then zt = g(yt) implies ||zj ^ a \\yt\\ = 0
for the same /. But then (yt, zt) = 0 for all / by the group property of T'.
The remaining assertions of Corollary 5.2 follow from those of Corollary
5.1 and from E.29).
The following two exercises give another proof of Lemma 5.1 based
on the methods of §§ X 8-10 for nonautonomous differential equations.
Invariant Manifolds and Linearizations 241
The main part of the proof is Exercise 5A(b), which leads to a com-
comparatively simple proof of Lemma 5.1 because it deals with maps of the
form Tn = Sn° Sn_t ° ¦ ¦ ¦ ° Su where Sk depends on k, rather than with
Tn = Ta.. . o r. cf parts (^ and (e) of Exercise 54
Exercise 5.3. (a) For n = 1,2,..., let f —>¦ Sn? be a continuous map
of the I = (I1,. . . , ?d) space R = Rd into itself and Tn = Sn° Sn_t ° • • • °
St. Let S be a compact and Kt, K2, . . . closed f-sets such that Sn(R —
Kn_^ c R - Kn, and Kn n Tn(S) is not empty for n = 2, 3, Then
there exists a point f 0 e S such that TJ0 e Kn for n = 1, 2, . . . . (b) Let
? be a nonsingular constant d x d matrix, F(g) a continuous vector-
valued function for all | such that F(?) = 0 for large || |||. Then ?->¦ E? +
F{?) maps the f-space onto itself (i.e., the equation Eg + F(g) = r\ has at
least one solution f for every r\ e Rd).
Exercise 5.4. (a) Let A, C be matrices as in Lemma 5.1. For n = I,
2,. . ., let Yn(y, z), Zn(y, z) be continuous functions for all (y, z) which
which vanish for large ||y|| + ||z||. Let Sn denote the map
S«: (y, z)^(^y + Yn(y, z), Cz + Zn(y, z)),
Tn the map Tn = Sn° Sn_x ° • • • ° Slt and P the projection P(y, z) = z of
the (y, z)-space onto the z-space. For a fixed y0, show that
is a map onto the z-space. (b) Let Yn, Zn satisfy
||rn(y,z)||,||Zn(y,z)|| ^(\\y\\ +\\z\\)d,
where 0 < 46 < c — a. Let K be the cone ||z|| ^ ||y|| and, for a fixed
2/0, S the sphere S = {(y, z): y = y0, \\z\\ ^ ||yo||}. Show that Sn(R - K)
<=¦ R — K and that there exists a z = z{n) such that P ° Tn(y0, z(n)) = 0,
hence (y0, z(n)) e S and K n T^S) is not empty. Consequently, there
exists a (y0, z0) e ,S such that (yn, zn) = Tn(y0, z0) e K for n = 0, 1, . . . .
(c) Show that if (yn, zn) = Tn(y0, z0) e K for n = 0, 1, . . . , then ||zj| <
II2/JI < (a + 2d)n H2/0II , where a + 26 < |(« + c); but if
(y«> zn) = 2"n(yo, zo) t K
for some n (hence for all large n), then, for large n,
lly.ll > Kll ^ (const.)(c - 2S)n > 0, where c - 26 > J(a + c).
(d) In addition to the conditions of parts (a), (b), and (c), assume that
fi, = r,, Zn satisfy
for all y, z, 2/*, z*. In terms of a sequence (yn, zn) = Tn(y0, z0), introduce
the maps '
Sn*-- (y, *) — (Ay + Yn*(y, z), Cz + Zn*(y, z)) for n = I, 2,. . . ,
242 Ordinary Differential Equations
where
Yn*(y, z) = Yn(y + yn,z + zn) - Zn(yn, zn);
Zn*(y, z) = Zn(y +yn,z + zn) - Zn(yn, zn);
so that if (yn*,zn*) = Tn(y0*,z0*), then Sn*(yn*-»„,*„*-«») =
GC+i - Vn+i> z*+i ~ ZJ- Show that »o in part (b) uniquely determines
z0; in fact, if (yn, zn), (yn*, zn*) e K for n = 0, 1 then
Ik* - z»ll ^ ll»«* - y.ll ^ « = o, l
(e) Assume the conditions of (a), (b), (c), (d) and, in addition, that Yn(y, z),
Zn(y, z) are of class C1. Let z0 = g(«/0) be the unique z0 in (b). Show that
g(«/0) is of class C1 [and that the partial derivatives of g vanish at y0 = 0
if the partials of yn,Zn vanish at (y,z) = @,0)]. In fact, let T** =
S** ° 51**! o .. . o s**, where S** is the linear map
(ii, v) - (Au + (dyYn)u + (dzYn)v, Cv + (dvZn)u + CZZ»
with the Jacobian matrices dVfZYn, Zn evaluated at (yn-V zn-i); let e3- =
@,. . . , 0, 1,0 0) be the vector with the kth component ef = 0 or
e/= 1 according as k^j or k=j. Then (un,vn) = (dyn(yo)ldyoj,
dzn(y^jdyoj), n = 0, 1 exists and is the unique sequence satisfying
(«„, vn) = T**(u0, v0), u0 = e» and ||cn|| ^ ||iin|| for n = 0, 1 (/)
Deduce Lemma 5.1 from part (e) with the choice Sx = S2 = • ¦ ¦ = T.
6. Existence of Invariant Manifolds
A consequence of Corollary 5.2 is the following:
Theorem 6.1. In the differential equation
F.1) ? = E?-
let F(f) be of class C1 and F@) = 0, dsF@) = 0. Let the constant matrix
E possess d (>0) eigenvalues having negative real parts, say, dt eigenvalues .
with real parts equalto ol{, where ax < • • • < ar < 0andd± + • • • + dr = d,
whereas the other eigenvalues, if any, have non-negative real parts. If
0 < e < — ar, then F.1) has solutions f = ?(t) ^ 0 satisfying
F.2) IIf(Oil e<Lt -*0 as t-+co
and any such solution satisfies
F.3) lim r1 log 111@11 = <x( for some i.
Furthermore, for sufficiently small e > 0, the point f = 0 and the set of
points ? on solutions |@ satisfying lim n1 log |||@ll ^ <*i for a fixed
i [or lim sup r1 log |||@ll < 0] as t -»¦ oo constitute a locally invariant
C1 manifold S{ [or Sr] of dimension d1 + • ¦ • + dt [or dx -\ • + dT = d].
Invariant Manifolds and Linearizations 243
It will be clear that the proof has the following consequence.
Corollary 6.1. Let F(g) be of class C1 for small |||||, F@) = 0, and
d(F@) = 0. Then F.1) has a solution ? = ?(t) & 0 satisfying F.2) for
some OOi/ and only if E has at least one eigenvalue with negative real
part.
For another proof of Theorem 6.1 and a generalization to nonautono-
mous systems, see §§ X 8, 11.
Proof. If "lim" is replaced by "lim sup," the last part of Theorem 6.1
follows from the normalizations of § 4 'and from Corollary 5.2 with
a = ai» /? = «i+i [or a = ar, /? = 0]. This argument also shows that, as
lim inf r1 log |||@|| < ai+1 implies that lim sup r1 log |||@H ^ a,,
for i = 1 r with ar+1 interpreted as 0. Hence lim sup r1 log || |@H =
a.{ for some i implies that lim inf = lim sup.
Remark. We have similar results for the solutions f(/) ^ 0 satisfying
111@11 e~€t-+0 as /->— oo. This follows by replacing / by the new
variable —/, so that F.1) becomes
and applying Theorem 6.1 to this equation.
The arguments used to obtain Theorem 6.1 and Corollary 5.2 give
Theorem 6.2. Let E, F(f) be as in the last theorem. In addition, let E
have e(>0) eigenvalues with positive real parts. Let ?t = ?(/, f0) be the
solution of F.1) satisfying ?@, f0) = |0 and V the corresponding map T':
?t = ?(t, f0). Let € > 0. Then there exists a map R of a neighborhood of
? = 0 in the ?-space onto a neighborhood of the origin in the Euclidean
(u, v, w)-space, where dim u = d, dim v = e, dim u + dim v + dim w =
dim |, such that R is of class C1 with a nonvanishing Jacobian and RVR
has the form
ut = e lu0 + U(t, u0, v0, w0),
F.4) RT'R-1: wt = ep^w0 + W(t, u0, v0, w0),
vt = eQtv0 + V(t,uo,vo,wo);
U, V, W and their partial derivatives with respect to u0, v0, w0 vanish at
(u0, v0, w0) = 0. Furthermore, V = 0, W = 0 if v0 = 0, w0 = 0; and
U = 0, W=0ifu0 = 0,w0 = 0; finally \\ep\\ < 1, ||e-°|| < 1 and the
eigenvalues of ePo are of absolute value 1.
It can be remarked that the change of variables R:? -> (u, v, w) trans-
transforms F.1) into differential equations of the form
F.5) u' = Pu + F±(u, v, w), w' = Pow + F2(u, v, w),
v'=Qv+ F3(u, v, w),
244 Ordinary Differential Equations
where Fu F2, F3 are o(\\u\\ + \\v\\ + \\w\\) as (u, v,w)^0 (but Flf F2, F3
need not be of class C1); F2 = 0, F3 = 0 if v = 0, w = 0; and Fx = 0,
F2 = 0 if u = 0, w = 0.
The condition V = 0, W = 0 when u0 = 0, w0 = 0 [or U = 0, W = 0
when u0 = 0, w0 = 0] means that the ^-dimensional plane v0 = 0, w0 = 0
[or e-dimensional plane m0 = 0, w0 = 0] are locally invariant manifolds.
When E has no eigenvalues with real part 0, then the variables w, w0 are
absent; in this case, the manifold u0 = 0 [or v0 = 0] that consists of the
solution arcs which tend to 0 as / ->¦ oo [or / ->¦ — co] is called the stable
[or unstable] manifold of F.1) through f = 0.
7. Linearizations
In the differential equation
G.1) ? = E? + F{?),
suppose that no eigenvalue of E has a vanishing real part. The remarks
concerning F.5) suggest the question as to whether or not there is a C1
change of variables .R: f —>¦ ? with nonvanishing Jacobian in a neighborhood
of | = 0 which transforms G.1) into the linear system
G.2)
?' =
in a neighborhood of ? = 0. In general the answer is in the negative if
dim | > 2; see Exercises 7.1 and 8.1-8.2. A discussion of this problem
is given in the Appendix of this chapter.
Exercise 7.1. Let f, r\, ? be real variables and consider the system of
three differential equations:
?' = of, V = (a - y)n + ef?, ?' = -y?,
where a > y > 0 and e ^ 0. Show that there is no map R: (f, r\, ?) ->
(m, v, w) of class C1 with nonvanishing Jacobian from a neighborhood of
(|, ^, ?) = 0 onto a neighborhood of (u, d, w) = 0 transforming the given
differential equations into the linear system
u = a.u, v' = (a — y)v, w' = —yw.
See Hartman [21].
For a topological, rather than a C1, map J?, we have:
Theorem 7.1. Suppose that no eigenvalue of E has a vanishing real part
and that F(f) is of class C1 for small |||||, F@) = 0,3fF@) = 0. Let
T':?t = r)(t, |0) andL*:^ = eE( ?„ *e the general solution of G.1) and G.2),
respectively. Then there exists a continuous one-to-one map of a neighbor-
neighborf ? 0 / / RT'R1 L(
/«
py
hood of ? = 0 on/o a neighborhood of ? = 0 rac/j //w/ RT'R-1 = L
particular, R:? -^-? ma/M solutions o/G.1) near I = 0 on/o solutions of
G.2) preserving parametrizations.
Invariant Manifolds and Linearizations 245
Thus the topological structure of the set of solutions of G.1) in a
neighborhood of f = 0 is identical with that of the sblutions of G.2) near
? = 0. This is no longer true if some eigenvalues of E have vanishing
real parts. For in this case, G.2) has closed solutions paths arbitrarily
near ? = 0, but G.1) need not have closed solution paths near f = 0;
cf. Exercise VIII 3.1. Theorem 7.1 will be proved in § 9.
8. Linearization of a Map
Instead of the problem of linearizing a group of maps T\ the correspond-
corresponding question involving one map Twill be considered first.
Lemma 8.1. Let A, C be non-singular constant matrices, where A is a
d x d matrix, C an e x e matrix, and
(8.1) a = \\A\\ < 1 and \\c = ||C!! < 1.
Let T:(y0, z0) -> (yu zx) be a map of the form
(8.2) T: yx = Ay0 + Y(y0, z0), zx = Cz0 + Z(y0, z0),
where Y, Z are functions of class C1 for small \\yo\\, \\zo\\ which vanish
together with their Jacobian matrices at (y0, z0) = 0. Then there exists a
continuous, one-to-one map
(8-3) R:u = Q>(y,z), v =Y(y,z)
of a neighborhood of (y, z) = 0 onto a neighborhood of (u, v) = 0 such that
R transforms T into the linear map
(8.4)
RTRr1 =
= Au0, vx = Cv0.
Remark. In view of Lemma 2.1, it can be supposed that Y, Z are of
class C1 for all (y0, z0) and satisfy D.12)-D.13), where 0O, s0 are arbitrarily
small positive numbers. It will be shown in this case that if 60 > 0 is
sufficiently small, then R can be chosen so that it is a continuous, one-to-one
map of the (y, z)-space onto the (u, i>)-space, that <&(y, z) = y andTB/, z) =
z for large ||^|| and ||z||, respectively, and that (8.4) holds for all (u0, v0).
These facts will be useful in the proof of Theorem 7.1.
The following exercises give positive and negative results concerning
the existence of linearizing maps R which are smoother than (8.3) in Lemma
8.1; see also the Appendix.
Exercise 8.1. (a) Using the example T: x1 = ax, y1 = ac(y + exz),
z1 = cz, where 0 < c < 1 < a, ac > 1, and e > 0, show that if R is any
linearizing map, then R and R'1 are not of class C1. See Hartman [21].
(b) Let dim y = dim u ^ 2. Let the map T:y1 = Ay + Y(y) be of class
C2 for small ||y||; Y and its first order partials vanish at y = 0; A is a
246 Ordinary Differential Equations
constant matrix having no eigenvalue of absolute value 0 or 1. Show that
there exists a map R:y -*¦ u of class C1 of a neighborhood of y = 0 onto
a neighborhood of u = 0 with nonvanishing Jacobian such that RTR~X is
the linear map RTRr1:^ = Au. See Hartman [25, Part 3]. (c) Using the
example T: xx = e2x + y2, ft = ey, where x, y are real variables and
e > 0 is fixed, show that R in part (b) cannot always be chosen of class C2
even if Tis analytic and \\A\\ < 1. See Sternberg [3, p. 812].
Exercise 8.2. Contractions, (a) Let dim y = dim u = d be arbitrary.
Consider a map T:yx = Ay + Y(y), where A is a nonsingular d X d
matrix such that \A\ < 1 and Y(y) is of class C2 for small ||y|| with
y@) = 0, dy y@) = 0. Show that there exists a map R:y-+u of a neigh-
neighborhood of y = 0 onto a neighborhood of m = 0 such that R@) = 0,
/? is of class C1 and has a nonvanishing Jacobian, and RTR'1 is the linear
map RTR-1: ux = Au. See Hartman [20]. [Note, that by Exercise 8.1 (c),
there may not exist an R of class C2 even if Y{y) is analytic in y.] (b) In
part (a), let ax,. .., a.d be the eigenvalues of A and suppose that (*)
a3. ?? a^a™2.. . a™" for any set of non-negative integers (mx,.. . , md)
satisfying 1 < ~Lmk ^ n, where n is an integer such that n > log |a}-|/log
K| for 1 ^j,k < d. Let Y(y) be of class Cn. Then ? in part (a) can be
chosen of class Cn. Also, if (*) holds for all sets of non-negative integers
(mx,. . ., md) satisfying ~Lmk > 1 and Y(y) is of class C°° [or analytic],
then R can be chosen of class C°° [or analytic]. See Sternberg [3] and
Appendix to this chapter.
Proof of Lemma 8.1. (a) Since A is nonsingular, there exists a number
ax < 1 such that
\\Ay\\ ^ ax \\y\\ for all y; e.g., ljax = M~i||.
Let (yx + Aft, zx + Azx) = T(y0 + Ay0, z0 + Az0). Then
|| Aft - AAyo\\,
so that
(8.5) H^Aft - Ai/
(8.6)
By (8.6), T is one-to-one if ax — 20O > 0 and, by D.13), it is onto the
G/i> Zi)-space; cf. Excercise 5.3(b). Thus Thas a C1 inverse of the form
||Aft|| + IIA^H ^ (fll - 20o)(||At/o|| +
ft, 2,),
Zx(yx, zx),
(8.7) T 1:y0 = A *ft +
where
(8.8) Yx, Zx and their Jacobian matrices vanish at (ft, zx) = 0.
Invariant Manifolds and Linearizations 247
By (8.5)-(8.6),
(8-9) H^^Zill^i for all (ft, zx),
if 6X = flo/a^ai - 20O). Finally, by D.13)
(8.10) Yx = 0, Zx = 0 if Ilftf+H^ll2^^2
if sx = sja. Thus 71, 71 satisfy analogous sets of conditions.
In what follows, it will be supposed that 0O > 0 is so small that
(8-11) 0o<|minBa, 1 - a, c - 1),
Bx < i minB/c, 1 - 1/c, l/fll - 1).
(b) The map (8.3) will first be determined so as to satisfy
(8.12) LR=RT,
i.e., the relations
(8-13) A O(y, 2) = <&(^y + y(y, 2), Cz + Z(ft z)),
(8.14) CY(y, z) = W(Ay + Y(y, z), Cz + Z(y, 2)).
It will be shown that the functional equation (8.14) has a continuous
solution T(y, z) defined for all (y, z). To this end, define the successive
approximations
(8-150) Y0(ftz) = z,
(8.15n) Yn(y, z) = C^Yn_x(Ay + Y(y, z), Cz + Z(y, 2))
for « = 1, 2, It is clear that Yn(t/, z) is a continuous function for all
(y, 2). These functions satisfy
(8-16) Wn(y,z) = z if ||z||^0
for n = 0, 1, .... This follows by a simple induction since Z{y, z) = 0
if ||2|| > s0 and ||Cz|| ^ c||z|| ^ ||z||.
In order to prove the convergence of these approximations, define
(8.17) Y»(y, z) = Yn(y, z) - Yn_x(y, z) for « = 1, 2, . ..,
so that
(8-18) YKy, 2) = C^Zfo, 2),
(8.19) Y-(y, z) = C-^-i^y + y(ft z), Cz + Z(y, 2))
for « = 2, 3, .... It will be verified by induction that there exist positive
numbers r (< 1), M, d such that
(8.20) ||T"(ft 2)|| < Mrn(||i/|| + ||z||)' for all ft z.
248 Ordinary Differential Equations
Since ||Ylto, 2)|| = (floAOfllvll + INI) f°r lltfll + M = 2*« and
= 0 for ||i/|| + ||z|| = 2i0, it is clear that (8.20) holds for n = 1 if
0 < E < 1 and Mr = Q0{2s^-sjc. Assume that (8.20) holds if n is
replaced by n — 1. Then (8.19) shows that
Since c > 1, it is possible to choose <5 > 0 so small that (c + 0o)*/e < 1.
Hence (8.20) holds for all n if r = (c + do)s/c and Mr = 60Bs0I-ilc.
Consequently, Y(t/, z) = lim Tn(y, 2), w->-oo, exists uniformly on
bounded (y, z)-sets, and satisfies
(8.21) Y(y, z) = z for ||z|| = *„
and (8.14). Similarly, there exists a continuous function <t>(y, z) for all
(y, z) satisfying
(8.22) Oft/, z) = y for ||y|| = ^
and (8.13). For, in view of (8.7), the functional equation (8.13) can be
written
yx + Yx(yx, zx), C~\ + Zx(yx, zx)) = A-^SXylt zx)
which is analogous to (8.14) since both \\A\\ and HC!! are less than 1.
(c) It next will be shown that R is one-to-one. Suppose the contrary,
so that there exists a pair of points (t/10, z10), B/20. Z2o) such that
but (t/10, z10) ?* B/20. Z2o)-
(8.23) R(yxo, z10) = -RB/2o. ^20)
For « = 0, ± 1,. .. and; =1,2, put Tn(yj0, zi0) = (yin, zjn). The relation
(8.12) and inductions for n _ 0 and for n _ 0 show that
(8.24) R(yln, zln) = R(y2n, z2n) for n = 0, ± 1,...,
and since T is one-to-one,
(8.25) {yln, zln) * (y2n, z2n) for n = 0, ±1, ....
Corollary 5.1 associates a manifold S+:z = g(y) with Tand a manifold
S~:y = h(z) with T~x, where S± are of class C\ have only the origin in
common, and are transversal there. Suppose first that (y10, z10) e S+;
i.e., z10 = ^(t/10). Then (yln, zln) ^ 0; hence R{yln, zln) -» 0 as n -» 00.
It follows that 0/ao, Z20) e -5+. otherwise \\(y2n, z2n)\\ -»- 00 as «-»- 00, by
Corollary 5.1. But (8.21), (8.22) lead to the contradiction \\R(yln, zXn)\\ =
ll*(yi«. Z2M)II ^ 00 as n ^ 00. Thus A/^, z^0) e S+ for j = 1, 2 and so
(»*.2*) = 0 or ||to*»,2*«)ll ^ oo as n^ -00 [since S+ n 5~ contains
only (y, z) = 0]. But zjn = g(yin) is bounded by Remark 2 following
Lemma 5.1. Hence \\(yiKi zln)\\ ->¦ 00 implies that ||^B|| -»- 00 asn -»¦ — 00.
Invariant Manifolds and Linearizations 249
Here, (8.22) and (8.24) show that yln = y2n for large — n and also
tin = gto/m) = g(V2n) = Hn- This contradicts (8.25).
Hence if (8.23) holds, then {ylo,zjO)$S+ for j = \, 2. Similarly,
B/30. zjo) $ S~ for y =1,2. But then
(8.26) ||z3.J|^oo as «^oo for j =1,2.
This follows from the proof of Corollary 5.1 which shows that vjn =
zin ~ g(yin) satisfies H^JI = (c - 2d)n \\Vj0\\ * 0. By Remark 2 following
Lemma 5.1, however, g{y) is bounded for all y, so that (8.26) holds.
Similarly
(8-27) ||y,n
00
as n —*¦ — oo
for j= 1,2.
Thus the enumeration of the points (yin, zjn) for n = 0, ±1, ... can be
changed so that ||y,0|| _ sx forj =1,2. Hence (8.22) and (8.23) show that
(8-28) yxo = y20, z10 ^ zw.
Define the numbers
(8.29) an = ||yln - y2n\\, ft, = ||zln - z2n\\
for n = 0,1,. . ., so that a0 = 0, /30 j* 0. Thus the case n = 0 of
(8-30) an = /*„
holds. Assume that (8.30) holds if n is replaced by n — 1. By (8.2) and
the fact that {yln, zjn) = T(yjn_x, zin_J, it follows that
=
=
=
=
This gives (8.30) for all n = 0. The relations (8.21), (8.24), and (8.26)
imply that zXn = z2n for large n i.e., ft, = 0 large n. By (8.29) and (8.30),
it follows that an = ft, = 0 for large n. This contradicts (8.25) and shows
that R is one-to-one.
(d) R maps the (y, z)-space onto the («,u)-space. In fact, since R is
continuous and one-to-one, its restriction to a compact set C, R :C^>-R(C),
is a homeomorphism. Thus, by Brouwer's theorem on the invariance of
domain, R maps open sets onto open sets. (For a proof of this theorem of
Brouwer, see Hureiwicz and Wallman [1, pp. 95-97].) Consequently, the
/{-image of the (?/, z)-space is open. In order to see that it is also closed,
consider a sequence {yx, zx), (y2, z2),. . . such that lim R(yn, zn) = (u0, v0)
exists. Then the sequence (yx, zx), (y2, z2), ... is bounded, for otherwise it
follows from (8.21), (8.22) that either the u- or u-coordinate of R(yx, zx),
^CV2» zd> ¦ ¦ ¦ is not bounded. Since this contradicts R{yn, zn) —>- (u0, v0)
as n -* 00, there is a point (y0, z0) satisfying R(y0, z0) = (u0, v0).
250 Ordinary Differential Equations
(e) Thus the map R has continuous (single-valued) inverse R denned
on the entire (u, u)-space, and so (8.4) is a consequence of (8.12). This
proves the lemma.
9. Proof of Theorem 7.1
In the proof of the theorem, it will be supposed that E has tf eigenvalues
with negative real parts and e eigenvalues with positive real parts. (The
case when d = 0 or e = 0 is easier and is contained in the general case by
the addition of dummy components to f.) After preliminary normaliza-
normalizations, it can be supposed that T* is of the form D.10) for all (t, y0, zQ),
where D.11) holds, D.12) and D.13) hold for 0 ^ t <i 1, and A = ep,
C = eQ satisfy the conditions in Lemma 8.1.
It will also be supposed that 0O 's s0 small that Lemma 8.1 and the
Remark following it are applicable to T = T1. Denote by Ro the corre-
corresponding map supplied by Lemma 8.1, so that R^Rq1 = L. Put
(9.1)
Then
(9.2)
R - P™
Jo
TS ds.
-* ds\ r
If s — t is introduced as a new integration variable, say s, the last integral
becomes
In the first of these integrals, the integrand can be written as
L-SROTS = L-1-3R0T°+1
since L = RoJ^Rq1. Thus (9.2) becomes
(9.3) L*R = ( I L-SROTS ds) T* = RT.
Thus, in order to complete the proof it is sufficient to verify that R is
one-to-one and maps the (y, z)-space onto the (u, u)-space. To this end,
write R and Ro as
(9.4) R: u = U(y, z), v = V(y, z); R0:u = U0(y, z), v = V0(y, z),
where
(9.5) U0(y, z) = y if ||y|| ^ slt V0(y, z) = z if ||:|| ^ s0.
Invariant Manifolds and Linearizations 251
It follows from (9.1) and the validity of D.13) for 0 ^ t ^ 1 that there
exist positive constants s2, s3 such that
(9.6) U(y,z) =
if
^ s3, V(y, z) = z if
The arguments of parts (c) and (d) of the last section and the case
t = 1 of (9.3) show that R is one-to-one and maps the (y, z)-space onto
the (u, u)-space. Thus Theorem 7.1 follows from (9.3).
10. Periodic Solution
Lemmas 5.1, 8.1 will now be applied to the study of solutions in the
neighborhood of a periodic solution of an autonomous system:
A0.1)
r =
Lemma 10.1. Let /(?) be of class C1 on an open set containing f = 0.
Let | = rj(t, f0) be the solution o/A0.1) satisfying rj(O, f0) = fo- Suppose
that y(t) = t)(t, 0) is periodic of least periodp > 0. [Thus y(t) ^ const.,
f(y(t)) 7* 0 and rj(t, f0) exists on an open t-interval containing [0,p] if
|| foil is sufficiently small.] Let n be the hyperplane -n: ?•/(()) = 0 orthogonal
to the curve (?:? = y(t) at f = 0. Then there exists a unique (real-valued)
function t = t(|0) of class C1 for small ||fo|| such that t@) =p and
rj(t, f0) e 77 when t = t(|0); i.e.,
A0.2)
= 0.
Roughly speaking, if a solution starts at f0 near 0, then at a time
7 = r(f0) near /?, the solution meets the hyperplane tt ; see Figure 2.
Proof of Lemma 10.1. This is an immediate consequence of the implicit
function theorem. The equation t)(t, f0) -/@) = 0 is satisfied if t = p,
f0 = 0. Also the derivative rj'(t, f0) "/@) =/(>?(?, f0)) "/@) at f0 = 0
is/G@) -/@). At t =p, it becomes |/@)|2 * 0 since y(/>) = y@) = 0. This
gives the lemma.
If we consider small ||fo||, |0 g tt, then
A0.3) T: |j = rj(r(i0), |0)
is a map from one neighborhood of f = 0 on tt into another. The meaning
of T is clear; the solution f = r\(t, f0) starts for t = 0 at f0 gtt and
^! = T(?o) is the first point (t > 0) where the solution f = 77G, f0) again
meets n. The applicability and consequences of Lemmas 5.1, 8.1 will be
considered. Roughly, we can expect the following type of results: On the
one hand, if the Jacobian matrix of this map at f0 = 0 has a norm less
than 1, then f = y(t) is orbitally asymptotically stable (in the sense that
if ?° is sufficiently near a point of the curve 1?:i = y(t), then the solution
252 Ordinary Differential Equations
f = t)(t, f °) "tends" to <€ as t -» oo, i.e., is in an arbitrarily small neighbor-
neighborhood of *€ for large i). On the other hand, if dim f = m and the Jacobian
matrix at f0 = Oofthismaphas onlyc?« w — 1) eigenvalues with absolute
value less than 1 and (m — 1) — d with absolute value greater than 1, then
the set of f0 e tt for which f = j?(f, f „) "tends" to eS, as f -> oo, constitutes
a ^-dimensional manifold S.
nit,
In order to calculate the eigenvalues of the Jacobian matrix of the map
T:?o -*¦ fi at f0 = 0, consider f0 arbitrary for a moment (i.e., not subject
to f0 e - The matrix #(f, f0) = dif)rj(t, f0) satisfies
A0.4) tf'(*, fo) = 3*/fo0. &))#('. fa), H@, f0) = /•
In particular, for f0 = 0,
A0.5) H'(t, 0) = dJ(,Y(t)Wt, 0), fl@, 0) = /.
Note that H(t, 0) is a fundamental matrix of A0.5) and that the coefficient
matrix, d(f(y(t)), is periodic of period p. It follows from the Floquet
theory in § IV 6 that H(t, 0) has a representation of the form
A0.6) H(t,0) = K(t)eDi, where K(t + p) = K(t)
is a periodic matrix function and D is a constant matrix. In particular,
T@) = /? and #(/>) = K@) = I imply that
A0.7) H(ri0),0) = H(p,0) = eDv.
The characteristic roots (= eigenvalues) eu e2, . . . , em of the matrix
H(p, 0) = zDp are called the characteristic roots of the periodic solution
? = y(t). Note that ^ ^ 0 since H(p, 0) is nonsingular. Correspondingly,
/7 log ex, /7 log e2,. . . are called the characteristic exponents, so that
only the real parts of the characteristic exponents of f = y(t) are uniquely
Invariant Manifolds and Linearizations 253
defined. The characteristic exponents, modulo 2-ni, are eigenvalues of D
(and D is not unique).
When | is subjected to a linear change of coordinates, I = Nt, it is
readily verified that H(t, 0) is replaced by NH(t, O)^, so that the set of
characteristic roots are not changed.
Lemma 10.2. Let /(f) be as in Lemma 10.1, dim f = m, and T the
map f0 -*¦ fj in A0.3), w/jere f0, fx e tt. Le? e^ . .., em be the characteristic
roots of i, = y(t). Then one of these, say, em is 1 and elt . . ., em_1 are the
eigenvalues of the Jacobian matrix of the map T at f0 = 0 e -n. In fact, if
the coordinates in the ?-space are chosen so that
A0.8) /@) = @, . . . , 0, 1)
and 77: fm = 0, then the last column of H(p, 0) is @, . . . , 0, 1) and the
(m — 1) X (m — 1) matrix obtained by deleting the last row and column
from H(p, 0) = eDp is the Jacobian matrix of the map T at f0 = 0.
Proof. It will first be verified that H(p, 0) satisfies
A0.9) H(p,0)f@)=f@);
i.e., X = 1 is an eigenvalue of H{p, 0) and/@) is a corresponding eigen-
eigenvector. Note that y'(t) =f(y(t)). If this relation is differentiated with
respect to t, it is seen that f = y'(t) is a solution of linear initial value
problem
A0.10) > f' = dj(y(t))i, 1@) = /@) =/@).
Since H(t, 0) is a fundamental matrix for this linear system reducing to /
at t = 0, y'(t) = H(t, 0)/@); cf. § IV 1. For t = p, this relation becomes
A0.9).
Thus, if A0.8) holds, then
A0.11) the last column of H(p, 0) is @,. . . , 0, 1).
The Jacobian matrix of the map A0.3), without the restriction ?0 e tt, is
d^M^ol fo)] = V'(r, f0) d^r + H(t, f0).
At f0 = 0, this becomes
3ft,fottfo), fa)]ft,-o = /@) ^0t@) + H(p, 0).
The first term on the right is the matrix (/'(O) 9t@)/9|00, so that the
first m — 1 rows are 0 by virtue of A0.8). The last term is H(p,0).
Consequently, the lemma follows.
11. Limit Cycles
The remarks of the last section make it clear that, in a study of the
solutions of A0.1) near f = y(t), the real parts of the (nontrivial)
254 Ordinary Differential Equations
characteristic exponents of f = y(t) play a role similar to that of the real
parts of the eigenvalues of E in a study of solutions of A.2) near f = 0.
Theorem 11.1 Let fig) be of class C1 on an open set and let A0.1)
possess aperiodic solution f = y(t) of (least)periodp > 0. Let dim f = m
awrf /e? ?Ae raz/ /xzrta of m — \ characteristic exponents of f = y(?) 6e
negative, say, less than a < 0. TAen ?Aere exw/s1 a 5 > 0 a«c? a constant L
with the property that for each f ° o« ?Ae o/?e« se? dist (f °, ^) < 5, wAere
^"i | = y(?), 0 _ ? ^/?, there is an asymptotic phase t0 such that the
solution f = r\(t, |°) o/
A1.1) f =
sa?/s/zes
A1.2)
= 0.
In particular, f = y(f) w a //w/7 cjc/e awe? is asymptotically, orbitally stable.
Proof. As before, let y@) = 0. The assumptions make it clear that
after a linear change of the variables f, it can be supposed that the map
A0.3) is of the form
S(f) f0 e tt,
where S(f0) is of class C1 for small ||fo|| and vanishes together with its
Jacobian matrix at f 0 = 0 and tnat a = Mil < *"• Thus> if IIfoil is
sufficiently small, say, ||fo|| < e, then HfJ = e« ||fo||. Hence fn = Tnf0
is defined for «= 1, 2,... and ||fB|| < e™ ||fo|| -* 0 as « -* oo.
Since the solutions of A1.1) are unique, it follows from Theorem V 2.1
that if e > 0 is arbitrary, then there exists a d = de > 0 with the property
that if dist <&, f °) < 6, then there exists a least positive value t = t°(|°)
such that r?t, f°) exists for 0 = r ^ r° and >?(t0, f°) e tt, ^(A f°)|| < e.
Let fo = ^(-r0' f°) e w- Put t1 = T(fo) + t° and, for n = 2, 3, . . .,
rn = T(fn_!) + t"-1, so that ?„ = *?(rn, |°). Clearly, r@) =/> implies
that r(f J ->/» and Tn/«/? -> 1 as n -> oo. Actually, ?0 = lim (t" — «/>)
exists as « -*¦ oo and there is a constant Lx such that
A1.3) \rn - (np + to)\ < Lie"n ||foil •
In order to see this, note that
- (n - l)p)\ = HSnr.0 -P\-
Since t@) =p and t(|) is of class C1, kd^j) —/>| is majorized by
Lo llfn-ill = -V1"1 IIfoil for a suitable constant Lo. Thus the existence
of a tl satisfying A1.3) with Lx = L0/(l - e") follows.
Since ij(r, 1°) is of class C\ \\V(t + rn, |°) - y(OH = ||i#, O -
>?(?, 0) | = L2 ||fB|| ^ I^e*" II foil for some constant L2 and 0 = / ^ />.
Invariant Manifolds and Linearizations 255
for 0 < / = p.
Also, the boundedness of /(f) in a vicinity of # implies that \\t)(t +
rn, f°) - V(t + np + t0, fO|| < Ije™ ||fo|| by virtue of A1.3). Hence
\\V(t + np+ t0, f°) - y(OII = (L2 + L3)exn |
If t + np is replaced by t, it follows that
l|i?0 + *o, f°) - 7@11 ^ (i, +
for «/? ^ ? < (n + \)p and t = 0, 1,. . . . This proves the theorem.
Theorem 11.2. Let dim f = m,f(?) be of class C1 on an open set, and
let A0.1) possess a periodic solution f = y(t) of least period p > 0. (i) 7%e«
f/jere exwtt a solution f = f@ ^ y(?) defined for large t and satisfying
A1.4) dist(«',KO)ert-*-0 as t^oo,
/or some e > 0, w/jere <€:? = y(t), 0 < t ^ p, if and only if ? = y(t)
has at least one characteristic exponent with negative real part, (ii) If
f = y(t) has exactly d (^ m — 1) characteristic exponents with negative
real parts, then the set of points f near <? on solutions f = f(f) o/A0.1)
satisfying A1.4) /or some e>0 constitutes a (d + \)-dimensional C1
manifold S. (iii) 7/" a? /eas? owe characteristic exponent has positive real
part, then f = y(?) « «of orbitally stable, (iv) If0<d<m—lin (ii) awrf
(w — 1) — rf characteristic exponents have positive real parts, then S can
also be described as the set of points f near <? on solutions ? =
satisfying
A1.5) dist (*", f@)-*¦ 0, as t-
andjor
oo
A1.6)
dist
for 0 = ? < oo
/or some sufficiently small e > 0.
It will be clear from the proof and from Exercise 5.1 that the C1 assump-
assumption on/and assertion on S can be weakened somewhat or strengthened
either to analyticity or to Cm, 2 _ m ^ oo. In case (iv), there is, of
course, an analogous (m — cO-dimensional manifold corresponding to
f-> — oo which intersects 5 transversally along (€; S and the corre-
corresponding (m — J)-dimensional manifold are called the stable and unstable
manifolds ofS.
Proof. Only the proof of (ii), when 0 < d < m — 1, will be indicated.
It can be supposed that y@) = 0 and that A0.8) holds. After a linear
change of variables in n [i.e., in the (f1, .. ., f "-^-subspace], it can be
supposed that T in A0.3) is of the form E.1), where Y, Z are of class C1
for small ||yj, ||zj, and Y, Z and their Jacobian matrices vanish at
(.Vo. 20) = 0. Here dim y0 = d, (y0, z0) = (f1,. .., fm-i), a = \\A\\ < \
and \/c = || C!! < 1 + 0, where 0 > 0 is arbitrarily small. Let y = g(z)
256 Ordinary Differential Equations
be the manifold supplied by Lemma 5.1. Then Tn(y0,z0) is defined for
n = 1, 2, ... and || 7^(t/0, zo)|| tends to 0 exponentially as n -* oo if and
only if z0 = g(y0).
If || fo|| is small and f0 e tt, there is a (t/0, z0) such that f0 = (t/0, zo> 0),
where the last 0 is the real number 0. Consider the set of f-points given
by I: | = t](t, f0), where f „ = Oo, giVo), 0) g tt and 0 < ? ^ r(|0)-
It will be left to the reader to verify that the subset S of 2 in a small
open vicinity of ^ satisfies the assertion of the theorem.
Remark. From Lemma 8.1, we can deduce the topological nature of
the set of solutions of A0.1) near <? when the real parts of the nontrivial
m — 1 characteristic exponents of f = y(t) are not 0.
Exercise 11.1. Under the conditions of Theorem 11.2(i), let f(f) be a
solution of A0.1) satisfying A1.4) for some e > 0. Show the existence of
numbers t0 and c> 0 such that U(t + t0) — y(t)\\ ect-*0 as t ->¦ oo.
Theorem 11.3. Let fig) be of class C1 ow aw o/?ew set and such that A0.1)
possesses a periodic solution f = y(?) of (least) period p > 0. Pm?
A1.7)
= f "tr
Jo
ds.
If A > 0, f/jew | = y(?) is not orbitally stable. If dim 1 = 2 awe? A < 0,
then f = y(?) is exponentially, asymptotically, orbitally stable.
Proof. For arbitrary m = dim f, A0.5) implies that
det H(t, 0) = exp tr dj(y(s)) ds;
Jo
cf. Theorem IV 1.2. Thus
A1.8) det H(p, 0) = eA,
so that eA is the product exe2. . . em. When A > 0, then \ek\ > 1 for some
k and when m = 2, it can be supposed that e2 = 1 and ex = eA. Thus
Theorem 11.3 follows from Theorems 11.1, 11.2.
Exercise 11.2. Let dim f = 2,/(f) be of class C1 on a simply connected
open set E, and tr 9|/(f) = div/(f) ^ 0. Then A0.1) does not possess a
periodic solution (^ const.) nor a solution f = ?(t), — oo < t < oo, such
that f(± °°) = lim f(?) as t —*¦ ± oo exist, are equal, and f(± oo) e E.
APPENDIX: SMOOTH EQUIVALENCE MAPS
12. Smooth Linearizations
As pointed out earlier, Theorem 7.1 and Lemma 8.1 become false if
"the continuity of R and R~v' in the assertions is replaced by the assertion
thai 'R, R'1 are of class C1." This and the following two sections concern
Invariant Manifolds and Linearizations 257
the existence of smooth linearizing maps R, i? under additional
hypotheses.
Theorem 12.1. Let n > 0 be an integer [or n = oo]. Then there exists
an integer N=N(n)^2 [or N = oo] with the following properties:
If T is a real, constant, non-singular d X d matrix with eigenvalues yx,..., yd
such that
A2.1) Yr:
¦¦Y7d
^yk for k = 1, . . ., d and 2 ^ 1 m^ N
for all sets of non-negative integers mx,. . . ,md and if, in the map
A2.2) T: Sx = rf + 3A),
3(D is of class CN for small ||f|; satisfying 3@) = 0, 9{S@) = 0,
f/jere exwta a map R: t, = Zo(|) of class Cn for small
A2.3) Z0@) = 0, 3fZo(O) = /,
A2.4)
Note that A2.1) implies that \yk\ ^ 1 for k = 1,.. ., d for, since T is
real, its eigenvalues are real-valued or occur in pairs of complex conjugates
e.g., if \yx\ = 1 and y2 = yx = \\y2, then yxy22 = yx). When T* is the
"group" of maps associated with the differential equation G.1) and Tin
A2.2) is T = T1, then T = eE; so that if ex,. .., ed are the eigenvalues
of E, then y3- = exp et and A2.1) is replaced by mxex + • • • + mded ^ ek.
Exercise 12.1. Formulate an analogous theorem concerning the
linearization of the differential equation G.1) and prove it by using
Theorem 12.1 and the device involving (9.1) in obtaining Theorem 7.1 from
Lemma 8.1.
Remark. The proof of Theorem 12.2 supplies a choice of N(n) or,
equivalently, of X = N — n (probably, far from the best choice): Let
0 < a < a < 1 be such that the eigenvalues of Y satisfy 0 < a < min
(ly*l> l/ly*l) < a < 1 for jfc = 1,. . ., rf. (In particular, in a suitable
coordinate system, the norms of V, P are less than c = I/a; furthermore
P = A or P = diag [A, B], where the norms of A and B~x satisfy a < \A\,
\B~l\ < a.) Then N can be chosen to be n + X if X is an integer such that
" ' 1 t. a (d+ n - l)l
1, where An * = ^—! —
vn+2
n\(d- 1)!
is the number of partial derivatives of order n of a function of d variables;
cf. A4.29) and steps (/) and (m) in the proof of Theorem 12.2 in § 14.
Theorem 12.1 will be obtained from a more general result, Theorem
12.2. The fact that Theorem 12.1 is contained in Theorem 12.2 is implied
by the following lemma which depends on a simple formal calculation.
258 Ordinary Differential Equations
Lemma 12.1. Let N ^ 2 be an integer [or N = oo]. Let Y be a real,
constant, nonsingular matrix with eigenvalues satisfying A2.1) and let
S(f) be of class CN for small ||f|| satisfying 5@) = 0, 9jS@) = 0. Then
there exists a map Rx: ? = Zx(?) of class CN for small ||f|| satisfying
Zx@) = 0, 9^@) = /; and S^l) zw
A2.5) t^^i^ + s^o,
where Tx = R^R^1, has the property that all partial derivatives of SX(Q
of order ^ N vanish at ? = 0.
A generalization of Theorem 12.1 involves the "equivalence" of two
maps T, Tx without the assumption that either is linear:
Theorem 12.2. Let A2.5) be a map of class CN for small ||f||, where
2 ^ N ^ oo aw<5? S^O) = 0, SjE^O) = 0, a«<5? suppose that the eigenvalues
yk ofY satisfy \y\k ^ 0, 1. Let n > 0 be an integer. Then there exists an
integer X = X(n) > 0 depending only on n and T with the following property:
IfN ^ n + X(n) and Tin A2.2) is of class CN such that all partial derivatives
of E(f) — Sid) of order ^ N vanish at f = 0, r/zew r/zere ex/ste a map
R: ? = Z0(f) o/c/om Cyo' jma// ||f || satisfying A2.3)
A2.6)
= Tx.
In the proof of this theorem, the definition of R will not depend on n,
thus R is of class Cn for every applicable n. In particular, if N = oo, then
ReC°. Also, the proof will show that for a given « > 0, the assumption
that 5, Sx are of class C* can be relaxed to the assumption that S, Sx
are of class Cn+1 and that each partial derivative of E(f) — Sid) of
order A: ^ « + 1 is majorized by const. ||f II*0"* for a fixed No ^ « + A(«),
in which case /? e Cn is such that each partial derivative of R? — f of
ordery ^ w is majorized by const. ||f H*0^.
Lemma 12.1 will be proved in the next section and Theorem 12.2 in
§ 14. The proof of the following theorem which is the analogue of
Theorem 12.2 for differential equations depends on a simple modification
of the proof of Theorem 12.2 and will be left as an exercise; see Exercise
14.1.
Theorem 12.3. In the differential equation
A2.7) C' = EC + F^Q,
let FtiQ be of class CN for small ||?||, where 2 ^ N ^ oo and F^O) = 0,
d^i@) = 0, and suppose that no eigenvalue of E has a vanishing real part.
Let n > 0 be an integer. Then there exists an integer X = X(n) > 0
depending only on n and E with the following property: If N ^ n + X(n)
and F(?) in
A2.8) f = ?f )
Invariant Manifolds and Linearizations 259
is of class CN such that all partial derivatives of jF(f) — FX(D of order
^ N vanish at 1 = 0, then there exists a map R: ? = Z0(f) of class Cn
for small ||f || satisfying A2.3) and transforming A2.7) into A2.8).
The remarks following Theorem 12.2 on the smoothness of S, Ex have
analogues concerning the smoothness of F, Fx. In particular, the analogue
of the last part of the remark concerning R implies the following result on
asymptotic integration:
Corollary 12.1. Under the conditions of Theorem 12.3, there is a one-to-
one correspondence between solutions ?(t) of A2.8) and solutions ?(?) =
R?(t) o/A2.7) satisfying g(t), ?(*)->¦ 0 as t ->¦ oo [or — oo]; furthermore
U(t) - ?@H ^ const. ||?@||* as t^ co [or -oo].
Exercise 12.2. (a) Let T and Tx in A2.2) and A2.5) be maps of class
C00 for small ||f || such that the eigenvalues yk of T satisfy \yk\ ^ 0, 1 and
that E@) = 0, 9jS@) = 0, S^O) = 0, djS^O) = 0. Let To, T10 denote
the (not necessarily convergent) Taylor expansion for Tf, T^f at the origin.
Then there exists a map R: ? = Zo(|) of class C00 for small ||||| satisfying
A2.3) and 7\ = RTR-1 if and only if there exists a formal power series
map Ro: ? = f + • • • such that formally T^o = ^o^o- (This assertion
is a consequence of Theorem 12.2 and Lemma 13.1) The question of the
existence of Ro depends on the solvability of certain linear equations;
cf., e.g., the proof of Lemma 12.1. (b) Formulate the analogue of (a)
in which the maps T, Tx are replaced by the differential equations A2.7),
A2.8).
13. Proof of Lemma 12.1
Case 1 B _ N < oo). After a suitable linear change of variables, it
can be supposed that F is in a form similar to a Jordan normal form
except that the subdiagonal elements are 0 or € > 0, where € will be
specified below; cf. § IV 9. Thus the transformation A2.2) can be written
in terms of components as follows:
A3.1) T: tt = yg + e,** + I... I c{^{i) + o(U\\N),
wherey = 1,. . ., d and e} is 0 or €. In A3.1) and below, (/) represents a
<5?-tuple (/i,.. ., id) of non-negative integers; |z| = ix + • • • + id; and
f (i> is the product f"> = (f1)**.. . (?*)'«.
The map Rx will be determined so that each component of Rx? is a
polynomial in (f1, . .., ?*), say,
A3.2) Rx:V=^+l-..lr\m)^m) for j = 1, . . . , d.
2SHSJV
The object is to determine Rx so that R^R^g = Tf + o(||||hv), or,
equivalently,
A3.3) UxTf = TR^ + o(U\\N) as f —0.
260 Ordinary Differential Equations
Up to terms of order o(||f H*), they-th component of the left side of A3.3) is
yf + e,^1 +2 4^> + 2rim)U(yJ*+ ej^ + J, c^f,
(j) (m) a-1 (j)
while that of the right side is
(m)
(m)
Thus A3.3) is equivalent to
||
- n
+
+1 <»>«"') J
Comparing coefficients gives a linear system of equations for r(m).
It is easy to see that in view of A2.1) these linear equations uniquely
determine the numbers /fm) for j = 1,. . ., d and 2 _ \m\ ^ N if e1 =
¦ ¦ ¦ ea = 0. In fact, the main part of the factor of r(m) on the left (i.e., the
part of lowest order in f) is (ys — U >C)f(m), so that we can successively
determine r(m) first for \m\ = 2, then \m\ = 3, etc. This shows that if
e1 = ¦ ¦ ¦ = ed = 0, then the determinant of the matrix of coefficients of
the unknown r(m) is not zero.
It follows that if e > 0 is sufficiently small, where e3- is 0 or e, then the
matrix of coefficients is nonsingular. This proves Lemma 12.1 when
2 = N< oo.
Remark. For the purpose of treating the case N = oo, note the
following corollary of the proof just completed: if 2 ^ N < oo, then
there exists a unique map R1 of the form A3.2) satisfying the conclusions
of Lemma 12.1. Actually, the proof shows that Rt is unique after a certain
linear change of variables (which leaves the form A3.2) unchanged) and
hence is unique before the linear change of variables.
Case 2 (N = oo). Since T is of class C00, it has a formal (not neces-
necessarily convergent) Taylor expansion at f = 0;
f()f(" for 7 = 1 d,
where P = (yjk). The proof just completed and the remark following it
show that there is a unique formal power series map
A3.4)
such that
t(m)
|m|S2
= H^ formally.
for j = 1, ...,d
Invariant Manifolds and Linearizations 261
In order to complete the proof, grant the following fact for a moment:
Lemma 13.1. Corresponding to any formal power series
A3.5)
2, r
|m|=0
<»»>
with real coefficients, there is a function r(f) of class C00 having A3.5) as its
formal Taylor development.
For if Lemma 13.1 holds, a desired C00 map /?! is obtained as Rx? =
(r\?),..., /¦"(?)) in which rJ(f) is of class C00 and has the formal Taylor
series on the right of A3.4). Thus in order to complete the proof of
Lemma 12.1, it only remains to verify Lemma 13.1.
Proof of Lemma 13.1. Let 0@ be a real-valued function of class C00
such that O(?) = 1 for \t\ = 1/4, 0 < 0@ < 1 for 1/4 < \t\ < 3/4, and
0@ = 0 for \t| ^ 3/4. It is easy to see that
= I r
(m)S
m
\( 2 \r{{)\)
\|<l = |m| /
A3.6)
|m|=0
is a uniformly convergent series for ||f || _: 1 and represents a C00 function
with the desired property. The (w)-th term is zero unless
in which case the (w)-th term of A3.6) is majorized by
3
II2 <
II => II ^ V^^ i - VII' ' ll " ll =^ . , ,
4m1\ . . . md\
if \m\ ^ 2. Thus the series in A3.6) is uniformly convergent for ||f || _ 1
and r@) = r@). Similarly, the series in A3.6) can be differentiated for-
formally any number of times to give a uniformly convergent series and
. . . d(id)m" = r{m) at | = 0. This proves Lemma 13.1.
14. Proof of Theorem 12.2
In what follows it is supposed that T, Tx are of class CN and that
there exists a constant cN such that
A4.1) \\Dim)(T- T?\\ ^vllfU"-"-! for small ||?||
and \m\ = 0, 1,. . . , N, where D(m) = d^jd{^)mi. . . d(^)md. In the
following only ||f || < r0 < 1 is considered, so that U\\> ^ U\\* if; > k.
In view of the normalizations of §4 and of the results of Exercise 5.1, it
is no loss of generality to suppose that T, 7\ are defined on the entire
f-space, are one-to-one, and reduce to the linear map Pf for large ||f||;
that P = diag [A, B] and f = (f_, f+); that Tis of the form
T: f,_ = AS_ + y(f_ l+), f1+ = 5f+
262 Ordinary Differential Equations
that y@, f+) = 0 and Z(f_, 0) = 0, i.e., that
A4.2) M+: f_ = 0 and M-. f + = 0
are invariant manifolds; and that there exist constants
A4.3) 0 < a < a < 1
such that
A4.4) a||f||_^||7?||_^fl||f||_, a||n||+^||f||+^
where || f± || = ||f||±. These inequalities will also be used in the equivalent
form
^ a
ll- ^ UW- ^ a \
Both sets of inequalities are illustrated in Figure 3 when dim f_ = dim f + = 1.
Figure 3.
Let #+, ^°, A;- denote the f-sets
A4.5) K+:U\\_^U\\+; K°: ||f||_
K-. U\\- >
Thus A0 is a conical hypersurface. Let Kj = T'K0 for j = 0, ± 1, . . .,
so that T~jKj= A0 and, in view of A4.4), Kj c a:+ if; ^ 0 and A3 c a;-
ify < 0. Let 2° denote the f-set between K° and A1 including K° but not
1, i.e.,
In addition, let g3' = Pg° the corresponding set between A3 and Ki+1
including K> but not Ki+1, so that
A+, T-1! ^ A"+ - A0};
as y'-»±oo,
cf. Figure 4. It is clear that
Invariant Manifolds and Linearizations 263
Figure 4.
264 Ordinary Differential Equations
that the sets Q1 are pairwise disjoint, that K+ — M+ is the union of Q1
for/ ^ 0, and that K~ - M~ is the union of Q* for/ < 0. For f $ M+ U
M~, let k(?) denote the unique integer k such that f e 2*.
For r > 0, let S(r) denote the sphere ||f|| < r, ^(r) = A± n 5(r),
#°(r) = #° n S(r), and Q\r) = G3 n 5(r). For a given r0 > 0, there
exists an rt = rjfo) > 0 such that 0 < rt < r0 and that if f e G*^) for
A: > 0 [or A: < 0], then T| e S(r0) for / = 0, 1, .. ., k [or -/ = 0,
The proof will be divided into steps (a)-(m). For brevity, most discus-
discussions will be given for f e K+; the obvious analogous statements and
discussions for f e Kr will not be given, but will be used.
(a) Ifk(?) ^ 0, then, for j = 0, . . .,,
A4.6) a'llf
A4.7) \\T-i?\\ < 2 ||T-^||+ ^ 2a3' ||f||+ < 2a> ||f||.
This is clear from A4.4) and from the fact that T~^ e K+ for/ = 0, . . . ,
*(?)¦
F) Let jfc(f) ^ 0-
A4.8)
In order to verify this, let r\ = T~m)Z, so that rj e Q° <= K+ but T*?
A;-. Then, by A4.4),
am U\\+ ^ \\T-lrj\\+,
where A: = k(?); so that A4.8) follows from H^IU < \\rj\\+ and || T^rjW^
(c) There exist constants c < 1 and k > 1 such that if k(?) > k, then
A4.9) Hr-^H <c||f||.
[In fact, c can be chosen arbitrarily close to a if k is sufficiently large.]
By A4.4) and A4.8),
||r^fll2 < a ||f||_2 + a2 U\\+2 < (orV*'** + a2) ||l||+2.
Thus if k is so large that a~2a4(C + a2 < 1, the result follows with c =
(aa4" + a2I^.
(d) Let DTifll ^C || III /or ||f|| ^ r0 anc? to r/IJ^eSW for
/= 0, 1, . . . ,h. Then
A4.10) || 7Vf - T^ll ^ C3 || | - 97II for / = 0,. . ., h.
Put ?(j) = T/l — rj3??, so that ?(/) = T^ij— 1) for / = 1,. . ., A.
Hence ||^(/)|| ^ C|K(y- 1)|| and A4.10) follows.
Invariant Manifolds and Linearizations 265
(e) Let r0 > 0 be fixed, rx = ri(r0) > 0 as before part (a). Let C ^ 1
Ae a constant such that || J?||, |17\^H < C ||f|| /or ||f|| ^ r0. Le? A, N be
integers such that
A4.11) X ^ 2, CaA < 1 and AT ^ X
and let there exist a constant cN such that A4.1) holds for ||f || < r0. Then
there exists an r2 = r2(A), 0 < r2 < rl5 wA that (i) for j = 0, . . ., jfc(!),
A4.12) llVr-^H^r! for |eX+(r2)-M+;
a«G? (ii) 7/jere exww a constant CN such that
A4.13) ||T^r-*^ - III ^ C,v Ulir for f g K>2) - M+.
Proof. Let >? e /sT+^j) and suppose that T% Txjr\ are denned for/ =
0, 1, .. ., h for some h ^ k(g). Put
A4.14) ?(/) = Tx*n - T\
Then ^(/) = TATi-ifj - V^r,] + (T, - T)V~\, i.e.,
Since ^@) = 0, an easy induction gives
i=0
Hence, by A4.1),
for j =
Choose j? = r-*<*>f, so that
A: =
By A4.7), for/ = 0, . . ., h,
K(j)\\^2»t
Hence, if Cal < 1,
so that
A4.15)
for/ = 0, ...,h.
From the definition of ?(j) and t),
hence
if ClV
= CU) +
266 Ordinary Differential Equations
This inequality, A4.7), and A4.15) make assertion (/) clear. The definition
of l(j) and the choice; = k(?) in A4.15) give A4.13).
(/) Definition of R. Let Ri0) be a map of the closure of Q° into the
|-space such that Rm is of class CN, reduces to the identity / on K° and
to 717~1 on K1, and satisfies an inequality of the form
A4.16) || D(m)(R@)? - l)|| ^ dN U\\N-lm] for m = 0, . . . , N,
small Hill, and some constant dN. Define a map R(k) on Qk, the closure of
Qk, by putting
A4.17) K(wf = TfR^T'^ for ? ? Qk
and A; = 0, ±1, .... If|^M±, put
A4.18) K? = Rik)? for ? ? Qk.
The conditions i?@) = 717 on K\ Ri0) = / on K° imply that R is con-
continuous at | ? AT1 and hence for all ? <? Af*. It will also be supposed that
Ri0) is chosen so that
A4.19)
is of class Cv for ? ? M±.
Such a choice of Rm is clearly possible; it suffices to choose i?@)? = ?
for 0 # ? near AT0, and Rw? = 717? for 0 # ? near AT1.
A possible construction of R@) is as follows: Note that if 0 # ? ? AT1 =
m then a2 ^ ||f ||_/||f ||+ ^ a2 and if 0 # ? e Q\ then a2 ^ ||f ||_/||f ||+ ^
1. Let 0 < a2 < 6l < €2 < 1, so that the set {?: 6l ||f ||+ ^ ||f ||_ ^ eJf ||+,
? # 0} is interior to g° while the set {|: a2 |||||+ ^ ||l||_^ |||||+} contains
g°. Let <D(/) be of class C00 for a2 < / < 1 such that <&(/) = 0 for
a2 < / ^ ex and $(/) = 1 for €2 < / < 1. For 0 # ? ? g*0', put
u\\
imi
and i?@)| = 0 if | = 0. Note that i?*01 = / on K°, Rw =
and
on K1
from which A4.16) readily follows,
(g) 77ie ma/> R satisfies, for ? ?
A4.20)
This is clear from A4.17) and A4.18), for if f e Qk, then 7? ? Qk+1 and
Invariant Manifolds and Linearizations
Assertion (e) remains valid if A4.12), A4.13) are replaced by
267
for
A4.21) ll^
respectively. Note that
for |?A:+(r2)-M+,
g a i|R(O)r-w^ -
The right side is majorized by 0 dN || T~k{i)^\\N in view of A4.16) and hence
by2NdN0 U\\Nak{i)N in view of A4.7). Assuming that C ^ 1, it follows
that, for; = 0, . . ., *@,
U7y.R(O)r-*(^ - T^r-^'f|| ^ 2* dN(CaN)m u\\N ^ 2N dN u\\N.
In view of (e), this makes the validity of (h) clear.
The object of the remainder of the proof is to obtain estimates for the
derivatives of i?? — ? for ? ^M±. To this end introduce the following
notation:
To avoid confusion with other superscripts, the coordinate index of a vector
will be written before, instead of after, the vector symbol; e.g.,/ = 0/ .. ., df)
and I = Cl,..., aS). For a function/^), let/, = d/^f). If (a) = (aj,.. ., ad)
is a rf-tuple of non-negative integers, write/(a) = Dial)f = d1"//^1!)ai... dCS)"*.
Let/(a) ° ??,/¦ ° t? denote the value of the corresponding derivative of/evaluated
at the point I = r\. Similarly for a map, say T, the abbreviations r,-| or Tix)$
meana(r»/aei)or(ri)(a).
It will be shown that if X = X(ri) in (e) is sufficiently large, then for a
suitable constant ClV(|a|)
A4.22) ||(*f - f)te)|| ^ C^(|a|) llff-l-"
for ? ? /T+(r2) — M+ and |a| ^ n. The proof will be by induction on |a
The relation A4.21) corresponds to the case |a| =0.
For 0 # | ? Qk, define
A4.23)
By A4.17),
hence
for 0
? Q
k+1;
or
A4.24)
where
A4.25)
for 0
268 Ordinary Differential Equations
The chain rule of differentiation gives
m=l h*
Repeated differentiation leads to a formula of the type
A4.26) '*&¦"?= I 2 (%,
where the second summation is over all ^-tuples /S = (j31;. . ., j3d) with
|/5| = |a|; 0>a^ is a product of |a| factors of the form "(T-1)^; W*'** is a
polynomial (independent of k) in the components of T1(|3) ° Rik)T~1^,
(T-%)? for |/5| <: |a| and /?$ ° T^ for |y| < |a|. Note that the poly-
polynomial in Y*1*'* does not depend on k; the dependence on k arises from
its arguments Tm ° R^T~H and R$ ° J-^.
If the second term on the right of A4.24) is written as T^T^1^ =
Jj IT-1!;, where /is the identity operator, it follows that the analogue of
A4.26) holds, i.e.,
KT.T-1)^ = 22 (%m ° T~^)(mIw ° T^"-' +I"",
where ^>a is obtained by replacing Tm ° R^T*1^ Rk(Y) ° J-1! by Tm °
J-1!, /(y) o J-1!, respectively, inT2''51'*. Consequently, A4.24) implies that
A4.27) 'S^1' = 2i + In + Y'""'*- Xfi"+ %*?
where
2i = 2 2 CTi
A4.28) t -rf|"-1
In = I 2 (Tu
m=l |^|=|«|
2 2 2 \hTlT,
ft-l m=l I/J| = |«|
wA/cA w defined for 0 # | ? 2*+1(^2) Aoj a bound L\a\ independent of k =
0, 1, .... For, if C is a bound for the first order derivatives of T^,
7-1! on Hill < r0, then |0>a^| ^ C|a| and \\Tlm° RmT-^\\ ^ C. Hence a
majorant for the sum in (i) is
A4.29) LW = C1+'^2 2 1, where 2 1 = Ku=^ I,'°j " ^ -
Remark. By considering sufficiently small r0, the number C, hence
L|a|, can be made to depend only on the norms of V and F.
Invariant Manifolds and Linearizations 269
(j) There exists a number Co such that
¦I < Co ||T-^\\N-M for 0 ^ |oc| ^ N and
^ r0.
2 2 \hF{^\<C0\\T
»-i IP|-I«I
By A4.25), a differentiation of F? gives
f,? = 2 K^i - T)m o
Repeated differentiations show that A-f(/j)l is a polynomial in AG1 — T)M°.
T~H and "(J-1)^, for |y| ^ |/5| and w = 1, . . ., d, and that each term
contains a factor of the first type. Since there exists a constant Co', such
that
for 0 ^ |y| ^ AT and small ||?||, the assertion (j) follows.
(k) There exists a number Co such that
for 0 ^ |a| ^ iV,
^ r0
and A: = 0,
If C is a bound for the second order derivatives of 7\? for small ||?||, the
mean value theorem of differential calculus shows that
( 2
M/>l-l
In view of A4.21), the assertion (jfc) holds with Co = C'C|a| rf(S l)CN.
(/) Le/ 1 ^ n ^ AT — A. Let the conditions of (e) hold and, in addition,
let X be so large that Lvax < 1 for v = 1, . . . , n, where Lv is given in
part (i). Then there exists a constant CN(\cc\) such that A4.22) holds for
? ? K+(r2) - M+ifO^ |a| ^ n.
Proof. Introduce the abbreviation
A4.30) S(^ = 2 2 lmS$fl-
m=l l/JI-n
The assertion to be proved can be cast in the form
A4.31) S*""'! = CN{n) |||||jV-" for ^ ^ n,
0#|?g\r2), and A: =0, 1, ....
The proof will be by induction on n. The case n = 0 is contained in A4.21)
in assertion (A). Assume 0 < n ^ N — X and the desired result for
0,1,..., n- 1.
It will first be shown that there is a constant CN0(n), independent of k,
such that
A4.32) |Y''"'*- ?'¦«! < C^o(n) lir-1!!!*-" for |oc| = n
270 Ordinary Differential Equations
and 0 ^ | e gfc+ifo). Note that the descriptions of?i'"-k and Y^ follow-
following A4.26) imply that the expression on the left of A4.32) is majorized by
const. { 2 HT1W • RWT^ ~ Tm ° 7^|| + 2 \\S\*\ • 7^
||
where "const." is independent of k. Since N > n, the argument used in
part (Jfc) shows that there exists a constant KN(n), depending on n but
independent of k, such that the first sum does not exceed KN(n) \\T ^|| .
Hence the existence of Cm(n) in A4.32) follows from the induction hy-
hypothesis.
In view of A4.27) and (i), (j), (k), A4.32), and A4.30),
A4.33) S<* + 1'M>? ^ LnS^T'^ + [CN0(n) + 2Col IIT-H\\N~n.
It is clear from A4.16) that there exists a constant such that
S(o,«)? ^ const. ||f||*-» for 0 # ? 6 QVi)-
It follows from A4.33) and a simple induction on k that similar inequalities
hold if S@'n), Q° are replaced by S<*n>, ?>*. Thus if k is given in step (c),
then, for the finite set of fc-values, k = 0, 1, .. . #c, there exists a constant
Cv(«) such that
A4.34) S<k'n)^CN(n)U\\N-n for 0 # I e fi*(n).
A4.35) CiV(n) ^ i 1 _ L^_n ¦
Note that, by (c), Z^c*"" ^ Z.»cA < 1 if *c is (fixed) sufficiently large.
An induction on k will now be used to show that A4.34) holds for all
fc = 0, 1, Assume that A4.34) holds for some k ^ k. Then, by
A4.33) for 0# 16 Qk+1(r2),
'I < [LnCN(n) + Cm(n) + 2C0] \\T-1^\\N-n.
But || T-1^ ^cU\\ by (c), so that the right side is majorized by
[L Cv(n) + Cvo(/0 + 2Co]^-M 11111*"". Since A4.35) implies that the
factor of M\\s-" is at most CN(n\ the inequality A4.34) holds if k is
replaced by k + 1. This completes the induction on k and also on n.
(m) Let the conditions of(e) and (I) hold. Then R can be defined on M±
so that R is of class C" on ||?|| ^ r2.
Proof. It has been shown that R, considered on K+(r2) - M+, is of
class CN and that its partial derivatives of order ^ n are bounded if
N - n^X. Without loss of generality, it can be supposed dim M+ <
d — 1 (e.g., by increasing d, by adding dummy coordinates to ?+). Thus
points of K+(r2) - M+ which are near can be joined by short rectifiable
paths in K+(r2) - M+. It follows that R and its partial derivatives of
Invariant Manifolds and Linearizations 271
order ^ n — 1 are uniformly continuous in K+(r2) — M+. Hence R has
an extension to K+(r2) of class Cn-1. This proves (m).
In view of (g) and the continuity of R, A4.20) holds on ||?|| < r2.
Since R is close to the identity for small ||?||, R has an inverse of class
C"-1. This proves Theorem 12.2. [We should remark that X depends
only on Y; cf. (c) and the Remark following (;).]
Exercise 14.1. Prove Theorem 12.3. To this end, treat the "group"
of maps T', 7/ instead of the differential equations and the problem of
finding a suitable R satisfying RT* = Txl R. For ? ? M±, define f(?) so
that 7-((*>? ? A:0. Put Rt = 7y«r-««f. The restriction of this R to
Q° gives an Rm as in (/). It is only necessary to verify A4.16); but this
can be deduced from (e).
Notes
The idea of reducing the study of the behavior of solutions of ordinary differential
equations to the study of maps is due to Poincare; cf. e.g., VII Appendix. In particular,
in connection with periodic solutions of autonomous systems, see Poincare [3, IV] or
[5, III, chap. XXVII]. In the latter context, he introduced the concept of invariant
manifolds of a map (actually, "invariant curves" in his case). Poincare's fundamental
idea of studying maps in the theory of differential equations was further exploited by
G. D. Birkhoff and has led to a large body of research associated with the name of
"dynamical systems" or "topological dynamics."
sections 5-6. In the case where y0, z0 are 1-dimensional and the map T is analytic,
Lemma 5.1 is due to Poincare [3, IV, pp. 202-204] or [5, III, chap. XXVII]. The
corresponding result where 7" is C1 is due to Hadamard [2] (who, however, did not show
that his invariant curve is of class C1) and when 7" is C1 (or as in Excerise 5.1), it is due
to Sternberg [1]. D. C. Lewis [1] extended Hadamard's method to the case of arbitrary
dimension when A, C have simple elementary divisors, carrying out the details in the
analytic case. Lemma 5.1 was stated by Sternberg in [3] where he indicated a rather
incomplete sketch of the proof based on the successive approximations E.13). The
proof in the text, using these approximations, is taken from Hartman [20] (cf. [28]).
(The proof contained in Exercises 5.3, 5.4 may be new. These exercises give a simple
proof of the assertion in Exercise 5.4F) due to Coffman [2]; a weaker existence
assertion and the uniqueness statement in Exercise 5.4(rf) are contained in Perron [13];
Exercise 5.3F) is in Coffman [2].). The analogous result for differential equations
(i.e., Theorem 6.1) is due to Lyapunov [2, p. 291], under conditions of analyticity,
and to Coddington and Levinson [2, p. 333] in the nonanalytic (and, more general,
nonautonomous) case; cf. § X 8 and the reference there to Petrovsky [1].
section 7. Theorem 7.1 with F, R, Rr1 analytic was proved by Poincare in 1879
(see [1, pp. xcix-cx]) under the assumptions that the elementary divisors of E are simple
and that the eigenvalues Xx,...,XdoiE lie in an open half-plane, Re e'eA > 0, of the
complex A-plane and (*) Xt ^ wi^i + ¦ ¦ ¦ + md\d for all sets (m1; . . . , md) of non-
negative integers satisfying mx + ¦ ¦ ¦ + md > 1. For an analogous result for smooth,
but nonanalytic F, R, R1, when Re A, < 0, see Sternberg [3]; cf. Exercise 8.2F).
For Re Xt < 0, Fe C2 and R, R-^EC1, but without Diophantine conditions of type
(*), see Hartman [20]; cf. Exercise 8.1 and 8.2(a). When it is not assumed that
A, A,, lie in an open half-plane Re e'eA > 0, but Re A,- jt 0, the problem for analytic
272 Ordinary Differential Equations
F, R, R has been considered by C. L. Siegel [2] under conditions stronger than (*).
For smooth, but nonanalytic, F, R, R *, see Sternberg [4], Ise and Nagumo [1], and
Chen [1]; cf. Appendix. Theorem 7.1 is due to Grobman [1], [2] and to Hartman [28],
[21] with different proofs.
sections 8-9. The 1-dimensional case of Lemma 8.1 (i.e., «/0 is 1-dimensional and
z0 is absent) with T, R, R'1 analytic goes back to Abel [1, II, pp. 36-39]; Schroder [1],
[2]; and Koenigs [1], [2]; see Picard [3, chap. IV]. For a similar treatment of the
1-dimensional, smooth (but nonanalytic) case, see Sternberg [2]. The case of a con-
contraction (i.e., j/0 of arbitrary dimension d and z0 missing) with T, R, R-1 analytic was
treated by Leau [1] under the assumption that the eigenvalues a,, . . ., aa of A satisfy
otj jt a™ia™2 . . . a™<i for all sets of non-negative integers (mx,. . ., md) subject to
»!, + •••+ md > 1. Lemma 8.1, as stated, is due to Hartman [21], [28].
The papers of Sternberg [3], [4] and Hartman [20], [21], [28] mentioned in connection
with § 7 are relevant here and, in fact, deal principally with maps. The device involving
(9.1) which permits the deduction of Theorem 7.1 on differential equations from Lemma
8.1 on maps is due to Sternberg [3] and is used in these related papers. Sternberg [3]
also considers the question of normal forms for RTRr1 different from the linear ones
when there are relations a, = a™ia™2 . . . a™<i and generalizes results of Lattes [1],
[2] on 2-dimensional analytic maps. For related papers, see also Sternberg [5], Moser
[1], C. L. Siegel [3], and Chen [1]; cf. Appendix. A recent important paper of J.
Moser [3] is related to the critical case not considered here when some eigenvalues of
the linear part of T have absolute value 1 and concerns the problem of the existence of
closed invariant curves.
sections 10-11. As mentioned earlier, the principal results of these sections are due
to Poincare in the analytic C-dimensional) case. Their validity in more general cases
depends on the extensions of Poincare's results on maps given in the earlier sections of
the chapter.
appendix (sections 12-14). Theorem 12.1 is due to Sternberg [4] and its general-
generalization, Theorem 12.2, to Chen [1]. (Particular normal forms other than linear ones
had also been considered by Lattes [1], [2] and by Sternberg [3].) The proof in the text
for Theorem 12.2 is based on Chen [1] which, in turn, is a modification and simplification
of Sternberg [4]. Chen's improvement consists essentially of noting that Sternberg's
procedure is valid without first using the result in Sternberg [3] to obtain a linearization
on the invariant manifolds. An admissible choice of A = N — n in Theorem 12.1 is
given by Sternberg [3] for the case of contraction maps; for n = 1 and the general case
of differential equations, see Ise and Nagumo [1]. Exercise 12.1 is due to Chen [1].
Chapter X
Perturbed Linear Systems
This chapter is concerned with methods for the asymptotic integrations of
differential equations g' = Eg + F(t, g) which can be considered as
perturbations of linear systems with constant coefficients g' = Eg.
The first section of the chapter concerns the simple but important case
E = 0. Since a very easy argument, which has wide applications, gives the
desired result in this case, it seems worth isolating it.
One of the most important methods to be used for an arbitrary E is
based on a simple topological principle, discussed in §§ 2-3. This principle
has wide applications beyond the scope of this chapter. A very different
method for obtaining results analogous to those of §§ 13 and 16 is discussed
in Part III of Chapter XII.
In this chapter, for convenience and generality, we shall allow the
components of g to be complex-valued, so that a linear change of
coordinates permits the assumption that E is in a suitable normal form;
cf. § IV 9. Correspondingly, if glt ?2 are two vectors, then g^ • g2 denotes
the scalar product ? |/|2*.
*
1. The Case E = 0
This section concerns the equation
where Fis "small" in a suitable sense. The main results are the following:
Theorem 1.1. Let F(t, ?) be continuous for t ^ 0, \\g\\ < <5(^ oo) and
satisfy
where y>(t) is a continuous function for t > 0 such that
A.3)
273
f(t)dt < oo.
Jo
274 Ordinary Differential Equations
If || fo|| is sufficiently small, say
A.4) Uo\\ exp T
Jo
dt < d,
then a solution ?(/) of (I.I) satisfying 1@) = ?0 exists for t ^ 0. Further-
Furthermore, ;/?(/) is a solution of (I.I) for large t, say t ^ /0, /
A.5)
?„ = lim ?@
«->OO
..-.j and ?«, # 0 unless ?(/) = 0.
In other words, the solutions of A.1) for large / behave like the solutions
of ?' = 0, namely, like constants. Theorem 1.1 has the following extension:
Theorem 1.2. Let F(t, ?) be as in Theorem 1.1, and let ?00 be an arbitrary
vector such that
A.6) ||?J| e:
Then A.1) has at least one solution ?(/)/or / > 0 satisfying A.5). If, in
addition, F(t, ?) satisfies the following type of Lipschitz condition
A.7) PUW-^/.WII^vMllfi-f.ll.
then for a given ?„, there is at most one solution ?(/) o/" A.1) which exists
for large t and satisfies A.5).
The last part of Theorem 1.2 states that condition A.7) establishes a
one-to-one correspondence between solutions ? = ?00 = const, of
?' = 0 and solutions of A.1), with the understanding that \\^x\\ is suffi-
sufficiently small when d < oo.
Proof of Theorem 1.1. Multiply A.1) scalarly by ?, so that A.1), A.2)
imply that
A.8) |?-?'l^v@ll?ll2.
Since d ||?(/)||2/rf/ = 2 Re ? • ?', a quadrature gives
A.9)
exp - | I y>(s) ds
^ II 1@II ^ II f(to)II exp
if ?(/) exists on a /-interval containing / and /0, where / = /0. In particular,
if t0 = 0 and ?@) = ?0 satisfies A.4), then ?(/) exists for / ^ 0.
More generally, if ?(/) exists for / ^ /0, then it is bounded,
A.10) ||f(/)|| ^ ||?(/„)II M(/o), where M(/o) = exp
Hence, A.1), A.2) show that ||f'@li ^ v@ ll^^o)!! ^Oo)- Consequently,
J
Perturbed Linear Systems 275
/) dt is absolutely convergent and so the limit A.4) exists. In fact
A.11) ||?(/) - ?J| < ||?(/0)|| M(to)j\(s) ds for / ^ /0.
Note that the inequality A.9) shows that ?(/) = 0 if and only if ?(/)
vanishes at some point /0. When ?(/) ^ 0, the first inequality in A.9)
implies that ?„ # 0. This proves Theorem 1.1.
Proof of Theorem 1.2. Consider first the existence assertion. For a
given /0 ^ 0, let ? = ?(/, /0) be a fixed solution (which is not necessarily
unique) of the initial value problem
A.12) i' = F(t,i), ?0o) = ?oo-
Since A.9) holds for any / at which ?(/, /0) exists, it follows from A.6) and
?(/0) = ix, that ?(/, /0) exists for / ^ 0. Also, ||?(/, /0)|| ^ ||f _ || M@) < d,
where M@) is defined in A.10). Hence A.2) shows that ||?'(/, /0)|| <
f(t) ||?J| M@) for all / ^ 0; thus, for 0 ^ / ^ /0,
A.13)
U(t, t0) - U\ ? M(Q)
ds.
In particular, the family of functions ?(/, /0) are uniformly bounded and
equicontinuous on every bounded /-interval. Hence there exists a sequence
/i < t2 < .. . of /0-values such that /„ —*¦ oo as n —> oo, and
|(/) = lim ?(/, /„)
exists uniformly on every bounded /-interval. Furthermore ? = ?(/)
is a solution of A.1). Putting /0 = /„ in A.13) and letting n -*¦ oo, with /
fixed, gives
A.14)
«r
V(.s) ds.
This implies A.5) and completes the existence proof.
Uniqueness will now be proved under the assumption A.7). Let
I = ?i(/), ?2@ be two solutions of A.1) for large /, say / ^ T, satisfying
A.5). Let ?(/) = fx@ - ?2(/). Then A.1) and A.7) give A.8), hence A.9)
for /0 > / ^ T. If / is fixed and /0 -*¦ oo in A.9), it follows that ?(/) = 0
since ?(/0) -»• 0 as /0 -*¦ oo. This completes the proof of Theorem 1.2.
The majorant y>(t) ||f|| in A.2) involving a factor ||?|| is convenient in
Theorems 1.1 and 1.2 only to assure that certain solutions exist for / ^ 0.
A simpler result involving existence for "large /" is given in the following
exercise.
Exercise 1.1. Let F(t, ?) be continuous on a product set {/ ^ 0} X D,
where D is a bounded open ?-set. Let F satisfy \\F(t, ?)|| ^ y@> l = 0
276 Ordinary Differential Equations
and |?Z), for some continuous function v@ satisfying A.3). (a) Let
?0 ? D. Then there exists a number T, depending only on dist (?0, dD)
and the function v@> such that if t0 ^ T, then a solution ?(/) of A.1)
satisfying ?(/0) = ?o exists for / ^ T. Furthermore, any solution ?(/) of
A.1) for large / has a limit ?«, as /— oo. F) Let ?«, ? Z>. Then there
exists a number 7, depending only on dist (?«,, dD) and the function
^(Z), such that A.1) has a solution ?(/) for / ^ T satisfying A.5).
Exercise 1.2. Show that Theorem 1.1 and the first part of Theorem 1.2
remain valid if F(t, I) is continuous for / ^ 0 and all ?, if ?0 and ?«, are
arbitrary, and if A.2) is replaced by
where y>(t) is as in Theorem 1.1 and f(r) is continuous for r ^ 0 and
satisfies
A.3')
Cx dr
= oo.
J <p(r)
Theorem 1.1 and 1.2 have corollaries for the case that A.1) is re-
replaced by
where A(t) is a continuous d X d matrix. Here, solutions of A.15) should
be compared with
Let Z(t) be a fundamental matrix for A.16), so that the change of variables
A.17) t = Z(t)S
transforms A.15) into
A.18) C=Z-\t)<Kt,Z(t)?)'
Thus an application of Theorems 1.1 and 1.2 to A.18) gives
Corollary 1.1. Let A(t) be continuous for t^Oand Z(t) a fundamental
matrix for A.16). Let G(t, Q be continuous for t ^ 0 and all ? and satisfy
A.19) \\Z-\t)G(t,Z(t)i)\\^f(t)Uh
where y>(t) is as in Theorem 1.1. Let ?(/) be a solution of A.15) on some
t-interval. Then ?(/) exists for t ^ 0,
A.20) f« = lim Z~\t%(t)
exists and ?„, # 0 unless ?(/) = 0; conversely, for a given ?„, there is a
solution t,(t) o/A.15) satisfying A.20).
Perturbed Linear Systems 277
When Z(t) is bounded for / ^ 0, we can formulate a corresponding
result when G(t, ?) is only defined for / ^ 0, ||?|| < 6 < 00. In addition,
we can obtain an analogue of the uniqueness assertion of Theorem 1.2.
When A(t) = A is a constant matrix, Corollary 1.1 takes the following
form :
Corollary 1.2. Let G(t, ?) be continuous for t ^ 0 and all ? and satisfy
A.21)
^ y>{t) \\?\\,
where y>(t) is as in Theorem 1.1. Let ?(/) be a solution of
A-22) C = At+ G(t, 0
on some t-interval. Then t,(t) exists for t ^ 0,
f, = lim e-Att(t)
exists and ?x # 0 unless ?(/) = 0; furthermore, if ?„ is given, there is a
solution o/A.22) for t ^ 0 satisfying A.23).
Exercise 1.3. Formulate theorems related to Corollaries 1.1 and 1.2
as Exercises 1.1 and 1.2 are related to Theorems 1.1 and 1.2.
Generally, a result of the type given by Corollary 1.2 is only convenient
when e±M are bounded for / ^ 0. For example, suppose that d = 2 and
A = diag [1, —1], so that eM = diag [e\ e~']. Then, if Corollary 1.2
is applicable, A.22) has a solution of the form ? = e\\ + o(l), o(l)) as
t —>¦ co, but not necessarily a solution of the form ? = e~'(o(l), 1 + o(l))
as / —> co. Furthermore the hypothesis A.21) can be very severe for the
type of conclusion stated in Corollary 1.2. The results obtained in the
remainder of this chapter are much better, under less stringent conditions,
for the situation just described.
Exercise 1.4. Suppose that A.1) is a linear homogeneous system, say
A.23)
f = G(t)?,
where G(t) is a continuous matrix for / ^ 0. The system A.23) will be
said to be of class (*) if (i) every solution ?(/) of A.23) has a limit ?x as
t —> co, and (ii) for every constant vector gx, there is a solution ?(/) °f
A.23) such that ?(/) -* f«, as t — 00. (a) Show that A.23) is of class (*)
if and only if, for one and/or every fundamental matrix Y(t) of A.23),
Y^ = lim Y(t) exists as / -»¦ oo and is nonsingular (and that this is true if
POO
and only if Yx = lim Y(t) exists as / —>- 00 and tr G(s) ds converges,
possibly conditionally), (b) The system A.23) is of class (*) if and only if
the adjoint system ?' = — G*(/)? is class (*); cf. §IV 7. (c) The system
278 Ordinary Differential Equations
fee
A.23) is of class (*) if ||G(/)|| dt < co [or, equivalently, if G(t) =
(gjlc(t)) and |gJfc(/)| dt < co for ;", k = 1,..., d]. This is merely a
consequence of Theorems 1.1, 1.2. (d) Show that (c) has the following
corollary [which is a refinement of (c)]: The system A.23) is of class (*)
fco
if G0(t) = G (s) ds converges (possibly just conditionally) and either
r\\G(t)G0(t)\\ dt < oo or p\\G0(t)G(t)\\ dt < co.
2. A Topological Principle
Let y,f be J-dimensional vectors with real- or complex-valued com-
components and/(/, y) a continuous function defined on an open (/, «/)-set D.
Let O° be an open subset of D, dD° the boundary and D,0 the closure of
O°. Recall, from § III 8, that a point (/0, y0) e D n dD° is called an egress
point of D°, with respect to the system
B.1) y'=f(t,y),
if for every solution y = y(t) of A.1) satisfying the initial condition
B.2)
y(t0) = y0,
there is an € > 0 such that (/, y(t)) ? O° for t0 — e ^ / < /0. An egress
point (/0, y0) of ?2° is called a strict egress point of D° if (/,«/(/)) <? 0° for
/0 < / ^ /0 + € for a small € > 0. The set of egress points of Q° will be
denoted by De° and the set of strict egress points by ?2°e.
If U is a topological space and V a subset of U, a continuous mapping
¦n: U—>- V denned on all of U is called a retraction of U onto F if the
restriction n | F of tt to F is the identity; i.e., tt(m) ? F for all « e C/ and
tt(u) = d for all v e V. When there exists a retraction of C/ onto V, V is
called a retract of U. This notion can be illustrated by the following
examples, which will have applications.
Example 1. Let U be a J-dimensional ball ||y|| ^ r in the Euclidean
«/-space and F its boundary sphere \\y\\ = r. Then F is not a retract of U.
For if there exists a retraction tt : [/ ->¦ F, then there exists a map of U
into itself, y—>-— Tr(y), without fixed points, which is impossible by the
classical fixed point theorem of Brouwer; for the latter, see Hureiwicz
andWallman [1, pp. 40-41].
Example 2. Let C be the "cylinder" which is the product space of
a Euclidean sphere ||y|| = r and a Euclidean w-space, so that C =
{(y> u): \\y\\ = r, u arbitrary}. Let S be a section of C, say, S =
{(V, Wo): 112/11 ^ r, w0fixed}; see Figure 1. Then 5 C\ C = {(y,u0): \\y\\ = r,
Perturbed Linear Systems 279
w0 fixed} is a retract of C [as can be seen by choosing the retraction
w(y> M) = (V> Mo)]» but 5 n C is not a retract of S by Example 1.
Theorem 2.1. Let fit, y) be continuous on an open (t, y)-set D with the
property that initial values determine unique solutions o/B.1). Let D° be
Figure 1.
an open subset of D satisfying De° = D°e; i.e., all egress points ofD° are
strict egress points. Let S be a nonempty subset ofD° U De° such that S n
D° is not a retract of S but is a retract of De°. Then there exists at least
one point (t0, y0) e S n DP such that the solution arc (t, y(i)) of B.1), B.2)
is contained in D° on its right maximal interval of existence.
/ b
/
0
\
\ -6
y
/ /
/ /
s
\ \
\ \
Figure 2.
As an illustration, consider B.1) where y is a real variable and f(t,y)
is continuous on D: (t,y) arbitrary. Let DP be a strip \y\ < b, — co <
/ < co; see Figure 2. Thus the part of the boundary of D° in Q, i.e.,
dD° n Q, consists of the two lines y = ±b. Suppose that/(/, b) > 0 and
/(/, -b) < 0, so that De° = D°se = dD° n D. Let S be the line segment
S = {(/, y): t = 0, \y\ < b}. Then S n De° is the set of two points @, ±b)
and is a retract of De° but not of S. Thus it follows from Theorem 2.1,
that there exists at least one point @, y0), \yo\ < b, such that a solution of
B.1) determined by y@) = y0 exists and satisfies \y(t)\ < b for / ^ 0.
Proof of Theorem 2.1. Suppose that the theorem is false. Then for
Oo, J/o) e S — iie°, there exists a t1 = t^, t/0) such that t1 > t0 and the
280 Ordinary Differential Equations
solution y(t) of B.1), B.2) exists on t0 ^ / ^ tu (t, y(t)) ?ii° for t0 ^
/ < ^ and (/l5 y(t$) ? ii,,0 for / = tx. Define a map tt0 : 5 ->¦ ii/ as follows:
"oOo, y0) = On V(h)) if Oo, Vo)gS- &e° and 7ro(/o, «/o) = (/0, y0) if
Oo>2/o) e 5 n ii/. Since the solutions of B.1) depend continuously on
initial conditions (Theorem V2.1) and ii/ = ?l°e, it follows that tt0 is
continuous. In order to see this, let y(t) = rj(t, t°, y°) be the solution of
B.1) such that r](t°, t°, y°) = y°, so that r](t, t°, y°) is continuous. Suppose
that (t0, y0) ? S n ii°, and (t°, y°) is near (/0, y0), then rj(t, t°, y°) exists on
the interval [t°, t^, y0) + e] for some € and (t, t](t, t°, y°) ? ii° on
*° ^ / Ss tjito, y0) - € and (/, rj(t, t\ y°)) $ Q° if / = t^, y0) + e. Thus,
\'i(to,y°) ~ M>o,2/o)l < «, and so (tu ^(t0, y°), t°, y0)) is a continuous
function of (/°, y°); i.e., tt0 is continuous at (/0, y0). A similar argument
holds if (/0, yo)eSn ii/.
Let tx\ ii/ -»- S n ii/ be a retraction of ii,° onto S n ii,°. Then the
composite map 7T7t0 is a retraction of S onto S f\ Q,e°. The existence of
such a retraction gives a contradiction and proves the theorem.
Exercise 2.1. Let U be a topological space; Kl5 K2 subsets of U. The
set V1 is called a quasi-isotopic deformation retract of K2 in U if there
exists a continuous map tt: Vz x {0 ^ s ^ 1} —>¦ U such that (i) tt(v2, 0) =
d2 for v2 ? K2; (ii) 77A^, j) = t^ for 1^ ? Ft and 0 ^ j ^ 1; (iii) tt(v2, 1) ?
Vi for u2 e K2; and (iv) for fixed jon0^j< 1, 7r(v2, s) is a homeomor-
phism of V2 onto its image. Let/, Q, ii° be as in Theorem 2.1; Sx a
subset of iie°; S a nonempty subset of O° U Sx such that Sx is not a
quasi-isotopic deformation retract of S U Sx in O° U S,. Then there
exists at least one point (/0, y0) ? S n O such that the solution arc (/,«/(/))
of B.1), B.2) is either in QP on its right maximal interval of existence or
first meets dD.0 at a point of S — Sx.
3. A Theorem of Wazewski
The usefulness of Theorem 2.1 depends on suitable choices of 0,°.
One of the difficulties in the application is the determination of the set of
egress points. In some cases to be described, this difficulty can be overcome.
Recall from § III 8 that a real-valued function u(t, y) defined on an open
subset of Q. is said to possess a trajectory derivative u(t, y) at the point
(/0, y0) along the solution y(t) of B.1)-B.2) if u(t, y{t)) has a derivative at
t = t0; in this case,
C.1)
u(t0, y0) = Mt, y(t))]'t=k-
If y (hence /) has real-valued components and u(t, y) is of class C1, this
trajectory derivative exists and is
C.2)
u(t, y) = du/dt + (gradv w) •/,
Perturbed Linear Systems 281
where the last term is the scalar product of/ and the gradient of u with
respect to y.
When y has complex-valued components, a function u(t, y) is said to be
of class C1 if it has continuous partial derivatives with respect to / and the
real and imaginary parts of the components of?/. Write the &th component
yk of y as yk = ak + it*, where ak, rk are real, so that ak = (yk + yk)/2,
Tk _ (yk _ yk)j2i. This suggests the standard notation, du\dyk =
\\du\dak - i dujdrk] and du/dyk = \{du\dak + i dujdr"]. Thus if grads u =
(du/dy\ ..., du/dyd) and grad^ u = (du/dy\ ..., du/dyd), then C.2) should
be replaced by
C.2*) ii(t, y) = dujdt + (grad, u) ¦/ + (grad^ u) ¦/
as can be seen by writing B.1) as a system of 2d differential equations for
(a, t) = (ff\ . . ., &1, r\ . . . , A
An open subset ii° of ii will be called a (w, v)-subset of ii with respect
to B.1) if there exists an (arbitrary) number of real-valued'continuous
functions, u^t,«/),..., ufa, y), v^t,«/),..., vm(t, y), on ii such that
C.3) ii° = {(/, y): Uj(t, y)<0 and vk(t, y)<0 for all;, k}
and if C/a, Vp are the sets
Ux = {(t, y): ux(t, y) = 0 and u,(t, y) ^ 0, vk(t, y) ^ 0 for ally, /c},
C.4)
^ = {(', 2/): »,(/, 2/) = 0 and Uj{t, y) ^ 0, vk(t, y) ^ 0 for all j, /c},
then the trajectory derivatives iia, ^ exist on C/a, K^ and satisfy
C.5) ux(t,y)>0 for (/,2/)?J7a,
C.6) ;)//, 2/)<0 for (t,y)eVl),
respectively, along all solutions through (/, y). In this definition, either /
or m can be zero.
Lemma 3.1. Let f(t, y) be continuous to an open (t, y)-set ii and ii° a
(w, v)-subset of ii with respect to B.1). Then
I m
C.7) ii/ = Q». = U ?/. - U Kr
<x=l /J=l
Proof. It is clear that dii°n ii c (|J C/a) U (U V?). In addition, ii/ n
Vp is empty, for if (/0, y0) ? K^ and y(t) is a solution of B.1), B.2), then
C.6) shows that vp(t, y(t) > 0 for t0 — e ^ / < t0 for small € > 0, so that
(/, y(t)) $ ii°. Thus
C.8)
ii?e
n ii) -
U Ux - U ^,.
282 Ordinary Differential Equations
Let (/0> y0) c U Ua - U Vp. Then «,(/„, y0) < 0 and ^(/0, y0) < 0 for
all ;, k. By C.5), there is an € > 0 such that «a(/0,«/(/)) < 0 or > 0
according as t0 — e ^ / < t0 or t0 < / ^ /0 + 6 if Oo> 2/o) e ?4."/', 2/@) <
0 for t0 - e ^ t ^ /0 + e if ('0.2/o) ? ^ and vk(t, y{t)) < 0 for t0 - e ^
/ < /0 + « for all k. Hence (/0, y0) ? ?2°,; i.e., |J ?/„ - |J ^ c ?2°e. In
view of C.8), this proves the lemma.
Theorem 3.1. Let f(t, y) be continuous on an open (t, y)-set ?2 with the
property that solutions of B.1) are uniquely determined by initial conditions.
Let ?2° be a (u, v)-subset o/?2 with respect to B.1). Let S be a nonempty
subset of ?2° U ?2e° satisfying S O ?2e0 is not a retract of S but is a retract
of ?2e0. Then there exists at least one point (t0, y0) e S n ?2° such that a
solution arc (t, y(t)) o/B.1), B.2) is contained in ?2° on its right maximal
interval of existence.
This is a corollary of Theorem 2.1 and Lemma 3.1. Sometimes, the
requirement of the uniqueness of the solution of B.1), B.2) can be
omitted:
Corollary 3.1. Letf ?2, ?2°, S be as in Theorem 3.1. except that it is not
required that solutions of B.1) be uniquely determined by initial conditions.
But, in addition, let S be compact and let «,(/, y), vk(t, y) be of class C1
{with respect to t and the real and imaginary parts of the components ofy).
Then the conclusion of Theorem 3.1 is valid.
Proof. Let/i(/, y),f2(t,«/),... be a sequence of functions of class C1 on
?2 which tend to /(/, y) uniformly, as n -*¦ 00, on compact subsets of ?2.
Let ?2l5 ?22, ... be a sequence of open subsets of ?2, such that 5 <= ?21;
?2n has a compact closure ?\ <= ?2n+1, and ?2 = U ?2n.
By replacing fi,f2,... by a subsequence, if necessary, it can be supposed
that
dujdt + (grad,ux) ¦ fn + (grad;ux) ¦/„ > 0 on U. n?2n,
dofldt + (grad, Vf) ¦ fn + (gradj Vf) •/„ < 0 on Vfi n ?2n.
Thus if ?2n° = ?2° n ?2n, then ?2n° is a (u, u)-subset of ?2n with respect to
the system
C.9) y'=fn(t,y).
The set of (strict) egress points Q°ne of ?2n° is ?2/ O ?2n. Hence ?2^e nS =
Qe° O S is not a retract of S, but ?2°, nSisa retract of ?2?e <= ?2e°.
Thus by Theorem 2.1 there is a point (/„, yn) ? S, such that the solution
y = yn(t) of C.9) satisfying yn(tn) = yn is in ?2n° on its right maximal
interval of existence [/„, rn) relative to ?2n. If C.9) is considered on ?2,
instead of ?2n, let the right maximal interval of existence of yn(t) be
[tn, a>n), so that rn ^ con ^ co and rn < wn implies that (rn, yn(rn)) <=
5?2. n ?2.
Perturbed Linear Systems 283
Since S is compact, there is a point (/0, y0) on S which is a cluster
point of the sequence of points (tu yj, (t2, y2),.... By Theorem II 3.2,
there exists a solution y(t) of B.1), B.2) having a right maximal interval
of existence [/0, co) and a sequence of integers n(l) < nB) < ¦ ¦ ¦ such that
yn{k)(t)~*y{t) uniformly as, k -> co, on any interval [/0, /*] a [to,w).
It follows that (/,«/(/)) c Q° n ?2 for t0 ^ / < co. For suppose that
there is a /-value t°, t0 < t° < co, such that (/°, y(t0)) $ ?l°. Then, for
n = »(fc) and large k, (t°, yn(t0)) $ ?>, so that (f9, «/n(/0)) ? ?\°- Hence
rn < t° < wn for n = n(k) and large A:. By choosing a subsequence, if
necessary, it can be supposed that t = lim rn(k) exists as k -* co, so that
'o ^ t ^ ^° and (rn, «/n(rJ) -> (t, «/(t)) as n = n(A:) -* co. But this gives
a contradiction for (rn, yB(rJ) ? 3?2n n ?2, where n = n(k), cannot have
a limit point (t, «/(t)) ? ?2.
Since (/0, y0) ? S c ?2" n ?2,° and ?2,° = Qje, it is seen that (/0, y0) ? ?2°,
otherwise (/, y(t)) $ ?2° for /0 < / ^ /0 + € for some € > 0. By the same
argument, (/, y(t)) c ?2° for t0 ^ t < co. This proves the corollary.
4. Preliminary Lemmas
The theorems of § 3 will be illustrated by using them to obtain results
about the asymptotic integrations of
D.1) ? = E? + F(t,?),
where E is a constant matrix and F(t, |) is "small", say,
D-2)
and y>(t) is "small" for large /. In this section, we state the basic Lemmas
4.1, 4.2, 4.3. Their proofs are given in §§ 5-7 using the results of §3.
Theorems on the asymptotic integration of D.1) are stated in §§ 8, 11, 13
and 16 and are deduced, respectively, from Lemma 4.1 in §§ 9-10, from
Lemmas 4.1-4.2 in § 12, and from Lemmas 4.1-4.3 in §§ 14-15.
If .Ehas at least two eigenvalues with distinct real parts, we can suppose,
after a linear change of variables with constant coefficients, that E =
diag [P, Q], ? = (y, 2), E? = (Py, Qz), where dim y + dim 2 = dim ?,
the real parts of the eigenvalues plt p2, . . ., qx, q2, ¦ ¦ ¦ of P, Q satisfy
D.3)
Rep, ^ ft, Reqk>/t
for some number [i. We can also assume that P, Q are in a suitable
normal form (cf. § IV 9), so that for an arbitrarily fixed € > 0 and some c,
D.4)
0 < € < c,
284 Ordinary Differential Equations
we have the inequalities
D.5) Re y ¦ Py ^ (jt + e) ||y||», Re z • Qz ^ (^ + c) ||2||2.
Correspondingly, write D.1) in the form
D.6) y' =Py + F^t, y, z), z'= Qz + F2(t, y, z),
where F = (F1; F2). The initial conditions will be of the form
D.7) y(t0) = y0, z(t0) = z0.
When D.2) holds, D.5) and D.6) give
D.8)
Sometimes, it will be convenient to suppose that E = diag (Au A2, A3),
? = (x, y, z), E? = (Axx, A$, Aaz), where the eigenvalues an, cnj2, ... of
Aj satisfy
D.9) Re cclk < ix — c, Re o.2k = /x, Re cx.3k > /x + c,
where D.4) holds. Correspondingly, it will be supposed that
Re x • Axx ^ (jx - c) \\x\\% Rez-A*^(jt + c) \\z\\\
D.10)
|Rejr^y-iu||ylH ^e\\y\\2.
The initial value problem to be considered is
D.11) x' = Axx + Fu y = A2y + F2, z' = Asz + F3,
where F(t, f) = (F1; F2, FJ, and
D.12) x(t0) = x0, y{t0) = y0, z(t0) = z0.
When D.2) holds, D.10) and D.11) imply that
D.13) \Reyy'-tt \\y
Rez-z'>((z + c)\\z\\*-y,(t)U\\-\\z\\.
In what follows, x,y,z are (real or complex) Euclidean vectors;
? = B/,z) or ? = (z,2/,z) and F = (Fl3 FJ or F = (Flf F2, F3) are
Euclidean vectors in the corresponding product space. The first lemma
refers to D.6) and D.7); the last two, to D.11) and D.12).
Lemma 4.1. Let fx, e, c be constants andP, Q constant matrices satisfying
D.4)_D.5). Let F(t, ?) = (Fu F2) be continuous for /^0 and \\y\\,
Perturbed Linear Systems 285
INI < <H^ c0) and satisfy D.2), where f(t) > 0 is continuous for t ^> 0,
and
D.14)
/-C
=J
e-U-eHs-t) ds
converges, so that there exists a T ^ 0 such that
D.15) 5t(/)^ 1 if t ^ T.
Let t0 > TandO < \\yo\\ < d. Then there exists at least one z0, \\zo\\ < 6,
such that D.6)-D.7) has a solution y(t), z(t) satisfying
D.16) HOIK 5r(/) \\y(t)\\,
D.17) ||2/@II ^ Il2/oll exp [V. + € + 2v<s)] ds
Jto
on its right maximal interval t0 ^ / < w (^ 00). In particular, if the right
side of D.17) is less than dfor t ^ t0, then co = 00.
The last assertion is a consequence of Corollary I 3.1. The other parts
of Lemma 4.1 will be proved in § 5.
Lemma 4.2. Let [i, e, c be constants and A\, A2, A3 constant matrices
satisfying D.4) and D.10). Let F(t, g) = (Fu F2, F3) be continuous for
/ ^ 0 and \\x\\, \\y\\, \\z\\ < 6 (^ 00) and satisfy D.2). Let f(t) > 0 be
continuous for t ^ 0,
D.18) o(/)
= ) *y,(s)e
Jo
-{c-e)(t-s) ds and
r(/) = f
J«
converges, and let there exist a T^ 0
D.19) 7<r@ ^ 1 and 7r(/) < 1 if / ^ T.
Le/ t0 > T, ||aro|| < la{Q \\yo\\, 0 < ||yo|| < d. Then there exists at least
one z0, ||zo|| < d, such that D.11)-D.12) has a solution x(t), y(t), z(t)
satisfying
D.20) koii < 7<K/) llsKOII, llz@ll < M0 W0II.
D.21) H2/0II exp (\/x -e-3y>) ds< \\y(t)\\
Jto
exp (
on its right maximal interval of existence t0 ^ / < co (^ 00). In particular,
if the right side of D.21) w few /Aan dfor t ^ /0, /Ae« co = 00.
In applications of Lemmas 4.1 and 4.2, it is convenient to know when
D.22) a(t), t(/)-»-0 as f-^oo.
286 Ordinary Differential Equations
This is the case if
D.23) y(f)->-0 as f->-oo or ip(t) dt < 00;
or, more generally, if
D.24) sup(l + 5 — t)] y(r)dr^0 as f-»-oo.
s?i Jt
Holder's inequality shows that a sufficient condition for D.24) [hence, for
D.22)] is that
D.25)
f °°
\ip(t)\v dt < oo for some p = 1.
Actually, the next exercise states that if f ^ 0 and c — e > 0, then D.24)
is necessary and sufficient for D.22) to hold.
Exercise 4.1. Let y>(t) > 0 be continuous for t ^ 0 and c — e > 0.
(a) Show that D.24) implies D.22). In fact, if E@ denotes the "sup" in
D.24), then
a(t) ^ e-(<w)(*-T> f \(S) ds+[l + (c- e)-1] d(T),
Jo
r(t) < [1 + (c - e)] <5@ for 0 ^ T ^ r.
(Z>) Conversely, show that if either a(t)-^-Oor r(t) ->¦ 0 as t -> oo, then
D.24) holds.
Exercise 4.2. Let y, c — e be as in Exercise 4.1 and let D.25) hold for
/* 00 /* OD
some/?, 1^/7^2. (a) Show that a\t) dt < oo, Tv(t) dt < oo. F)
Conclude that yCOKO + r(t)] dt < oo.
Exercise 4.3. Show that Lemma 4.2 remains valid if e = e(t), c = c(t),
fi = ii(t) are continuous functions of t for t ^ 0 satisfying D.4), D.10)
and if D.18) is replaced by
a(t) =j\(s) exp (-JW) - <'')] dr) ds,
where it is assumed that the last integral converges and D.19) holds for
some T.
Lemma 4.3. In addition to the assumptions of Lemma 4.2, assume that
y(t) satisfies D.24) [so that D.22) holds]; that [i = 0, e = 0 in D.13);
that an equality of the form
D.26) Hy'll ^ wo(t) ||f||, i.e., ^ = 0, HF.H ^ v»o(O llfll,
Perturbed Linear Systems 287
holds, where VoW '^ fl continuous function for t ^ 0 satisfying
D.27)
finally, that
D.28)
oo;
<x> = oo in the assertion of Lemma 4.2,
D.29) x@, z@ -^- 0 as ? ->- oo and «/,„ =lim
exists, and yx ^ 0. Furthermore there exist d1 > 0, T ^ 0 and, for every
t0 > T, a positive constant S2(t0), such that if yx j? 0, x0 are given vectors
and HyJI < ^1; ||xo|| < E2 \\yx\\, then there exist y0 and z0 such that
D.11)-D.12) has a solution for t ^ t0 satisfying D.20) and D.29). [When
5 = oo, <5j can Z>e m/:en to be oo.]
Remark 1. The proof of this lemma will show that there exists a
constant C depending only on the integral of tpo(t) over t0 ^ t < oo such
that the solutions mentioned satisfy
f
Jt
\\x(t)\\ ^ C ||2/*|| o(t), \\z(t)\\ ^ C ||2/*
where ? > ?0 and y+ can be either y0 or y^,.
Remark 2. In the proof of the first part of Lemma 4.3, the inequalities
D.13) with ^ = 6 = 0 and D.26) need not hold for all \\S\\ < d. For in
view of D.21), the proof will involve only y satisfying Cj ||^0|| < ||y|| <
c2 \\yo\\, hence
D.30)
by D.20), where
D.31) cj
2/0II < IIIII < 3c2 ||yo||,
for 7=1,2.
Correspondingly, D.13) and D.26) need only be assumed when D.30)
holds. In the second part of Lemma 4.3, the same remains true if D.30)
is replaced by
D.32)
llyJI < llfll
These assertions permit the replacement of the assumptions D.2) and
D.26) in the derivation of D.13) by another type of hypothesis: For a
288 Ordinary Differential Equations
pair of numbers r, R satisfying 0 < r < R (^ oo), let there exist a con-
continuous function yrR(t) > 0 for t ^ 0 such that
D.33) \\F(t,m ?<prR(t) if r<\\?\\ <R,
r
<prR(t) dt < co.
for r< || f|| <R,
D.34)
Then D.33) implies that
D.35) \\F(t,e)\\?
which is the analogue of D.2) with
D.36) v@ = ^W)-
Notice that with this choice of y(t) and yo(O = w@> D-31) shows that
c,->- 1 as 71-- oo. Hence if r < ||yo|| < i?/3 [or r < ||yx|| < R/3], then
the first part [or last part] of Lemma 4.3 remains valid.
The case n ^ 0 can be reduced to fi = 0 by the change of variables
f = e"(? [when p > 0, it is necessary to assume that F(f, f) is defined for
f >0anda//f]:
Corollary 4.1. /« addition to the assumptions of Lemma 4.2, assume
that tp(t) satisfies D.24) [so that D.22) /zote] awrf f/wf an inequality of the
form
D.37) \\y' - fty\\ ^ yo(t) llfll; »•«•. ^2 = M ll^ll ^ Vo(O II f II.
/itofcfr, w/zere yo(O " a continuous function for t ^ 0 satisfying D.27). J/"
/* > 0, assume that d = oo (so that F is defined for t ^ 0 and all f)
f/ze assertions of Lemma 4.3 remain valid if D.29) is replaced by
D.38) <r"'a;@,
0
as t -* oo and
= lim
Exercise 4.4. Verify Corollary 4.1.
The condition A2 = pi can be replaced by the assumption that A2 is
a diagonal matrix (or has simple elementary divisors) and that all of its
eigenvalues have the same real part p:
Corollary 4.2. Let the assumptions of Corollary 4.1 hold except that
A2 = diag [p + iylt p + iy2, . . . ], where y1; y2,. . . are real numbers, and
D.37) is replaced by
D.39) ||y' — A2y\\ ^ tpo(t) ||?||.
assertions of Lemma 4.3 remain valid if D.29) is replaced by
*¦ oo and «/<„ = lim e~Asty(t).
as
D.40) e~" «@> e~"X0 -""
Note that the last part of D.40) means that the kth. component yk(t) of
y(t) satisfies e~(-ll+ivi')tyk(t) -^ yj* as t ->¦ co.
Perturbed Linear Systems 289
Exercise 4.5. Reduce Corollary 4.2 to Lemma 4.3 by the change of
variables (x, y, z) ->¦ (m, t', vc) given by x = e^K, y = e^2'y, 2 = e"'w.
Exercise 4.6. Let ? be a constant matrix with eigenvalues /l1;. .., Xd
such that Xx,. . ., Xx are simple eigenvalues with Re kt = 0 fory = 1,. . ., k
for some k, \ ^ k ^ d. Of the eigenvalues 4+1; . . . , XA, let m have
positive real parts, n negative real parts, where 0 ^ m, n ^ d — k and
m + n = d — k. Let G(t) be a continuous matrix for f > 0 such that
G(t) ->¦ 0 as f ->¦ 00 and the elements ^w@ are °f bounded variation for
fee
/ > 0 (i.e., |
J
tiable, let G(t)
«(OI < °°)- For example, if G@ is continuously differen-
/*00
0 as t -+ 00 and || G'@1| A < 00. For large f, the matrix
E + G(t) has k simple continuous eigenvalues ^(t),.. ., Xk(t) such that
lj(t) -»¦ A^ as f -»¦ 00; cf. Exercise IV 9.1. (a) Show that the linear system
?'=[?+ G(O]f has n linearly independent exponentially small solutions
as t -+ 00. (Z>) If Re kf(t) ^ 0 for j = 1, . . ., fc, then f' = [? + G(O]f
has h + k bounded solutions as t ->¦ 00. (c) If A; = 1, then there exists a
vector c 5^ 0 such that f' = [? + G(f)]f has a solution of the form
I Ai(
= (c + o(l)) exp I Ai(j) &as(-»- 00.
If f
Re A/j) rfj is bounded
fory = 1,.. ., k, then there exist linearly independent vectors c1;.. ., ck
such that ?'=[?+ G(O]f has solutions of the form
as
oo for j = 1, . .., k.
For applications of the corollaries of Lemma 4.3, see the exercises in
§ VIII 3. Further applications and extensions of Lemma 4.3 and its
corollaries are given in §§ 13-16
Exercise IX 5.4 gives an analogue of Lemma 4.1 for difference equations.
Exercises 4.8 and 4.9 to follow give analogues of Lemmas 4.2 and 4.3.
Exercise 4.7. Let R = Rd be the f = (f\ . . ., fd)-space. For n = 1,
2, . . . , let Sn be a map of R into itself and Tn = Sn ° Sn_! ° • • • ° Sv
Let S be a compact and Ko, KX,K2..., Koo, K10, K20,. . . closed sets of R
such that S <= Ko n Koo, Sn(R - K^) a R _ Kn, Sn{K^ n ^n_10) c
^o, and ^ n 7'nE') is not empty for n = 1, 2,. . . . Then there
exists a point f „e •S' sucn that TJ0 cz Kn r\ Kn0 for n = 1, 2,....
Exercise 4.8. Let ^, 5, C be square matrices satisfying
D.41) \\Ax\\ ?(p-c) \\x\\ , 0* - e) ||2/|| ^ ||52/|| ^(/i+ e \\y\\ ,
||Cz|| ^ (p + c) \\z\\,
where n > 0 and 0 < e < c. For n = 1, 2,. . ., let A^, Fn, Zn be
continuous, vector-valued functions defined for all (x, y, z) which vanish
290 Ordinary Differential Equations
for large ||*|| + \\y\\ + \\z\\ . Let Tn = Sn ° Sn_x ° • • • ° Su where Sn is
the map
D.42) Sn: xx = Ax + Xn(x, y, 2), Vl = By + Yn(x, y, z),
zx= Cz + ZJx, y, z).
(a) Let 0 < 0 < 1 and ||xo|| ^ 0 Il2/oll- Show that if 0 ^ d 5| (c - eH/6
and
D.43) HA-JI, || FJ|, ||Zn|| ^ (||*|| + ||2/|| + \\z\\)d,
then there exists a z0 such that (xn, yn, zn) = Tn(x0, y0, z0) satisfies
D.44) \\xj ^ 6 \\yj, \\zn\\ ^ 6 \\yj for n = 0, 1, ....
(b) Show that if 0 < n < 1 and
IIXJUIFJUIZJI =o(||ac|| + 112/11 + INI)
as (»,*, y, z) -*(oo, 0,0,0), then D.44) implies that ||*n||/||yn|| -+0 and
Exercise 4.9. Let A, C be matrices satisfying
\\Ax\\ < a \\x\\ and \\Cz\\ ^ c \\z\\ , where 0 < a < 1 < c.
Let B,Xn,Yn,Zn be as in Exercise 4.8, with d < (c - 1H/6 in
D.43). Let Tn = Sno Sn_t ° • • • ° Su where ?„ is given by D.42) with
B = I. Let yoO) + VoB) + • • • be convergent, and
\\Yn(x,y,z)\\ ^ Wo(n)(\\x\\ + \\y\\ + \\z\\) .
(a) Let (x0,2/0, 20) be such that (xn, yn, zn) = Tn(x0, y0, z0) satisfies D.44).
Show that
D.45) y«> =
exists, (b) In addition, assume that d < A — aH/3 and that 0 :g
3yo(«) < 1- Let x0>2/oo be given and satisfy ||aro|| ^ 0 Hy^H. Show that there
exists a (y0, z0) such that (a;n, yn, zn) = rn(x0,2/o, «o) satisfies D.44) and
D.45). (c) Formulate analogues of parts (a) and (Z>) when the matrix B
in D.42) is fil, fi ^ 0 (instead of /).
5. Proof of Lemma 4.1
In order to apply Corollary 3.1, let
Q = {(t, y,z):t>T; \\y\\, \\z\\ < d; (y, z) * 0},
Perturbed Linear Systems 291
It will be verified that Q° is a (u, y)-subset of Q determined by the one
function u = ||2||a - 25t2@ ||2/||2. Let
U={(t,y,z)eQ: u = 0}.
Since u = 2(Re z ¦ z' - 25r2 Re y ¦ y' - 25tt' ||«/||2), it follows from D.8)
that on U, where 5r(t) ^ 1, ||z|| = 5r(t) \\y\\ ^ ||y||, and ||f || ^ 2 ||y||, we
have
The last factor is positive since t satisfies the differential equation
(c-e)r - y> - T' = 0,
and y > 0. Thus Q° is a (w, y)-subset of Q and U = Q.e° = Q.°se.
Note that, by the definition of Q, the point (y, 2) = @, 0) is not in Q;
hence (y, 2) e Qe° implies that y ^ 0.
Let S = {(to, 2/0, 2): ||2|| ^ 5t00) ||2/ol|}. Thus S n Q,« = {(/„, 2/0, 2):
II2II = 5tOo) Il2/oll}- 51 is a ball, ||z|| ^ 5t(/0) II2/0II, and 5 n Qe° is its bound-
boundary and is not a retract of S. Since C/ = Qe°, the map 77: Qe° -> 51 n Qe°
given by Tr(t, y, 2) = (t0, yo,zr(to) ||y0IIM0 112/@11) is continuous [since
2/ 5"* 0 on Q.J> and t@ > 0] and hence is a retraction of Qe° onto S n Qe°.
The existence of 20 and a solution 2/@. 2@ of D.6)-D.7) satisfying D.16)
follows from Corollary 3.1.
Since D.15), D.16) imply that \\z(t)\\ ^ ||y(f)||, hence ||f@|| ^ 2 \\y(t)\\
the inequality D.17) is a consequence of D.8). This proves Lemma 4.1.
6. Proof of Lemma 4.2
This proof is similar. It depends on the choices
Q = {(t, x, y, z): t> T; \\x\\, \\y\\,\\z\\ <d;(x,y,z) * 0}
^°= {(t,x,y,z)eQ: u < 0, v < 0},
where
» = ||x||2 - A9a\t) \\y\\\
u = ||2f - 49r2@ H2/II2
Define the sets
U = {(t,x,y,z)eQ: u = 0, v ^ 0},
^ = {(t, x, y, 2) e Q: u ^ 0, v = 0}.
It is readily verified that D.13), D.19), and D.20) imply that ||f|| ^ 3 ||y||
and
w > 0 onU, v < 0 on F.
Hence Q° is a (w, y)-subset of Q and Qe° = U - V = {(t, x,y,z)eQ:
u = 0, v < 0}.
292 Ordinary Differential Equations
Choose S to be the set {(t0, x0, y0, z): \\z\\ < 7r(f0) ||yo||}. As above, it is
seen that S n D.J> is not a retract of S but is a retract of Qe°. Thus Lemma
4.2 follows from Corollary 3.1.
7. Proof of Lemma 4.3
Let (x0, y0, z0) and f@ = (x(t),2/@, «@) be as in Lemma 4.2. By D.19)
and D.20), ||f@|| ^ 3 \\y(t)\\. Hence D.26) gives ||y'|| ^ 3%@ \\y\\. It
follows from D.19), D.20), and D.28) that f(f) exists for t ^ f0- The nrst
part of D.29) follows from D.20) and D.22). The inequality ||y'|| ^
3vo(O 112/11 implies the existence of the limit yx and yx ^ 0 as in the proof
of Theorem 1.1.
The last part of Lemma 4.3 will not be deduced from Lemma 4.2 but
will be obtained from another application of Corollary 3.1. Let
Q = {(t,x,y,z): t> t0; \\x\\, \\y\\, \\z\\ < d; (x,y,z)^0},
Q° = {(t, x, y, z) eQ: ut < 0, u2 < 0, v < 0},
where m1; u2, v are defined by
G.1)
«2 =
- 49T2@
- 49<r2@
and t0 is a positive constant to be specified. Let Ua be the subset of Q
where ua = 0 and ut^ 0, v ^ 0, and K the subset of Q where v = 0 and
«!, m2 ^ 0. Then, as in the last section, u2 > 0 on U2, v < 0 on K. When
Mj, u2, v < 0, then ||f || < 3||y|| and
G.2)
G.3)
Since
- 2/ooH ^ 7
\\ya
M1 = 2rRe(y- yj-y' + 49
a simple calculation shows that on C/1;
*i ^ 14 || yo
Let T be so large that
8
Wo
7].
r
ds < 1 for f ^ T.
Perturbed Linear Systems 293
Thus, if t0 > T, it follows that wx > 0 on U^ In this case, Q° is a (m, v)-
subset of Q and D.J> = D.% is the subset of Q where ult u2 < 0, v < 0 and
either mx = 0 or u2 = 0.
Choose d2(t0) to be
foryeQ".
* y ds}, so
G-4) d2(to) = c(
Thus, by G.3), ||xo|| < 62(t0) \\yj\ implies that ||xo|| < a(t0)
Let S={(t0, x0, y, z): \\z\\ ^ 7r(t0) \\y\\, \\y - yj\ ^ 7 ||y [
that Jcfl»u Qe°. Topologically, 51 is a ball in the (y, z)-space. (If y, z
Figure 3.
are 1-dimensional, then S appears as the shaded area in Figure 3.) It is
clear that S n Q,e° is the subset of S on which a, = 0 or u2 = 0, so
that, topologically, S n Qe° is the boundary of S and is not a retract of 51.
On the other hand, S n Qe° is a retract of Qe° for a retraction n: D.J> ->¦
S n Q,o is given by tK/, «, 2/, 2) = (^0> *o, 2/°, «t(<o) Hj^IIMO II2/II), where
, 2/) is chosen so that y° — ym = cc(y — yx) and
2/° =
/•» / /*=°
a = rp ds \ tp ds.
Jt0 I Jt
(That S n Qe° is a retract of Qe° is geometrically easy to see, because
the projection (t, x, y, z) ->¦ (t, x0, y, z) of the (t, x, y, z)-space into the
(t, y, z)-space carries Qe° into a set which is topologically the boundary
of a "cylinder" with S n Qe° corresponding to a section f = t0; cf.
Example 2 of § 2.)
Thus by Corollary 3.1, there is a point (t0, xo,yo, z0) e S n Q° such that a
solution arc (?, x@,2/@, 2@) belonging to D.11)—D.12) remains in Q° on
its maximal right interval of existence [t0, a>). As in the argument at the
beginning of the proof of this lemma, co = 00 if d = 00 or if ||yx|| is
sufficiently small; in this case, D.20), D.29) hold.
294 Ordinary Differential Equations
8. Asymptotic Integrations. Logarithmic Scale
Consider again a system of the form
(8.1) ? = E? + F(t, 0
in which
(8.2) II*¦(',?) II ^W(OU\\
holds. In this section it will be supposed that f = (y, z), F = (Flt F2), and
E = diag [P, Q], so that initial value problems associated with (8.1) take
the form
(8.3) y' =Py + F^t, y, a), z' = Qz + F2(t, y, z),
(8.4) y(t0) = y0, z(t0) = z0.
The eigenvalues px,p2,... and qx,q2, ... of P and Q will be assumed to
satisfy
(8.5) Rept ^ ii, Reqk>/a
for some number fi.
Theorem 8.1. Let (8.1) be equivalent to (8.3) where the eigenvalues of
P, Q satisfy (8.5); F(t, f) is continuous and satisfies (8.2) for t ^ 0 and
\\y\\, \\z\\ < d (^ oo); and y@ > 0 « continuous for t ^ 0 and satisfies
sup A + s —
sit
- 0
as
oo.
When ft^O, assume that d = oo. 77zew f/am? exwf T ^ 0 aw^ >
tfwf/or «x?ry f0 ^ 71 anJ y0 satisfying \\yo\\ < dlt there is a z0 with the
property that the initial value problem (8.3)-(8.4) has a solution for t ^ t0
satisfying either (y(t), z{i)) = 0 or y(t) ^ Ofor t ^ t0 and
(8.6) IKOII = o(||y(OII) as t-+co,
(8.7) limsuprMogHfWH ^A*.
t->OO
If n in (8.7) is replaced by n + e > p, this follows at once from Lemma
4.1 [with S^tf,) = oo if d = oo]. Since a linear transformation of the
y-variables with constant coefficients does not affect (8.6) but permits an
arbitrary choice of e > 0, Theorem 8.1 follows. Assertions (8.6), (8.7)
will be improved in § 11 below.
Remark 1. This proof of Theorem 8.1 shows that if the y-variables
and 2-variables are each subjected to a linear transformation with constant
coefficients and tp(t) is replaced by const. y(t) for a suitable constant, then
it can be supposed that D.5) and D.8) hold. With these choices of co-
coordinates and \p, the inequalities D.16)-D.17) in Lemma 4.1 hold for any
solution 0/@, <0) ^ 0 of (8.3) satisfying (8.6)-(8.7).
Perturbed Linear Systems 295
Theorem 8.2. In addition to the conditions of Theorem 8.1, assume that
F satisfies the Lipschitz condition
(8.8)
that t0 is sufficiently large, and that \\yo\\ is sufficiently small. Then z0
and (y(t), z(t)) are unique andz0 = g(t0, y0) is a continuous function (in fact,
uniformly Lipschitz continuous on compact subsets of its domain of
definition).
If, in addition, Fis assumed to be smooth (say, of class Cm, m^. 1, or
analytic), then z0 = g(t0, y0) is of the same smoothness. Here, a function
of a vector with complex-valued components is said to be of class Cm if
it has continuous, mth order partial derivatives with respect to the real
and imaginary parts of its variables. In this terminology, the result for
FeCMs
Theorem 8.3. Let the conditions of Theorems 8.1,8.2 hold and let F(t, f)
have continuous, first order partial derivatives with respect to the real and
imaginary parts of the components of f. Suppose also that n < 0. Then
z0 = g(f0, y0) is of class C1. If, in addition, the partial derivatives of F with
respect to the real and imaginary parts of the components of f vanish at
f = 0/or all t, then the partial derivatives of g with respect to the real and
imaginary parts of the components ofy0 vanish at y0 = Ofor all t0.
The proofs in §§ 9 and 10 will show that Theorems 8.2 and 8.3 are
corollaries of Theorem 8.1, which is, in turn, an immediate consequence of
Lemma 4.1. For applications, note that the proofs of Theorems 8.1-8.3
imply the following remark.
Remark 2. Let e > 0 be fixed so small that iM+e<0ifi«<0 and
that Reqk> ft + e in (8.5). Then there exists a number pe > 0 with the
property that if the condition on ip(t) is relaxed to
(8.9)
A + s — t)-1 \p{r) dr ^ pe for large t and s ^ t,
then Theorems 8.1-8.3 remain valid if (8.6), (8.7) are replaced by the
single condition
(8.10) lim sup r1 log ||f(OH ^ ii + e.
Notice that the "smallness condition" (8.2) does not seem appropriate
if (8.1) is considered only for small f, e.g., if F(t, f) does not depend on t.
In this case, more natural conditions are
\\p(t,
(8.11)
11*11
and, of course, n < 0.
¦0 as (*, ?) — (oo, 0)
296 Ordinary Differential Equations
Corollary 8.1 Let the assumptions ofTheorem 8.1 hold except that(8.U)
replaces (8.2); also assume that d < oo and p<0. Then the conclusions of
Theorem 8.1 remain valid. If, in addition,
(8.12)
\\F(t, Si) - F(t,
0
as (t,
(oo,0,0)
II fi " fill
when Si # S2, then the conclusions of Theorem 8.2 hold in the following
sense: there exists a small <50 > 0 with the property that ift0 is sufficiently
large and \\yo\\ is sufficiently small, then there exists a unique z0 = g(t0, y0)
such that the solution S(t) = (y(t), z(t)) of (8.3)-(8.4) exists and satisfies
IIf(Oil = dofor t ^ t0 and the conclusions of Theorem 8.1; furthermore,
g(t0, y0) is uniformly Lipschitz continuous. Also, if F satisfies the smooth-
smoothness assumptions of Theorem 8.3, then the conclusions of Theorem 8.3 are
valid.
This generalizes the last part of Theorem IX 6.1 on the existence of
invariant manifolds. The other part will be generalized later in §11.
Corollary 8.1 follows from the Remark 2 by virtue of the fact that
(8.11) implies that, for every p > 0, there exist T ^ 0 and do>O such that
(8.13)
\\F(t, 0\\ < P U\\ for f>
and correspondingly, (8.12) gives
||F(f, Si) - F(t,S2)\\ ^ p || fi - fall for t^Tand\\Si\\,\\S2\\^S0;
furthermore, if e > 0 is sufficiently small, then n + e < 0 and (8.10),
(8.11) imply (8.6), (8.7).
For another deduction of the first part of Corollary 8.1 from Theorem
8.1, make the change of variables
(8.14) S = e-'X,
where 0 < a < — p. Then (8.1) becomes
?' = (? + a/)? + e"F(t, e~x%)
and p is replaced by p + a < 0. For the applicability of Theorem 8.1,
it is sufficient to verify the existence of a f(t) such that ip(t) ->- 0 as f -> oo
and
||ea'F(?, e~a'^)|| ^ ip(t) ||?|| for ||?|| < id.
Note that a > 0 and (8.11) imply that such a tp(t) is given by
f(t) = & ^p -
Exercise 8.1. This exercise involves a proof of the conclusions of
Theorems 8.1 and 8.2 by the method of successive approximations rather
Perturbed Linear Systems 297
than by the use of Corollary 3.1 (via Lemma 4.1). In view of the change
of variables (8.14) with a suitable a, there is no loss of generality in
assuming that p < 0. If S = (y(t), z(t)) is a solution of (8.3) satisfying
(8.7), then it is easy to see that
ft
(8.15)
•f
<t) = -|Vw-s)F2(s, y(s), z(s)) ds,
where it can be supposed that P, Q are such that
(8.16) ||ep'|| ^ e{"+c)i, \\e~Qt\\ ^ e-{"+c)t for t ^ 0.
Conversely, if S = (y(t), z(t)) is a solution of (8.15) satisfying (8.7), then
it is a solution of (8.3). Show, by the method of successive approximations,
that under the assumptions (8.2), (8.8), where y>(t) -»¦ 0 as t -»¦ oo, (8.15)
has a solution (for sufficiently large t0, small ||yo|| if 6 < oo) satisfying
y(to) = Vo and (8.6)-(8.7). Let the Oth approximations be yo(t) =
ent~k)y0, zo(t) = 0, and the nth approximation be obtained by writing
(y(s),z(s)) = (yn_1(s),zn_1(s)) on the right of (8.15) and (y(t),z(t) =
(yn(t),zn(t)) on the left. See Coddington and Levinson [2, Chapter 13].
This gives the existence Theorem 8.1 under the additional condition (8.8)
and the condition y(t) ->¦ 0. Theorems 8.2 and 8.3 can also be proved by
the considerations of the successive approximations, but note that
Theorems 8.2, 8.3 are deduced in §§ 9 and 10 essentially from Theorem 8.1.
(Despite the disadvantages of the method of successive approximations
in the present situation, this method has important applications in related
problems.)
9. Proof of Theorem 8.2
It can be assumed that D.4), D.5), and D.8) hold; cf. Remark 1 following
Theorem 8.1. In terms of the function a(t) in D.18), define
(9.1)
u(t,y,z)=25c%t)\\z\\*- \\y\\*.
It is readily verified (cf. § 5) that if D.8) holds, T ^ 0 is sufficiently large,
and t ^ T, then
(9.2)
it > 0 when u = 0.
Uniqueness. Suppose that (8.1) has two solutions f}@ = (y?t), zjii)),
where ;" = 1, 2, satisfying y/f0) = y0 and (8.7), but z^) ^ z2(t0). Put
= fi@ - Si(t) = (y(t),z(t)). Then (8.8) implies D.8) hence, by (9.2),
298 Ordinary Differential Equations
du(t, y(t), z{t))jdt > 0 if u(t, y(t), z(t)) = 0. Since y(t0) = 0, u(t0, y(t0),
z(toJ) > 0 and, consequently, u(t, y(t), z(tj) cannot vanish for t ^ t0. Thus
(9.3) \\y(t)\\ < 5ff@ ||z@|| for t ^ to.
It follows from D.8) and a(t) -»¦ 0 as t -»¦ oo (cf. Exercise 4.1) that
(9.4) lim inf r1 log ||z@ll ^ fi + c.
«->oo
But this contradicts ||z@|| ^ ||f@ll ^ IIWOII + IIWOII, since both
? = flf f, satisfy (8.7).
Continuity of z0 = g(to,yo). Let f0 ^ To, HyJ < ^(f,,), zx = g(f0, 2/i) and
WO be the corresponding solution of (8.1). Introduce new variables into
(8.1) denned by
(9.5) ? = ? - WO,
so that (8.1) becomes
(9.6) ?' = ?? + f(t, i + WO) - *"(', WO),
and, by (8.8),
(9.7) \\F(t, & + WO) - *"(', ?2 + W0)ll ^ Y»@ IISi ~ ^li-
^lilt follows from the part of Theorem 8.2 already proved that if \\ya — yj
is sufficiently small, then (9.6) has a unique solution ?@ which satisfies
?@ s 0 or lim sup r1 log || ?@|| ^ ju + e as t -+ oo, ?(/„) = (y, - yi,.. .),
and
(9.8) W0 - Zl@|| ^ 5r@ \\y2(t) - yi@||
for / > r0 if f,@ = S@ + WO = WO, *2@)- (The inequality (9.8) is
the analogue of D.16) is Lemma 4.1.) It follows that f = f a@ 's a solution
of (8.1) and that z2 = g(t0, y2). Thus t=toin (9.8) gives
(9-9) ||?(*o, 2/2) - f«o, 2/i)ll ^ 5tOo) ||y, - yi||.
Let f = f(f, f0,2/0) = (y('. '0,2/o)> z(t, t0, y0)) be the unique solution of
(8.1) supplied by Theorem 8.1 and the first part of Theorem 8.2. Thus
(9.10) ?(to,to,yo) = (yo,g(to,yo))-
The uniqueness of this solution implies that for tx ^ t0,
(9.11) f(t, t0, y0) = ?(t, tu y(h, t0, y0)).
In order to examine the continuity of g(t0, y0) with respect to t0, consider
?(*, t0, y0) - f(f, h, y0) for tt ^ t0 and small ||yj. In view of (9.11), this
difference can be written as f(f, tlt y(tu t0, y0)) — f(f, tu y0). The analogue
of (9.8) holds and at t = tt gives
l|z('i, h, tfih, 'o,2/o)) - g(h, 2/o)ll ^ 5t(^) ||2/(^, ?0, y0) - yo\\.
Perturbed Linear Systems 299
Since I = f(f, f0, y0) remains in a compact f-set for f0 = ' = '0 + U and
||«/0II small, it follows from (8.1) that W(t, t0, yo)\\ ^ M, if A/" is a bound
for Fon this set. Hence Hffo, f0, y0) - f(@, f0, yo|| ^ Af(rx - @), so that
i, h,
to,
, '0,2/o) - 2/oll ^
- t0).
Hence, for t0 ^ t± ^ t0 + \ and M = M(t0, y0),
(9.12) ||^(^, y0) - g(t0, yo)\\ < M(tl - to)[\
The inequalities (9.9), (9.12) complete the proof of Theorem 8.2.
10. Proof of Theorem 8.3
It will be shown that f(f, t0, y0) is of class C1; in particular, f(f0, f0, y0) =
(yo,g(to, 2/o)) is of class C1. The proof will be given as if all variables and
functions are real-valued. This is justified since a real system is obtained
by separating real and imaginary parts of (8.1); cf. the interpretation
dujdyk = Kdu/dff* - / dujdr*) if yk = <r* + i-r" mentioned after C.2).
Let e be a unit vector in the y-space, A#0a small real number. By the
Lemma V 3.1, the difference
A0.1)
0,2/0 + he) - g(f, t0,2/0)
satisfies a linear differential equation of the form
where, in view of the continuity of f (f, ?0, «/<,),
A0.3) E1{t,h,yo)^dsF(t,l-{t,to,yo)) as ^^0
uniformly on bounded ?-sets, and d^F denotes the Jacobian matrix of
F with respect to f. By the analogue of (9.8), the function A0.1) is bounded
by
[1 + 5r(f)] ll2/^' t<h Vo + H^ ~ y^' ?
The derivation of (9.8) and the analogue of the inequality D.17) in Lemma
4.1 show that this is at most
[1 + 5r@] exp I [fi + <= + 2W(s)] ds = r(t).
Hence, for fixed (to,yo), the family of functions A0.1) is uniformly
bounded and equicontinuous in t on bounded ^-intervals of t ^ t0. Thus
there exist sequences hlt h2,.. . such that hn->-0 and the corresponding
functions A0.1) tend to a limit ?@ = ?(*, t0, y0) uniformly for bounded
300 Ordinary Differential Equations
t (^ t0). This limit satisfies the linear system
A0.4) V = [E + dsF(t, ?(f, t0,
an initial condition of the form ?(*<,) = (e, 2*) for some 2*, and ||?@H ^
r{t). The last inequality implies that ?@ = 0 or
A0.5) lim sup r1 log |i4'@ll ^ /* + e-
By (8.8), ||3fF(f, ?)|| ^ y@- Then Theorems 8.1, 8.2 imply that if
f0 is sufficiently large, there is a unique 2* such that A0.4) has a solution
satisfying ?(f0) = (e, 2*) and A0.5). Consequently, the selection of the
sequence hlt h2, . . . is unnecessary and
A0.6)
lim
0, y0)
exists uniformly on bounded ^intervals and is the unique solution of A0.4)
satisfying A0.5) and ?(*„) = (e, 2*) for a unique 2*.
Hence ?(f, t0, y0) has partial derivatives with respect to the components
of y0. The continuity of these derivatives as functions of (t, t0, y0) follows
from A0.4) and arguments similar to those just used to prove A0.6).
The existence and continuity of d?(t, t0, yo)ldto follows by the arguments
in the proof of formula (V 3.4) in Theorem V 3.1.
Note that if d$F(t, 0) = 0, then A0.4) reduces for y0 = 0 to ?' = ??.
The only solutions ? = (y(t),z(t)) of this linear system satisfying A0.5)
have 2@ = 0; cf. § IV 5. Thus dyz{t, t0, 0) = 0 and, at t = t0, this gives
dVog(to, 0) = 0 and proves Theorem 8.3.
11. Logarithmic Scale (Continued)
The object of this section is to obtain improvements of the assertions
of Theorem 8.1 without adding additional assumptions on F. To this end,
let ? = (x, y, 2), E = diag [Alt A2, A3], and F(t, x, y, 2) = (F1( F2, F3), so
that initial value problems associated with (8.1) take the form
A1.1) x' = A1x+F1, y'=A2y+F2, 2' = A3z + F3,
A1.2) x(to) = xo, y(to) = yo, z(to) = zo.
It will be assumed that the eigenvalues an, ocj2,... of A, satisfy
A1.3) Rea1A.<iM, Re ac2k = /*, Re «3t > /(
for some number fi.
Theorem 11.1. Let (8.1) be equivalent to A1.1), where Ax, A2, A3 are
matrices satisfying A1.3), and let F = (F1; F2, F3), tp(t), d, and p be as in
Theorem 8.1. Then there exist d1 > 0, T^0 and, for every t0^ T, a
Perturbed Linear Systems 301
constant a(t0) > 0 such that if \\xo\\ < 7<r(f0) \\yo\\ and 0 < \\yo\\ < <31( then
there is a z0 with the properties that A1.1)-A1.2) has a solution for t > t0
satisfying y(t) ^ 0 and
A1.4)
A1.5)
iwoii. urn = o(\\y(t)\\)
as t
co,
lim r1 log || f(Oil =p.
i->oo
This theorem, which concerns certain solutions of (8.1), follows at once
from Lemma 4.2 (with d1 = oo when d = oo). Note that if fi is the least
[or greatest] real part of the eigenvalues of E (so that there are no x [or 2]
variables), a corresponding statement holds. In fact, this case is contained
in Theorem 11.1 since dummy 2; or 2 variables can be added to the system
(8.1), with suitable choices of Ax or A3 and Fx = 0 or F3 = 0. The next
theorem concerns all solutions of (8.1).
Theorem 11.2. Assume the hypotheses of Theorem 11.1 on F(t, ?). If
d = co, let ?0@ ^ 0 be any solution o/(8.1); and if d < 00, let ?0@ ^ 0
be a solution of (8.1) for large t satisfying
A1 6) limsuprMogH^OII <0.
i-»oo
Then the limit A1.5) exists and is the real part /a of an eigenvalue ofE. If, in
addition, coordinates in the %-space are chosen so that (8.1) is of the form
A1.1), where A1.3) holds, then ?@ = Mt), y{t), z(t)) satisfies A1.4).
It is clear that the first part of Corollary 8.1 has a similar improvement:
Corollary 11.1. Let the assumptions of Theorem 11.1 [or Theorem 11.2]
hold except that (8.2) is replaced by (8.11), and let d < 00, /j, < 0. Then the
conclusions of Theorem 11.1 [or Theorem 11.2] remain valid.
Exercise HA. (a). Consider the case of a linear system of differential
equations
(H.7) ?'=[?+G@]?,
where G(t) is a continuous matrix for f > 0 such that ||G@ll ^
where tp(t) is continuous and satisfies D.24). Let E = diag [A1(.. .,. UJ,
and let the real parts filt . . . , fid of A1( . . ., Xd be distinct. Then, for any;,
l^j^d, A1.7) has a solution ?@ = (?x@, ¦ ¦ •, ?d@) such that
?'@*0 for large t, |?*@| = o(|?'@l) as *->« for **;, and
r1 log I ?'@1-*¦/** as t--co. (b) Show that if y>(t) dt < 00, then
?J@ = [c + o(l)] exp Ijt, as f ->¦ 00, for some constant c^0.
Exercise 11.2. Let ? = diag [^1; ^2, ^3], where A^ is a square matrix
with eigenvalues atn, ocj2, . . . satisfying A1.3). Let G(t) be a continuous
matrix for t > 0 and identify A1.1) with A1.7), where ? = (x, y, 2) and
F{t, ?) = G@?. Suppose that ||G@ll ^ y@» where ip(t) is continuous
be a 1 x 1
302 Ordinary Differential Equations
and satisfies |y@lp dt < oo for some/?, 1 ^ p ^ 2. Let
matrix, consisting of the constant 2, Re I = /<; thus y is 1-dimensional.
Let f@ = (x(t),y(t),z(t)) be a solution of A1.7) satisfying A1.4), A1.5).
Show that there is a constant c j? § such that
2/@ = [c + 0@1 exp [2 + g(s)] ds,
Jo
where g(t) is the diagonal element of G(t) which is the coefficient of y in
the second equation of A1.1). Note that this equation is the form y' =
ky + ?>q}(t)xs + g(t)y + Zrk(t)z\ where (qlt qa,...,g, rx, r2,...) is a row
of G(t).
Exercise 11.3. Let f(t,y) be continuous and have continuous partial
derivatives with respect to the components of y on a (t, 2/)-domain and be
periodic of period p in t, f(t + p,y) = f(t, y). Let
A1.8) y'=f(Uy)
have a periodic solution y = y(t) of period p. Discuss the behavior of
solutions of A1.8) and y(t0) = y0, where (t0, y0) is near the curve (t, y(t)),
0 ^ t ^ p, on the basis of the following suggestions: Introduce the new
variables
A1.9) ? = 2/
Thus A1.8) becomes
which can be written as
A1.10) ?'=
where
P(t) = dyf(t, y) = $A at y = y(t),
A1.11)
#0, 0 =/(', ? + y@) -/(', y@) - A0?,
i*@ is a matrix function of period p and //(*, 0 is continuous and has
continuous partial derivatives with respect to the components of ?, and
H(t, 0) = 0, dcH(t, 0) = 0. The linear matrix initial value problem
A1.12) J' = P(t)J, J@) = I
has a solution which, by the Floquet theory in § IV 6, is of the form
A1.13) J(t) = K(t)eEt,
where K(t + p) = K(t) and ? is a constant matrix. The change of variables
A1.14)
Perturbed Linear Systems 303
transforms A1.10) into
(n.15) r
K~\t)H{t, K(t)Q.
Consider the application of the theorems of § 8 and of this section to A1.15)
to obtain generalizations of the results of §§ IX 10, 11. (Note that eB
need not have I = 1 as an eigenvalue in the situation here.)
12. Proof of Theorem 11.2
It will be shown that it is sufficient to consider the case of linear equa-
equations. Note that (8.2) implies that if the solution f = fo(f) of (8.1)
vanishes at one f-value, then it vanishes for all t. Hence fo(O ^ 0 for
large t, say t ^ t0. Define a matrix G(t) = (gik(t)) as follows: if F =
(F\ F\...), put
A2.1)
ii won1
for t ^ t0. Since f = fo(O is a solution of (8.1), it follows that it is a
solution of the linear system
A2.2) ? = (E+ G(t))l
Note that (8.2) and A2.1) imply that
Hence Theorem 11.2 is contained in the following:
Lemma 12.1 Let G(t) be a continuous matrix for t ^ 0 such that
A2.3) l|G@ll ^ f(t),
where tp{t) > 0 is a continuous function satisfying D.24). Let f = fo(O ^ 0
be a solution of A2.2). Then the conclusion of Theorem 11.2 holds.
Proof of Lemma 12.1. Let /i1 < fit < ¦ ¦ ¦ < p, denote the different
real parts of the eigenvalues of E. After a change of coordinates, it can
be supposed that E = diag [B^ B2, .. ., Bf], where the eigenvalues
Pik of Bj satisfy Re pik = rt. Correspondingly, let f = {ylt . . . , yf),
?? = (B&i, .. ., Btyt), and let A2.2) be written as
A2.4)
for
where Gjk(t) is a rectangular matrix and ||Gyt@|| ^ y(t).
304 Ordinary Differential Equations
If 1 ^ q </, f0 is sufficiently large, and yq0 ?> 0, then Theorem 11.1
implies that A2.4) has a solution f = (^(O, • • •, 2//@) satisfying
A2.5)
A2.6)
A2.7)
11^@11 = o(
if k<q, yQ(to) = y<,o,
as t^co for k
as
This solution, say f = ?q{t, t0, yQ0), is unique by Theorem 8.2. In fact, it
is unique even if A2.6), A2.7) are replaced by
A2.8) limsup v1 log ||f(OH ^
(->00
(cf. Remark 2 following Theorem 8.3). This uniqueness implies that
?„('. t0, yQo) is linear in yQ0 (for fixed t, t0, q).
With the understanding that f,(f, t0,0) = 0, it follows that there exist
unique y10,. . . , yf0 such that the given solution !-0(t) is of the form
A2.9) WO =2**0. <o. 2/,0)-
In fact, 2/io, • • • > 2//o are defined recursively as follows: if fo(O =
B/i@> • • • > 2/>@)> let 2/io = 2/iOo); then let «/20 = 2/2('o) - 2/12(^0). where
fi(«. to'Vio) = B/n@,2/i2(O> • • • > 2/i/@); etc.
Let q be the largest /-value such that yj0 ^ 0 in A2.9). It is clear that
fo(f) = (yi@,. . ., 2//0) satisfies A2.6), A2.7). This proves the lemma.
13. Asymptotic Integration
The object of this section is to study the asymptotic behavior of solu-
solutions f@ °f a perturbed linear systems
A3.1)
F(f, ?),
rather than the behavior of ||f(OII as in § 11.
Suppose that ? is in a Jordan normal form E = diag [7A),. .., Jig)],
where /(/) is an /*(/) X /*(/) matrix [as in (IV 5.15)-(IV 5.16)]. Thus
J(j) = 2(/L,j) + KMj), where Ih is the unit h X h matrix and Kh is 0 if
h = 1 or is the A x h matrix with ones on the subdiagonal and other
elements zero if h > 1. According as h = 1 or h > 1,
yg),
. , Fg\
A3.2) JU)y, = *y, or
where 2 = 2(j), y, = (y,\ . . ., *//), A = /*(/).
Correspondingly, it is supposed that ? = (ylt
), F = (F
and A3.1) is of the form
A3.3) y! = J{j)Vj + Fit, f)
Perturbed Linear Systems 305
for j = 1, . . . , f.
Let /< denote one of the numbers Re 2A),. . ., Re l{g). An index j
will be denoted by p, q, or r according as Re X{j) < fi, Re 2(j) = /< or
Re X{j) > /*. Put
A3.4) h^ = max h{q).
Let/0 be an integer and /? a number satisfying
A3-5) jo?h+-l and ^^1,
and %), A:(^r) integers, if any, such that
A3.6) 1 ^ %) ^ fc(?) ^ min (%), ^ and %) - l(q) ^ j0.
The next theorem concerns sufficient conditions for A3.3) to have a
solution with the following asymptotic properties as t —>¦ oo,
A3.7)
y? =
y? = o{
Jc—i
if
where c9* are constants,
A3.8)
2i= 2
and 2i = 0 if l(q), Jc(q) do not exist.
Note that if the o-terms are replaced by 0, then, since 1 < / ^ k in
2i, A3.7) becomes a solution of the linear system
A3.9)
i.e., y/ =
for j=l,...,g.
The choice of the range of summation l(q) ^ / ^ min (A:, k(q)) is dictated
by several considerations. On the one hand, results permitting / > k can
easily (but will not) be obtained as a consequence of Theorem 13.1; also
the first term in the first line of A3.7) is not significant unless / ^ @, hence
the choice i ^ min (k, k(q)) ^ min (k, /?) since k ^ h(q). On the other
hand, the condition / ^ l(q) means that the degree of the polynomial
2i cqitlc-'ij{k — i)! does not exceed the given /„.
Theorem 13.1. In the system A3.3), let J{j) be a Jordan block; cf.
A3.2). Let n = Re X{j)for some j. Let an index j = 1, . . . , g be denoted
by p,q or r according as Re A(/) < fi, Re A(/) = fi, or Re A(/) > /
306 Ordinary Differential Equations
define h+ by A3.4). Let j0 be an integer and ft a number satisfying
A3.10) 0 <y0^ A* - 1, ft+ja^h^; thus 0^1.
Let l{q), k(q) be integers {if any) satisfying A3.6). Let F{t, f) = (Fx,. .., Fg)
be continuous for t ^ 0 and all f, and satisfy
A3.11)
\\F(t, f) ^
where tp^t) > 0 is a continuous function such that
A3.12) t'+'o-1^) dt < oo.
Let m = JiP h(p) + ?9 [%) - k(q)]. For any w/ o/ constants cg\ l(q) <
A: ^ fc(<7), not all 0, /Aere craw an m parameter family of solutions f@ of
A3.3) defined for large t and satisfying the asymptotic relations A3.7) as
t-* oo.
The part of the assertion concerning "m parameter family of solutions"
means essentially that it is possible to specify a partial set of m "initial
conditions," as well as the asymptotic behavior A3.7) for f@; cf. the
statement following A4.15) in the proof of Theorem 13.1.
Remark 1. Consider a system of differential equations
A3.13)
rf = E«r, + F°(t, r,),
where E" is a constant matrix and F°(t, rj) is continuous for t ^ 0 and all
rj. Let L be a nonsingular constant matrix such that L^1E°L is a matrix
E = diag [/(I),. . . ,J(g)] in a Jordan normal form. Then the change
of variables r] = Lf reduces A3.13) to A3.1) [i.e., to A3.3)], where
F(t, |) = L^F^t. 14). The applicability of Theorem 13.1, or at least
the condition A3.11), can sometimes be verified without the knowledge
of L or the explicit reduction of A3.13) to A3.1). For it is clear that
\\F°(t,ri)\\ ^ v>i@MI implies that ||F(/,|)|| ^ cy^/) llfll if, e.g., c =
WL-H ¦ \\L\\.
Remark 2. The derivation of Theorem 13.1 from Lemma 4.3 will
show that the theorem remains valid if F(t, f) is defined only for / ^ 0,
HI! < d < oo if p < 0 (or [i = 0, h* = 1, and the constants |c,*| are
sufficiently small).
Theorem 13.1 has a partial "converse" dealing with all (rather than
certain) solutions f(f) of A3.1) satisfying
A3.14)
log ||?(OII-••/« as
(cf. Theorem 11.2):
Theorem 13.2. Let E = diag [J(l),.. ., J(g)] and F{t, f) be as in
Perturbed Linear Systems 307
Theorem 13.1 except that A3.12) is replaced by
A3-15) ^">~1y)J{t) dt < co, where ho> A +
(and Ao is no/ necessarily an integer). Let f@ ^ 0 be a solution o/A3.3)
satisfying A3.14). Then there exists constants ck, k = 1, . . ., h(q), not
all 0, such that ifj0 is defined by
A3.16) j0 = max [k(q) - k] for cgk * 0,
P = h —jo> and l(q), k(q) are the least, greatest integers (if any) satisfying
A3.6), then f@ satisfies the asymptotic relations A3.7) as / -»- oo.
Consequences and refinements of Theorem 13.1, 13.2 will be given in
§16; see also § XII 9.
14. Proof of Theorem 13.1
Change of Variables. In order to apply Lemma 4.3, make the linear
change of variables
given in terms of f = (Vl,. .., yg) and ? = (zly ...,zg)by the formulae
A4.2)
if
where 0 < e < 1, ?i is the sum over the z-range /(?) ^ i < min (fc, %))
as in A3.8), and 2n 's the sum over the other indices i on the range
1 ^ / < k, so that
A solution |@ of A3.3) satisfies A3.7) if the corresponding vector ?@,
defined by A4.1), satisfies
A4.3) z* = c* + o(ti-<1) for l(q) ^ i ^ A(«),
z3* = o(l) otherwise.
To clarify the meaning of A4.1) and to calculate the resulting differential
equation for ?, the map A4.1) will be given a decomposition of the form
A4-4) f = Q(t)i = D(t)Qo(t)L
to be described. This factorization is suggested by the fact that if t*~P in
the first formula of A4.2) is replaced by tk-1 (and written behind the sign
2n), then this formula becomes yg = eJ{-<i)tz
308 Ordinary Differential Equations
The change of variables f = Q0(t)w, w = (wu ..., wq), is given by
A4.5) St = eJU)twt if j =q, S, = e'"wj if j 5* q.
Thus A3.3) becomes
A4 6)
if j
where h = h(J). Finally, let D(t) be the diagonal matrix such that w =
D(t)i is given by
wk = zk if
= /c =
A4.7)
w* = tk~h* if k < l(q) or k > k(q),
if
If the resulting differential equation for ? is written as
A4.8) T = ?0? + Qr\t)F{t, QQ,
then the linear part ?' = Eot, is given by
zka' = 0 if /(?) ^ k ^ k(q),
A4.9) zkg' = (/J - fe)r V if fc < '(«) or fe > fcfo),
where /e(y) is the matrix obtained by replacing the ones on the subdiagonal
of J(j) by e; cf. § IV 9. The last part of A4.9) is easy to see if the trans-
transformation Wj-*-Zj is made in two steps wf-*-ziklek-1-*-t1~%"/e''~1.
Finally, replace the independent variable / by s, where
A4.10)
Thus A4.8) becomes
A4.11)
ds
so that ^ = t?.
ds
''Fit, QQ,
where the linear part of this equation is
^- = 0 if
ds
A4.2) -?-V-
ds
if k < l(q) or k >
Perturbed Linear Systems 309
Preliminary Existence Result. Suppose that there is a continuous
function fo(t) for large t such that
A4.13) \\Q-\t)F(t, Q0\\ ^ Vo(O U\\ for K\\ < & ^ «>,
A4.14)
f
< co.
The last condition is equivalent to tyio(t) ds < oo since ds = dt/t.
Then if e > 0 is sufficiently small, Lemma 4.3 is applicable to A4.11) if x
is a vector with components zj°, and zgk, k > k(q); y is the vector with
components zgk, l{q) ^ k _^ fc(^); and 2 is the vector with components
zr* and 2,*, A: < %). Note that A4.12) shows that there is a constant
c > 0 such that Re (zgk dzgk/ds) ^ —c |zff*|a or _: c |2,*|2 according as
k> k(q)^P or fc < l(q) ^ /3; also Re (z} ¦ dz'jds) <> -ct ||a3||2 or
_: c/ ||23-||a according asy = p (i.e., Re l(j) < ^a) ory = r (i.e., Re A(y) >
pi) if e > 0 is small and / > 0 is large.
Thus, by Lemma 4.3, A4.13)-A4.14) imply that if ck, l(q) ^k^ k(q),
are given constants, not all 0, then there exists a solution ?(/) of A4.11)
such that, as t —»- oo,
A4.15)
\t) = ck +
; = o(l)
for l(q)^k<k(q),
otherwise.
In fact, we can also specify a set of m initial conditions for ?: zvk(T) =
a*0 and a/CT) = a*0 for %) < A: = %) if T is sufficiently large and
|a*0|, |a*0| are sufficiently small numbers.
The Norms \\Q\\, ||2-1II- IR order to complete the proof, it remains to
show that the assumptions A3.11)—A3.12) imply A4.13)-A4.14) and that a
solution ?(f) of A4.8) satisfying A4.15) also satisfies A4.3). To this end,
it will first be verified that there exist positive constants c, c' such that for
large t,
A4.16) c' <; ||e(/)|| e-"'rJ» < c, c' <; \\Q-\t)\\ e"^w ^ c.
From A4.2), the norm of Q(t) is easily seen to be O{t^ttr), where
y = max \hn — ji, h(q) — l(q)] and the max refers to the set q. From
A3.6), h(q) - l(q) ^ j0 and, from A3.10), A, - /J < j0; hence \\Q(t)\\ =
O{e"ttir>) as t -»- oo. It is similarly seen that e"V» = O(|| g (Oil) as t —- oo.
This gives the first part of A4.16).
The factorization Q = DQ0 of Q into nonsingular matrices for t > 0
shows that Q exists and is Q-1 = Qq1D~1. The inverse map
A4.17)
310 Ordinary Differential Equations
is easily seen, from ? = Qow, w = Z>? in A4.5), A4.7), to be
if l(q) ^k^ k{q),
if k < l{q) or k
A4.18) ^ =
Thus, for large /, || Qr\t)\\ is bounded from above and below by a positive
constant times e^*ty, where y = max \fi — I, k{q) — 1]. Since %) — 1 ^
/3 - 1, by A3.6), the last part of A4.16) follows.
Completion of the Proof. In view of A3.11),
Hence, by A4.16),
(i4.i9) ii e^
Thus A3.12) implies that A4.13), A4.14) hold if fo(t) = c2^"-1 y>i@» and
so A4.8) has a solution ?(/) satisfying A4.15).
In view of the first part A4.9), the corresponding equations in A4.8) are
zkq' = (qk)th component of QrxF{t, Q?),
so that, by A4.18),
zf = e-
if Kq)^k<k(q),
zf ei\Fv
i=i {k — i)!
where Fq* is the (qi)th component of F. Hence,
zk> = O(e-"^-Vi@ 116^11) as t — co,
by A3.11). In view of A4.16) and the boundedness of ?(?) as t-+ co,
zkg' = O(tk+h~1f1(t)) as t->co.
Consequently, k ^ /3 shows that
A4.20) 2e*(/) - cgk = tk-'O (j~ f+1'-1y>1(.s) dsj.
This gives the first part of A4.3) and completes the proof of Theorem 13.1.
15. Proof of Theorem 13.2
This theorem can be reduced to the case of linear equations by the
device used at the beginning of § 12. Hence we can be suppose that
F(t, f) = G(t)§, where G(t) is a matrix satisfying ||G@ll ^ Vi@ and A3.1)
is replaced by
A5.1) ? = ES
Perturbed Linear Systems 311
Let q0 denote a fixed value of q and k0 an integer on the range 1 ^ k0 ^
h(q0). Then the equation A5.1) has a solution ?^@ satisfying, as t -» oo,
if q = «»• ^ = k ^ h^'
A5.2) yq\t) =
yq\t) = oie
yk(t) = oC
if g = 9o,
if q*q0,
if j^9,
^ fe < fe0,
<: k ^ h{q),
where y = h0 — h(q0) + k0 ^ 1. This follows from Theorem 13.1 with
/„, /3 replaced by A(^o) - k0, y = h0 - [h(q0) - k0] and the choice cgk = 1
or ck = 0 according as (qk) = (qoko) or (^fc) ^ (?ofco).
The set of solutions ?gk(t) is a set of ^h{q) linearly independent solutions.
Also if n = 2A(», then Theorem 8.2 implies that there are exactly n
linearly independent solutions ^(t),. . ., ?„(/) satisfying
lim sup/-1 log
and n + ^h(g) linearly independent solutions satisfying
limsuprMog
Hence if ?(/) ^ 0 is a solution of A5.1) satisfying A3.14), then there exist
constants cla.. ., cn and c/ such that
A5.3)
and that not all c9* are 0. It will be left to the reader to verify that this
implies Theorem 13.2.
16. Corollaries and Refinements
When the matrix E in Theorem 13.1 has simple elementary divisors
(e.g., when the eigenvalues of E are distinct) or even if h^ = 1, then
^* = Wo = 0' P = 1' and condition A3.12) reduces to
f
fyx(t) dt < oo for a = /3 - 1 ^ 0;
cf. Corollary 4.2. Here, the asymptotic formulae A3.7) reduce to
VM = eXMt[ca + o@], 2/XO = eA(9)io(r") for j ^ g.
For a fixed /„> the smallest admissible value of /5 in Theorem 13.1 is
/? = A* — /o in which case A3.12) becomes th*-1f1{t)dt< oo. A
312 Ordinary Differential Equations
larger choice of ft has the role of possibly increasing the number of signi-
significant terms in the asymptotic formulae A3.7) and of improving the error
terms. When A3.12) is strengthened to
A6.1)
f
< oo,
the maximal number of significant terms is possible. In this case, we have
Corollary 16.1. Let E = diag [/(I),. . ., J(g)], [i, h* be as in Theorem
13.1 and let F{t, f) be continuous for t ^ 0 and all f and satisfy A3.11),
where f^t) > 0 is a continuousfunction satisfying A6.1). Let? = fo(O ^ 0
be a solution of the linear system f = ?? such that r1 log ||f(OH -+p
as t-+ oo. Then A3.1) has a solution f@ satisfying |||@ — fo(OII <r"'->-0,
t-+ oo.
In this corollary, E is not required to be in a Jordan normal form (cf.
Remark 1 following Theorem 13.1). If it is, we can, in addition, assign a
partial set of 2A(p) initial conditions, yv(t0) = yv0 for sufficiently large t0.
Also, |@ satisfies the asymptotic relations A3.7), where l{q) = 1, k(q) =
Kq), cqk are suitable constants determined by ?0(t),j0 is defined by A3.16),
and /? = 2/j.,, — y0. This improves the asymptotic relation claimed in the
corollary.
The deduction of Theorem 13.1 from Lemma 4.3 shows that assumptions
A3.11), A3.12) can be weakened somewhat.
Corollary 16.2. Let assumptions A3.11), A3.12) of Theorem 13.1 be
relaxed to
A6.2) \\Q-Kt)F(t,Q(t)O\\^Wo(t)U\\,
or, more generally, to
A6.3) \\Q-W(t, Q(t)Q\\ ^ f(t) U\\
A6.4) e-'V-* \F,\t, 6@01 ^ Vo(O II?11 for i = 1,. . . , Kq),
where f = Q@? is given by A4.2) and f(t), y>0(t) are positive continuous
functions for t > 0 such that
A6.5)
A6.6)
sup A +
- 0 f
J*
y>(r) dr -*¦ 0 as t-
co,
the conclusions of Theorem 13.1 remain valid.
Exercise 16.1. By referring to Remark 1 following Lemma 4.3 and to
the proof of Theorem 13.1, find sharper estimates for the o-terms in A3.7)
under the conditions A6.3)—A6.6) of Corollary 16.2.
Remark 2 following Theorem 13.1 and Corollary 16.2 have important
consequences. For example, suppose that Fin A3.1) does not depend on
Perturbed Linear Systems
t, so that A3.1) can be written as
A6.7) !' = ?? + F(f),
where F(f) is defined for ||| || < 6 < oo and satisfies
A6.8) ||F(f)|| ^ Co llfll^9, 0 > 0, Co = const.,
or, more generally,
A6.9) ||F(|)|| <_CoJil_.
llogllfllT'
or even
A6-10) ||F(|)|| ^ ^(HHI) mi
313
v>Jo+ P,
where <p(p) is a nondecreasing function for 0 ^ p < 6 such that
A6.11) | p-1 |log p|y°+'-V(p) rfp < oo.
J+o
Then A4.16) and A6.10) imply that if p < 0 and ||?|| ^ 1, then
lie-1 ii • iiF(eon ^ ce-"V-v(c^*i
for large 7. Thus the analogue of A6.2) holds with
If p = ce^H'o is introduced as a new integration variable in the integral in
A6.11) and it is noted that dplp~ p dt and log p ~ pt as / -*¦ oo, then it
is seen that A6.6) is a consequence of A6.11).
Corollary 16.3. In A6.7), let E = diag [/(I),.. ., %)] be as in Theorem
13.1, let F(f) Ae continuous for ||||| < C (< oo) and satisfy A6.10),
9?(p) is a nondecreasing function of p satisfying A6.11). Lef ^a < 0.
7Ae conclusions of Theorem 13.1, wM A3.3) replaced by A6.7), remain
valid.
Exercise 16.2. By involving the Remark 2 following Lemma 4.3, show
that conditions A6.10), A6.11) in Corollary 16.3 can be replaced by
A6.12) I|F(M)|| ^ %(||f||) for U\\<6,
where <po(p) is a nondecreasing function of p, 0 < p < C, such that
A6.13) I p |log p |"+3° ^o(p) ^P ^ oo.
[This is somewhat more general than Corollary 16.3 for A6.12) implies
A6.10) with 9?0(p) = 9?(p)p. Although A6.10) is a consequence of A6.12)
with (p(p) = <po(p)/p, the monotony of <p0 does not imply that of <p.]
Analogously, we obtain the following consequence of the proofs of
Theorems 13.1 and 13.2.
314 Ordinary Differential Equations
Corollary 16.4. Let E, p, F, <p be as in Corollary 16.3 except that A6.11)
is replaced by
A6.14) J V1 Il°g Pi"*'1 <Ap) dp < oo, where h0 ^ hn
{and h0 need not be an integer). Then the conclusions of Theorem 13.2, with
A3.1) replaced by A6.7), are valid.
17. Linear Higher Order Equations
The results of §§ 4, 11, 13, 16 will be applied in this section to a linear
differential equation of order d > 1,
A7.1) u^ + [a1+p1(t)yd-1) + ---
+ K-i +Pa-i{t)V + [aa+pM« = 0,
for a real- or complex-valued function u. This will be viewed as a pertur-
perturbation of the equation
A7.2) M«" + a^-v + ¦¦¦ + ad_xu' + adu = 0
with constant coefficients. The characteristic equation for A7.2) is
A7.3) X" + a.X"-1 + ¦¦¦ + ad^X + ad = 0.
Equation A7.1) can be written as a linear system
(n.4) r =
for the J-dimensional vector ? = (u'
and R, G(t) are the matrices
f—a1 —a2 —a3
1 0
0 1
., mA», m@»), where u = m
@»
R =
aA
0
0
0
0
0
0
\ 0 0
0/
A7.5)
G(t) =
(-*
0
0
0
0
~Pa
0
0
¦ ¦ ¦ -Pi-i
0
0
-I
0
0
\o
o
Perturbed Linear Systems 315
Note first that if a1 = ¦ ¦ ¦ = ad = 0, then R is in the Jordan normal form
and consists of one Jordan block with X = 0 on its main diagonal. If,
the coefficients p^t), . .. ,pd(t) are small, then A7.1) can be considered
to be a perturbation of u(d) = 0 which has the linearly independent
solutions u = 1, t, . . ., f*. It will be verified that Corollary 16.2 has
the following consequence.
Theorem 17.1 In A7.1), let ax = • • • = ad = 0 and letp^t),. .. ,pd(t)
be continuous complex-valued functions for t^O satisfying
A7.6)
tk+* x \pk(t)\ dt < oo for some a ^ 0
and k = 1, ..., d. Then, for any j, 0 ^ j ^ d — 1, A7.1) has a solution
satisfying u(t) = (/3'//0(l + °(^a)) as t —*¦ oo, and this relation can be
"differentiated" d — \ times, i.e.,
««» =
A7.7) 0' - k
M<*> = o(y-*-*) for k = j + 1,. . ., d - 1.
It will be clear from the proof that, for a given j (rather than for any j)
on the range 0 ^ j ^ d — 1, a sufficient condition for the existence of a
solution satisfying A7.7) is that
k~d-i
<
Proof. Since R in A7.5) is in a Jordan normal form, A7.4) can be
identified with A3.3) if F(t, I) = G(t)?, where ? = (f1, . . ., f) and
|r* _ MC-*). in order to verify the conditions of Corollary 16.2, note that
the sets of p and r are vacuous and that there is only one q. Correspond-
Correspondingly, %) = 0 and h(q) = d. Let/0 = j be the index j in A7.7), /? = d —
j + a, and l(q) = k(q) = d — j. Thus 2i 'n A3.8) contains no terms if
k < d — j or exactly one term i = d— j if d—j^k^d. Also, let
cj = 1 or c9* = 0 according as i = d — J or ;' ^ d — j, so that the desired
asymptotic relation A7.7) is identical with (the first part of) A3.7).
Consider F(t, Q(t)?) = G(t)Q(t)i. Since only the first row of G(t) con-
contains nonzero elements, this can be written as F(t, QQ = (F1, 0, ..., 0),
where, by A4.1)-A4.2) and A7.5),
¦a+i
1
t^S-i (k-d+ j)! " *5i* *' ~" (fe - i)!
? = (z1,..., zd), and 2n is the sum over the set of i-values, 1 ^ / ^
and / j6 d—j. Consequently, ||Q-\t)\\ ^ ct^1 implies that
iif(/, eon ^
316 Ordinary Differential Equations
for a suitable constant c0 and large t. Since the coefficient of ||?|| is a
/•CO
function fa(f) satisfying fo(t) dt < oo by A7.6), Theorem 17.1 follows
from Corollary 16.2. ^
When all the roots of A7.3) are the same, say X, this can be reduced to
the situation of Theorem 17.1 by replacing u by the new dependent
variable v = ue~x%. In the other extreme case, when X is a simple root of
A7.3), we have
Theorem 17.2 Let A7.3) have a simple root, say X, and suppose that if
Xo is any other root, then Re X ^ Re Xo. Letpx{f),. . . ,pd(f) be continuous
functions for t ^ 0 satisfying
A7.8) I \pk(i)\ f dt < oo for some a ^ 0
and k = 1, . . ., d. Then A7.1) has a solution u(t) satisfying
A7.9) uik\t) = eu[xk + o(-\~\ for k = 0, . . . , d - 1
as t -> oo.
Proof. This is the simplest case of Corollary 16.2 when h(q) = 1. Let
j0 = 0, p = 1 + a. Let A7.4) be identified with A3.13) in Remark 1
following Theorem 13.1. Then
for some constant c. Thus A3.11) holds with yi^t) = cS \pk(t)\ and the
theorem follows from Corollary 16.2.
Consider the general case where A7.3) has a root, say X = 0, of multi-
multiplicity h, 1 -<h^d.
Theorem 17.3. LetX = 0bea root of{17.3) of multiplicity h,l ^h^d;
i.e., let ad_h+1 = • • • = ad = 0 and aa_h # 0; and suppose that if Xo ^ 0
w any o/Aer roof, /Aen Re Xo ^ 0. Let p^t),.. ., pd{t) be continuous func-
functions for t > 0
A7.10) I0/*-"***-! |ft@l d/ < oo for it = d - h + 1,. . . , d.
A7.11) ["* \ph(t)\ f dt < oo for k=\,...,d-h,
for some a ^ 0. TAen, /or any j, 0 ^j ^ h — 1, A7.1) Aas1 a solution
u(t) satisfying
A7.12)
U-k)l
u =
i-h+s-*) for k = O,...,j,
for k=j + 1, . . . , d- 1,
oo.
Perturbed Linear Systems 317
Exercise 17.1. Prove Theorem 17.3.
Exercise 17.2. Restate Theorem 17.3 when X = 0 is replaced by an
arbitrary X.
Theorems 17.1-17.3 depend on §§ 13, 16; we can also apply the results
of§ 11:
Theorem 17.4. Let X be a simple root o/A7.3) and suppose that ifX0 is
any other root, then Re Xo # Re X. Let p^t),. .. ,pd(t) be continuous func-
functions for t^O satisfying
A7.13) pk(t)-+O as f-*oo for k = 1, . . . , d
or, more generally,
A7.14) sup A -q- s - t)-1 \\pk(r)\ dr-*O as / -> oo
for k = 1,. . ., d. Then A7.1) possesses a solution u(t) ^ 0 for large t
such that
A7.15) ««*>(/) = u(t)[Xk + o(l)] for k = 1,. • •, d- 1 as t-+ oo.
Proof. It is sufficient to prove this theorem in the case that X = 0,
otherwise ue~xi is introduced as a new dependent variable in A7.1). Thus
ad = 0. Write A7.1) as the system A7.4), A7.5). Let Fbea constant
nonsingular matrix such that Y~XR Y = E = diag [/(I),. . . , J(g)] is in a
Jordan normal form. The first column of yean be taken to be @,. .., 0,1),
since this is an eigenvector of R belonging to the simple eigenvalue X = 0.
Thus /(I) is the 1 x 1 zero matrix and the diagonal elements X(j) of /(_/)
are such that Re X{j) #0 for j = 2, . . . , d. The change of variables
| = Yr\ reduces A7.4) to
A7.16) rj' = Erj + Y'1 G(t) Yr\.
If n = (rj\ . . . , rf), it follows from Theorem 11.1 that A7.16) has a
solution rj(t) # 0 such that rj\t) = oflr?1^)!) as / -* oo for k = 2,.. . , d.
The corresponding solution f(/) = Yr}{t) of A7.4), where f = (f1,.. .,
I"), satisfies ?*(*) = o(||d@l) as /-> oo for k=\,...,d- \. Since
M(/) = ?<*(/) and u{d~k) = |* for k = 1, . . . , d - 1, the relations A7.15)
follow.
It cannot be expected that condition A7.14) in Theorem 17.4 can be
improved. This is shown by the following exercise.
Exercise 17.3. (a) In the second order equation,
A7.17) u" - [X* + q{t)]u = 0,
let q(t) be continuous for t ^ 0 and Re X ^ 0. Show that a necessary
condition for A7.17) to possess a solution u(t) which does not vanish for
318 Ordinary Differential Equations
large / and satisfies u'ju —>¦ A as t —>¦ oo is that
A7.18) sup A + s + tf1
i
q(r) dr
-0
as t
oo.
(o) Prove that the necessary condition A7.18) in (a) is sufficient if A is a
positive number and q{t) is real-valued; see Hartman [5]. For a related
result, see Exercise XI 7.5.
Theorem 17.5. Assume the conditions of Theorem 17.4 with A7.14)
strengthened to
A7.19)
< oo for some p, 1 ^ p ^ 2,
for k = 1, . . ., d. Then a solution u{t) ^0 of A7.1) satisfying A7.15)
satisfies
A7.20) u@ = [c + o(l)] expj [X + g(s)] ds as t -+ oo,
c ^ 0 is a constant,
A7.21)
F' = dF\dX and F is the polynomial on the left of A7.3) [so that F\X) =
(X — X2). . . (A — Xd) ifX2 Xd are the roots of A7.3) distinct from X].
Proof. Write A7.1) as the system A7.4), A7.5) and make the change of
variables f = Yrj, where Y = Y@) is the constant matrix given in Exercise
IV 8.2 and having (A",..., A, 1) as its first column. Then A7.4) becomes
A7.16), where E = diag [/(I),. . ., J(g)] and /(I) is the 1 x 1 matrix X.
Since Y is a constant matrix, A7.19) implies that the pth power of the
absolute values of the elements of y^G^^areintegrableoverO ^ / < oo.
Hence, it follows from Theorem 11.1 that A7.16) has solutions r](t) such
that if r] = (r?1 rf), then ^(O * ° for large t and r)\t) = o(\r]\t)\)
as t—>- oo for _/ = 2,..., d. Furthermore, by Exercise 11.2, any such
solution satisfies
r,\t) =[c
exp
g(s)] ds as
oo,
where #(/) is the element in the first row and first column of Y 1G(t) Y.
In order to calculate g(t), note that since the first column of Y is
(A*-1 X, 1), the element in the first row, first column of G{t)Y is
—'ZXd-kplc{t). All elements of G(t)Y not in the first row are 0. Hence,
the upper left corner element of Y^Git) Y is —~ZXd-lcpi{t) times the corre-
corresponding element of Y*1. This element of Y^1 is the cofactor A of the
corresponding element of Y divided by det Y. If the distinct roots of
A7.3) and their multiplicities are X, XB) X(g) and 1, hB) h(g),
Perturbed Linear Systems 319
respectively, then
det r = n [A - X(j)]Mi) IT WO - XU)f(i)Mi);
i=2 2gf<i
see Exercise IV 8.2. The determinant which is the cofactor A has the same
form as det Y, except that X does not occur. It follows that A is the second
of the two products above. Hence
g = -~^i *"-% = - n [* - m-hu) i **-%,
aei x
i.e., A7.21) holds. The relations f = Yr], |* = m<"-*> and the fact that the
first column of Y is (A*-1 A, 1) completes trie proof of Theorem 17.5.
As an illustration of Theorem 17.5, consider the second order equation
A7.17) in which Re A ^ 0 and \q{t)\» dt < oo for some p, 1 ^ p ^ 2.
Then A7.17) has a pair of solutions satisfying
A7.22) u'~±Xu,
— q(s)~\
ds as t -»• oo.
Exercise 17.4. Let <?(/) be real-valued and continuous for / ^ 0,
) -»¦ 0 as t -*¦ oo, and <?(/) of bounded variation for / > 0 [e.g., let
q{t) be monotone or let q(t) have a continuous derivative such that
I |?'@l dt < oo]. Show that {a) u" + [1 + ?(?)]« = 0 has solutions u{t)
satisfying
u' ~ ±/m and m@ ~ exp ±i [1 + ^(s)]^ ds as / -> oo,
and that (o) m" + [—1 + q{t)]u = 0 has solutions u(t) satisfying
u' ~ ±u and m(/) ~ exp ± I [1 — q(s)fA
ds
t ^ oo.
(c) State an analogous result for A7.17) where A ^ 0 and it is not assumed
that A or<7(/) are real-valued; cf. Exercises XI 8.4(b).
Exercise 17.5. In the differential equation
A7.23)
u" -f(t)u = 0,
let /(/) be a continuously differentiable, complex-valued function for
/ ^ 0 such that
A7.24) Re/w(/)*0 and J°°|Re/H(OU/ = oo.
320 Ordinary Differential Equations
(a) Show that if f'\\f \ • |Re fA\ -* 0 as /-* oo, then A7.23) has solutions
satisfying
A7.25) u'~±fA(t)u as f-»oo.
(A) Show that if I IRe/^OI1^1 |/'@//@lp* < °° for some />» * =
/> ^ 2, then A7.23) has solutions satisfying A7.25) and
A7.26) u(t) ~f~y\t) exp ± f /"to <*¦ as / ^ oo,
(c) Show that if f'(t)IJ'Vi(t) is of bounded variation, i.e.,
J"
f'(.t)lfH(.t)->0 as?-* oo, then A7.23) has a pair of solutions satisfying
A7.25) and
A7.27) !<0~
as
For other results of this type, see § XI 9. For analogous results when
Re/1^ s 0, see Exercise XI 8.5.
Exercise 17.6. As a simple application of the last exercise, consider
Weber's equation
A7.28)
u" + tu' - 22m = 0,
where X is a constant, (a) By introducing the new independent variable
5 = i/2, deduce from Theorem 17.4 that A7.28) has a pair of solutions
Mo(O> Mi@ which do not vanish for large / and satisfy u0' ~ — tu0, u{ =
o^Mi) as t ->- oo. (b) Show that A7.28) has a pair of solutions w0, «i
satisfying w0 ~ r1 V2'2, ux ~ ?2A as /—oo. (c) Find asymptotic
relations for derivatives u of solutions u of A7.28) by differentiating A7.28)
and applying (b). (See also Exercise XI 9.7.)
Notes
For references and other treatments of the topics in this chapter, see Cesari [2] and
Bellman [4].
section 1. The main results, Theorems 1.1 and 1.2, are due to Wintner [3], [7], [8],
who gave the existence assertions essentially in the form stated in Exercise 1.2. Linear
cases, where F(t, I) = G{t)? for a matrix C(t), are much older; see Dunkel [1]. For
Exercise 1.1, see Hale and Onuchic [1]; cf. §XII 9. For Exercise 1.4, see Wintner [21].
section 2. Theorem 2.1 was formulated by Wazewski [5] and is a very useful tool
in the study of differential equations. Special cases of this theorem and the arguments
in its proof had been used earlier; cf. Hartman and Wintner [1] or Nemytzkil and
Perturbed Linear Systems 321
Stepanov [1, p. 93]. For another type of topological argument, useful for similar
purposes, cf. Atkinson [2]. Exercise 2.1 is due to Plis [1].
section 3. The results of this section are due to Wazewski [5].
sections 4-7. Lemmas 4.1 and 4.2 are related to results of Wazewski [6], Szmydtowna
[1], Lojasiewicz [1], and Hartman and Wintner [17], [19]. The proofs in the text are
adapted from those of Wazewski and his students just mentioned; for other proofs,
see the papers of Hartman and Wintner. Lemma 4.3 and applications were given in the
papers of Hartman and Wintner. Conditions of the type D.24) were introduced by
Hartman [5]. For Exercise 4.6, see Levinson [3] (for the part dealing with boundedness,
see Cesari [1]); an analogous result (see Exercise 17.4) on a second order equation was
given by Wintner [10]. See Cesari [2, pp. 38-42], for related results and references.
For results related to Exercises 4.8, 4.9 and applications, see Coffman [2].
sections 8-12. Results related to those occurring in § 8 for analytic systems are the
oldest in this chapter and go back to Poincar6 and to Lyapunov [2]. For particular
cases for linear differential equations, see Poincare [4] and Perron [2]. Cotton [1] and
then Perron [9], [10], [12] systematically investigated nonanalytic, nonlinear cases,
but under conditions heavier than those in the text. Their results depended on the method
of successive approximations. See also Bellman [1], who used fixed point theorems to
obtain an analogue of Theorem 8.1, and the references above for §§ 4-7 to Wazewski,
Hartman, and Wintner, etc. The relaxation of the condition "y(f) ->¦ 0 as / -* co" to
D.24) is due to Hartman [5] and to Hartman and Wintner [19]. A form of Theorem
8.2 involving stronger hypothesis and weaker assertions was given by Petrovsky [1].
The last two parts of Corollary 8.1 are proved in Coddington and Levinson [2, Chap. 13]
by the method of successive approximations; cf. Exercise 8.1. For another application
of a related method of successive approximations, see Lillo [1]. The comparatively
simple proofs in the text for Theorems 8.2, 8.3 and Corollary 8.1 are new. Theorem 11.1
is a slight improvement of a result of Lettenmeyer [2]. Theorem 11.2 is given by
Hartman and Wintner [19]. Results of the type in Exercise 11.1 go back to Bdcher [2]
and Dunkel [1]; cf. notes on §§ 13-16 below. Exercise 11.2 was first given by Hartman
[5] for the case of a second order equation (see Theorem 17.5 with d = 2) and generalized
to the situation in Exercise 11.2 by Hartman and Wintner [17].
sections 13-16. Results of the type in Theorem 13.1 were first given by Bdcher [2]
for a second order, linear equation. Using successive approximations similar to those
of Exercise 8.1, Dunkel [1] generalized Bocher's result to arbitrary linear systems A3.1),
where F(/, f) = G(OI, but his results are not as sharp as those given here. Theorems
13.1, 13.2 and their corollaries in § 16 are due to Hartman and Wintner [19]. The
proofs in the text, which take full advantage of Wazewski's principle of § 2, depend in
an essential way on the change of variables A4.1)-A4.2) similar to those introduced by
Hartman and Wintner [17] and simplified by Coffman [2]. See also Olech [1].
section 17. When a = 0, Theorem 17.1 is due to Bdcher [2] for d = 2 and, in a
weakened form, it is contained in Dunkel's result [1 ] for arbitrary d. For a = 0, it is
given by Faedo [1] and Ghizetti [2]. Theorem 17.2 and a less precise form of Theorem
17.3 with a = 0, are also contained in Dunkel [1]; Faedo [1], [2]; and Ghizetti [1].
Theorem 17.4 is a generalization of results of Poincar6 [4] and Perron [2] and is con-
contained in Hartman and Wintner [17]. For Exercise 17.3, see Hartman [5]. Theorem
17.5 for d = 2 is due to Hartman [5]; the result formulated in the text is new. For a
generalization of the case d = 2, see Bellman [3]. Exercise 17.4 is due to Wintner [10],
[13] and is contained in the more general result of Exercise 4.6; cf. also Exercise XI
8.4F). Results of the type in Exercise 17.5(a) go back to Wiman [1 ], [2]; for both parts
(a) and F), see Hartman and Wintner [17].
I
Chapter XI
Linear Second Order Equations
1. Preliminaries
One of the most frequently occurring types of differential equations in
mathematics and the physical sciences is the linear second order differential
equation of the form
A.1) u" + g(t)u'+ f(t)u =
or of the form
A.2) (p(t)u')' + q{t)u ¦-
Unless otherwise specified, it is assumed that the functions f(t), g(t), h(t),
and p{t) j? 0, q{t) in these equations are continuous (real- or complex-
valued) functions on some /-interval /, which can be bounded or un-
unbounded. The reason for the assumptionp(t) j± 0 will soon become clear.
Of the two forms A.1) and A.2), the latter is the more general since
A.1) can be written as
A.3) 0>@«')' + p(t)Rt)u = p(t)h{t),
if p{t) is defined as
A.4) p{t) = exp g(s) ds
for some aeJ. As a partial converse, note that if p{t) is continuously
differentiate then A.2) can be written as
u =
which is of the form A.1).
When the function p(t) is continuous but does not have a continuous
derivative, A.2) cannot be written in the form A.1). In this case, A.2)
is to be interpreted as the first order, linear system for the binary vector
322
A.5)
**'=¦*
Pit)
Linear Second Order Equations 323
h{t).
In other words, a solution u = u(t) of A.2) is a continuously differentiate
function such that p(t)u'{t) has a continuous derivative satisfying A.2).
When p{t) j? 0, q(t), h(t) are continuous, the standard existence and
uniqueness theorems for linear systems of §IV 1 are applicable to A.5),
hence A.2). [We can also deal with more general (i.e., less smooth) types
of solutions if it is only assumed, e.g., that \jp(t), q{t), h{t) are locally
integrable; cf. Exercise IV 1.2.]
The particular case of A.2) where p(t) = 1 is
A-6) u" + q{t)u = h(t).
When p{t) ^ 0 is real-valued, A.2) can be reduced to this form by the
change of independent variables
A.7)
A dt
ds = — ,
Pit)
i.e., s = — + const.
for some a eJ. The function 5 = s(t) has a derivative dsjdt = l/p(t) ^ 0
and is therefore strictly monotone. Hence 5 = s(t) has an inverse function
/ = t(s) defined on some ^-interval. In terms of the new independent
variable s, the equation A.2) becomes ¦
A.8)
p{t)h(t),
where t in p{t)q{t) and p(t)h(t) is replaced by the function t = t(s). The
equation A.8) is of the type A.6).
If g(t) has a continuous derivative, then A.1) can be reduced to an
equation of the form A.6) also by a change of the dependent variable
u ¦—¦ z defined by
A-9) M = zex
for some aeJ. In fact, substitution of A.9) into A.1) leads to the equation
A.10) z" + |/@ - &1 - i^y = h{t) exp
which is of the type A.6).
In view of the preceding discussion, the second order equations to be
considered will generally be assumed to be of the form A.2) or A.6). The
following exercises will often be mentioned.
324 Ordinary Differential Equations
Exercise 1.1. (a) The simplest equations of the type considered in this
chapter are
A.11) m"=0, u"-a2u = 0, u" + a2u = 0,
where a ^ 0 is a constant. Verify that the general solution of these
equations is
A.12) u = Cj + c2t, u = creai + c2e~at, u = cx cos at + c2 sin at,
respectively, (b) Let a, b be constants. Show that u = e** is a solution of
A.13) u" + bu' + au = 0,
if and only if X satisfies
A.14) X* + bX + a = 0.
Actually, the substitution u = ze~m [cf. A.9)] reduces A.13) to
z" + a*z = 0, a* = a- W.
Hence by (a) the general solution of A.13) is
A.15) u = e~m\c-L + c2t) or u = c^1* + c2eXit
according as A.14) has a double root X = \b or distinct roots Xlt X2 =
-\b ± (W - a)l/\ When a, b are real and \b2 - a < 0, nonreal
exponents in the last part of A.15) can be avoided by writing
A.16) u = e-w/2[c! cos (a - W)VH + c2 sin (a -
(c) Let [x be a constant. Show that u = tx is a solution of
A.17) u" + ?u = 0
if and only if X satisfies
A.18) X(X—l) + /x = 0, i.e., X = f ± (J -
Thus if ^ 7>* 1, the general solution of A.17) is
A.19) u = c^1 + c2tx\ ix?^\, and Al5 X2 = | ± A — /xIA.
If ^ is real and [i > \, the nonreal exponents can be avoided by writing
A.20) u = iA\cx cos (/x - if4 log t + c2 sin {[x - IJA log t].
Actually, the change of variables u = tv*z and / = es transforms A.17)
into
A.21)
71 + 0« " Dz = 0-
as
Linear Second Order Equations 325
Thus by (a) the general solution of A.17) is
A.22) u = tV\cx + c2log t) or u = cxtki + c2ix*
according as /x, = J or ^ # J.
Exercise 1.2. Consider the differential equation
A.23) u"+q(t)u = 0.
The change of variables
A.24) t = es and u = ?«a
transforms A.23) into
A.25) ^5 +
_ ±1 = o,
4/2J
where / = e\
For a given constant ^a, consider the sequence of functions
r2(i + ^ log-
denned by t*[qn(t) - 1/4/2] = q^s) if / = e\ so that ?n(/) = r2[J +
?n-i(log 0] or
9»@ = r2 [i 2[ (ft log, ?j+ ^ (H log, ^] for n ^ 1,
[ (
l°gi t = log /, log,/ = log (log^!/), and the empty product is 1. If
q(t) =qn{t), n > 0, in A.23), then the change of variables A.24) reduces
A.23) to the case where /, qn(t) are replaced by s, qn^(s). In particular, if
/x is real and q = qn(t), n ^ 0, then real-valued solutions of A.23) have
infinitely many zeros for large t > 0 if and only if fx > |.
2. Basic Facts
Before considering more complicated matters, it is well to point out the
consequences of Chapter IV (in particular, § IV 8) for the homogeneous
and inhomogeneous equation
B-1) QKOt/y + q(t)u = 0,
B.2) (p(t)Wy + q(t)w = h{t).
To this end, the scalar equations B.1) or B.2) can be written as the binary
vector equations
B.3) x' = A(t)x,
0
B.4)
y' = A(t)y +
M0I,
326 Ordinary Differential Equations
where x = (x1, x2),y = (y1, y2) are the vectors x = (u,p(t)u'),y = (w,p(t)w')
and A(t) is the 2x2 matrix
B.5)
A(t)
Pit)
Unless the contrary is stated, it is assumed that p(t) j* 0, q(t), h(t), and
other coefficient functions are continuous, complex-valued functions on a
^-interval J (which may or may not be closed and/or bounded).
(i) If toeJ and u0, u0' are arbitrary complex numbers, then the initial
value problem B.2) and
B.6) w(t0) = u0, w'(t0) = u0'
has a unique solution which exists on all of/; Lemma IV 1.1.
(ii) In the particular case B.1) of B.2) and m0 = u0' = 0, the correspond-
corresponding unique solution is u(t) = 0. Hence, if u(t) ^ 0 is a solution of B.1),
then the zeros of u(t) cannot have a cluster point in /.
(iii) Superposition Principles. If u(t),v(t) are solutions of B.1) and
Ci, c2 are constants, then cxu{t) + c2v(t) is a solution of B.1). If wo(t) is a
solution of B.2), then w^t) is also a solution of B.2) if and only if u =
wi(t) — wo(t) is a solution of B.1).
(iv) If u(t), v(t) are solutions of B.1), then the corresponding vector
solutions x = (u(t), p(t)u'(t)), (v(t), p(t)v'(t)) of B.3) are linearly independent
(at every value of t) if and only if u(t), v(t) are linearly independent in the
sense that if clt c2 are constants such that c^t) + c2v(t) = 0, then
Cl = c2 = 0; cf. § IV 8(iii).
(v) If u(t), v(t) are solutions of B.1), then there is a constant c, depending
on u(t) and v(t), such that their Wronskian W(t) = W(t;u,v) satisfies
B.7)
u(t)v'(t) - u'(t)v(t) = —
This follows from Theorem IV 1.2 since a solution matrix for B.3) is
u(t) v(t)
X(t)
p(t)u'(t) p{t)v\t\
det X(t) = p(t)W(t) and tr A(t) = 0; cf. §IV 8(iv). A simple direct
proof is contained in the following paragraph,
(vi) Lagrange Identity. Consider the pair of relations
B.8)
(pu')' +qu=f, ipv')' + qv=g,
Linear Second Order Equations 327
where f = f(t), g = g(t) are continuous functions on /. If the second is
multiplied by u, the first by v, and the results subtracted, it follows that
B.9) [p(uv' - «'»)]' = gu -fv
since [p(uv' - u'v)]' = u(pv')' - v(pu')'. The relation B.9) is called the
Lagrange identity. Its integrated form
B.10) [p(uv' - u'v)]: = (\gu - fv) ds,
Ja
where [a, t] <= /, is called Green's formula.
(vii) In particular, (v) shows that u(t) and v(t) are linearly independent
solutions of B.1) if and only if c # 0 in B.7). In this case every solution
of B.1) is a linear combination cxu(t) + c2v(t) of u(t), v(t) with constant
coefficients.
(viii) If p(t) = const, [e.g., p(t) =1], the Wronskian of any pair of
solutions u(t), v(t) of B.1) is a constant.
(ix) According to the general theory of § IV 3, if one solution of u(t) ^ 0
of B.1) is known, the determination (at least, locally) of other solutions
v(t) of B.1) are obtained by considering a certain scalar differential equa-
equation of first order. If u(t) ^ 0 on a subinterval /' of /, the differential
equation in question is B.7), where u is considered known and v unknown.
If B.7) is divided by u2(t), the equation becomes
B-U) V" P(t)u\t)
and a quadrature gives
B.12) v(t) = clU(t) + cu{t) f -^— ,
Ja P(S)U\S)
if a, teJ'; cf. § IV 8(iv). It is readily verified that if clt c are arbitrary
constants and a, t eJ', then B.12) is a solution of B.1) satisfying B.7) on
any interval J' where u(t) ^ 0.
(x) Let u(t), v(t) be solutions of B.1) satisfying B.7) with c # 0. For a
fixed seJ, the solution of B.1) satisfying the initial conditions u(s) = 0,
p{s)u'(s) = 1 is c-^kCsXO — u(t)v(s)]. Hence the solution of B.2)
satisfying w(t0) = w\t0) = 0 is
B.13)
w(t) = c-1 [u(s)v(t) - u(t)v(s)]h(s) ds;
cf. § IV 8(v) (or, more simply, verify this directly). The general solution of
B.2) is obtained by adding a general solution c^t) + c2v(t) of B.1) to
B.13) to give
B.14) w(t) = w(o|c, - c-11 v(s)h{s) ds\ + v(t)\c2 + c! u(s)h(s) ds .
328 Ordinary Differential Equations
If the closed bounded interval [a, b] is contained in J, then the choice
t0 = a, c1= c~x v(s)h(s) ds and c2 = 0
Jo
reduces B.14) to the particular solution
B.15) h>@ = c-tfvit) f u(s)h(s) ds + u(t) \ v(s)h(s) ds\
L Ja Jt J
This can be written in the form
B.16) w(t) = \hG(t, s)h(s) ds,
Ja
where
G(t, s) = c-MO"t*) if a <j s < t,
B.17)
G(t, s) = <rhi(f)v(s) if t <k s ^ b.
Remark. If h(t) is (not necessarily continuous but) integrable over
[a, b], then w(t) is a "solution" of B.2) in the sense that w(t) has a con-
continuous derivative w' such that p(t)w'(t) is absolutely continuous and
B.2) holds except on a t-stt of measure 0.
Exercise 2.1 Verify that if a, E, y, d are constants such that
xu(a) + Pp(a)u'(a) = 0, yv(b) + dp(b)v'(b) = 0,
then the particular solution B.15) of B.2) satisfies
a.w{a) + pp(a)w'(a) = 0, yw(b) + dp(b)w'(b) = 0.
An extremely simple but important case occurs if p = 1, q = 0 so that
B.1) becomes w" = 0. Then u(t) = t — a and v(t) = b — t are the
solutions of B.1) satisfying u{a) = 0, v(b) = 0, and B.7) with c = a — b.
Hence
B.18)
w{t) =
a — b
- 0 | (s - a)h(s) ds + (t - a) \ (b - s)h(s) ds
Ja Jt
is the solution of w" = h(t) satisfying w(a) = w(b) = 0.
Exercise 2.2. Let [a, b] <= /. Show that most general function G(t, s)
defined for a < s, t < b for which B.16) is a solution of B.2) for a ^t<b
for every continuous function h(t) is given by
G(t, s) =
*=1 i=\
if a ^ s <: t,
if t^s<b,
Linear Second Order Equations 329
where A = (ajk), B = (bjk) are constant matrices such that
B- A =
0 1
-1 0
and «! = u(t), u2 = v(t) are solutions of B.1) satisfying B.7) with c j? 0.
In this case, G(t, s) is continuous for a ^ s, t ^ b.
Exercise 2.3. Let a (and/or b) be a possibly infinite end point of /
which does not belong to /, so that p(t), q{t), h{t) and u(t), v(t) need not
have limits as t —>¦ a + 0 (and/or t —>¦ b — 0). Suppose, however, that
h, u, v have the property that the integrals in B.15) are convergent (possibly,
just conditionally). Then B.15) is a solution of B.2) on /. [This follows
from the derivation of B.15) or can be verified directly by substituting
B.15) into B.2).]
(xi) Variation of Constants. In addition to B.1), consider another
equation
B.19) (po(t)w')' + qo(t)w = 0,
where po(t) ^ 0, qo(t) are also continuous in /. Correspondingly, B.19) is
equivalent to a first order system
B.20) y' = A0(t)y,
where
B.21) y = (u,Po(t)u') and A0(t) = [
Let uo(t), vo(t) be linearly independent solutions of B.19) such that
B.22) Y(t)=[
\/>o"o PoV o /
is a fundamental matrix for B.20) with det Y(t) = 1; i.e.,
/>o(*W - "o'fo) = 1-
Hence
B.23) ( ,
-/>o"o "o
Consider the linear change of variables
B.24)
x = Y(t)y =
\/>o"o V1 + PoVo 2/7
for the system B.3). The resulting differential equation for the vector y is
B.25) y' = C(t)y, where C(t) = Y~\t){A{t) - A0(t)]Y(t);
330 Ordinary Differential Equations
cf. Theorem IV 2.1. A direct calculation using B.5), B.21), B.22), and
B.23) shows that
B.26,
I 4- (a — an)l
2
\P PJ' \-u'» -UO'VO'/ \-U0z -U0V
In the particular case, po(t) = p(t), so that B.19) reduces to
B.27) (pw')' + qow = 0,
the matrix C(t) depends on uo(t), vo(t) but not on their derivatives. Here,
B.1) or equivalently B.3) is reduced to the binary system
B.28)
2
-u02 -uovo
Exercise 2.4. In order to interpret the significance of y, i.e., of the
components y1, y2 ofy in B.28) for a corresponding solution u(t) of B.1),
write B.1) as (pw')' + qow = h(t), where w = u(t), h = [qo(t) - q(t)]u(t).
Then it is seen that the solution u(t) of B.1) is of the form B.14) if c = 1
and u(t), v(t) are replaced by uo(t), vo(t). Using B.24), where p = p0 and
x is the binary vector (u(t),p(t)u'(t)), show that the coefficients of uo(t),
vo(t) in this analogue of formula B.14) are the component y1, y2 of the
corresponding solution y(t) of B.28).
(xii) If we know a particular solution uo(t) of B.27) which does not
vanish on /, then we can determine linearly independent solutions by a
quadrature [cf. (ix)] and hence obtain the matrix in B.28). Actually this
desired result can be obtained much more directly. Let B.27) have a
solution w(t) j? 0 on the interval J. Change the dependent variable from
u to z in B.1), where
B.29)
u = w(t)z.
The differential equation satisfied by z is
w(pz')' + Ipz'w' + [{pw')' + qw]z = 0.
If this is multiplied by w, it follows that
B.30) (pwh')' + w[(pw')' + qw]z = 0
or, by B.27),
B.31) (pwH1)' + w\q - qo)z = 0;
i.e., B.29) reduces B.1) to B.30) or B.31). Instead of starting with a
differential equation B.27) and a solution w(t), we can start with a
function w(t) j? 0 such that w(t) has a continuous derivative w'(t) and
p(t)w'(t) has a continuous derivative, in which case ?„(/) is defined by B.27),
Linear Second Order Equations 331
so q0 = —(pw')'lw. The substitution B.29) will also be called a variation
of constants.
(xiii) Liouville Substitution. As a particular case, consider B.1) with
pit) = i,
B.32) u" + q(t)u = 0.
Suppose that q(t) has a continuous second derivative, is real-valued, and
does not vanish, say
B.33) ±q(t)>0, where ±=sgn?@
is independent of t. Consider the variation of constants
B.34) u = w(t)z, where w = \q(t)\-Vl > 0.
Then B.32) is reduced to B.30), where p = 1, i.e., to
B.35)
A change of independent variables t —»¦ s defined by
dt
B.36)
transforms B.35) into
B.37)
where
ds =
\d\
lA
5q'2
and the argument of q and its derivatives in B.38) is t = t{s), the inverse of
the function s = s{t) defined by B.36) and a quadrature; cf. A.7). In
these formulae, a prime denotes differentiation with respect to t, so that
q = dqldt.
The change of variables B.34), B.36) is the Liouville substitution. This
substitution, or repeated applications of it, often leads to a differential
equation of the type B.37) in which/(j) is "nearly" constant; cf. Exercise
8.3. For a simple extreme case of this remark, see Exercise 1.1 (c).
(xiv) Riccati Equations. Paragraphs (xi), (xii), and (xiii) concern the
transformation of B.1) into a different second order linear equation or
into a suitable binary, first order linear system. (Other such transfor-
transformations will be utilized later; cf. §§ 8-9.) Frequently, it is useful to
transform B.1) into a suitable nonlinear equation or system. In this
direction, one of the most widely used devices is the following: Let
B.39)
,. _ P(t)W
332 Ordinary Differential Equations
so that r' = (pu')'ju — p^ipu'juf. Thus, if B.1) is divided by u, the result
can be written as
B.40)
r' + — + q(t)r = 0.
Pit)
This is called the Riccati equation of B.1). (In general, a differential
equation of the form r = a(t)r2 + bit)r + c@, where the right side is a
quadratic polynomial in r, is called a Riccati differential equation.)
It will be left to the reader to verify that if u{t) is a solution of B.1)
which does not vanish on a ^-interval /'(¦=/), then B.39) is a solution of
B.40) on/'; conversely if r = r(t) is a solution of B.40) on a ^-interval /'
(c J), then a quadrature of B.39) gives
J'risjds
J i
B.41)
u = c exp
a nonvanishing solution of B.1) on /'.
Exercise 2.5. Verify that the substitution r = u'fu transforms
u"+g(t)u' +f(t)u = 0
into the Riccati equation
r' + r*+g(t)r+f(t) = 0.
(xv) Priifer Transformation. In the case of an equation B.1) with real-
valued coefficients, the following transformation of B.1) is often useful
(cf. §§ 3, 5): Let u = u(t) ^ 0 be a real-valued solution of B.1) and let
B.42)
p2u'2f > 0,
w = arc tan — .
pu'
Since u and u cannot vanish simultaneously a suitable choice of <p at
some fixed point t0 e/and the last part of B.42) determine a continuously
differentiable function (pit). The relations B.42) transform B.1) into
B.43)
B.44)
<p' = cos2 <p + q(t) sin2 <p,
Pit)
P' = - lit) —\P sin <p cos <p.
The equation B.43) involves only the one unknown function <p. If a
solution <p = <pit) of B.43) is known, a corresponding solution of B.44)
is obtained by a quadrature.
An advantage of B.43) over B.40) is that any solution of B.43) exists on
the whole interval / where p, q are continuous. This is clear from the
relation between solutions of B.1) and B.43).
Linear Second Order Equations 333
Exercise 2.6. Verify that if i{t) > 0 is continuous on / and is locally of
bounded variation (i.e., is of bounded variation on all closed, bounded
subintervals of/) and if u = uit) ^ 0 is a real-valued solution of B.1),
then
B.45)
p = (TV + p2u'T > 0, <p = arc tan —
pu'
and a choice of <p(t0) for some t0 e / determine continuous functions
pit), (pit) which are locally of bounded variation and
B.46) d(p = I - cos2 (p + - sin2 (p) dt + (sin q> cos q>) d(log t)
\p t /
B.47) d(log p) = — I" — "I sin 9? cos 9? df + (sin2 cp) rf(logt),
The relations B.46), B.47) are understood to mean that Riemann,
Stieltjes integrals of both sides of these relations are equal. Conversely
(continuous) solutions of B.46)-B.47) determine solutions of B.1), via
B.45). Note that if qit) > 0, p{t) > 0, and qii)p{i) is locally of bounded
variation, then the choice t@ = p1Ait)q'Ait) > 0 gives q\r = r\p =
p'A\q'A and reduces B.45) and B.46), B.47) to
B.48) p = ipqu2 + p2u'SA > 0, q> = arc tan^JL ,
^'
and
B.49)
B.50)
aA
d <p = \ dt + (| sin <p cos <p) d(log pq),
P
d(log p) = A sin2 <p) di\ogpq).
3. Theorems of Sturm
In this section, we will consider only differential equations of the type
B.1) having real-valued, continuous coefficient functions p(t) > 0, qit).
"Solution" will mean "real-valued, nontrivial (^ 0) solution." The
object of interest will be the set of zeros of a solution uit). For the study of
zeros of uit), the Priifer transformation B.42) is particularly useful since
m(/0) = 0 if and only if q>(t0) = 0 mod n.
Lemma 3.1. Let uit) ^ 0 be a real-valued solution of B.1) on t0 ^ t ^ t°,
where pit) > 0 and qit) are real-valued and continuous. Let u(t) have
exactly n (^ 1) zeros tl < t2 < • ¦ ¦ < tn on t0 < t ^ t°. Let q>(t) be a
continuous function defined by B.42) and 0 ^ q>iQ < 77. Then (p(tk) = kn
and tp(t) > kir for tk < t ^ t°for k = 1, . .., n.
334 Ordinary Differential Equations
Proof. Note that at a f-value where u = 0, i.e., where <p = 0 mod n,
B.43) implies that <p' = p(t) > 0. Consequently <p(t) is increasing in the
neighborhoods of points where <p(t) = jn for some integer j. It follows
that if t0 ^ a _ t° andjn < <p{a), then <p(t) >jn for a < t < t°; also if
y77 _ <p(a), then <p@ <J77 for t0 ^ t < a. This implies the assertion.
In the theorems of this section, two equations will be considered
"')' + ?,(*)« = 0, 7=1,2,
C.1;)
where />/0> <7;@ are real-valued continuous functions on an interval /,
and
C.2)
and qx(t) = q2(t).
In this case, C.12) is called a Sturm majorant for C.10 on / and C.10 is a
Starm minorant for C.1a). If, in addition,
Pi(t) > pJit) > 0 and
C.30
or
C.32)
holds at some point t of/, then C.12) is called a sfncf Sturm majorant for
C.10 on/.
Theorem 3.1 (Sturm's First Comparison Theorem). Let the coefficient
Junctions in C.1,) be continuous on an iritervalJ:t0 _¦ t ^ f° and let C.12)
be a Sturm majorant for C.10- ?ef" = "i@ &0bea solution o/C.10 and
/ef Mi@ toe exacf/y n (=1) zeros t = tx < t2 < ¦ ¦ • < tn on t0 < t ^ t°.
Let u = u2(t) ^Obea solution o/C.12) satisfying
C.4)
p2(t)u2'(t)
at t = t0. (The expression on the right [or left] of C A) att = tois considered
to be +oo ifu2(t0) = 0 [or ufa) = 0]; in particular, C.4) holds at t = t0
ifui(to) = °-) Then u2(t) has at least n zeros on t0 < t ^ tn. Furthermore
u2(i) has at least n zeros on t0 < t < tn if either the inequality in C.4) holds
at t= t0 or C.12) is a strict Sturm majorant for C.10 on t0 _ f _ tn.
Proof. In view of C.4), it is possible to define a pair of continuous
functions ^(O, 9>2@ on t0 =¦ t ^ t° by
C.5)
Then the analogue of B.43) is
and 0 ^
C.6,.)
<p/ = cos2 <p} + <?/0 sin2 <pt = fj (t,
Linear Second Order Equations 335
Since the continuous functions f{t, <p}) are smooth as functions of the
variable <p;, the solutions of C.6) are uniquely determined by their initial
conditions. It follows from C.2) that f^t, <p) ^f2(t, <p) for t0 _ t _¦ t° and
all q>. Hence the last part of C.5) and Corollary III 4.2 show that
C.7)
^ <p2(t) for *„<*<
In particular, <?>i0n) = nn implies that nn ^ <p2(tn) and the first part of the
theorem follows from Lemma 3.1.
In order to prove the last part of the theorem, suppose first that the
sign of inequality holds in C.4) at t = t0. Then (p^) < <p2(t0). Let <?>2o(O
be the solution of C.62) satisfying the initial condition <?>2oOo) = •PiOoX so
that <p20(t0) < <P2(t0)- Since solutions of C.62) are uniquely determined by
initial conditions, <?>2o(O < 9?a@ for t0 ^ t ^ t°. Thus the analogue of
C.7) gives 9^@ _¦ ^20@ < ^(O. and so <p2(tn) > rnr. Hence u2(t) has n
zeros on t0 < t < tn.
Consider the case that equality holds in C.4) but either C.30 or C-32)
holds at some point of [t0, tn]. Write C.62) as
where
2 = — cos2 <p2 + qx sin2 <p2 + e(t),
Pi
., /I 1\ 2 , ^ • 2 ^
e@ = l~ 1 cos <p2 + (q2 — <?0 sin2 cp2 ^ 0.
\P2 Pi/
If the assertion is false, it follows from the case just considered that
9>i@ = "M0 for t0 _: r _ tn. Hence, ^'@ = <p2'(t) and so e@ = 0 for
to = t = tn- Smce sin <p2(t) = 0 only at the zeros of w2@> it follows that
^2@ = ?i@ f°r {o = t^tn and that (p^1 — pf1) cos2 <p2 = 0. Hence,
Pt\t) — Pix(t) > 0 at some t implies cos <p2(t) = 0; i.e., u2 = 0. If C.30
does not hold at any t on [t0, tn], it follows that C.32) holds at some t and
hence on some subinterval of [t0, tn]. But then u2 = 0 on this interval,
thus ip2u2')' = 0 on this interval. But this contradicts q2(t) # 0 on this
interval. This completes the proof.
Corollary 3.1 (Sturm's Separation Theorem). Let C.12) be a Sturm
majorant for C.1 x) on an interval J and let u = ufa) ^ 0 be a real-valued
solution o/C.ly). Let ux(t) vanish at a pair of points t = tu t2 (>?0 of J.
Then u2(t) has at least one zero on [tlt t2]. In particular, ifpx = p2, qx = q2
and m1( u2 are real-valued, linearly independent solutions o/C.10 = C.12),
then the zeros ofux separate and are separated by those of u2.
Note that the last statement of this theorem is meaningful since the
zeros of m1( m2 do not have a cluster point on /; see § 2(ii). In addition,
"i(')> w»@ cannot have a common zero t = tx; otherwise, the uniqueness of
336 Ordinary Differential Equations
the solutions of C.1!) implies that ux(t) = cu2(t) with c = «i'(*i)/M2'('i)
[so that u^t), u2(t) are not linearly independent].
Exercise 3.1. (a) [Another proof for Sturm's separation theorem
whenp^) = p2(t) > 0,q2(t) ^ q^t).] Suppose that u^t) > Ofor^ < t <
t2 and that the assertion is false, say u2(t) > 0 for ^ < f ^ t2. Multiplying
C.10 where u = ux by u2 and C.12) where u = u2 by Mi, subtracting, and
integrating over [f1( t] gives
/>@(«i'w2 - Wi') ^0 for ^ < f 5j f2,
wherep =pi =p2; cf. the derivation of B.9). This implies that («i/m2)' ^
0; hence uju2 > 0 for h < t ^ f2 (Z>) Reduce the case /^(f) ^ />2@ to the
case px{t) = p2(t) by the device used below in the proof of Corollary 6.5.
Exercise 3.2. (a) In the differential equation
C.8)
u" + q{t)u = 0,
let q(t) be real-valued, continuous, and satisfy 0 < m ^ q(t) ^ M. If
u = u(t) ^0 is a solution with a pair of zeros t =tu t2(> tj), then
Tr\mA ^ t2 — tx ^ njM1A. (b) Let q(t) be continuous for t^.0 and
<7(f) ->• 1 as t ->¦ oo. Show that if m = w(f) ^ 0 is a real-valued solution of
C.8), then the zeros of u(t) form a sequence @ <) tx < f2 < ... such that
tn — tn__1 —>¦ 77 as n —*¦ oo. (c) Observe that real-valued solutions u(t) ^ 0
of A.17) have at most one zero for f > 0 if fi ^ J and have infinitely
many zeros for t > 0 if fi > J. In the latter case, the zeros cluster at
t = 0 and t = oo. (i) Consider the Bessel equation
C.9) v" + V- H
where /i is a real parameter. The variations of constants u = t'Av trans-
transforms C.9) into
C.10) u" + (l — -\u = 0, where a =// — {.
Show that the zeros of a real-valued solution v(t) of C.9) on t > 0 form a
sequence tx< t2< ... such that fn — tn_x ->• tt as n ->• oo.
Theorem 3.2 (Sturm's Second Comparison Theorem). ,4jswwe the
conditions of the first part of Theorem 3.1 and that u2(t) also has exactly n
zeros on t0 < t ^ t°. Then C.4) /10W5 at t = t° (where the expression on
the right [or left] of C.4) at t = t° is taken to be + co if u2(t°) = 0 [or
u^t0) = 0]). Furthermore the sign of inequality holds at t = t° in C.4) if
the conditions of the last part of Theorem 3.1 hold.
Proof. The proof of this assertion is essentially contained in the proof
of Theorem 3.1 if it is noted that the assumption on the number of zeros of
Linear Second Order Equations 337
u2(t) implies the last inequality in rnr ^ y^t0) ^ <p2(t°) < (n + l)n. Also,
the proof of Theorem 3.1 gives (p^t0) < <p2(t°) under the conditions of the
last part of the theorem.
4. Sturm-Liouville Boundary Value Problems
This topic is one of the most important in the theory of second order
linear equations. Since a full discussion of it would be very lengthy and
since very complete treatments can be found in many books, only a few
high points will be discussed here.
In the equation
D.1 A)
[q(t) + X\u = 0,
let p(t) > 0, q(t) be real-valued and continuous for a ^ t ^ b and X a
complex number. Let a, E be given real numbers and consider the problem
of finding, if possible, a nontrivial (^0) solution of D.U) satisfying the
boundary conditions
D.2) u(a) cos a - p(a)u'(a) sin a = 0, u(b) cos E - p(b)u'(b) sin E = 0.
Exercise 4.1. Show that if X is not real, then D.U) and D.2) do not
have a nontrivial solution.
Exercise 4.2. Consider the following special cases of D.1/1), D.2):
D.3) u" + hi = 0, w@) = m(tt) = 0.
Show that this has a solution only if X = (n + IJ for n' = 0, 1, . . . and
that the corresponding solution, up to a multiplicative constant, is
u = sin (n + \)t.
It will be shown that the results of Exercise 4.2 for the special case D.3)
are typical for the general situation D.1 A), D.2).
Theorem 4.1. Let p(i) > 0, q(t) be real-valued and continuous for
a ^ t ^b. Then there exists an unbounded sequence of real numbers Xo <
Xx < . . . such that (i) D.1 A), D.2) has a nontrivial (^0) solution if and
only ifX = Xnfor some n; (ii) if X = Xn and u = un(t) ^ 0 is a solution of
D.UJ, D.2), then un(t) is unique up to a multiplicative constant, and un(t)
has exactly n zeros on a < t < b for n = 0, 1,. .. ; (iii) if n j± m, then
D.4)
«»@«,
Ja
it) dt = 0;
(iv) if X is a complex number X j± Xnfor n = 0, 1,. . ., then there exists a
continuous function G(t, s; X) = G(s, t; X) for a ^ s, t ^ b with the
property that if h(t) is any function integrable on a ^ t ^ b, then
D.51) (p(t)w')' + [q(t) + X]w = h(t)
338 Ordinary Differential Equations
has a unique solution w = u(t) satisfying
D.2') w(a) cos a — p(a)w'(a) sin a = 0, w(b) cos C — p(b)w'(b) sin E = 0
D.6)
also G(t, s; X) is real-valued when X is real; (v) if X = Xn and h(t) is a
function integrable on a ^ t ^ b, then D.5AJ, D.2') has a solution if and
only if
D.7)
Ja
(t)h(t) dt = O;
in this case, if w(t) is a solution ofD.5Xn), D.2'), then w(t) + cun{t) is also
a solution and all solutions are of this form; (vi) if the functions un(t) are
chosen real-valued (uniquely up to a factor ±1) so as to satisfy
D.8)
(bun\t)dt=l,
Ja
then uo(t), Mj@, ¦ • • form a complete orthonormal sequence for L2(a, b);
i.e., if h(t) e L2(a, b), then h(t) has the Fourier series
D.9)
and
D.10)
where c
n = \
Ja
h(t)un(t) dt
f
Ja
K0-Ickuk(t)
dt-^O
as n
00.
If h{t) is not continuous in (iv) or (v), then a solution of D.5/1) is to be
interpreted as in the Remark in § 2(x).
Note the parallel of the assertions concerning the solvability of D.5A),
D.2') with the corresponding situation for linear algebraic equations
(XI — L)w = h, where L is a d X d Hermitian symmetric matrix, / is the
unit matrix and w, h vectors: (XI — L)u = 0 has a solution u j^ 0 if and
only if X is an eigenvalue Xx, . . . , Xd of L; Xx, . . . , Xd are real; if X j± Xn,
then (XI — L)w = h has a unique solution w for every h; finally, if X = Xn,
then (XI — L)w = h has a solution w if and only if h is orthogonal (i.e.,
u ¦ h = 0) to all solutions u of (XI — L)u = 0.
Proof. This proof will only be sketched; details will be left to the
reader.
On (/) and (it). In view of Exercise 4.1, it suffices to consider only real
X. Let u(t, X) be the solution of D.1 X) satisfying the initial condition
D.11) u(a) = sin a, p(d)u'(d) = cos a,
so u(t, X) satisfies the first of the two conditions D.2). It is clear that
Linear Second Order Equations 339
D.1/1), D.2) has a solution (^ 0) if and only if u(t, X) satisfies the second
condition in D.2).
For fixed X, define a continuous function q{t, X) of t on [a, b] by
u(t, X)
D-12)
cp(t, X) = arc tan
<p(a, X) = a.
Then y(t, X) has a continuous derivative satisfying
D.13) <p'=~ cos2 cp + [q(t) + X] sin2 <p,
<p(a) = a;
cf. § 2(xv). If follows from Theorem V 2.1 that the solution 9? = q{t, X)
of D.13) is a continuous function of (t, X) for a ^ t ^ b, — 00 < X < 00.
The proof of the Sturm Comparison Theorem 3.1 shows that y(b, X) is an
increasing function of X. Without loss of generality, it can be supposed
that a satisfies 0 ^ a < n. Note that
D-14)
b, X)
00
as
00.
In order to see this, introduce the new independent variable defined by
ds = dtjp(t) and s(a) = 0, so that D.1 X) becomes
D.15) u + p(t)[q(t) + X]u = 0,
= t(s),
ds
If M > 0 is any number, X > 0 can be chosen so large that p(t)[q(t) +
X] ^ M2 for a ^ t ^ b. Sturm's Comparison Theorem 3.1 applied to
shows that if n is arbitrary and M is sufficiently large, then a nontrivial
real-valued solution of D.15) has at least n zeros on the s-interval, 0 ^
s ^ dt/p(t); i.e., y(b, X) ^ n if X > 0 is sufficiently large by Lemma 3.1.
Ja
It will be verified that
D.16) <p(b,X)-+0 as
— 00.
By Lemma 3.1, q>(b, X) ^ 0. Let -X > 0 be so large thatp(t)[q(t) + X] <
-M2 < 0. The solution of
u - M2u = 0
satisfying the analogue of D.11), where a = 0 andp = 1, is
u(s) = sin a cos/i Ms -\ cos a sin/i Ms.
The analogue of <p(t, X) is
y>(s, M) = arc tan , y@, M) = a.
u(s)
340 Ordinary Differential Equations
For any fixed s > 0,
• 0
u(s)
hence y>(b0, M)—*-0 as M—*- oo, where b,
as M —»¦ oo;
.„ = [dtipit).
Ja
By Sturm's Com-
parison Theorem 3.1, <p(b, X) ^ y>(b0, M). This proves D.16)
The limit relations D.14), D.16) and the strict monotony of <p(b, X) as
a function of X show that there exist Xo, Xx,... such that
<p(b, Xn) = p + tin for n = 0, 1, . . .,
where it is supposed that 0 < (i ^ n. Furthermore <p(b, X) j? (i mod n
unless X = Xn. This implies (i) and (ii).
On (Hi). In order to verify (iii), multiply D.1 An) by um, D.Um) by un,
subtract and integrate over a ^ t ^ b\ i.e., apply the Green identity
B.10) to/ = -Xnun(t), g = -Xmum(t).
On (iv). See § 2(x) and Exercise 2.1. Choose u = u(t, X), and v(t) as a
solution of D.1 X) satisfying the second condition in D.2).
On (v). Suppose first that D.5AJ, D.2') has a solution w = w(t).
Apply the Green identity B.10) in the case where q is replaced by q +
Xn,f= h, w = u, v = un, g = 0 in B.8) in order to obtain D.7).
Conversely, assume that D.7) holds. Let u(t) = un(t) and let v(t) be a
solution of D.UJ linearly independent of un(t), say p(t)[uv' — u'v] =
c?±0. Then B.15) is a solution of D.5AJ. Furthermore w(t) satisfies the
first of the boundary conditions in D.2') since u = un does; cf. Exercise
2.1. On the other hand, D.7) and B.15) show that w(b) = w'(b) = 0.
Hence w(t) is a solution D.5/1) satisfying the boundary conditions D.2').
On (vi). Although the assertion (vi) is the main part of Theorem 4.1,
it is a consequence of elementary theorems on completely continuous,
self-adjoint operators on Hilbert space. For the sake of completeness, the
proof of the necessary theorems will be sketched and (vi) will be deduced
from them. A knowledge of Fourier series (involving, e.g., Bessel's
inequality, Parseval's relation, and the theorem of Fischer-Riesz) will be
assumed. In order to minimize the required discussion of topics on Hilbert
space, some of the definitions or results, as stated, will involve redundant
hypotheses.
Introduce the following notation and terminology:
D-17) (f,g) = fhf(t)g(t) dt, 11/11 = (/, ff ^ 0,
Ja
where/ g e L\a, b). Thus |(/, g)\ ^ \\f\\ ¦ \\g\\ and \\f+g\\ ^ ||/|| + ||g||
by Schwarz's inequality. A sequence of functions/^O.^Oi • • •ln L2(a, b)
will be said to tend to/@ in L2(a, b) if ||/n - f\\ -+ 0 as n -+ 00. They will
Linear Second Order Equations 341
be said to tend to f(t) weakly in L2{a, b) if the sequence H/J, ||/2||,... is
bounded and, for every <p(t) e L2{a, b), (/,, <p) —>¦ (f, <p) as n —>¦ oo. (In this
last definition, the condition on H/J, ||/2||,... is redundant but this fact
will not be needed below.) A subset H of L2{a, b) is called a linear manifold
if/ g eHimplies that c-J + c2g e Hfor all constants cx, c2 and it is called
closed if/, e # for w = 1,2,... ,/e L\a, b) and ||/n -/|| -> 0 as n -^ oo
imply that/e //. A linear manifold H of L2(a, b) will be called weakly
closed if/n e // for n = 1, 2,... ,/e L2(a, fe) and/, —>-/weakly as n —»¦ oo
imply that /e //. (The fact that the notions of "closed" and "weakly
closed" are equivalent for linear manifolds will not be needed here.)
Lemma 4.1. Letfltf2,. . .be a sequence of elements ofL2(a, b) satisfying
||/n|| ^ 1. Then there exists an f(t) e L2(a, b) and a subsequence fnW(t),
/n<2)@> • • • °f the given sequence such that ||/|| ^ 1 andfnU) —-f(t) weakly
Proof. Without loss of generality, it can be supposed that [a, b] =
[0, n]. Thus each/n(f) has a sine Fourier series
/n@~2cn* sin kt,
k=l
where, by Parseval's relation, J \cnk\2 = \\fn\\2 ^ 1. It follows from
k
Cantor's diagonal process (Theorem I 2.1) that there exists a sequence of
integers 1 < n(l) < wB) < . . . such that
D.18) ck = lim cnU)k exists as j —>¦ oo for k = 1, 2,. ...
Note that
Hence S \ck\2 ^ 1 and so, by the theorem of Fischer-Riesz, there exists an
f{t) e L2(a, b) such that
It follows from D.18) that (fn{j), <p) —>¦ (/, <p) asy—>- 00 holds if <p = sin kt
for k = 1, 2,.... Hence it holds for any sine polynomial p(t) = ax sin t +
• • • + am sin mt. For any <p(t) e L2@, 77), there exists a sine poly-
polynomial p(t) such that ||<p — p\\ is arbitrarily small and |(/n(j) —/ <p)\ ^
UnU) ~f,P)\ + \(fnU) ~f,P ~ <P)\, While |(/n(i) ~ f, P ~ 9>)\ ^ ll/»(,) "
fW'Wp — 9>\\ = 2 \\p — <p\\- Hence the lemma follows.
Lemma 4.2. Let G be a self-adjoint, linear operator defined on a weakly
closed linear manifold H of L2(a, b) satisfying (Gh, h) = 0 for all h e H.
Then Gh = 0/or all h e H.
342 Ordinary Differential Equations
To say that G is a linear operator on H means that to every he H there
is associated a unique element w = Gh e H and that if wt = Ghj for
j= 1,2, then c^ + c2w2 = G{c1h1 + c2h2) for all complex constants
Ci, c2. The assumption that G is self-adjoint means that (Gh,f) = (h,Gf)
for all/, he H.
Proof. If/, /i e L\a, b) and c is a complex number, then 0 = (G(h + cf),
h + cf) = 2 Re c(Gh,f) since (G/,/) = (Gh, h) = 0. By the choices c = 1
and c = /, it follows that (Gh,f) = 0. On choosing/ = Gh, it is seen that
Gh = 0.
Lemma 4.3. Let G be a completely continuous, self-adjoint linear
operator on a weakly closed linear manifold H of L2(a, b) and let Gh ^ Ofor
some h e H. Then G has at least one (real) eigenvalue fi ^ 0; i.e., there
exists a (real) number //^0 and an h0 e H, h0 ^ 0, such that Gh0 = fih0.
A linear operator G on H is called completely continuous if hn,h e H
and /«„-»• h weakly as n -*¦ oo imply that \\Ghn — Gh|| -*¦ 0 as n ->- oo.
Proof. It follows from Lemma 4.1, the complete continuity of G, and
the fact that H is weakly closed that G is bounded, i.e., that there exists a
constant C such that \\Gh\\ ^ C for all h e Hsatisfying \\h\\ < 1.
By Schwarz's inequality, \(Gh,h)\ ^ \\Gh\\ ¦ \\h\\ ^ Cif \\h\\ ^ 1. Hence
sup (Gh,h) and inf (Gh,h) for all \\h\\ ^ 1 exist and are finite. Since
Gh j? 0 for some heH, it follows from Lemma 4.2 that at least one of
these two numbers is not zero. For the sake of definiteness, let /i =
sup (Gh, h) j? 0. The choice h = 0 shows that /i ^ 0, hence /i > 0.
It will be shown that there exists an h0 e H such that (Gh0, h0) = /x and
\\ho\\ ^ 1. For there exist elements hlt h2,... in //such that ||/ij ^ 1 and
(G/in, /iJ->// asn->oo. In view of Lemma 4.1, we can suppose that
there exists an h0 e l?(a, b) such that hn -*¦ h0 weakly as n ->- oo and
\\ho\\ ^ 1. Since //is weakly closed, hoeH. The complete continuity of
G shows that \\Ghn — Gho\\ -> 0 as n -> oo. Also (G/i0, /i0) = (Ghn, hn) +
2 Re (G(h0 - hn), hn) + (G(h0 - hn), h0 - hn). From the boundedness
of G and Schwarz's inequality, we conclude, by letting w —>- oo, that
(Gh0, h0) = ix.
Note that /i j? 0 implies /i0 7^ 0. Also, since /i > 0, it follows that
||AJ = 1, otherwise (Gh, h) = /i/||/io||2 > p for h = fco/ll^oll and ||/i|| = 1.
In order to verify that Gh0 = fih0, let h be any element of H satisfying
||/i|| = 1 and (/i0, /i) = 0. Let h€ = (/i0 + e/i)/(l + c2I^ for a real €, so that
||/ij|2 = 1. Then the function
c, /ic) = A+
Re
of € has a maximum at e = 0 and hence Re (Gh0, h) = 0. Since /i can be
replaced by ih, it follows that (Gh0, h) = 0 for all /i ? // satisfying (/i0, /i) =
0. In particular, (Gh0, h) = 0 if /i = G/i0 — juh0. This implies that
Linear Second Order Equations 343
/i2 = ||G/io||2 and hence \\Gh0 - /i/ij2 = ||G/io||2 - 2(Gh0, h0) + /i2 = 0.
This proves G/i0 = ph0 and completes the proof of the lemma.
Completion of Proof of (vi). A standard theorem on Fourier series
implies that (vi) is false if and only if there exist functions h(t) e L\a, b),
|| A || t* 0, having zero Fourier coefficients (h, un) = 0 for n = 0, 1, . .. .
Suppose, if possible, that (vi) is false and let //denote the set of all elements
h(t) e L\a, b) satisfying (h, un) = 0 for n = 0, 1, Then H is a
weakly closed, linear manifold in L\a, b) and contains elements h ^ 0.
Choose a real number X ^ Xn for n = 0, 1,. .. . Then D.6) defines a
linear operator G, w = Gh, on L\a, b). This operator is self-adjoint since
6 fb
(Gh,f) = G(t, s; X)h(s)f(t) ds dt = (h, Gf)
Ja Ja
follows from the fact that G(f, s; /I) is real-valued and G(f, s; X) = G(s, t;X).
Also G is completely continuous. In order to verify this, let hn—>-h
weakly as n -*¦ 00 and wn = Ghn, w = Gh, then
wn(t) - w(t) = \G(t, s; X)[hn(s) - h(s)] ds = (hn - h, G(t, • ; X))
Ja
tends to 0 as n ->- 00 for every fixed t. Furthermore, by Schwarz's
inequality
|w»@ - KOI2 ^ 2C \\G(t, s; X)\2 ds < const.
if II/1JI2 ^ C, ||/i(s)||2 < C. Thus ||wn - w||2 = J \wn(t) - w@l2 A -» 0 as
n ->- 00 by Lebesgue's theorem on dominated convergence. [Actually, by
Theorem I 2.2, wn(t) -*¦ w(t) as n -*¦ 00 uniformly for a ^ t ^ 6 since it is
easily seen that the sequence w1( w2, . . . is uniformly bounded and
equicontinuous.]
Finally, note that if he H, then w = Gh is in H. In fact, (h, un) = 0
implies that (w, un) = 0 as can be seen by applying the Green identity
B.10) to u = un,f= —Xnun, v = w, g = —Xw + h. Thus the restriction
of G to the weakly closed linear manifold //gives a completely continuous,
self-adjoint operator on H.
From D.5A) and D.6), it is seen that h ^ 0 implies that w = Gh ^ 0.
Since H contains elements h j? 0, Lemma 4.3 is applicable. Let Gh0 =
/i/i0, where hoe H, \\ho\\ = 1, /* ?? 0. Thus, if w0 = G/i0, it follows from
D.5X), D.6) that u = wo(O ^ 0 is a solution of D.1 X - l//i) satisfying the
boundary conditions D.2). Hence, by part (i), there is a non-negative
integer k such that X — l//i = Xk and w0 = cwt for some constant c j? 0.
But this contradicts (w0, un) = 0 for n = 0, 1, . .. and proves the theorem.
344 Ordinary Differential Equations
Exercise 4.3. Let po(t) > 0, ro(t) > 0 and qo(t) be real-valued con-
continuous functions on an open bounded interval a < t < b. Let /l0 < y^ <
.... Suppose that
D.19) (/>o(O"')' + [qo(t) + Kro(O]u = 0, n = 0, 1,...,
has a (real) solution un(t) on a < t < b having at most n zeros and such
that the limits lim un(t)/u0(t), t —*¦ a and t-*-b, exist and are not zero.
(a) Show that if Pl(t) = l/ro(O"o2(O > 0, r^t) = l//>o(O"o2(O > 0 and
9i@ = -hlPo(t)UoKt), then vn(t) = po(uou'n+1 - wo'wn+i) is a solution of
D.20) (pi(t)v'Y + [fr@ + 4+i''i@]f = 0> n = 0, 1,. ..,
having at most n zeros on a < f < 6 and such that the limits lim vn(t)j
yo(O> t —>¦ a and t -+b, exist and are not zero, (b) Show that there exist
positive continuous functions ao(t), ax(t),. . ., a*_i@ on a < t < b, such
that uo(t), . . ., uk_1(t) are solutions of the &th order linear differential
equation
D.21) (a*_i. .. {a2[«i(«o")']'}' ...)'= 0.
Exercise 4.4 (Continuation), (a) Let />„, /¦„, ^0, An, un be as in Exercise
4.3. Let a <*!<•••< ^+i < b and a1;.. ., at+1 be arbitrary numbers.
Then there exists a unique set of constants c0, .. . , ck such that
D.22) couo(tj) + ¦¦¦+ ckuk(t,) = a, for j = 1, . . ., k + 1.
Use induction on k (for a// systems u0, uu .. .) or use Exercise IV 8.3.
[This result is, of course, applicable to (real-valued) un(t) in Theorem 4.1.
If the functions p0, r0, q0 have derivatives of sufficiently high order, then the
interpolation property D.22) can be generalized, as in Exercise IV 8.3(d).]
(b) Let a < t0 < ¦ ¦ ¦ < tn < b. Then D(t0, ...,tn) = det («,(**)), where
j, k = 0, . . . , n, is different from 0. (c) Let c0, . . . , cn be real numbers
and Un(t) = couo(O +¦¦¦ + cnun(t). Then UJt) = 0 if UJt) vanishes at
n + 1 distinct points of a < t < b, and if Un(t) ^ 0 vanishes at n distinct
points, then it changes sign at each, (d) Every real-valued continuous
function v(t) orthogonal to u0, . . ., un on [a, b] (i.e., vut dt = 0 for
Ja
j = 0, . . ., n) changes sign at least n + 1 times, (e) For any choice of
constants cm, . . ., cn, the function cmum(t) + ¦ ¦ ¦ + cnun(t) changes sign
at least m times and at most n times, where m ^ n.
5. Number of Zeros
This section will be concerned with zeros of real-valued solutions of an
equation of the form
E.1) u" + q(t)u = 0.
Linear Second Order Equations 345
Theorem 5.1. Let q(t) be real-valued and continuous for a ^ t ^ b.
Let m(t) ^OAea continuous function for a ^ t ^ b and
E.2)
Ym =
m(t)
for a < t < b.
(t - aXb - t)
If a real-valued solution u(t) ^ 0 o/E.1) has two zeros, then
E.3) fbm(t)q+(t) dt > ym(b - a),
Ja
where q+{t) = max (q(t), 0); in particular,
E.4) \\t - a)(b - i)q+(t) dt > b - a.
Ja
Exercise 5.1. Show that the inequality E.3) is "sharp" in the sense
that E.3) need not hold if ym is replaced by ym + e for e > 0.
Proof of Theorem 5.1. Assume that E.1) has a solution (^ 0) with two
zeros on [a, b]. Since q+{t) ^ q{t), the equation
E.5) u" + q+(t)u = 0
is a Sturm majorant for E.1) and hence has a solution u(t) ^ 0 with two
zeros t = a, /? on [a, b]; cf. Theorem 3.1. Since u" = —q+u, it follows
that
03- a)u@ = (/? - t)j\s - o.)q+(s)u(s) ds + (t - a)J'oS - s)q+{s)u{s)ds;
cf. Exercise 2.1, in particular B.18). Suppose that a, /? are successive
zeros of u and that u{t) > 0 for a < t < /?. Choose t = t0 so that m(Oo =
max u(t) on (a, /?). The right side is increased if u(s) is replaced by
u(t0). Thus dividing by u(t0) > 0 gives
/? - a < (/? - 0 f(s - «)q+(s) ds + (t
- a) f' @ -
Jt
s)q+(s) ds,
where f = f0. Since fi — t ^ ft — s for f^s and t — a. ^ s — a for
5^ f,
E.6)
- a < OS -
Ja.
s)(s — a.)q+(s) ds.
Finally, note that (t - a)(Z> - 0/F - a)^(t - a)(/S - r)/(/S - a) for
a^a^t^ft^b; in fact, differentiation with respect to /S and a shows
that 0 — «)(/S — O/(/S — «) increases with /S if r > a and decreases with
a if / ^ /5. Hence E.4) follows from the last display. The relation E.3)
is a consequence of E.2) and E.4). This proves Theorem 5.1.
346 Ordinary Differential Equations
Since (t — a)(b — t) ^(b — aJ/4, the choice m(t) = 1 in Theorem 5.1
gives the following:
Corollary 5.1 (Lyapunov). Let q{t) be real-valued and continuous on
a ^ t ^ b. A necessary condition for E.1) to have a solution u(t) j? 0
possessing two zeros is that
E.7)
+(t) dt
b-a
Exercise 5.2. Let q{t) ^ 0 be continuous on a ^ ( ^ A and let E.1)
have a solution u{t) vanishing at t = a, b and u(t) > 0 in (a, b). (a) Use
E.7) to show that q(t) dt > 2M/A, where M = max u(t) and A =
J-6 Ja
u(t) dt. (b) Show that the factor 2 of M/A cannot be replaced by a
a
larger constant.
Exercise 5.3. (a) Consider a differential equation u" + g(t)u + f(t)u =
0 with real-valued continuous coefficients on 0 ^ t ^ b having a solution
u(t) # 0 vanishing at t = 0, b. Show that
b <j\b - t)f+(t) dt + max {J\ |g| dt, j\b - t) \g\ dt\ .
(b) In particular, if \g\ < Mx and |/| ^ M,, then 1 < Mxb\2 + M2b2l6.
But this inequality can easily be improved by the use of Wirtinger's
inequality u2 dt ^ (b/nJ u'2 dt (which can be proved by assuming
Jo Jo
b = 77, expanding m into a Fourier sine series, and applying Parseval's
relation for u, u'). Show that 1 < Mxb/n + M2b*/n2. (c) The result of
part (b) can further be improved to 1 ^ 2Mxb/n2 + M2b2/n2. See Opial
[3]. (d) An analogous result for a <ith order equation, d ^ 2, is as follows:
Let the differential equation um + ^(O"'*' + ' " ' + Pa(t)u = ° have
continuous coefficients for 0 ^ t ^ b and a solution n(f) ^ 0 with <i
zeros on [0, b]. Let |/>/f)l ^ ^- Then 1 < Mxb + Af2A2/2! + • • • +
When ^@ = <7+@ is a positive constant on [0, T], the number N of
zeros of a solution (^ 0) of E.1) on @, T] obviously satisfies
E.8)
T -M
q+(t)dt\
where the last inequality follows from Schwartz's inequality. It turns out
that a similar inequality holds for nonconstant, continuous ^@:
Corollary 5.2. Let q(t) be real-valued and continuous for 0 ^ t ^ T.
Linear Second Order Equations 347
Let u(t) 5^ 0 be a solution ofE.\) and N the number of its zeros on 0 <
t < T. Then
E.9)
N <-
(t)dt) +1.
Proof. In order to prove this, let N ^ 2 and let the N zeros of u on
@, T] be @ <) tx < t2 < ¦ ¦ ¦ < tN (^ T). By Corollary 5.1,
4
if « = („ f = ffc+1,
E.10)
\y
dt >
— M
for k = I,. .., N — 1. Since the harmonic mean of N — 1 positive
numbers is majorized by their arithmetic mean,
i ,V-1 i -1-1
— 2 —-—I <
— 1 *-i tk+1 — tkj N — 1 fc-i
Thus adding E.10) for k = 1,..., N - 1 gives
ty — tx
hence E.9).
Exercise 5.4. Show that N also satisfies
E.11)
:\ tq
Jo
N < tq+(t) dt + 1.
To this end, use E.3) with m(t) = t — a in place of E.7).,
Note that if q(t) is a positive constant, then
77iV- '
An analogous inequality holds under mild assumptions on nonconstant q:
Theorem 5.2. Let q(t) > 0 be continuous and of bounded variation on
0 ^ t < T. Let u(t) 5^ 0 be a real-valued solution of E.1) and N the
number of its zeros onO < t ^ T. Then
E.12)
Jo 4 Jo q(i)
Proof. In terms of u(t) define a continuous function cp{t) by
<p(t) = arc tang *¦*'" , 0 < <p@) < 77.
Then [cf. Exercise 2.6; in particular B.49) where p(t) = 1 ]
<P(T) = <p@) + qA(t) dt + - sin2<p@ d(logg).
Jo 4 Jo
348 Ordinary Differential Equations
By Lemma 3.1, N is the greatest integer not exceeding f(T)ln, so that
ttN ^ q>(T) < tt{N + 1). This implies E.12).
Exercise 5.5. (a) Let q(t) be continuous on 0 ^ / ^ I. Let u(t) ^ 0
be a real-valued solution and N the number of its zeros on 0 < ( ^ T.
Show that
WN - T\ <: n + |1 - <z(OI dt.
Jo
(b) If, in addition, q(t) > 0 has a continuous second derivative, then
-\Tq\t)dt <„ + {
Jo Jo
5q'2
dt.
Corollary 5.3. Let q(t) > 0 be continuous and of bounded variation on
[0, T]for every T > 0. Suppose also that
E.13)
as
co;
e.g., suppose that q(t) has a continuous derivative q'(t) satisfying
E.14) q'(t) = o(q3A(t)) as t-*oo.
Let u(t) ^ 0 be a real-valued solution o/E.1) and N(T) the number of its
zeros on 0 < t ^ T. Then
E.15) nN(T)~\ qA(t)dt as T-> oo.
Jo
This is clear from E.13) and the formula E.12) in Theorem 5.2. It
should be mentioned that if, e.g., q is monotone and q(t) -»¦ oo as t -»¦ oo,
then E.14) imposes no restriction on the rapidity of growth of q(t) but is a
condition on the regularity of growth. This can be seen from the fact that
the integral
rTq'dt
J 7^'?
q»(T)
+ const.
tends to a limit as T—> oo; thus, in general, q'\qA is "small" for large t.
The conditions of Corollary 5.3 for the validity of E.15) can be lightened
somewhat, as is shown by the following exercises.
Exercise 5.6. (a) Let q(t) > 0 be continuous for t ^ 0 and satisfy
E.16)
sup
\\og q(t)lq(s)\
\r) dr
¦0 as s -»¦ oo.
Let u(t) ^0 be a solution of E.1) and N(T) the number of its zeros on
0 < t ^ T. Then E.15) holds, (b) Necessary and sufficient for E.16) is
Linear Second Order Equations 349
the following pair of conditions: qiyi dt = oo and q(t + cq~^{t))j
q(t) -»¦ 1 as t -»¦ oo holds uniformly on every fixed bounded c-interval on
— oo < c < oo.
Exercise 5.7. Part (b) of the last exercise can be generalized as follows:
Let q(t) > 0 be continuous for t ^ 0. Let m(t) > 0 be continuous for
t > 0 and satisfy [w@/w(^)]±1 ^ C(f/s)y for 0 < .s < t < oo and some
pair of non-negative constants C, y. Necessary and sufficient for
sup
Js
¦0
as s ¦
oo
1 + m(q(r)) dr
as min
is that m(q(t)) dt = oo and that q(t + clm[q(t)])lq(t) -»¦ 1
t + clm(q(t))] —*¦ oo holds uniformly on every bounded c-interval on
— oo < c < oo.
An estimate for N of a type very different from those just given is the
following:
Theorem 5.3. Let p(t) > 0, q(t) be real-valued and continuous for
0 _ t ^ T. Let u(t), v(t) be real-valued solutions of
E.17) (pu1)' +qu = O
satisfying
E.18) p(t)[u'(t)v(t) - u(t)v'(t)] = c>0.
Let N be the number of zeros ofu(t) on 0 < t ^ T. Then
dt
E.19)
nN -
Jo p(t)[u\t) + v\t)]
Proof. Let a be an arbitrary real number. Consider the solutions
m*@ = u(t) cos a + v(t) sin a, v*(t) = —u(t) sin a + v(t) cos a
of E.17). They satisfy
E.20) m2 + v2 = m*2 + v*2, p[u*'v* - u*v*'] = c > 0.
Choose a so that m*@) = 0 and let N* be the number of zeros of u*(t) on
0 < t < T.
Since E.20) implies that «*, v* are linearly independent, they have no
common zeros. Hence it is possible to define a continuous function by
u*(t)
E.21)
f(t) = arc tan
v*(t)
and
= 0.
This function is continuously differentiable and, by E.20),
c .. „
E.22)
9/@ =
p@[«2@ + v\t)]
350 Ordinary Differential Equations
Hence <p(t) is increasing; also cp{t) = 0 mod v if and only if u(t) = 0.
Thus N* is the greatest integer not exceeding <p(T)JTr and a quadrature of
E.22) gives
= 1
Sturm's separation theorem implies N* ^ N ^ N* + 1, thus E.19)
follows.
Exercise 5.8. Let/?@, ^@, "@. "@. and N be as in Theorem 5.3 and,
in addition, let <j<0 ^ 0. Show that
E.23) nN-~tT q^-
< 277
(If q > 0, the relations E.19) and E.23) are particular cases of "duality" in
which (u,u',q,dt) are replaced by (pu , — u, \jq,qdt); cf. Lemma XIV 3.1.)
Exercise 5.9. (a) Let q(t) be continuous for t ^ 0. Using E.9) and
E.19), show that if all solutions of u" + q{i)u = 0 are bounded, then, for
large t,
E.24)
1 C
-
t Jo
q+(s) ds ^ const. > 0.
Replacing u, v in E.19) by u/e, ev, show that if, in addition, a nontrivial
solution u(t) -»¦ 0 as t -»¦ oo, then
E.25)
q+(s) ,
oo
as f ¦
oo.
(b) Let ?@ ^ 0 for t^ 0. Using E.9) and E.23), show that if the first
derivatives of all solutions of u" + q{t)u = 0 are bounded, then, for large t,
1 C*
E.26) - q+(s) ds ^ const.
t Jo
If, in addition, u'(t)->-0 as (^-co for some solution u(t) ^ 0, then
E.27)
1 f
-
f Jo
q+(s) ds^O as t -> oo.
(c) Generalize (a) [or F)] for the case when u" + qu = 0 is replaced by
(/?«')' + <jw = 0 and the assumption that solutions [or derivatives of
solutions] are bounded is replaced by the assumption that all solutions
satisfy u(t) = O(l/O@) [or u'(t) = 0A/0@)], where 0@ > 0 is con-
continuous.
6. Nonoscillatory Equations and Principal Solutions
A homogeneous, linear second order equation with real-valued
coefficient functions defined on an interval / is said to be oscillatory on J
Linear Second Order Equations 351
if one (and/or every) real-valued solution (^ 0) has infinitely many zeros
on /. Conversely, when every solution (^ 0) has at most a finite number
of zeros on /, it is said to be nonoscillatory on J. In the latter case, the
equation is said to be disconjugate on J if every solution (^ 0) has at most
one zero on /. If t = co is a (possibly infinite) endpoint of / which does
not belong to /, then the equation is said to be oscillatory at t = <x> if one
(and/or every) real-valued solution (^ 0) has an infinite sequence of zeros
clustering at t = co. Otherwise it is called nonoscillatory at t = co.
Extensions of many of the results of this section to higher order equations
or more general systems will be indicated in §§ 10, 11 of the Appendix.
Theorem 6.1. Let p(t) > 0, q(t) be real-valued, continuous functions on
a t-intervalJ. Then
F.1)
(p(t)u'y + q(t)u = 0
is disconjugate on J if and only if for every pair of distinct points tu t2eJ
and arbitrary numbers ult u2; there exists a unique solution u = u*(t) of
F.1) satisfying
F.2) u*(t1) = u1 and u*(t2) = m2;
or, equivalently, if and only if every pair of linearly independent solutions
u(t), v(t) of F.1) satisfy
F.3) u{tMh) ~ <h)v{h) * 0
for distinct points tu t2 eJ.
Proof. Let u(t), v(t) be a pair of linearly independent solutions of
F.1). Then any solution u*(t) is of the form u* = Cjm@ + c2v(t). This
solution satisfies F.2) if and only if
cMh) + <?>('i) = "i> cMt2) + c2v(t2) = u2.
These linear equations for c1; c2 have a solution for all ult u2 if and only if
F.3) holds. In addition, they have a solution for all ult u2 if and only if
the only solution of
+
= 0,
+ c2v(t2) = 0
is Cj = c2 = 0; i.e., if and only if the only solution u*(t) of F.1) with
two zeros t = tu t2 is u*{t) = 0.
Corollary 6.1. Let p{t) > 0, <jr(O be as in Theorem 6.1. Ifj is open or is
closed and bounded, then F.1) is disconjugate on J if and only if F.1) has a
solution satisfying u(t) > 0 on J. If J is a half-closed interval or a closed
half-line, then F.1) is disconjugate on J if and only if there exists a solution
u(t) > 0 on the interior ofJ.
The example u" + u = 0 on J: 0 ^ t < tt shows that, in the last part
of the theorem, there need not exist a solution u(t) > 0 on J.
352 Ordinary Differential Equations
Exercise 6.1. Deduce Corollary 6.1 from Theorem 6.1 (another proof
follows from Exercise 6.6).
Exercise 6.2. Let p(t) > 0, q(t) ^0 be continuous on an interval
[•to
J: a ^ t < co (^ oo) such that dt/p(t) = co, then F.1) is disconjugate
on J if and only if it has a solution u(t) such that u(t) > 0, u'(t) ^ 0 for
a < t < co.
A very useful criterion for F.1) to be disconjugate is a "variational
principle" to be stated as the next theorem. A real-valued function rj(t)
on the subinterval [a, b] of / will be said to be admissible of class Ax(a, b)
[or A2(a, b)\ if (i) r\(a) = r)(b) = 0, and (iij) r/(t) is absolutely continuous
and its derivative r/'(t) is of class L2 on a ^ t ^ b [or (ii2) rj(t) is contin-
continuously differentiate and p(t)r)'(t) is continuously difFerentiable on
a ^ t ^ b]. Put
F.4) I(ri; a, b) = (pr)'2 - qrj2) dt for rj ? Ax(a, b).
Ja
If rj is admissible A2(a, b), the first term can be integrated by parts and it
is seen that
f"
F.5) I(ri; a, b) = — *?[(/»?')' + <m\ dt for r\ ? A2(a, b).
Ja
Theorem 6.2. Let p(t) > 0, q(t) be real-valued continuous functions on a
t-interval J. Then F.1) is disconjugate on J if and only if, for every closed
bounded subinterval a^t^bofJ, the functional F.4) is positive-definite
on Ax(a,b) [or A2(a, b)]; i.e., I(rj;a,b)^0 for rjsA^b) [or rj ?
A2(a, b)] and I(rj; a,b) = 0 if and only ifr) = O.
The "only if" half of the theorem is stronger for Ax(a, b) and the "if"
half is stronger for A2(a, b).
Proof ("Only if"). Suppose that F.1) is disconjugate on a ^ ( ^ i.
Then, by Corollary 6.1, there is a solution u(t) > 0 on a ^ ( ^ 4. If
ri(t) ? Ax(a, b), put ?@ = rj(t)ju(t). Then
F.6)
, ;a,b) = f'
Ja
u'2 - qu2)
'««')] dt.
An integration by parts [integrating u' and differentiating (pu')t,2] shows
that the first term is
Ja
\2pu'2dt =
'uu'] dt.
The integrated terms vanish since rj(a) = rj(b) = 0 imply that
= 0. The last two formula lines and ?2u[(pu')' + qu] = 0 give
F.7) I(n;a,b) = [pu%'2dt for V = ut, ? Ax{a, b).
Ja
,
Linear Second Order Equations 353
It is clear that /(??; a, b) ^ 0 and /(??; a, b) = 0 if and only if t,(t) = 0.
This proves the "only if" part of the theorem.
Proof ("If"). Suppose that /(»?; a, b) is positive definite on A2(a, b)
for every [a, b] <= J. Let rj(t) be a solution of F.1) having two zeros
t = a,bej. It will be shown that rj(t) = 0. In fact t](t) ? A2(a, b); thus
F.5) holds. Hence, /(»?; a, b) = 0 because »? is a solution of F.1). Since
F.4) is positive definite on A2(a, b), it follows that t](t) = 0. This implies
that F.1) is disconjugate on / and completes the proof of the theorem.
Exercise 6.3. Suppose that / is not a closed bounded interval. Show
that, in Theorem 6.2, F.1) is disconjugate on / if I(rj; a, b) ^ 0 for all
[a, b] <= J and all r\ ? A2(a, b).
Exercise 6.4. Deduce Sturm's separation theorem (Corollary 3.1)
from Theorem 6.2.
If P is a constant positive definite Hermitian matrix, then there exists
a positive definite Hermitian matrix P1 which is the "square root" of P
in the sense that P = P12 = P1*p1; cf. Exercise XIV 1.2. An analogue
of this algebraic fact will be obtained for the differential operator
L[r,] = -
Note that F.5) can be written as
'y - q(t)r,.
I(t] ;a,b) = (L[rj\, rj) for
cf. D.17). Also, F.7) can be written as
? A2(a, b);
Ja U
In addition to the quadratic functional F.4), consider the bilinear form
'(>h, rJ; a, b) = (p^ V - OTi^) dt
Ja
for rjlr r\2 ? A^(a, b). If r]1 ? A2(a, b), an integration by parts shows that
/(?h, r]2;a,b)=-\ j?2[(pj?j')' + q(r]i)] dt = (Lfoj], r/2).
Ja
If u(t) is a solution of F.1) and u(t) > 0 on [a, b], it is readily verified
that, for r\x, r\2 ? A2(a, b) and ?j = j^/m, ?2 = »?/m,
or
P
?i,»?2;alA)= Pu%%2' dt.
Ja
*,,.; «, *) = f6 P()?/M " ^
u(t)
354 Ordinary Differential Equations
Thus if the first order differential operator Lx is defined by
F.8) y
then it follows that
F.9)
for
(«, *)¦
Consequently, if L [i.e., F.4)] is positive definite on A%(a, b), so that there
exists a positive solution u(t) > 0 of F.1) on [a, b], then formally
L = L^L^
In fact this relation is not only formally correct but is correct in the
following sense:
Corollary 6.2. Let p(t) > 0, q{t) be continuous on J and let F.1) have a
solution u(t) >0onJ. Let L1 be defined by F.8) and LX* its formal adjoint
cf. § IV 8 (viii). Then
u(t)
L[rj\ =
/or a// continuously dijferentiable functions r\for which p(t)r{ is absolutely
continuous (i.e., for all rjfor which L[rj\ is usually defined).
This can be deduced from the identity F.9) or, more easily, by a
straightforward verification. See Appendix for generalizations of this
result.
Theorem 5.3 and its proof have the following consequence.
Theorem 6.3. Let p(t) > 0, q(t) be real-valued and continuous on a
t-intervalJ. Then J is nonoscillatory on Jif and only if every pair of linearly
independent solutions u(t), v(t) of F.1) satisfy
L
dt
< oo.
j P(t)(M2 + \v\2)
Furthermore, F.1) is disconjugate on J if and only if
*» jdt
u -\~ v )
for every pair of real-valued solutions u(t), v(t) satisfying p(u'v — uv')
c 9± 0 and every interval [a, b] <= J.
Linear Second Order Equations 355
If / is a half-open interval, say J:a ^ t < co (< oo) and F.1) is non-
nonoscillatory at t = co, then F.1) has real-valued solutions u(t) for which
dtjpu2 is convergent and solutions for which it is divergent. The
latter type of solution will be called a principal solution of F.1) at t = co.
Theorem 6.4. Let p(t) > 0, q(t) be real-valued and continuous gn J:
a ^ t < co (< oo) and such that F.1) is nonoscillatory at t = co. Then
there exists a real-valued solution u = uo(t) of F.1) which is uniquely
determined up to a constant factor by any one of the following conditions in
which Ui(t) denotes an arbitrary real-valued solution linearly independent
ofu0(t): (i) m0, Mj satisfy
F.10)
(ii) m0, Mj satisfy
ujjt)
J
0
and
as
J
(iii) ifTeJ exceeds the largest zero, if any, ofu0(t) and ifu±(T)
Uj(t) has one or no zero on T < t < co according as
0, then
F.120) ^
or
F.12,) ^>^
holds at t = T; in particular, F.12j) holds for all t (eJ) near co.
It is understood that in F.10) and F.11) only /-values exceeding the
largest zeros, if any, of u0, Mj are considered. A solution uo(t) satisfying
one (and/or) all of the conditions (i), (ii), (iii) will be called a principal
solution of F.1) (at t = co). A solution u(t) linearly independent of uo(t)
will be termed a nonprincipalsolution of F.1) (at t = co). In view of F.10),
F.11), the terms "principal" and "nonprincipal" might well be replaced
by "small" and "large." The expressions "small," "large" will not be
used in this context because of the relative nature of these terms. Consider,
e.g., the equations u" — u = 0, u" = 0 and u" + u/4t2 = 0 for t ^ 1.
Examples of principal and nonprincipal solutions at t = 00 for the first
equation are u = er* and u = e'; for the second, u = 1 and u = t; for
the third, u = t1-4 and u = t'A log t; cf. Exercise 1.1. The proof of
(ii) will lead to the following:
Corollary 6.3. Assume the conditions of Theorem 6.4. Let u = u(t) ^ 0
be any real-valued solution of F.1) and let t = T exceed its last zero. Then
F.13)
7?
Jt p(s)u
ds
p(s)u\s)
356 Ordinary Differential Equations
is a nonprincipal solution of F.1) on T^.t<co. If, in addition, u(t) is a
nonprincipal solution o/F.1), then
F.14)
f'0 ds
mo(O = «@ 2
J* P(s)m!(s)
is a principal solution on T ^ t <. co-
Proof of Theorem 6.4 and Corollary 6.3
On (/). Let w(r), v(t) be a pair of real-valued linearly independent
solutions of F.1) such that
F.15) p{u'v - uv') = c 5* 0.
If T exceeds the largest zero, if any, of v(t), then F.15) is equivalent to
F.16)
\vl pv
for T ^ t < co. Hence u/v is monotone on this r-range and so
F.17) C = lim —
exists if C = ± oo is allowed.
It will be shown that u, v can be chosen so that C = 0 in F.17). If this
is granted and if u(t) is called uo(t), then (i) holds. In fact, a solution u±(t)
is linearly independent of uo(t) if and only if it is of the form u^t) =
coMo(O + civ@ and Cj 7^ 0; in which case, C = 0 implies that Mj =
[q + o(l)]y(r); thus u0 = o(mj) as r-» <o.
If C = ±oo in F.17) and if m, d are interchanged, then F.17) holds
with C = 0. If |C| < oo and if u(t) — Cv(t) is renamed u(t), then F.15)
still holds and F.17) holds with C = 0. This proves (i).
On (ii). Note that F.16), F.17) give
ds
T p(s)v (S)
whether or not \C\ = oo or \C\ < oo. If u, v is a pair m0, Mj, so that
C = 0, then F.1 lj) holds. If m, v is a pair Mj, m0, so that C = ± co, then
F.110) holds.
On Corollary 6.3. Note that if u(i) is a solution of F.1) and u(t) j? 0
for T ^ t < en, then F.13) defines a solution w^r) linearly independent of
Linear Second Order Equations 357
u(t) and that the same is true of F.14) when the integral is convergent;
see §2 (ix). By (i), this implies Corollary 6.3.
On (Hi). Since u0, u1 can be replaced by — m0, — ult respectively, with-
without affecting the zeros of Mj or the inequalities F.12), it can be supposed
that
F.18)
mo(O>O for T^t<co and u1(T)>0.
Multiplying F.12) by M0(r)w1(r) > 0 shows that the case F.15), where
(m, v) = (uu m0) holds with c < 0 or c > 0 according as F.120) or F.12j)
holds. Hence Mi(O/Mo(O~*" T °° as t —» co according as F.120) or F.12j)
holds. Since M1(r)/w0(r) > 0 and, by the Sturm separation theorem, ux
has at most one zero on T < t < co, the statement concerning the zeros
of Mj on T < t < co follows.
It remains to show that property (iii) is characteristic of a principal
solution; i.e., if uo(t) has the property (iii) for every solution u^t) linearly
independent of uo(t), then uo(t) is a principal solution. In particular F.12j)
holds for t (eJ) near co. Consequently |mo(OI = const. |u1(r)| for t—> co.
This is a contradiction if uo(t) is not a principal solution and ux(t) is
chosen to be a principal solution.
Exercise 6.5. Assume (i) that the conditions of Theorem 6.4 hold;
(ii) that F.1) has a nonvanishing real-valued solution for (a ^) T ^ t < co;
and (iii) that uOr(t) is the unique solution of F.1) satisfying uOr(T) = 1,
uOr(r) = 0, where T < r < co; cf. Theorem 6.1. (a) Show that uo(t) =
lim wOr(O exists as r —>¦ co uniformly on compact intervals of / and is the
principal solution of F.1) at t = co satisfying uo(T) =1. (b) Show that
(a) is false if condition (ii) is relaxed to the condition that F.1) is
disconjugate on T ^ t < co.
Exercise 6.6. Let p(t) > 0, q(t) be real-valued and continuous func-
functions such that F.1) is disconjugate on a f-interval /having t = co (^ oo)
as right endpoint. Let uo(t) be a principal solution of F.1) at t = co.
Then uo(t) ^Oon the interior of/.
Sturm's comparison theorem implies that "q(t) ^ 0 on J" is sufficient
for F.1) to be disconjugate on /. In this case, we can give some additional
information about a principal solution.
Corollary 6.4. Let p(t) > 0, q(i) ^ 0 be continuous on J : a ^ t < co.
Then F.1) has a principal solution satisfying
F.19)
«0@ > 0, ko'(O ^
for a^
and a nonprincipal solution u^t) such that
F.20) m,@ > 0, m,'@ > 0 for a ^ t < co.
358 Ordinary Differential Equations
Exercise 6.7. (a) In Corollary 6.4, the conditions F.19) uniquely
determine uo(t), uP to a constant factor, if and only if
F.21)
rji = 00 or _r
J p(t) J
q(t)dt = oo.
(b) Assume the first part of F.21). Using Corollary 9.1, show that a
principal solution in Corollary 6.4 satisfies uo(t) -»¦ 0 as t -»¦ co if and only
if-
= co.
J J
For generalizations, related results, and a different proof of Corollary
6.4, see XIV §§1,2.
Proof. Assume first that p(t) = 1, so that F.1) is of the form
F.22)
u" + q{t)u = 0,
where q < 0. Hence the graph of a solution u = u{t) of F.22) in the
(t, w)-plane is concave upwards when u(t) > 0. Let u(t) be the solution of
F.22) determined by u(a) = 1, u'{a) = 1. Then u = u(t) has a graph
which is concave upward for a ^ t < co. In particular, u(t) > u(a) = 1,
k'@ ^ t/(a) = 1; so that m@ ^ 1 + f. Thus dt\u\t) is convergent,
and so u(f) is a nonprincipal solution of F.22). By Corollary 6.3,
t u(s)
is a principal solution of F.22). Differentiating this formula gives
t u\s) u(t)
Since u'(t) is nondecreasing,
«„'(*) ^ f V(s) -?- - -L = - lim -J- ^ 0.
Jt u (s) m@ s->o> m(s)
This gives F.19). The case p{t) > 0 can be reduced to the case/?(f) = 1
by the change of independent variables A.7). This completes the proof.
Exercise 6.8. Give a proof of the part of Corollary 6.4 concerning
uo(t) along the following lines: Let a < T < co and let uT(t) be the solution
of F.1) satisfying uT(a) = 1, uT(T) = 0; cf. Theorem 6.1. Show that
ua(t) = lim uT(t) exists as T —*¦ co uniformly on compact intervals of
[a, co), is a principal solution of F.1) and satisfies F.19); cf. Exercise 6.5.
Corollary 6.5. In the two differential equations
F.23,)
Linear Second Order Equations 359
where / = 1, 2, let p^t) > 0, qt{t) be real-valued and continuous on J : a ^
t < co; let F.232) be a Sturm majorant for F.23j), i.e.,
F.24) Pi^p2>0 and qx < q2;
let F.23a) be disconjugate [so that F.23j) is also]. Let u2(t) ^ 0 be a real-
valued solution of F.232). Then F.23j) has principal and nonprincipal
solutions, ulo(t) and uxl(t), which satisfy
F 25")
PlU'w
for all t beyond the last zero, if any, of u2(t).
The rough content of this corollary is that the principal [nonprincipal]
solutions of F.22j) are smaller [larger] than the principal [nonprincipal]
solutions of F.232). If px = p2 and u2, m10, mu are normalized by suitable
constant factors, F.25) implies that m10 ^ u2 ^ wn for t near co.
Exercise 6.9. In Corollary 6.5, the principal solutions m10 of F.23j)
satisfy u\0{q2 — qx)dt < 00. In particular, if q ^ 0 in F.1), then a
principal solution m0 of F.1) satisfies m02 \q\ ds < 00.
Proof.
Case 1 (/»! = p2). Suppose that u2(t) > 0 for T ^ t < co. Make the
variations of constants u = u2z in F.23j). Then F.23j) is transformed
[cf. B.31) of § 2 (xii)] into
F.26) (Piu22z')' + u2\qi - q^z = 0,
where qx — q2 ^ 0 and
F.27) U- = Hi
u u2
By Corollary 6.4, F.26) has solutions zo(t), zx(t) satisfying z0 > 0, z0' ^ 0,
and zx > 0, zx > 0 for T ^ t < co. The desired solutions of F.23j) are
«io = M22o- 1 = 2i-
Cai'e 2 (px ^ p2). The function r = p2u2'\u2 satisfies the Riccati equa-
equation r + r2/p2 + q2 = 0 belonging to F.23a); cf. § 2 (xiv). This equation
can be written as
2
2
F.28)
= 0,
where ft = q2 + A/p, - 1/aH2m27m2J ^ ^ ^ ^. But F.28) is the
Riccati equation belonging to
F.29) (/>!«')' + <7o« = 0,
360 Ordinary Differential Equations
which is a Sturm majorant for F.23j). In addition, F.29) has the solution
[cf. §2(xiv)]
satisfying
f7r\, f7p,«,'\ ,
u = exp I — \ ds = exp I i-L-L I ds
JTXpJ JT\p1u2/
Thus application of the Case 1 to F.23j), F.29) gives the desired result.
Exercise 6.10. In the differential equations
F.30,) u"+g}.(t)u'-f(t)u = 0,
where j = 1,2, let^-, g, be continuous for 0 ^ r < co (< oo); let 0 ^
/i@ ^/2@ and gi(t) ^ ?2@; let u^t) be a solution of F.30j) satisfying
Wj@) = 1 and u^t) > 0, «/(<) ^ 0 for 0 ^ t < w; cf. Corollary 6.4.
Then F.302) has a solution u2(t) satisfying «2@) = 1, m2'@ = 0 and
0 < «2@ ^ Mj(r) for 0 ^ t < co [in fact, satisfying «2@) = 1 and
0 ^ /"i ^ 1, (w2/"i)' ^ 0 for 0 ^ t < co].
The following is a "selection" or "continuity" theorem for principal
solutions:
Corollary 6.6. Let px(t), p2(t),. . . ,pjf) and qx(t), q2(t),..., qjf) be
continuous functions for a ^ t < co satisfying
F.31,) Pj(t) > 0, qs{t) 5; 0 for a ^ t < co and / = 1, 2, .. ., 00
and
F.32) Pj(t)^Pa3(t), q&)-+qJs) as ;-> co
uniformly on every closed interval of a ^ t < co. For 1 ^/ < 00, let
uj0(t) be a principal solution of
F.33,) (piuj + qj{t)u = 0
satisfying F.19) a«d
F.34) w(a) = 1.
Then there exists a sequence of positive integers j{\) </B) < • ¦ • such that
F.35) moo@ = lim M,0(r), where ; = ;(«)>
exists uniformly on every closed interval of a ^ < < a> a«<i w a solution of
F.33M) satisfying F.19) a«rf F.34).
Of course, a selection is unnecessary (i.e., /(«) = « is permitted) if
F.33^) has a unique solution satisfying F.19) and F.34); cf. Exercise 6.7.
Linear Second Order Equations 361
Exercise 6.11. This corollary is false if the condition qs{t) ^ 0 is
replaced by the assumption that F.33,) is nonoscillatory and F.19) is
deleted from both assumption and assertion.
Proof. Let un{t) be the solution of F.33,) determined by
F.36) un(a) = 1, /?,(a)M;'1(a) = 1.
Then F.20) holds and un(t) is a nonprincipal solution of F.33,); cf. the
proof of Corollary 6.4. Hence, by Corollary 6.3, the principal solution
w,0(r) of F.33,) satisfying F.34) is given by
Ca ds
F.37) C,u,0@ = wyi(f)J —— for a <! t < co,
where
ds
F.38)
« Pi(s)"n(s)
Differentiation of F.37) gives
o ^ c,«;o(o = u'n(t)
so that, if t = a,
0 > pAaWJ
Thus the sequence/>,(a)MJ0(a), y = 1, 2, . . ., is bounded if
F.39) C, ^ const. > 0 for ; = 1, 2, . . .
In order to verify F.39), note that F.36) and the assumption on F.32)
imply that un(t)-*uKl(t) as j—*- 00 uniformly on closed intervals of
a < t < co. Thus, by F.38),
C,
ds
PjU2n Ja
ds
as j -»¦ 00
for any fixed T, a < T < co. This implies F.39).
Since the sequence of numbers uj0(a) = 1 and u'i0(a) for j = 1,2,...,
are bounded, there exist subsequences which have limits. If y'(l) <
y'B) < • ¦ • are the indices of such a subsequence and
1 = lim uj0(a), u'x0 = lim u'j0(a) for j = j(n) -»¦ 00,
then the assumption on F.32) implies F.35) uniformly on every interval
[a, T] <= [a, co), where «„(() is the solution of F.33^) satisfying u^{a) = 1,
uj{a) = u'k0. The solution uK(t) clearly satisfies F.19) and F.34). This
proves Corollary 6.6.
362 Ordinary Differential Equations
7. Nonoscillation Theorems
This section will be concerned with conditions, necessary and/or
sufficient, for
G.1)
u" + q{t)u = 0
to be nonoscillatory. In view of the Sturm comparison theorem, the
simplest (and one of the most important) sufficient conditions for G.1)
to be nonoscillatory [or oscillatory] is for G.1) to possess a nonoscillatory
[or oscillatory] Sturm majorant [minorant]. For example, if q(t) _ 0 [so
that u" = 0 is a Sturm majorant for G.1)], then G.1) is nonoscillatory. If
q(t) = fxf2, then G.1) is nonoscillatory or oscillatory at t = oo according
as I* = i or P > i; see Exercise l.l(c). This gives the following criteria:
Theorem 7.1. Let q(t) be real-valued and continuous for large t > 0.
If
G.2) - oo < lim sup t2q(t) < -
(->00 4
or oo _ lim inf t2q(t) > -
L (->00
then G.1) is nonoscillatory [or oscillatory] at t = oo.
If, e.g., t2q(t) -> | as t -* oo, then Theorem 7.1 does not apply. In this
case, Exercise 1.2 shows that G.2) can be replaced by
-co ^ lim sup t* log2 t
\q(t) ~ ^l
or
co ^ lim inf t* log2 t \q(t) - ^
< 4
> i
In fact, the sequence of functions in Exercise 1.2 gives a scale of tests for
G.1) to be nonoscillatory or oscillatory at t = 00.
The criterion given by Sturm's comparison theorem can be cast in the
following convenient form:
Theorem 7.2 Let q(t) be real-valued and continuous for J : a _ t <
ft) (^ 00). Then G.1) is disconjugate on J if and only if there exists a
continuously differentiable function r(t)for a < t < co such that
G.3) r' + r2 + q(t) < 0.
Exercise 7.1. Formulate analogues of Theorem 7.2 when J is open or
J is closed and bounded.
Remark. It is clear from § 1 that analogues of Theorem 7.2 remain
valid if G.1) is replaced by an equation of the form ipu')' + qu = 0
or u" + gu +fu = 0 provided that G.3) is replaced by the corresponding
Linear Second Order Equations 363
Riccati differential inequality r + r2/p + q < 0 or r' + r2 + gr +/^ 0,
respectively.
Proof. First, if G.1) is disconjugate on J, then G.1) has a solution
u = uo(t) > 0 for a < t < co; see Corollary 6.1. In this case, r = u0'lu0
satisfies the Riccati equation
G.4)
r' + r2 + q(t) = 0
for a < t < ft). This proves the "only if" part of the theorem.
If there exists a continuously differentiable function r(t) satisfying
G.3), let qo(t) = 0 denote the left side of G.3) for a < t < co, so that
r + r2 + q — q0 = 0. Then
u" + [q(t) - qo(t)] u = 0
is a Sturm majorant for G.1) on a < t < co and, by § 2 (xiv), possesses the
ft
positive solution u = exp r(s) ds, where a < c < co. This shows that
Je
G.1) is disconjugate on a < t < co. In order to complete the proof, we
must show that if u^t) ^k 0 is the solution of G.1) satisfying u^a) = 0
and u^{a) = 1, then u^t) ^ 0 for a < t < co. Suppose that this is not
the case, so that Mj^o) = 0 for some t0, a < t0 < co. Since Mj changes
sign at t = t0 and solutions of G.1) depend continuously on initial
conditions, it follows that if e > 0 is sufficiently small, then the solution
of G.1) satisfying u(a + e) = 0, u'(a + e) = 1 has a zero near t0. This
contradicts the fact that G.1) is disconjugate on a < t < co and proves
the theorem.
Exercise 7.2. (a) Using the Remark following Theorem 7.2, show that
if, in the differential equations
G.5,) u"+gi(t)u' +f(t)u=O,
where/ =1,2, the coefficient functions are real-valued and continuous on
J:a < t < co (_ oo) such that
G-6) gi(t)<gjt), Mt)<f2(t)
and if G.52) has a solution u(t) satisfying u > 0, u _ 0 for a < t < co,
then G.5j) is disconjugate on /. [For an application in Exercise 7.9, note
that the conditions on G.52) hold if G.52) is disconjugate on J,f2(t) = 0
and exp — ?2(.s) ds\ dt = oo; cf. Exercise 6.2.] (b) Let f(t) be
continuous and g(t) continuously differentiable real-valued functions on
a = f <b. Then
u" + g(t)u'+ f(t)u = 0
364 Ordinary Differential Equations
is disconjugate on [a, b] if there exists a real number c such that
fit) ~ cg\t) + c(c - \)g\t) ^ 0
for a ^ t < b.
Corollary 7.1. Let q(t) be real-valued and continuous on J:a ^ t < a>,
C a constant, and
G.7) Q(t) = C - \q(s) ds.
Ja
If the differential equation
G.8) u + 4Q2(t)u = 0
is disconjugate on J, then G.1) is disconjugate on J.
Exercise 7.3. Show that this corollary is false if the 4 in G.8) is
replaced by a constant y < 4.
Proof of Corollary 7.1. In the Riccati equation G.4) belonging to G.1),
introduce the new variable
G.9) P = r-Q,
so that p' = r' + q, and G.4) becomes
G.10) p + p2 + 2QP + Q2 = 0.
Since 2Qp ^ p2 + Q2, a solution of
G.11) p' + 2(p2 + Q2) = 0
on some interval satisfies
G.12) p' + p2 + 2QP + Q2^ 0.
The differential equation G.11) can be written as
G.13) o'+<r2 + 4Q2 = 0 if a = 2p.
Finally, G.13) is the Riccati equation for G.8).
Thus if G.8) has a solution u(t) > 0 on J, then a = u'/u satisfies G.13).
Hence p = ^satisfies G.12) and/* = p + Q is a solution of the differential
inequality G.3) on /. In virtue of Theorem 7.2, this proves the corollary.
Exercise 1A. A counterpart of Corollary 7.1 can be stated as follows:
Let q(t) be real-valued and continuous for 0 ^ t ^ b. Let a be fixed,
0 ^ a < b. Suppose that
Q(t) = f.
Jo
q(s) ds
has the properties that Q(t) ^ 0 for a ^ t ^ b and that if z(t) is a solution
of z" + Q2(t)z = 0, z'(a) = 0, then z(t) has a zero on a < t ^ b. Then a
solution m@ of G.1) satisfying m'@) = 0 has a zero on 0 < t ^ b.
I
Linear Second Order Equations 365
One of the main results on equations G.1) which are nonoscillatory at
t = oo will be based on the following lemma.
Lemma 7.1. Let q(t) be real-valued and continuous on 0 ^ t < oo with
the property that G.1) is nonoscillatory at t = oo. Then a necessary and
sufficient condition that
G.14)
holds for one (and/or every) real-valued solution u(t) ^ 0 o/G.1) is that
G.15) lim- [\q(s)ds \ dt = C exists
T->aoTJo \Jo /
(as a finite number).
Remark. For the application of this lemma, it is important to note
that the proof will show that condition G.15) can be relaxed to
G.16) lim inf- f ( fq(s) ds) dt > - oo.
In other words, when G.1) is nonoscillatory at t = oo, then G.16) implies
G.15); in fact, it implies the stronger relation
G.17)
TJo
Jo
q(s) ds
dt-+O as T-
oo.
Exercise 7.5. Let q(t) be as in Lemma 7.1. Show that
t
G.18) -->-C
as t
oo
holds for one (and/or every) real-valued relation u(t) ^ 0 of G.1) if and
only if
G.19)
sup
q(s) ds
0
as t
oo.
[Note that G.19) holds if, e.g., q(t) -+ 0 as t ^ oo or f \q(s)\v ds < oo for
some y ^ 1.] J
Proof. Suppose first that G.14) holds for a real-valued solution
u(t) ^ 0 of G.1). Let t = a exceed the largest zero, if any, of u(t). Put
r = u'ju for t ^ a, so that r satisfies the Riccati equation G.4). A
quadrature gives
G.20)
r(t) + !'r\s) ds = r(a)-l'q(s)ds
Ja Ja
366 Ordinary Differential Equations
for / ^ a. Then G.14) implies that G.20) can be written as
G.21) r(t) - r\s) ds = C - \ q(s) ds,
where C = r(a) - f °°r%s) ds + \ q(s) ds. By G.14),
Ja JO
/ -> 0 as T
oo.
Hence G.21) implies G.17) [by virtue of the inequality (a + /?J ^
2(a2 + ft2) for real numbers a, 0]. Since Schwarz's inequality [cf. G.22)]
shows that G.15) is a consequence of G.17), it follows that G.15) is
necessary for G.14).
In order to prove the converse, assume G.16), that u(t) & 0 is any real-
valued solution of G.1), and that u(t) > 0 for / ^ a. Then G.20) holds
for r = u'fu and a quadrature of G.20) gives
[
t Ja
t Ja \Ja
= - r(a)(t - a) - - ! Y f q(a) da) ds.
t t Ja \Ja I
The assumption G.16) implies that the right side is bounded from above.
Suppose, if possible, that G.14) does not hold, then the second term on the
left tends to oo as / -»¦ oo, thus
- - ("'r(s) ds^- ['( [r\a) da) ds for large /.
/ Ja 2/ Ja \Ja I
Schwarz's inequality implies
G-22)
tJa
1 C*
- r(s) ds
tJa
and, consequently,
4/ r\s)ds^ r\a)da\ ds\ for large t.
Ja \Ja \Ja I I
This can be written as
4tS' ^ S2, where S(t) = ( r\a) da\ ds^-co
Ja \Ja I
as / -*¦ oo. A quadrature gives
const. — ^ log / for large /.
S(t)
Linear Second Order Equations 367
This contradiction shows that the hypotheses that G.14) fails to hold is
untenable and proves the theorem.
Theorem 7.3. Let q(t) be real-valued and continuous for 0 ^ / < oo.
A necessary condition for G.1) to be nonoscillatory at t = oo is that either
G.23)
lim inf—
T-oo T Jo
q(s) ds I dt = — oo
or that G.15) holds [and, in the latter case, G.17) holds].
It follows, e.g., that if q(t) ^ 0, then, in order for G.1) to be non-
f°°
oscillatory at / = co, it is necessary that q(t) dt < oo. In fact, as is
J /*00
seen from Exercise 7.8, it is necessary that tyq(t) dt < oo for every
y< 1. J
Proof. Suppose that G.1) is nonoscillatory at / = oo and that G.23)
fails to hold, so that G.16) holds. The validity of G.17) must be verified.
But this is clear from the proof of Lemma 7.1 which shows that, on the one
hand, G.16) implies G.14) for every real-valued solution u(t) ^ 0 of
G.1) and, on the other hand, that G.14) for some solution assures G.17).
Exercise 7.6. Let q(t) be as in Theorem 7.3 and, in addition, satisfy
G.24) ?(*)-»¦ 0
as
or, more generally, G.19). Then a necessary condition for G.1) to be
nonoscillatory at / = oo is that either
q(t)dt^-
Jo
oo
as T—>• oo
G.25)
or that
/"oo /*T
G.26) ^(/) dt = lim #(/) d/ converges
Jo r-»ooJo
(possibly conditionally).
Exercise 7.7. (a) Give examples to show that G.15) in Theorem 7.3 is
compatible with each of the possibilities
G.27) lim sup- ( \ q(s) ds) dt is — oo, finite,
t-oo TJo \Jo I
(b) Show that if, in Theorem 7.3, q(t) is half-bounded or, more generally,
if there exists an e > 0 such that
or +oo.
rs
q(a) da is half-bounded for 0 ^ / < oo, 0 ^ 5 ^ e,
then a necessary condition that G.1) be nonoscillatory is that either
G.15) or G.25) hold. See Hartman [11].
368 Ordinary Differential Equations
Changes of variables in G.1) followed by applications of Theorem 7.3
(and its consequences) give new necessary conditions for G.1) to be non-
nonoscillatory. This is illustrated by the following exercise.
Exercise 7.8. (a) Introduce the new independent and dependent
variables s = tr, y > 0 and z = t(r~1)l2u, and state necessary conditions
for the resulting equation and/or G.1) to be nonoscillatory at / = oo.
(b) In particular, show that if q(t) ^ 0 and G.1) is nonoscillatory at
,.00
/ = oo, then ^"'qi^dt < oo for all y > 0.
The next result gives a conclusion very different from G.17) in Theorem
7.3 in the case G.15).
Theorem 7.4. Let q(t) be as in Theorem 7.3 such that G.1) is non-
nonoscillatory at t = oo and G.23) does not hold [so that G.15) does]. Then
G.28) pexp i-y f *g(s) ds\ dt = oo for 0 < y ^ 4,
where
G.29) Q(t) = C-ftq(s)ds.
Jo
In applications, interesting cases of this theorem occur if G.26) holds,
so that
G.30)
Jt
It is readily verified from q(t) = fjijt2, t = 1, that the " in G.28) cannot
be replaced by a larger constant. It is rather curious that the proof of
Corollary 7.1 and Theorem 7.4 depend on the inequality 2Qp _ p2 + Q2.
In the proof of Corollary 7.1, this inequality is used to deduce G.12) from
G.11); in the proof of Theorem 7.4, it is used to deduce
G.31)
from G.10).
Proof. Let u = u(t) ^ 0 be a real-valued solution of G.1) and suppose
T is so large that«(/) ^ 0 for / _ T. Since it is assumed that G.15) holds,
the relation G.14) holds. Thus if r = u'ju, a quadrature of the corre-
corresponding Riccati equation gives G.21) as in the proof of Lemma 7.1.
Rewrite G.21) as r(t) = p(t) + Q(t), where
G.32)
= 1 r\s)ds.
Since p = — r2 = -(p + QJ, the equation G.10) holds. This gives
G.31). In particular, if Q(t) ^ 0, then
G.33) p + yQp = 0 for 0 < y ^ 4.
Linear Second Order Equations 369
Note that if Q < 0, then G.33) holds since p ^ 0, p ^ 0. Hence
G.33) holds for / ^ T. Since the result to be proved is trivial if q(t) = 0
for large /, it can be supposed that this is not the case. Hence r ^ 0 for
large / and so, p(t) > 0. Consequently, G.33) gives
G.34) p(t) ^ p(T) exp (-y? Q(s) dsj for / ^ T.
Suppose, if possible, that G.28) fails to hold, then G.32), G.34) show
that
r
p(t) dt < oo, hence tr\i) dt < oo
r
holds for r = u'/u, where u(t) ^ 0 is an arbitrary real-valued solution of
G.1). It will be shown that this leads to a contradiction. To this end, note
that
Thus Schwarz's inequality gives
Consequently there exist constants c0, c such that |w(/)| ^ c0 exp c(log t)l/i
for large /. It follows that
= GO
for all real-valued solutions (^ 0) of G.1). This contradicts the existence
(Theorem 6.4) of nonprincipal solutions and completes the proof.
Exercise 7.9. In the differential equations
G.35,) u" + qlt)u = 0,
where j =1,2, let q(t) be real-valued and continuous for large / and such
that
exo = ("iM ds = iim fa
Jt T-a>Jt
converges (possibly conditionally), \Q1(t)\ ^ Q2@, and G.352) is non-
nonoscillatory at / = oo. Show that G.35j) is nonoscillatory at / = oo.
8. Asymptotic Integrations. Elliptic Cases
In the next two sections, we will consider the problem of the asymptotic
integration of equations
(8.1) u"+q(t)u = 0,
370 Ordinary Differential Equations
where q(t) is continuous for large /. Except for the last part of this section,
the main interest will center around the situations where the coefficient
q(t) is nearly a constant or (8.1) can be reduced to this case. The last part
of this section (see Exercises 8.6, 8.8) deals with bounds for |w'| when
q(t) is bounded from above.
When q(t) is a constant, say A, and A is real and positive, then the solu-
solutions are, roughly speaking, of the same order of magnitude. On the
other hand, if A is not real and positive, then essentially there is one small
solution, as / ->¦ oo, and the other solutions are large. These facts indicate
that different techniques will be needed when q(t) is nearly a constant A,
and A is or is not real and positive. In this section, the first case will be
considered.
Theorem 8.1. In the differential equations (8.1) and
(8.2)
+ qo(t)w = 0,
let q(t), q9(t) be continuous, complex-valued functions for 0 ^ / < oo
satisfying
(8.3)
J°°|w@l2
MO - <z(OI df <
for every solution w(t) of (8.2). Let u9(t), v9(t) be linearly independent
solutions of (S.I). Then to every solution u(t) of (S.I), there corresponds at
least one pair of constants a, /S such that
u(t) = [a + o(l)K(/) + [P + o(l)]v0(t),
(8.4)
u'(t) = [a + o(l)]u0'(t) +[j3 +
as t-*- oo; conversely, to every pair of constants a, /S, there exists at least
one solution u(t) o/(8.1) satisfying (8.4).
Note that for a given u(t), (8.4) might hold for more than one pair of
constants (a, /S). This is true, e.g., if vo(t) = o(u9(t)) as / ->¦ oo.
An interesting aspect of Theorem 8.1 is the fact that the main condition
(8.3) does not involve the derivatives w'(t) of solutions w(t) of (8.2). This
advantage is lost if (8.1) or (8.2) is replaced by a more complicated equation
as in the Exercise 8.4 below.
Proof. It can be supposed that det Y(t) = 1, where
Y(t) =
Write (8.1) as a first order system x' = A(t)x for the binary vectors
x = (u, u'); cf. B.5). Then the variations of constants x = Y(t)y reduce
the system x' = A(t)x, say to y' = C(t)y, in B.28); cf. §2(xi). Thus
Linear Second Order Equations 371
Theorem 8.1 follows from the linear case of Theorem X 1.2, cf. Exercise
X1.4.
Corollary 8.1. Let q(t) be a continuous complex-valued function on
0 ^ / < oo satisfying
(8.5)
Then if a, /} are constants, there exists one and only one solution u(t) of
(8.1) satisfying the asymptotic relations
u = [a + 00)] cos /+[/? + o(l)] sin /,
(8.6)
w' = -[a + 0A)] sin /+[/? + o(l)] cos /.
The relations (8.6) can also be written as u = d cos [/ + y + o(l)],
u' = —dsin[t + y + o(l)] as /-> oo for some constants y and d.
Exercise 8.1. Show that if a, /} are constants, there exists a unique
solution v(t) of the Bessel equation t2v" + tv' + (t2 — [*2)v = 0 for / > 0
such that u(t) = t1Av(t) satisfies (8.6) as / -> oo.
Exercise 8.2. Show that the conclusion of Corollary 8.1 is correct if
(8.5) is relaxed to the following conditions in which/(/) = 1 — q(t): the
integrals
?o(O =j( f(s) ds, gl(t) =j^ f(s) cos 25 ds, g2(t) = J °/(s) sin 2s ds
= lim
as T^ co I and I \gk(t)f(t)\ dt < oo for k = 0, 1, 2.
Exercise 8.3. (a) Let q(t) be a positive function on 0 ^ / < oo possess-
possessing a continuous second derivative and such that
(8.7) 1*7^@ dt = oo and f
J J
dt < oo.
qVlu = [a + o(l)] cos qA(s) ds + [0 +
Jo
loq 4q
Then the assertion of Corollary 8.1 remains valid if (8.6) is replaced by
o(l)] sin f *qA(s) ds,
Jo
sin I'q^s) ds + [0 + o(l)] cos f 'qA(s) ds.
Jo Jo
(b) Show that (8.7) in (a) holds if 0 ^ a > -J, q(t) ^ const. > 0 for
0 ^ t < co, and/(r) = q*(t) ^ 0 has a continuous second derivative such
/*00
that |/"(r)| dt < co. [In fact, for the validity of the conclusion of (a), it
(qAu)'q-'A [a +
372 Ordinary Differential Equations
can be merely supposed that/(/) has a continuous first derivative which is
of bounded variation on 0 ^ / < oo, i.e., \df'(t)\ < °°; e.g'.,/'(/) is
monotone and bounded. This last refinement follows from the first part
by approximating q(t) by suitable smooth functions.]
Exercise 8.4. (a) In the differential equations
(8.8,) (p/)' + r,W + qtu = 0, / = 0, 1,
let Pj(t) t? 0, qt(t), r}(t) be continuous complex-valued functions for
0 ^ / < oo such that
w\2\qi-q<>\ ¦
¦r
rods
Po
(8.9)
f
Pi Po
dt < co,
< co,
dt < co
hold for all solutions w = vv(/) of (8.8O). Let mo(/), i'o(O be linearly in-
independent solutions of (8.8O). Then to every solution «(/) of (8.8^, there
corresponds at least one pair of constants a, /S such that (8.4) holds;
conversely, if a, /S are constants, then there is at least one solution «(/) of
(8.8j) satisfying (8.4). (o) In the differential equation (8.1), let q(t) ^ 0 be
a continuous, complex-valued function for 0 ^ / < oo such that q(t) is
/ r°° \
of bounded variation over 0 < / < oo i.e., \dq\ < oo ; c0 =
\ Jo I
lim<7(/), / -»¦ oo, is a positive constant; and the solutions «(/) of (8.1) are
bounded (e.g., if^(/) is real-valued or, more generally, \lm q(t)\ dt <oo,
then solutions u(t) and their derivatives u'(t) are bounded). Let a, /S be
constants. Then (8.1) has a unique solution «(/) satisfying, as /->¦ oo,
w = [a
(8.10)
w' = [a + o
cos f'$*(s
Jo
^ sin f V^
Jo
sin fg*
Jo
ds,
+ \fi +
cos f '^
Jo
ds,
where q1^ (t) is any fixed continuous determination of the square root of q.
Exercise 8.5. Let /(/) be a nonvanishing (possibly complex-valued)
function for / > 0 having a continuous derivative satisfying
4s <»,
and y2 ^ 1.
Linear Second Order Equations 373
Suppose further that
(8.11) exp± i\fA 1 -^—¦¦) dr are bounded
J \ 16/ /
as / ->¦ oo. Then the differential equation
(8.12) u" +/(/)« = 0
has a pair of solutions satisfying, as / —»¦ oo,
u ~f-y\t) exp ± ij f*\\ - -L
u'~[-y±i(l-y2IA]flAu.
dr,
Here all powers of /(/) that occur can be assumed to be integral (positive
or negative) powers of a fixed continuous fourth root/^(/) of/(/). Condi-
Condition (8.11) is trivially satisfied if /(/) is real-valued and positive and
0 <72< 1.
The object of the next exercise is to obtain bounds for derivatives of
solutions of (8.1) or, more generally, the inhomogeneous equation
(8.13)
u"+q(t)u =/(/).
Exercise 8.6. Let q(t), /(/) be continuous real-valued functions on
0 ^ / ^ /0. Let the positive constants e, 1/0 > 1, C be such that
(8.14)
and
(8.15)
f
Ja
q(r) dr^
^
if b -a <-
~ C
and 0 ^ a < b < t0. [The inequality (8.15) holds, e.g., is q(t) ^ C2/0
forO^/^/0.] Let u = u(t) be a real-valued solution of (8.13). Consider
the case
uu' ^ 0 at / = T, 0 ^ T^ t9 - 6/C, S= T+ d/C, U = T+ e
or the case
mm' ^ 0 at / = T, 6/C ^ T < t0, S = T - 0/C, U = T - e.
(a) Show that, in either case,
1
(8.16) \u\T)\ ^
1 - 0
\f\dr
+ max
374 Ordinary Differential Equations
I Cu
(b) Show that (8.16) holds if \u(U)\ is replaced by 2^ \u\ dr . (c) Part
(a) implies that if 0/C ^ / ^ t0 - 0/C, then I Jt
(8.i7) iM'(oi ^ ^ /;; i/i *+co(e) j>(,+Mi,
where C0(e) = max (C/(l - 0), C + 1/e). (d) Put
(8.18) /•(/) = |m'@I + C|m(/)|-
Show that there exists a nondecreasing function K(A) for 0 < A <
min @, 1 - 0)/C such that
(8.19) r(i\ < K(AMr(a} + rib) + \ \f\dr\ for a^t^b
if b - a = A and 0 ^ a < b ^ /0.
The results of the last exercise can be extended to an equation of the
form
(8.20) u" + p(t)u'+ q(t)u =/(/);
see Exercise 8.8. In fact, the results for (8.20) can be derived from those
on (8.13) by the use of the lemma given in the next exercise which has
nothing to do with differential equations.
Exercise 8.7. Let h(i) ^ 0 be of bounded variation and g(t) continuous
on an interval a < / < b. Then
(8.21)
f
Ja
b Cf>
h(t) dg(t) ^ (inf h + var h) sup dg(t),
aSoc<^S6 Ja
where the integrals are Riemann-Stieltjes integrals and var h denotes the
total variation of h(t) on a ^ / ^ b.
Exercise 8.8. Let/>(/), q(t),f(t) be continuous real-valued functions on
0 ^ / ^ /0 and «(/) a real-valued solution of (8.20). Let 1/0 > 1, C be
positive constants such that (8.15) holds. Consider the two cases in
Exercise 8.6, with (8.14) replaced by
(8.22) 0 < e ^ djCE2, where E = exp |p(r)| dr
(a) Then parts (a), (b) of Exercise 8.6 hold if \u'(T)\ in (8.16) is replaced by
\u'(T)\jE. (b) Part (c) of Exercise 8.6 holds if |m'@I in (8-17)is replaced by
\u'(t)\IE, where
(8.23) 0 < e ^ 0/C?2 and
ft+e/c
E = max exp |p(r)| dr for 0 ^ t ^ t0 - d/C.
Linear Second Order Equations 375
(c) Part (d) of Exercise 8.6 holds if A is restricted to 0 < A <
min @, 1 - 0)/C?2, where E is denned in (8.23).
9. Asymptotic Integrations. Nonelliptic Cases
Asymptotic integrations of u" + q(t)u = 0, where q(t) is "nearly" a real,
but not positive constant, can be based on Chapter X as were the results
of the last section. Instead a different technique will be used in this section;
this technique takes greater advantage of the special structure of the
second order equation
(9.1) (j,(t)u'Y + q(t)u = 0.
This equation is equivalent to a binary system of the form
(9.2) v' = p(t)z, z' = y(t)v
in which the diagonal elements vanish. [This system cannot be reduced
to an equation of the form (9.1) unless either /?(/) or y(t) does not vanish.]
The main results on (9.1) will be based on lemmas dealing with (9.2).
A system of the form (9.2) on 0 ^ / < co (< oo) will be called of type Z
at / = co if . . .. . .
z(co) = hm z(t) exists as t-*-co
for every solution (v(t), z(t)), and z(co) ?? 0 for some solution. It is easy
to see that (9.2) is of type Z if and only if there exist linearly independent
solutions (Vj(t), z^t)), j = 0, 1, such that lim z9(t) = 0 and \im z^t) = 1.
Lemma 9.1. Let /?(/), y(t) be continuous complex-valued functions for
0 ^ / < co (^ oo). Suppose that
(9.30 JV(OI dt < oo; (9.32)
or, more generally, that
/*co FT
(9.4X) y(t)dt = \im y(t) dt
J T-xo J
{possibly conditionally) and that
(JW)I ds\
dt < oo
exists
(9.4,)
r
y(r) dr
5)| T(s) ds < oo, where T(s) = sup
sSi<co I Jt
Then (9.2) is of type Z.
Unless /?(/) = 0, the condition (9.32) implies (9.3!). If the order of
integration is reversed, it is seen that (9.32) is equivalent to
\y(t)\ dt} ds < <x>.
This shows that (9.3) implies (9.4). Lemma 9.1 has a partial converse.
376 Ordinary Differential Equations
Lemma 9.2. If fi(t) and y(i), where 0 ^ / < w (^co), are continuous
real-valued functions which do not change sign (i.e., /S > 0 or ^ ^ 0 and
y-^Oory <0)and if (9.2) is of type Z, then (9.31)-(9.31!) hold.
Exercise 9.1. Generalize Lemma 9.1 to the case where (9.2) is replaced
by a d-dimensional system of the form v'' = yi(t)vi+1, where y = 1,. .. , d
and vd+1 = v1.
Proof of Lemma 9.1. Two quadratures of (9.2) give
(9.5) v(t) = | P(s)z(s) ds + clt c1 = v{T),
JT
(9.6) z(t) = \y(s) | P(rXr) dr ds + Cl y(s) ds
JT JT JT
c2, c2 = z(T).
On interchanging the order of integration, the last formula becomes
(9.7) z(/) = fVw) | Y(s) ds dr + cA y(s) ds + c2.
JT Jr JT
If / g T, then T ^ r ^ / and the definition of T in (9.42) imply that
(9.8)
t
y(s) ds
ds
+
y(s)
Consequently
\z(t)\ ^ 2!' \/3(s)\ T(s)\z(s)\ ds + C,
Jr
where
(9.9) C = 2 |Cl| rG0 + |c2|.
By Gronwall's inequality (Theorem III 1.1),
(9.10) KOI ^ C exp 2 f |/J(s)| r(s) Js ^ C exp 2 f "|/5(s)| T(s) rfs
for r < / < co. Hence (9.42) implies that z(t) is bounded. The relations
(9.7) and (9.42) then show that z(oS) = lim z(t) as / -»¦ co exists.
The limit z(co) is obtained by writing t = co in (9.7). In order to show
that z(co) 9^ 0 for some solution of (9.2), choose the initial conditions
Cl = v(T) = 0 and c2 = 2(T") = 1 in (9.5), (9.6). Thus C = 1 in (9.9)
and (9.10) and so (9.7), (9.8), and (9.10) give
\z{co) - 1| ^
dr} exp
s) ds.
Since the right side tends to 0 as T^>-co, it follows that if T is sufficiently
near to co, then 2(w) ?* 0. This proves Lemma 9.1.
Linear Second Order Equations 377
Proof of Lemma 9.2. Let (v(t), z{t)) be a solution of (9.2) such that
z(co) = 1. It can also be supposed that v(T) = 0 for some T. Otherwise
it is possible to add to (v(t), z(t)) a suitable multiple of a solution (vo(t),
zo(t)) ¦? 0 for which z(co) = 0. In fact vo(t) = 0 cannot hold, for then
(9.2) shows that z9(t) = z9(co) = 0.
Thus c1 = 0 in (9.6) and z(co) = 1 shows that (9.32) holds (since /S, y do
not change sign). If ,?(/) ^ 0 for / near co, (9.3j) follows. If, however,
/?(/) = 0 and (9.2) is of type Z, then (9.3j) holds when y does not change
signs. This completes the proof.
Let (v(t), z(t)), (Vl(t), ZjCO) be solutions of (9.2). Then
(9.11) Zi(OK') — vi(t)z@ = co
is a constant. This follows from Theorem IV 1.2 (or can easily be verified
by differentiation). If zx(t) j± 0 and (9.11) is multiplied by y{i)\z^{i), it is
seen from (9.2) that (z/z,)' = coy/z^, and hence there is a constant Cj such
that
(9.12)
3@ =
ids
if z1 ?? 0 on the interval [T, t]. Similarly, if v1 9* 0 for [T, t] then
/' and
(9.13)
Jt Vl\s) '
Conversely, if Zj ^ 0 [or ^ j± 0] in the /-interval [T, t], then (9.12) [or
(9.13)] and (9.11) define a solution (v(t), z(t)) of (9.2).
Exercise 9.2. Suppose that (9.2) is of type Z and that (v^t), z^t)) is
a solution of (9.2) satisfying z^co) =1. (a) Show that
f-2<2*_lim f'2<f
J Zj @ r-co J Zj
y(t)Jt
exists
and that (9.2) has a solution (t;0(/), z9(t)) in which
Jt z? (s)
for / near co. F) If (t;(/), z(t)) is any solution of (9.2), then
v(t) = C|i + I \p(s)\ as] as t^-co.
If, in addition, (9.3) holds, then (9.14) satisfies
(9.15)
as t-> co.
378 Ordinary Differential Equations
c = lim vo(t) exists as / ->¦ co, and
(9.16) po(/) = c + o(j"\p(s)\ja\y(r)\ dr ds\
Also, if y(t) is real-valued and does not change signs, then (9.15) can be
improved to
(9.17) z?t)~j°y(s)ds.
Lemma 9.3. Let pit), y(i) be as in Lemma 9.1. In addition, suppose that
Pit) ^ 0 and that
[a,
(9.18) Pit) dt = oo.
Then (9.2) has a pair of solutions iv,it), zfa)) for j = 0, 1, satisfying, as
t->-co,
(9.190)
^O^1. Z9 = °(
(9.190
This has a partial converse.
Lemma 9.4. Z,e/ /?(/), y(/) 6e continuous real-valued functions such that
Pit) ^ 0 satisfies (9.18) anrf y(/) rfoes no/ change signs. Let (9.2) toe a
solution satisfying either (9.190) or (9.19j). T«en (9.3) woWs [so that (9.2)
was solutions satisfying (9.190) and (9.190].
Exercise 9.3. Prove Lemma 9.4.
Proof of Lemma 9.3. By Lemma 9.1, (9.2) has a solution (v^i), «i(/))
such that z^co) = 1. Thus the first part of (9.190 follows from the first
equation in (9.2). Note that
(9.20)
= const. —
tends to const, as t->-co by (9.18). Consequently, the integral Cj =
/3(y) ds/v^is) is absolutely convergent (for T near co). It follows from
Jt
(9.13) with the choice c0 = 1, that (9.2) has a solution (y, z) = (y0, z0)
satisfying (9.11) with c0 = 1 and
Then v0 ~ 1 follows from the first part of (9.190 and (9.20). Letting
Linear Second Order Equations 379
iv, z, c0) = (v0, z0, 1) in (9.11) and solving for z0 gives the last part of
(9.190). This completes the proof.
Theorem 9.1. Let pit) be a positive and qoit) a real-valued continuous
function for 0 ^ / < co such that
(9.21) ipit)x')' + qoit)x = 0
is nonoscillatory at t = co and let xo(/), x^t) be principal, nonprincipal
solutions o/(9.21); cf § 6. Suppose that qit) is a continuous complex-valued
function satisfying
(9-22) JVo(O*i(OI • kit) - qoit)\ dt < oo
or, more generally,
(9.230 [°iq ~ qoW dt = lim C iq - qo)xo2 dt exists,
(9.23s)
car(s)ds_
J pis)xo\s)
< co, where
= sup
- qo)xo2 dr
(9.1) has a pair of solutions wo(/), u^t) satisfying, as t -*¦ co,
(9.24) «, - x,,
(9.25) V_ul = V^i +
i ^ 1011
forj = 0, 1.
Exercise 9.4. Verify that if ^(/) is real-valued, qit) — qoit) does not
change signs, and (9.1) has a solution «,</) satisfying (9.24)-(9.25) for
either; = 0 ory" = 1, then (9.22) holds.
Condition (9.22) in Theorem 9.1 should be compared with (8.3) in
Theorem 8.1. The analogue of (8.3) is the stronger condition
r
•\q—qo\dt< co since x0 = oix1) as /->-co.
Remark. It will be clear from the proof of Theorem 9.1 that if qoit)
is complex-valued but has a pair of solutions asymptotically proportional
to real-valued positive functions xo(/), x^t) satisfying (9.22) [or (9.23)] and
dsjpx^ < co, ds/px02 = co, x1 ~ xA ds\px*, then Theorem 9.1
remains valid.
Exercise 9.5. Let pit) ^ 0, qit), qoit) be continuous complex-valued
functions for 0 ^ / < co (^ oo) such that (9.21) has a solution x^t) which
does not vanish for large / and satisfies
C" dt CT
;— = lim as T-*- co exists
J PitWit) J
380 Ordinary Differential Equations
and
J
4ol
dt < oo, where T(t) = sup
C" dr
Js DX,2
Then (9.1) has a pair of nontrivial solutions wo(/), ux(t) such that
w0 = o(l«il), «i ~ *i
as
¦ co.
Proof of Theorem 9.1. The variations of constants u = xo(t)v reduces
(9.1) to
(9.26) 0»o2»')' + xo*(a - 9o> = 0
for/ near w; cf. B.31). Write this as a system (9.2), where
(9.27)
z = px0 v ,
p = —,, y = - V(? - ?<>)•
It will be verified that Lemma 9.3 is applicable. Note that condition
(9.18) holds since xo(t) is a principal solution of (9.21); Theorem 6.4. A
nonprincipal solution xx(t) of (9.21) is given by
(9.28) xx(t) = xo(t) f d\ = *0@ [pis) ds
J p(s)xo-(s) J
and any other nonprincipal solution is a constant times [1 + o(l)]*i@ as
t-+oi; Corollary 6.3. The condition (9.4) is equivalent to (9.23).
Thus Lemma 9.3 is applicable. Let (v0, z0), (vu zx) be the corresponding
solutions of (9.2) and w0 = xovo, Mj = x^ the corresponding solutions of
(9.1). Then the first part of (9.19,) fory = 0, 1 gives (9.24) fory = 0, 1.
Note that u = xov implies that pu'/u = pxo'lxo + Pv'lv> s0 that> by (9-27)>
/If1 \
pu'/u = pxo'lxo + z/x02v. Since zo/vo = o11 / J P(s) ds\, the casey = 0 of
(9.25) follows. Also, zx\x^ = [1 + o{\)\\xA /J(j) ds = [I + o^jx^
and, from (9.28), px1'/x1 = pxo'lxo + l/x^. Consequently, the case
j = 1 of (9.25) holds. This proves the theorem.
Corollary 9.1. In the equation
(9.29) u" - q(t)u = 0,
let q(t) be a continuous complex-valued function for large t satisfying
(9.30) t \q(t)\ dt < co
Linear Second Order Equations 381
or, more generally,
(9.31) Q(t) = q(s) ds = lim exists and sup \Q(r)\ dt < oo
Jt T->*> Jt J (gr<oo
Then (9.29) has a pair of solutions uo(t), ux(t) satisfying, as t —»¦ oo,
(9.32)
(9.33)
Conversely, ifq(t) is real-valued and does not change signs and if (9.29) has
a solution satisfying (9.32) or (9.33), then (9.30) holds.
The first part of the corollary follows from Theorem 9.1, where (9.29)
and x" = 0 are identified with (9.1) and (9.21), respectively. The latter has
the solutions xo(t) = 1, xx{t) = t. [Under the condition (9.30), the
existence of w0, Wj is also contained in Theorem X 17.1.] The last part of
the corollary follows from Lemma 9.4 or Exercise 9.4.
Corollary 9.2. In the equation
(9.34)
u" - [A2 + q(t)]u = 0,
let X > 0 and q(t) be a complex-valued continuous function for large t
satisfying
(9.35)
or, more generally,
(9.36)
f
J\@l
rr
= lim
f
dt < oo
exists and
e2kt sup
r dr
dt < oo.
Then (9.34) has solutions uo(t), u^t) satisfying
(9.37)
w0 ~ —
Conversely, ifq(t) is real-valued and does not change signs and if (9.34) has
a solution uo(t) or ur(t) satisfying the corresponding conditions in (9.37),
then (9.35) holds.
The first part follows from Theorem 9.1 if (9.34) and x" — X2x = 0 are
identified with (9.1) and (9.21), respectively. The latter has solutions
•*"o@ = e~u, xY(t) = eu. [Under condition (9.35), the existence of u0, Mj is
also implied by Theorem X 17.2.]
382 Ordinary Differential Equations
Exercise 9.6. Let q(t) > 0 be a positive function on 0 ^ t < co
possessing a continuous second derivative and satisfying
7^@ dt = co and
f
< oo.
Then w" — q(t)u = 0 has a pair of solutions satisfying
q/xu ~exp ± q (s) ds, q~~ (q 4w)''—' ±q 'u
as t ->• oo. (Compare this with Exercise X 17.5.)
Exercise 9.7. Find asymptotic formulae for the principal and non-
principal solutions of Weber's equation
u" + tu' -2Xu = 0
(where A is a real number) by first eliminating the middle term using the
analogue of substitution A.9) and then applying Exercise 9.6 to the
resulting equation; cf. Exercise X 17.6.
Corollary 9.3. In equation (9.29), let q(i) be a continuous complex-
valued function for large t such that Q(t) in (9.31) satisfies
fco pT
Q(t) dt = lim exists as T-> co.
Then a sufficient condition for (9.29) to have solutions u9(t), u^i) satisfying
(9.38) M0(f)~l, u9'(t) = o(l),
(9.39) «i@~'> «1'(f)~l,
as t —*¦ oo, is that
(9.40) pt \Q(t)\2 dt < co.
This condition is also necessary ifq(t) is real-valued.
Proof. It is easily verified that
(9.41) xo(f) = exp i-jfQis)ds^
is a solution
(9.42) x" - [q(t) + Q\t)]x = 0.
One of the conditions on Q implies that lim xo(t) exists as t ->¦ co and is
not 0. Correspondingly, the solution of (9.42) given by
is asymptotically proportional to t, as t -*¦ co.
Linear Second Order Equations 383
Thus (9.42) has solutions asymptotically proportional to the (positive)
functions 1, t. Hence if (9.29) and (9.42) are identified with (9.1), (9.21),
respectively, and if (9.40) holds, the Remark following Theorem 9.1 shows
that the conclusions of that theorem are valid. Consequently (9.29) has
solutions w0, Mj satisfying w0~x0, «1~a;1 as (->• oo. The analogues of
(9.25) are
Since Q(t)->-0 as t ->¦ co, it is clear that certain constant multiples of
w0, «! satisfy (9.38), (9.39). The last part of the theorem follows from the
fact that q + Q2 > q when q is real-valued; cf. Exercise 9.4.
By the use of a simple change of variables, a theorem about (9.29) for
"small" q(t) can be transcribed into a theorem about (9.34) for "small"
q(t), and conversely:
Lemma 9.5. Let q(t) be a continuous complex-valued function for large t.
Then the change of variables, where X > 0,
(9.44) u = ve
transforms (9.34) into
s = BX)-1e2Xt
(9.45)
d2v _«« , x
— -e iXiq(t)v = 0;
ds
while the change of variables
(9.46) u = t'Av = esv, s = | log t
transforms (9.29) into
(9.47) — - [1 + 4t2q(i)]v = 0.
Exercise 9.8. Verify this lemma.
Exercise 9.9. (a) Let X > 0 and q(t) be a continuous complex-valued
function for large t such that
f* oo /* T
Qx(t) = q(s)e~iXs ds = lim
Jt T-oo Jt
[XQx(t)e2Xt dt = lim f * exists and f °
J T-oo J J
exists,
exists and | \Q(t)\2 eiu dt < co.
Then w" — [A2 + q(t)]u = 0 has a pair of solutions satisfying, as t -*¦ oo,
U
384 Ordinary Differential Equations
ib) Let qit) be a continuous complex-valued function for / ^ 0 such that
t2^1 \qit)\v dt < oo for some p on the range 1 ^ p ^ 2. Then w" —
<7(/)« = 0 has a solution satisfying
5 and — = o I -
and a solution satisfying
f-
r M 1
w ~ / exp 55E) ds and — ~ - ,
as t->¦ oo.
APPENDIX: DISCONJUGATE SYSTEMS
10. Disconjugate Systems
This appendix deals with systems of equations of the form
A0.1) [P(ty + R(t)x]' - [R*(ty - Q(t)x] = 0
or, more generally, systems of the form
A0.2) x' = A(t)x + B(t)y, y' = C(t)x - A*(t)y.
Here x, y are d-dimensional vectors; A(t), B(t), C(t),P(t), Q(t), R(t) are
d X d matrices (with real- or complex-valued entries) continuous on a
/-interval /. The object is to obtain generalizations of some of the results
of § 6. The difficulty arises from the fact that the theorems of Sturm in
§ 3 do not have complete analogues.
In dealing with A0.1), it will usually be assumed that
A0.3) P = P* and Q = Q*,
A0.4) detP^O.
If the vector y is denned by
A0.5) y = P(ty + R{t)x,
then A0.1) is of the form A0.2), where
A0.6) A = -P-iR, B = P~\ C = -Q-
so that
A0.7) B = B* and C = C*,
A0.8) det 5^0.
Linear Second Order Equations 385
The motivations for the assumptions A0.3), A0.4) are the following:
Condition A0.4) makes the system A0.1) or, equivalently, A0.2) non-
singular in the sense that the usual existence theorems for initial value
problems are applicable. Condition A0.3) makes A0.1) formally "self-
adjoint" in the sense that if — L[x] denotes the vector on the left of A0.1),
whether or not L[x] = 0, then we have Green's formula
J
Ja
(L[x] ¦ z - x ¦ L[z]) dt = [x(Pz' + Rz) - (iV + Rx) ¦ z]ab
for all suitable vector functions x(t), z(t).
In particular, if x(t), xo(t) are solutions of A0.1), so that L[x] = 0,
L[x0] = 0, then
* • (P*9 + Rx0) — (iV + Rx) ¦ x0 = const.
When this constant is 0, the solutions x, x0 will be called conjugate solutions
of A0.1).
A system A0.1) is called disconjugate on J if every solution x(t) ^ 0
vanishes at most once on /. Correspondingly, a system of the form A0.2)
will be termed disconjugate on J if, for every solution (x(t), y(t)) ¦? 0, the
vector x(t) vanishes at most once on /.
Instead of A0.2), it will be more convenient to deal with the matrix
equations
A0.9) U' = A(t)U + B(t)V, V = C(t)U - A*(t)V,
where U, V are d X d matrices. Note that ?/(/), V(t) is a solution of
A0.9) if and only if x(t) = U(t)c, y(t) = V(t)c is a solution of A0.2) for
every constant vector c. Thus all solutions of A0.2) are determined if we
know two solutions (?/(/), V(t)), (U0(t), V0(t)) of A0.9) such that
/?/(/) U9(t)\
A0.10) det Y(t) * 0, where Y(t) =
\V(t) V0(t) I
is a 2d X 2d matrix. In fact, y(/) is a fundamental matrix for the system
A0.2)
(i) If ?/(/), VQ) is a solution of A0.9) and if A0.8) holds, then VQ)
is determined by ?/(/); in fact,
BVit) = U' - AU.
A0.11)
(ii) If A0.7) holds and (?/(/), VQ)), (t/0(/), Ko(/)) are solutions of A0.9),
then
A0.12) U*V9 - V*U0 = K9,
where K9 is a constant matrix. This is readily verified by differentiating
386 Ordinary Differential Equations
A0.12). If Ko = 0, the solutions (U, V), (Uo, Vo) of A0.9) will be called
conjugate solutions.
(iii) In particular, if Uo = U, Vo = V, then U*V - V*U is a constant.
When this constant matrix is 0, then
A0.13) U*V=V*U; i.e., (U*V)* = U*V is Hermitian.
In this case, the solution (U, V) of A0.9) is called self-conjugate,
(iv) Let (U(t), V(t)) be a solution of A0.9) such that
A0.14) det U(t) * 0
on some f-interval and let
A0.15) K= U*V-V*U.
Consider the "variation of constants"
A0.16) Uo = UZ,
where ([/„, Vo) is a solution of A0.9) satisfying A0.12). Then
Vo = U*~\K0 + V*UZ)
or, by A0.15),
A0.17)
I
Vo = VZ + U*~\K0 - KZ).
Since Uo' = AU0 + BV0 and U' = AU + BV, it follows from A0.16),
A0.17) that
Z' = -{U-^
Thus, if T(t) is the fundamental solution of the homogeneous part of this
equation, satisfying T(s) = I,
T(s) = I,
A0.18) T' = -{
then solutions Z are of the form
Z =
i +
where Kx is a constant matrix; see Corollary IV 2.1. Correspondingly,
by A0.16),
Conversely, it is readily verified that if A0.7)-A0.8) and A0.14) hold, then
A0.19) and A0.11) determine a solution of A0.9) satisfying A0.12).
Linear Second Order Equations 387
(v) If, in this discussion, ([/, V) is a self-conjugate solution of A0.2),
so that K = 0 in A0.15), then T(t) = I and A0.19) reduces to
A0.20) U0(t) = [/(oUi + (CU^BU*-1 dt\ Ko\ .
The corresponding solution (U0(t), V0(t)) of A0.2) is self-conjugate if and
only if KM = KO*KX; i.e., {KX*KO)* = KX*KO.
Let (U0(t), V0(t)) be self-conjugate, det U0(t) ^ 0, and det Kx ^ 0.
Interchanging ([/, V), ([/„, Vo) in A0.12) changes the sign of Ko. Thus
the argument leading to A0.20) gives
A0.21)
U(t) =
since U0(s) = U(s)Kx.
Theorem 10.1 Let A(i), B(t), C(t) be continuous on J. Then A0.2) is
disconjugate on J if and only if, for every pair of points t = tx, t2eJ and
arbitrary vectors xx, x2, the equation A0.2) has a (unique) solution (x(t),
y(t)) satisfying x(tx) = xx, x(Q = x2.
The proof will be omitted as it is similar to that of Theorem 6.1. It
does not depend on the structure of A0.2) and applies equally well to any
system x' = A(t)x + B(t)y, y' = C(t)x + D(t)y, where A, B, C, D are
continuous.
In order to avoid an interruption of the proofs to follow, we now prove
a lemma (which has nothing to do with differential equations).
Lemma 10.1 (F. Riesz). Let AX,A2, . . . be Hermitian matrices satisfying
Then A = lim An exists as n —*¦ oo.
If A, B are two Hermitian matrices, the inequality A > B [or A ^ B]
means that A — B is positive definite [or non-negative definite]. To say
that "A = lim An exists" means that "Atf, A2r\, ... is convergent for
every fixed vector rj."
Proof. Note that if A ^ 0 and f, r\ are two arbitrary vectors, then the
generalized Schwarz inequality
\AS ¦ rj\* ^ (AS ¦ Wl ¦ V)
holds. In order to see this, let e be real and f€ = f + e(A? • r\)r\, so that
0 ^ A(t ¦ $c = A? • ? + 2e \A? ¦ V\2 + e2 \A? • V\2 (Ar, ¦ r,).
Since the right side is a quadratic polynomial in e which is non-negative for
all real e, its discriminant is non-negative. This fact gives the desired
inequality.
388 Ordinary Differential Equations
Let Amn = An — Am where n > m, so that Amn ^ 0. Hence the
generalized Schwarz inequality implies that
Pmnfll* = \AmnV * AmnV\2 ^ (AmnV " V) (ALnV ' AmnVl
Since 0 ^ Amn ^ /, it follows that the norm of Amn is at most 1, and
so {Alnr, ¦ Amnrj) <: \\r)\\* and
The sequence of numbers Anr\ ¦ r\, for n = 1, 2, . . . is nondecreasing and
bounded, hence convergent, so that Anr] — Amrj is small for large m and n.
Therefore, lim Anr\ exists, as was to be proved.
Theorem 10.2. Let A(t), B(t) = B*(t), C(t) = C*(t) be continuous on J
and let B(t) be positive definite, (i) IfJ is a half-open bounded interval or a
closed half-line, then A0.2) is disconjugate on J if and only if there exists a
self-conjugate solution (U(t), V(t)) o/A0.9) such that det U(t) ^ 0 on the
interior of J. (ii) IfJ is a closed bounded interval or an open interval, then
A0.2) is disconjugate on J if and only */A0.9) has a self-conjugate solution
(U(t), V(t)) satisfying det U(t) ^OonJ.
Proof of (i). For the sake of definiteness, let / = [a, co), co ^ oo. Let
Y(t) in A0.10) be the fundamental matrix for A0.2) such that Y(t) is the
identity matrix at t = a. In particular, U(a) = I, V(a) = 0, and U0(a) = 0,
V0(a) = I; thus both (U(t), V(t)), (U0(t), V0(t)) are self-conjugate solutions
of A0.9), since A0.13) and the analogue for Uo, Vo hold at t = a. The
general solution of A0.2) vanishing at t = a is given by x = U0(t)c,
y = V0(t)c for an arbitrary vector c. If A0.2) is disconjugate on /, then
det U0(t0) t^o for a < t0 < co. Otherwise there is a c0 ^ 0 such that
x = U0(t)c0 vanishes at t = a and at t = t0. But then x(t) = 0. Hence
y = V0(t)c0 = 0 since det B ^ 0.
Conversely, let (U(t), V(t)) be a self-conjugate solution of A0.9) such
that det U(t) 5* 0 on a < t < co. Define a solution U0(t), V0(t) of A0.9)
by A0.20), where a < s < co, Kx = 0, Ko = I. Thus U0(s) = 0. Since
B, hence U^BU*'1, is positive definite, it follows that det U0(t) ^ 0 for
s <t < co. Clearly, if (x(t), y(t)) ^ 0 is a solution of A0.2) such that x
vanishes at t = s, then it is necessarily of the form (x, y) = (U0(t)c,
V0(t)c) and x does not vanish for s < t < co.
It remains to show that if (x(t), y(t)) 5^ 0 is a solution of A0.2) such
that x(a) = 0, then x(t) ^ 0 for a < t < co. To this end, it suffices to
show that if a < s < co, then there exists a self-conjugate solution
(^o(O» F0(f)) of A0.9) such that U0(a) = 0 and det U0(t) 5* Ofora < t ^ s.
Put
''dt
Linear Second Order Equations 389
so that, by the analogue of A0.20), Ult V1 = B~\U^ - A Ux) is a self-
conjugate solution of A0.9). Since B is positive definite, the factor
{. . .} of U(t) in the last formula is negative definite for a < t ^ s. Hence
det Ux{t) 5* 0 for a < t < s. By the analogue of A0.21),
U(t) = ?/!«{-/ -JV^t/f
Consequently,
I-I + U^BU^dt) [-1 - U^BUf-1 dt) = I.
\ J s / \ J s I
Since the first factor is negative definite for a < t ^ s, the second factor
(which is the inverse of the first) is negative definite. Hence
S(t) = -ju^BU^dt =jUT1BU*~1dt
is positive definite for a < t ^ s, increases with decreasing t (in the sense
that S(r) — S(t) > 0 for a < r < t ^ s), and S(t) ^ /. It follows from
Lemma 6.1 that
S(a) = lim S(t) = lu^BU*'1 dt exists.
For a < t ^ s, put
which can also be written in the form
uo(t) = i/i(
Hence, by A0.20), Uo, Vo = B-\U0' — AU0) is a self-conjugate solution
of A0.9). It is clear that U0(a) = 0 and det U0(t) ^ 0 for a < t ^ s. This
completes the proof for the case of [a, co).
Proof of (ii). If / is a closed bounded interval [a, b], we can extend the
definitions of the continuous A(t), B(t), C(t) to an interval J' = [a — d,
b + 6] = /, so that B = B*, C = C*, and B(t) is positive definite. It is
clear that if d > 0 is sufficiently small, then A0.2) is disconjugate on
[a — S, b + S] if and only if it is disconjugate on /. Since [a, b] <=
[a — d, b + 6), the case / = [a, b] of the theorem follows from the case
[a, w) just treated.
Consider / open. The arguments used in the case [a, co) show that if
A0.9) has a self-conjugate solution (U0(t), V0(t)) with det Uo 5* 0 on /,
then A0.2) is disconjugate on J. The converse is implied by Exercise 10.6.
390 Ordinary Differential Equations
Consider the functional
A0.22) I(r,; a, b) = !\[P(t)v' + R(t>]]" V' + [R*W - OHM • f]} dt,
Ja
where the matrices P, Q, R are determined by A0.6). A vector function
r)(t) on a subinterval [a, b] of / will be said to be admissible of class
Ax(a, b) [or A2(a, b)] if (i) r)(a) = r)(b) = 0, and (iix) t](t) is absolutely
continuous and its derivative t]'(t) is of class L2 on a ^ t ^ b [or (ii2)»?(?)
is continuously differentiable and Prf + Ry is continuously differentiable
on a ^ f ^ 6]. If r)(t) ? ^2(a, 6), then an integration by parts [integrating
rf and differentiating P(t)r)' + R(t)f]] gives
A0.23) I(ri; a, b) = - [\{Pr]' + Rrj)' - (R*v' - Qn)\' V dt.
Ja
If —L[x] denotes the vector on the right of A0.1), whether or not L[x] = 0,
then
A0.24)
C
i; a,b) = \
Ja
L[rf\ • r\ dt for r) ? A2{a, b).
Theorem 10.3. Let A(t), B(t) = B*(t), C(t) = C*(t) be continuous on J,
and B(t) positive definite. Then A0.2) is disconjugate on J if and only if for
every closed bounded interval [a, b] in J, the functional /(»?; a, b) is positive
definite on Ax(a, b) [or A2{a, b)]; i.e., /(??; a, b) ^ 0 for every r\ ? Ax(a, b)
[or r\ ? A2(a, b)] and /(??; a, b) = 0 if and only ifrj = 0.
Proof ("Only if"). The proof of "only if" is similar to that of the
"only if" portion of Theorem 6.2. Suppose that A0.2) is disconjugate on
/. Then, if [a, b] <= /, there is a self-conjugate solution (U(t), V(t)) of
A0.9) such that det U(t) ^Oon [a, b].
For a given r,{t) ? Ax{a, b), define ?(f) by r,{t) = U(t)i(t). Using the
analogue V = PU' + RUof A0.5) and the analogue V = R*U' - QU of
A0.1), it is seen that the integrand of A0.22) is
r + VQ ¦ ([/'? + t/O + {V'l + R*UO ¦ Ul
Since PUt,' • U'l = U? • PU1 Z and R*U? • U? = U? • RU(, the integrand
can be written as
e +
+ vi ¦ u'l + vr, ¦ ui' +
In addition, the second term is V*U? ¦ t, = U*Vt,' • I = Vt,' • Ui since
(C/, V) is self-conjugate and satisfies A0.13). Hence the integrand in
A0.17) is PUt; ¦ U? + {VI ¦ UQ', and so
A0
L25) I(ri; a, b) = \\pVC • UI') dt if r, = Ut, ? A
Ja
x{a, b)
Linear Second Order Equations 391
and l(a) = i(b) = 0. Since B(t) is positive definite, P = B~l is also
positive definite. Hence I(f], a,b)^0 and /(?;; a, b) = 0 if and only if
{HO.
Proof ("If"). This proof is identical to the corresponding part of the
proof of Theorem 6.2 and will be omitted.
Exercise 10.1 (Jacobi). Let AH(a, b) denote the set of vector-valued
functions on [a, b] such that (i) r](a) = r)(b) = 0; (ii) r](t) is continuous
ona ^ t ^ b; and (iii) the interval [a, b] has a decomposition (depending
on rj) a = t0 < t1 < • • • < tm = b, such that r\ is continuously differenti-
differentiable and Pr\ + Rr\ is continuously differentiable on each interval
[tj, tj+1] for j=0,...,m-l. Thus Ax\a, b] => A,A[a, b] => A2[a, b].
Let A(t), B(t), C(t) be as in Theorem 10.3 and suppose that / is not a
closed bounded interval. Show that A0.2) is disconjugate on / if and only
if I(ri; a,b)^.O for all r\ ? A3/i(a, b) and all [a, b] <= J.
Exercise 10.2. Let P^t), Q^t) be continuous d x d matrices for
; = 1, 2 on J:a < t ^b such that (i) P2 = P2* and Q2 = Q2* are
Hermitian; (ii) 0 < P2 ^ RePx and Re gx ^ g2 if the inequality P2 > 0
means that P2 is positive definite and, e.g., P2 ^ Re Px means that
P*(i)ri-ri^Re[Px(t)ri-ri\ for all constant vectors r\; finally (iii) let
(Pnx')' + Q& = 0 be disconjugate on /. Then (P^x')' + Qxx = 0 is
disconjugate on /.
If i(t), r\(t) are of class L2 on [a, b], introduce the "scalar product"
-r
•/a
Thus, in this notation, A0.24) can be written as /(??; a, b) = (L[rj], rf) for
r\ ? A2(a, b). Correspondingly A0.25) and r\ = Ut, imply that
A0.26)
where
A0.27)
V) = (Ljlrj], LJr,)) for V e A2(a, b),
L1[v]=P1A(t)U(t)(U-\t)r,)'
and where PX/i(t) is the unique positive definite Hermitian matrix such
that (P1A(t)J = P(t), so thatP(f) is continuous on/; cf. Exercise XIV 1.2.
The bilinear relation corresponding to A0.26) is
A0.28)
for
r,2 ?
Formally, then, L = LX*L. Actually, this is correct in the following
sense: A0.27) can be written as
A0.29)
Lx[rj\
- P*(t)U'(t)U-\t)r),
392 Ordinary Differential Equations
for (t/^1)' = —U^U'U'1 as can be seen by differentiating the relation
U'W = I. The formal adjoint of L1 is therefore
A0.30) LSW = -(PiA(t)rj)' - U*-\t)U«(t)P*{t)ri,
cf. IV § 8(viii); i.e.,
A0.31) Vfo] U*-\U*PiArj)'.
Corollary 10.1. Let A, B, C be as in Theorem 10.3. Let A0.9) have a
self-conjugate solution (U(t), V(t)) such that det U{t) ^ 0 on J. Then
L = L1*L1; i.e.,
A0.32) L [t]] = -{Prf + Rr,)' + (R*y - Qv)
is given by
A0.33) L [tj] = L.HLM}
for every continuously differentiable t](t) such that Pr\ + Rr\ is continuously
differentiable on J.
This can be deduced from A0.28) or, more easily, by straightforward
differentiation (using the relations V = PU' + RU, V = R*U' - QU).
Theorem 10.4 Let A(t), B(t) = B*(t), C(t) = C*(t) be continuous on
J:a < t < <w (^ oo) and let B{t) be positive definite. Let A0.2) be dis-
conjugate on J and (U(t), V{t)) a solution o/A0.9) such that det U(t) ^ 0
on s ^ t < co for some s e [a, co), and let T(t) be defined by A0.18). Then
A0.34) Ss(t) = [*T-\r)U'lBU*~l dt
J s
is nonsingular for s < t < a> and
A0.35) lim Sj\t) = M exists,
where M depends on s and the matrix function U(t).
If, in Theorem 10.4, M = 0, the solution (U(t), V(t)) of A0.9) will be
called a principal solution of A0.9). It will turn out that a principal
solution is necessarily self-conjugate. In this case T^) = /, and M = 0
can be expressed as
A0.36)
in the sense that
JV,
BU*~l dt-^co as t
-co,
-1 dt c
¦ oo
as t
*a>
uniformly for all unit vectors c. [Compare A0.36), where B = P, and
the definition F.11) for principal solution in § 6.]
Linear Second Order Equations 393
Theorem 10.5. Let A, B, C and J be as in Theorem 10.4. (i) Then A0.9)
possesses a principal solution (U0(t), V0(t)). (ii) Another solution (U(t),
V(t)) is a principal solution if and only if(U(t), V(tJ) = (?/0@-^i» ^o(O-^i).
where Kx is a constant nonsingular matrix, (iii) Let (U(t), V(t)) be a
solution o/A0.9). Then the constant matrix Ko in A0.12) is nonsingular if
and only if det U(t) ^ Ofor t near a> and
A0.37)
as
in which case M in A0.35) is nonsingular.
The proofs of Theorems 10.4 and 10.5 will be given together.
Proof of Theorem 10.5 (i). This proof is essentially contained in the
proof of Theorem 10.2(i). Since A0.2) is disconjugate on /, there exists a
self-conjugate solution (U^t), V^t)) of A0.9) such that det U^t) ^ 0 for
a < t < a>. Let a < a < co and put
A0.38)
u*(t)
= t/i@{/ +j*UT1BU*1
-'dt].
exists
Then U2, V2 = B~\U2 - AU2) is a self-conjugate solution of A0.9),
det U2{t) 5* 0 for a ^ t < co, and
A0.39) t/x@ = t/2(o{/ -^U^BUt'1 dt},
by A0.21). Thus
(i + ftUT1BU*'1dt) (i -j'u^BUt'dt) = /.
As in the proof of Theorem 10.2(i), it follows that
/•co
A0.40) M2=limS@= U^BUt^
<-*co Jx
if S(t) is defined by
A0.41) S(t) = ) U^BUt^dt.
J a
The limit M2 is nonsingular since M2 > S(t) and S(t) is positive definite
for a < t < co.
In terms of U2(t), define
A0.42) C/0@ = U2(t) ["u^Bl/?-1 dt,
Jt
or, equivalently,
A0.43) U0(t) = Ut(t) [m2 - jy^BUf1 dt\ ;
394 Ordinary Differential Equations
therefore, Uo, Vo = B-\U0' — AU0) is a self-conjugate solution of A0.9)
and det U0(t) ^ 0 for a < t < co. Thus, by A0.21),
Consequently,
and hence
71 + !'uo1BU*o-1 dt\ ir^BUt'1 dt\ = /,
'u^BU^1 dt\ tr^BUt-1 dt\=I + o(l)
as t-*-u),
so that
¦ 0 as t -> co.
Thus (U0(t), V0(t)) is a principal solution of A0.9); cf. A0.35).
Proof of Theorem 10.4. Let (U(t), F(f))be a solution of A0.9) such that
det 1/@ 5* 0 for s ^ f < co. Thus the matrix S?t) in A0.34) is defined
for t ^ s.
It will first be shown that det S$(t) ^ 0 for s < t < to. Put
[/°@ = U(t)T(t)SJLt)
and F°= fi^t/0' - ^l/»). This djefines a solution (C/°, F°) of A0.9);
cf. A0.19). Suppose that det Ss(t0) = 0 for some t0 > s, then there is a
constant vector c0 ^ 0 such that a;@ = U°(t)c0 vanishes at t = s, t0.
Since A0.2) is disconjugate on /, it follows that U°(t)c0 = 0. Hence
S,(t)c0 = 0, and so Ss'(t)c0 = 0. Since S,' = T^U^BU*-1 is nonsingular,
this is a contradiction and shows that det Ss@ ?* 0 for s < t < co.
It will next be shown that the limit A0.35) exists. Let (U0(t), V0(t)) be
the principal solution of A0.9) just constructed, so that U0(t) is given by
A0.42) in terms of a self-conjugate solution (U2(t), V2(t)) with det U2(t) ^ 0
for a ^ t < co. Let a ^ s < r < w and consider the function
A0.44) UOr(t) =
It is clear that (UOr, VOr), where FOr = fi^X^o'r ~ ^^or) 's a solution of
A0.9), for UOr(t) can be written as
UOr{i) = [/
cf. A0.19). It follows that if ^r is the constant matrix
A0.45) Kr=U*VOr-V*UOr,
Linear Second Order Equations 395
then, by the analogue of A0.19), where U0(s) = Uis)^,
UOr(t) = U(t)T(t){U-\s)UOr(s) + Sr(t)Kr}.
Since UOr(t) = 0 when t = r, it is seen that, in the last line, {...} = 0
when t = r. Hence
A0.46) S;\r)=-KrU-r\s)U(s).
It is clear from A0.42) and A0.44) that UOr(t)-> U0(t) as r -> co; hence
Kr -> Ko as r -> oo, where
A0.47) *„= t/*F0- F*[/o.
Thus
A0.48) M = lim S~\r) exists and is -^0[/~1(s)L/(s).
r-*co
This proves Theorem 10.4.
Proof of Theorem 10.5 (ii). Let (U(t), V(t)) be a principal solution of
A0.9). Let s be such that det U(t) ^ 0 for s ^ t < to and that the limit
M in A0.35) is 0. In view of A0.48), M = 0 holds if and only if Ko = 0
in A0.47). Since(C/0, Vo) is self-conjugate, it follows that Vo = U*^V0*U0.
Hence Ko = 0 gives
= Vn(t)U-\t)U(t) for f ^ s.
Let #! = t/o^)^^)- Then (U(t\V(t)) satisfies the initial conditions
A0.49) U(t) = UJfiK* V{t) = F0@^
at t = s. Hence A0.49) holds for all t. This proves (ii).
Proof of Theorem 10.5 (iii). Let (U(t), V(t)) be a solution of A0.9) and
let Ko be given by A0.47). Then
u(t) =
Since A0.36) holds with U replaced by Uo, it is clear from A0.36) that if
Ko is nonsingular, then U(t) is nonsingular for t near co. In this case,
/ = U-\t)U0(t)\u-0\s)U(s) - {Jv-fBUt-1 dtj Ko
and A0.37) follows from A0.36), where U is replaced by ?/„.
Conversely, if U(t) is nonsingular for t near co and A0.37) holds, then
the last formula line shows that Ko is nonsingular. Thus M is nonsingular
by virtue of A0.48). This completes the proof of Theorem 10.5.
Exercise 10.3. State an analogue of Corollary 6.3.
Exercise 10.4 (Analogue of{6.\ 1,) in Theorem 6.4). Let the conditions
396 Ordinary Differential Equations
of Theorem 10.5 hold. Let (U(t), V(t)) be a self-conjugate solution of
A0.9) such that det U(t) ^ 0 for t near co and the limit M in A0.35) is
nonsingular. [Note that T(t) = I.] Let c / 0 be a constant vector and
x(t)= U(t)c a solution of A0.1). Then [P (t)x(t) ¦ x(t)]~x dt < oo.
Exercise 10.5 {Analogue of Exercise 6.5). Let the conditions of Theorem
10.5 hold and let a < T < to. Let (UOr(t), VOr(t)) be the solution of
A0.9) satisfying UOr(T) = I, UOr(r) = 0; cf. Theorem 10.1. Then
lim (UOr(t), VOr(t)) = (U0(t), V0(t)) exists as r->cu and is a principal
solution.
Exercise 10.6. Let/be an open interval; A(t), B(t) = B*(t), C(t) =
C*(t) continuous on /; and B(t) positive definite. Let A0.2) be dis-
conjugate on /. Let co (^ oo) be the right endpoint of/and [a, co) <= /.
Let (U0(t), V0(t)) be a principal solution of A0.9) on [a, co). Then
det U0(t) /0on/.
Exercise 10.7 (Analogue of (Hi) in Theorem 6.4). Let A, B, C, and / be
as in Theorem 10.4. Let (U0(t), V0(t)) be a principal solution of A0.9) and
let det U0(t) ^ 0 for a ^ t < co. Let (U(t), V(t)) be a self-conjugate
solution of A0.9) satisfying U(a) =? 0. Let A = Vo(a)Uo\a) - V(a)U~\a),
so that A = A*. Then A ^ 0 (i.e., Ax • x < 0 for all vectors x) if and
only if det U(t) 5* 0 for t ^ a.
11. Generalizations
The methods of the last section are applicable to more general situations
which will be indicated here. The material of the last section can be
considered from the following point of view: First, what are conditions
(necessary and/or sufficient) for the functional I(t]; a, b) in A0.22) to be
positive definite for all [a, b] <= / on certain classes of functions Ax(a, b)
or A2(a, bI Second, if this is the case, what are some consequences for
the solutions of the corresponding Euler-Lagrange equations A0.1) or
their Hamiltonian form A0.2)?
In this section, a similar problem is considered, but the assumption that
P is positive definite is relaxed and the classes Ax(a, b), A2(a, b) are replaced
by more restricted classes of function rj(t). In particular, it will be required
that the competing functions rj(t) satisfy certain side conditions, namely,
certain linear differential equations (as in Bolza's problem in the calculus
of variations).
Let P(t) = P*(t), Q(t) = Q*(t), R(t) be continuous d x d matrices on
/ and consider the functional
A1.1)
; a,b) = \ [(PV'
¦ r,'
dt.
Linear Second Order Equations 397
In addition, consider a set of e first order linear differential equations
A1.2) M(t)V' + N(t)V = 0,
where M(t), N(t) are e x d, e < d, matrices of complex-valued continuous
functions on /. It will be assumed that the (d + e) x (d + e) Hermitian
matrix
A1.3)
(P(t) M*(t)\ . .
P0(t) = 1S nonsingular.
\M(t) 0
A1.6) Po\t) =
where E = ?*, G = G*,
In particular, det P0(t) ^ 0 and the rank of M(t) is e.
For the variational problem A1.1) subject to the side conditions A1.2),
the Euler-Langrange equations are
A1.4) (Px + Rx + M*z)' - (R*x - Qx + N*z) = 0,
A1.5) Mx' + Nx = 0,
where 2 is an e-dimensional vector. [The derivation of A1.4) and signifi-
significance of 2 need not concern us here.]
The matrix inverse to A1.3) is of the form
(E(t) F*(t)\
\F(t) G(t)J
E is a d x d, G an e x e, and Fane x d matrix. Introduce the variables
A1.7) y =P(t)x +R(t)x + M*z.
Then equations A1.5), A1.7) can be written as P0(x', 2) = (y — Rx, —Nx)
or (x1, 2) = P^(y - Rx, -Nx); i.e., x' = Ey - (ER + F*N)x and
2 = Fy - (FR + GN)x. Hence A1.4), A1.5) become
A1.8) x' = A (t)x + B(t)y, y' = C(t)x - A*(t)y,
where
A1.9) A = -(ER + F*N), B = E,
C= -Q- R*ER - R*F*N - N*FR - N*GN.
In particular, A1.8) implies A1.5).
The assumptions on P, Q, R will be P = P*, Q = Q*,
A1.10) P(t)r)-r]>0 whenever r) ^ 0 and M(t)t] = 0.
Correspondingly, B = B*, C = C*, and, by A1.3) and A1.6),
A1.11) B(t)M*(t) = 0; BPr) = r\ if Mr] = 0;
B(t) is non-negative definite and of rank d — e (for BPrj ¦ Pr\ = Pr\ • r\ if
Mt] = 0).
398 Ordinary Differential Equations
There is a complication for the system A1.8) which did not arise in the
last section. If (x(t), y(t)) is a solution of A1.8), it does not follow that
y(t) is determined by x(t). This difficulty will be avoided by an extra
assumption:
A1.12) A1.8) is identically normal,
where A1.12) means that if (x(t), y(t)) is a solution of A1.8) such that
x(t) = 0 on some subinterval of J, then x(t) = 0, y(t) = 0 on J.
Notions of "disconjugate," "conjugate solutions of A0.9)," "self-
conjugate solutions of A0.9)" can be denned as before.
Exercise 11.1. Verify the validity of analogues of Theorem 10.1 and
Theorem 10.2. (As in the proof of Theorem 10.2, postpone one half of the
proof of the statement concerning open J.)
Let the classes A^a, b) [or A2(a, b)] of vector functions r/(t) be denned
as before with the additional condition: (iii) r/(t) satisfies the differential
equations A1.2) except for a f-set of measure 0.
Exercise 11.2. Verify the validity of the analogue of Theorem 10.3.
For the proof, note that if (x(t), y(t)) is a solution of A1.8), then r\ = x(t)
is a solution of A1.2). Hence if (U(t), V(t)) is a solution of A0.9), then
MU' + NU = 0. Thus if r)(t) is a solution of A1.2) and r\ = [/?, then
M(Ut,') = 0, so that PUt,' ¦ U? = 0 only if rj = 0.
Exercise 11.3. If P(t) is of rank d — e [so that Pit) is non-negative
definite], let Pl/i(t) denote the unique, non-negative Hermitian square
rootofP(f); thus P1A(t) is continuous; cf. Exercise XIV 1.2. If P{t) is of
rank ^ d — e, let E0(t) be the orthogonal projection of the vector space
onto the null manifold of M(t) and let P*(t) = (E0(i)P{t)E0(t))^.
Actually, P{t) can be replaced by E0(t)P(t)E0(t) in A1.1)-A1.12), since
Eor) = r\ if Mr/ = 0. Using P1A(t), state and prove the analogue of
Corollary 10.1. [An additional condition for the validity of A0.33) will
bethat ??@ satisfy A1.2).]
Exercise 11.4. Verify the validity of the analogues of Theorem 10.4
and 10.5 with an analogous definition of principal solution.
As an example and application, consider a formally self-adjoint
differential equation of order Id for a scalar function u,
A1.13) L{u} = 0,
where
k=Q k=l
where po(t), ¦ ¦ ¦, Pui*) are real-valued functions on an interval J, i =
A1-15)
(-!)%*(') >0,
Linear Second Order Equations 399
and p2k(t), P2k-i(t) nave & continuous derivatives. For a given function
u(t) of class Cu on J, define a vector y = (y1,..., yd) by
A1.16) y* =
* = (-
2
-\{J-k)
for fc = 1,.. ., d, where p2d+1 = 0. The operator L{u) = 0 is termed
formally self-adjoint because of the validity of the Green relation
\
Ja
\l{u}v - u
where z = (z1,. . . ,zd) belongs to v as y belongs to u. In particular if
u, u0 are solutions of A1.14), then
d
I
i[uo{k-1)yk-yoku{k-1)] = const,
where y0 belongs to u0. When this constant is 0, the solutions u, u0 will be
called conjugate. If u = u0 and u, u are conjugate solutions, then u is
called a self-conjugate solution. [When p1= p3 = ¦ ¦ • = p2d_1 = 0, so
that A1.14) is a real equation, then all real-valued solutions are self-
conjugate.]
Consider the functional
A1.17)
{u;a, b) = [\{u}udt.
A formal integration by parts (ignoring the integrated terms which vanish
if u = u' = • ¦ • = w"*-1' = 0 at t = a, b) gives
•bl d
A1.18) I{u;a,b} = \°\i(-irpu\u<
Ja \k=0
- 22 (-OViImiJ^') dt
k=l )
k=l
Then /(??; a, b) = I{u; a, b} is of the type A1.1) for r/ = (u, u',. .., m'"
and the conditions A1.2) become
A1.19) r)t'-r)*+1 = 0 for ; = 1,..., d - 1.
Here
P = diag [0, . . . , 0, (- \fpul Q = diag [/>„, -Pl, ...,(-1)*-1/
R = /diag [-/>!,/>„, . .. , (-!)%_!];
400 Ordinary Differential Equations
also M, N are the (d — 1) x d matrices
'1 0 ... 0 0\
0 1 ... 0 0
M =
\0 0
1 0/
N =
@ 1 0
0 0 1
\0 0 0
°\
0
1/
Correspondingly, A is the matrix with ones on the superdiagonal, diag-
diagonal elements @,.. ., 0, — ip^d-ilp2d), and other elements zero; B =
diag [0, . . ., 0, (— lYIpia]; and C is the Hermitian matrix with superdiag-
superdiagonal elements (ipu —ip3,. . ., (— 1)<*~1//>2*-i)> diagonal elements (p0, —p2,
..., (-l)rf-2/>M_4, (-l)d-1/'2d-2 + \P2d-1\VP2d), and the elements not on
the main diagonal and superdiagonal are zero.
When (x(t), y(t)) is a solution of A1.8), then x{t) = (w, «',..., w"*'),
where u is a solution of A1.13) and the components of y are given by
A1.16). Conversely, if u is a solution of A1.13), this choice of a;, y gives a
solution of A1.8). The condition A1.15) assures A1.3), A1.10), and
A1,12).
Note that if (U(t), V(t)) is a solution of A0.9) and the kth column of
f 3) h
U(t) is (uk, uk
where u = uk is a solution of A1.13), then
det U(t) is the Wronskian of the solutions wx, . . ., wd. The solution
(U(t), V(t)) is self-conjugate, if and only if u}, uk are conjugate solutions
of A1.13) for;, k= 1,. . ., d.
Exercise 11.5. State the analogue of Theorem 10.3, specifying the
classes A-^a, b), A2(a, b) in terms of scalar functions u.
Consider finally the analogue of Corollary 10.1; cf. Exercise 11.3.
Since P(t) = diag [0,..., 0, (— l)"p2d], the matrix PVi(t) is P'A =
diag @,. . ., 0, \pM\Vi). The vector r](t) satisfies the analogue of A1.2) if
and only if r](t) is of the form r\ = (v(t), v'(t),.. . , v^-^t)) for a scalar
function v(t). In this case, A0.29) is a vector of the form @,..., 0, L^v}),
where L^v} = a.0(t)v{d) + ¦ ¦ ¦ + ad(t)v is a differential operator of order
d and ao(f), . . ., a.d(t) are continuous complex-valued functions. In fact,
it is clear from A0.29) that ao(f) = \pM{t)\'A > 0. It is also clear that
Li{v} = 0 if v = uk and (uk, uk', . . ., wjj?-1*) is the kth column of U(t).
Since the Wronskian of uu .. ., ud is det U(t) ^ 0, it follows that v =
«!,. . ., ud are d linearly independent solutions of Lx{v) = 0.
Consequently, if W(wu . . . , w}) denotes the Wronskian of the j
functions wu . . ., w3, then
A1.20)
W(Ul
Linear Second Order Equations 401
In order to see this, note that the expression on the right is a differential
operator of order d with leading coefficient I/^OI^ and solutions u =
«!,..., ud. Thus if either side of A1.19) is written as a linear homogeneous
system r\ = Q.(t)r) for r\ = (v, v', . . ., I?1*) in the obvious way, then the
system has a fundamental solution U(t) and so D. = U'U-1. This proves
the identity A1.20).
Exercise 11.6. If A1.13) has solutions «i@, ¦ ¦ •. "<?@ on /, which are
pairwise conjugate (and self-conjugate) and which have a nonvanishing
Wronskian W(ux, . . ., ud) ^ 0 on /, then L{u) = Lx *{Li[w]} for all
functions u of class Cu on /. Here, if L^u} = 2 <x-k(t)u{h), then
L!*{m} = 2 (- l)*{afr@w}w, where the sum 2 is over 0 < k ^ d.
Notes
sections 1 and 2. See notes on relevant portions of Chapter IV. For the substitution
B.34) in §2 (xiii), see Liouville [1, II, pp. 22-23]. The substitution r = u'/u, which
transforms B.32) into a Riccati equation, was used in special cases by Euler (circa 1765)
and Liouville A841) for rfth order linear differential equations. For the transformation
B.42) in § 2(xv), see Prufer [1, p. 503].
section 3. The results of this section are due to Sturm [1]; see Bocher [1], [3]. The
proofs in the text are suggested by Priifer's work [1] and are given in detail by Kamke
[3]. The proof for Sturm's separation theorem (Corollary 3.1) given in Exercise 3.1(a)
for the case^ij = p2 goes back to arguments of Sturm; a similar proof for the general
case is due to Picone [1].
section 4. Theorem 4.1 goes back to the work of Sturm [1] and Liouville [1]. The
proof of (i)-(iii) in the text follows that of Prufer [1], The proof of (vi) is based on
results of Hilbert and E. Schmidt on integral equations with "Hilbert-Schmidt" kernels;
cf. Riesz and Sz.-Nagy [1, pp. 239-242]. For useful and interesting results on the asymp-
asymptotic behavior of the eigenvalues Xn, see Borg [1] and references there to Weyl. For a
complete characterization of spectra of singular boundary value problem in terms of
zeros of solutions, see Hartman [6] and Wolfson [1]. For Exercises 4.3 and 4.4, see
Prufer [1].
section 5. Theorem 5.1 is due to Hartman and Wintner [9] and generalizes Corollary
5.1, which is an interpretation of a result of Lyapunov [1]. The proof of Theorem 5.1
in the text is that of Nehari [1]. That the factor 4 in E.7) cannot be increased was
first proved by van Kampen and Wintner [1]. The proof in Hints for Exercise 5.1 is
given in Hartman and Wintner [9] and is adapted from Borg [2]. For Exercise 5.2,
see Hartman and Wintner [10]. For Exercise 5.3(a), see Hartman and Wintner [21];
for part (c), see Opial [3]; part (d) is a slight improvement of a result of de la Vallee
Poussin [1] (cf. Sansone [1,1, p. 183]; see also Nehari [2]). Corollary 5.2 and Exercise
5.4 are due to Hartman and Wintner [5]; for generalizations, see Hartman [7]. Theorem
5.2 is a result of Hartman and Wintner; see Hartman [7, p. 642]. Corollary 5.3
is a result of Wiman [1]; for the generalization in Exercise 5.6(a), see Hartman and
Wintner [2]. For Exercises 5.6F) and 5.7, see Hartman [18]. For Theorem 5.3, see
Milne [1]; cf. Hartman and Wintner [5]. Generalizations of Exercise 5.9(a) and F)
to binary systems of first order were obtained by Petty [1] by different methods.
section 6. The use of the term "disconjugate" here is suggested by Wintner [20].
Theorem 6.2 is a classical result in the calculus of variations (Jacobi, Weierstrass,
402 Ordinary Differential Equations
Erdmann); cf. Bolza [I, chap. 2 and 3] or Morse [2, chap. 1]. The proof in the text is
based on Clebsch's [1] transformation of the second variation; cf. Bolza [1, p. 632].
Corollary 6.2 is a particular case of a result of Heinz [2]; cf. Exercise 11.6. The notion
of a "principal solution" for a disconjugate equation was introduced by Leighton and
Morse [1 ]; cf. Leighton [1 ] for the use of the term "principal." The proof in the text
of Theorem 6.4 is adapted from Hartman and Wintner [18, Appendix]. Corollary 6.4
is a particular case of a theorem of A. Kneser [2] on second order (not necessarily
linear equations); cf. Chapter XII, Part I in this book. The proof in the text is that of
Hartman [3]; Kneser's proof is similar to the one suggested in Exercise 6.7. For
Corollary 6.5, see Hartman and Wintner [6, p. 635]. For Corollary 6.6, see Hartman
and Wintner [3]. (For an application of Theorem 6.4 and Corollary 6.5 to differential
geometry, see E. Hopf [I].)
section 7. Theorem 7.1 is due to A. Kneser [1]. The remark following Theorem 7.1
on the use of the functions in Exercises 1.2 is due to Hille [1] and to Hartman [4].
Theorem 7.2 was given by Wintner [20]. For Corollary 7.1 and Exercise 7.4, see
Hartman [9] and [25], respectively. Exercise 7.2F) is a result of Hartman and Wintner
[20] and generalizes a result of Picard [4, p. 8]. Hartman [10] contains Lemma 7.1,
Theorems 7.3-7.4, and Exercises 7.5-7.8. Theorem 7.4 is a generalization of a result of
Wintner [20]. Exercise 7.8 is related to results of Wintner [9], [15], [20]; Hille [1]; and
Leighton [2]. Exercise 7.9 may be new (it was first given by Hille [1 ] under the additional
assumption that qx(t) ^ 0, q2(t) ^ 0 and then by Wintner [24] under the conditions
0 ^ Qi{t) ^ C2@; the proof suggested in the Hints is much simpler than the proofs
of these authors). For some results related to this section, see Wintner [14], Zlamal [1],
Olech, Opial and Wazewski [1], and Opial [4]. For a study of zeros of solutions of
certain fourth order equations, see Leighton and Nehari [1 ].
section 8. Theorem 8.1 is a variant of a result of Wintner [10]. Corollary 8.1 is a
result of Bocher [2]. For Exercise 8.2, see Prodi [1]. For Exercise 8.3, see Wintner [12].
Exercise 8.4F) is due to Wintner [10] and sharpens a result of Cesari [1]. For Exercise
8.5, see Hartman and Wintner [17]; for related results, see Atkinson [1] and references
there. For Exercises 8.6 and 8.8, see Hartman [25]. For Exercise 8.7, see Ganelius
[1]; this result was first used in connection with linear, second order differential
equations by Brinck [1].
section 9. The general procedure in this section is suggested by unpublished notes
of Hartman and Wintner. Lemma 9.1 is related to results of Wintner [13], [17] on a
second order equation. Analogues of Lemmas 9.1 and 9.2 for (9.1) when^>(?) > 0 and
q{t) ^ 0 go back to Weyl [1]. Theorem 9.1 is an unpublished result of Hartman and
Wintner. The first part of Corollary 9.1 is a result of Bocher [2] under the condition
(9.30) and of Wintner [17] under condition (9.31). Similarly, the first part of Corollary
9.20 is due to Bocher [2] under condition (9.35) and to Hartman and Wintner [12] under
condition (9.36). For Exercise 9.6, see Wintner [12]. For Corollary 9.3 and Exercises
9.7 and 9.8, see Hartman and Wintner [12], where analogues and generalizations are
given. For results related to this section, see Opial [2], [5], [6]; Rab [1]; Zlamal [1].
section 10. The use of the term "conjugate solutions" is the same as that suggested
by von Escherich [1] for the case of real systems A0.1); the analogous Lagrange
relation [see displays following A0.8)] on which this definition is based is due to
Clebsch [1]. In relation to A0.18)-A0.19), see Kaufman and R. L. Sternberg [1],
Barrett [1], and Reid [3]. The remarks above concerning Theorem 6.2 are applicable
to Theorem 10.3 in the case of real matrices P, Q, R. For the complex case, see Reid
[2]» [3], [5], [6]. The proof in the text is based on Clebsch's transformation of the
second variation. Exercise 10.1 is a special case of Jacobi's classical theorem on
i
Linear Second Order Equations 403
conjugate points. Exercise 10.2 whenPx = Px* and Qx = Qx* is the simplest case of a
result of Morse [1]; see Hartman and Wintner [22], where Ph Qt are assumed real.
Corollary 10.1 is suggested by Heinz [2]; cf. Exercise 11.6. The proof in the text is
much simpler than that of Heinz.
The concept of a "principal solution" for systems A0.1) with R(t) = 0, was intro-
introduced by Hartman [12] who proved an analogue of Theorem 10.5 dealing however,
with only self-conjugate solutions (V(t\ V(t)). The definition of principal solution in
the text, and Theorems 10.4, 10.5 are due to Reid [3]. Although a principal solution in
Reid's sense turns out to be self-conjugate and hence identical with a principal solution
in Hartman's sense, the handling of nonprincipal solutions is more convenient by
Reid's definition. The proof in the text for the existence of principal solutions [Theorem
10.5(i)] follows Hartman [12]. Reid's existence proof is outlined in Exercise 10.4.
The proofs of Theorem 10.4 and the other parts of Theorem 10.5 are based on Reid
[3]. Lemma 10.1 and its proof are due to F. Riesz; cf. Riesz and Sz.-Nagy [1]. For
Exercise 10.3, see Hartman [12].
section 11. The results of this section given in Exercises 11.1-11.5 are due to Reid
[3]; see also Sandor [1] and Reid [6] for related results and generalizations. For
Exercise 11.6, see Heinz [2] (and, for d = 1, cf. Brinck [1]).
Chapter XII
Use of Implicit Function and
Fixed Point Theorems
Many different problems in the theory of differential equations are solved
by the use of implicit function theory—either of the classical type or of a
more general type involving fixed point theorems and/or functional
analysis. This will be illustrated in this chapter. Part I deals with the
existence of periodic solutions of linear and nonlinear differential equations.
Part II deals with solutions of certain second order boundary value prob-
problems. In Part III, a general abstract theory is formulated. Use of this
general theory is illustrated by an application to a problem of asymptotic
integration.
Although Parts I and II are applications of the general theory of Part III,
there are several reasons for giving them separate treatments. The first
reason is the importance and comparative simplicity of the situations
involved. The second reason is that Parts I and II serve as motivation for
the somewhat abstract theory of Part III. The third and most important
reason is the fact that, as usual, a general theory in the theory of differential
equations only provides a guide for the procedure to be followed. Its use
in a particular situation generally involves important problems of ob-
obtaining appropriate estimates in order to establish the applicability of the
general theory.
Two general theorems will be used. The first is a very simple fact:
Theorem 0.1. Let D be a Banach space of elements x, y,. . . with norms
\x\, \y\ Let 7*0 be a map of the ball \x\ ^ p in D into D satisfying
\T0[x) - T0[y]\ ^e\x- y\for some 6, 0 < 6 < 1. Let m = \T0{Q]\ and
m ^ p{\ —6). Then there exists a unique fixed point x0 of To, i.e., a
unique point x0 satisfying T0[x0] = x0. In fact, x0 can be obtained as the
limit of successive approximations x1 = T0[0],x2 = T0[x1],x3 = T0[x2], ....
Remark. If To maps the ball \x\ ^ p into itself, then the condition
m < p(l — B) can be omitted.
Exercise 0.1. Verify this theorem and the Remark.
404
Use of Implicit Function and Fixed Point Theorems 405
A much more sophisticated fixed point theorem is the following:
Theorem 0.2 (Tychonov). Let D be a linear, locally convex, topological
space. Let S be a compact, convex subset o/X) and To a continuous map of
S into itself. Then To has a fixed point x0 e S, i.e., T0[x0] = x0.
The following corollary of this will be used subsequently.
Corollary 0.1. Let X> be a linear, locally convex, topological, complete
Hausdorjf space {e.g., let ?> be a Banach or a Frechet space). Let S be a
closed, convex subset of D and To a continuous map of S into itself such
that the image T0S of S has a compact closure. Then To has a fixed point
xoeS.
Theorem 0.2 was first proved by Schauder under the assumption that
D is a Banach space and this case of the theorem is usually called "Schauder's
fixed point theorem." For a proof of Theorem 0.2, see Tychonov [1].
Parts I and II will use the cases of Corollary 0.1 when D is the Banach
space C°, C1. Part III will use the case when X) is a simple Frechet space,
namely, the space of continuous functions on/:0^f<<w(^oo) with
the topology of uniform convergence on closed intervals in /.
Corollary 0.1 is obtained from Theorem 0.2 in the following way: Let
D, S, To be as in Corollary 0.1. Let S^. be the closure of 75, so that S^. is
compact. Also 5j <= 5 since 5 is closed. Under the assumptions on D,
the convex closure of 5j (i.e., the smallest closed convex set containing
5j) is compact since 5j is. (This is an immediate consequence of Arzela's
theorem in the applications below; cf., e.g., the Remark following the
proof of Theorem 2.2.) Let 5° denote this convex closure of Sv Since 5
is convex 5° <= s. Thus To is a continuous map of the convex compact
5° into itself (in fact, 75° <= T0S c ^ c 5°) and the corollary follows
from Theorem 0.2.
Part III will depend on the "open mapping theorem" in functional
analysis. This theorem will be used in the following form:
Theorem 0.3 (Open Mapping Theorem). Let Xu X2 be Banach spaces
and To a linear operator from X1 onto X2 with a domain ?&(T0), which is
necessarily a linear manifold in Xlt and range &(T0) = X2. Let To be a
closed operator, i.e., let the graph ofT0, @(T0) = {(xly Tox^) : x1 e@(T0)} be
a closed set in the Banach space X1 x X2 = {(xlt x2) ; x1 e Xu x2 e X2}
with norm \(xlt z2)| = max(|a;il, \x2\). Then there exists a constant K
with the property that, for every x2 ? X2, there is at least one x1 ? @(T0)
such that Tf$cx = x2 and \x^ ^ K\x2\. [Inparticular, when To is one-to-one,
so that x1 is unique, then \x,\ ^ KIT^] holds for all x1 e@(T0).]
For a proof of the open mapping theorem in the form that "if P is a
continuous, linear map from a Banach space X to another Banach space
Xt with domain 3(P) = X and range @t(P) = X2, then P maps open
sets into open sets," see Banach [1, pp. 38-40]. Theorem 0.3 results by
406 Ordinary Differential Equations
applying this to the projection map P : @(T0) -*¦ X2, where P(x1, Tox^) =
Tpe-L and noting that a sphere about the origin in @(T0) has a /"-image
which contains a sphere about the origin in X2.
As a motivation for the procedures to be followed consider the problem
of finding a solution of the differential equation
@.1)
y'=f\t,y)
in a certain set S of functions y(t). Write this differential equation as
y' = A(t)y +f(t, y), where f(t, y) =/"(?, y) - A(t)y,
for some choice of A(t). Suppose that for every x(t) e S, the equation
@.2) y' = A(t)y+f(t,x(t))
has a solution y(t) e S. Define an operator To : S -*¦ S by putting y(t) =
T0[x(t)], where y(t) eS is a suitably selected solution of @.2). It is clear
that a fixed point yo(t) of 70 [i.e., 70[?/0(?)] = 2/0@1 is a solution of @.1)
in 5.
For the applicability of the theorems just stated, it will be assumed that
S is a subset of a suitable topological vector space D. It will generally be
convenient to introduce another space 33 and two operators L and 7\.
The operator L is the linear differential operator L\y] = y' — A(t)y, so
that g(t) = L[y(t)] if
@.3) y' = A(t)y + g(t).
It will also be assumed that if x(t) e S, then g(t) = f(t, x{t)) is in 23 and
7\ : S->-23 is defined by g(?) = 7\[a:(?)]. Investigations of 70 are then
reduced to examinations of the linear differential operator L and of the
nonlinear operator 7\.
The applicability of Theorem 0.1 can arise in the following type of
situation: Suppose that $8, D, are Banach spaces and that |g|s, \y\^
denote the norms of elements g e $8, y e X>, respectively. Assume that for
every g{t) e 58, the equation @.3) (i.e., L[y] = g) has a unique solution
2/(?) 6 5 c J), that y(?) depends linearly on g(t), and that there exists a
constant K such that |y| j, ^ K |g|s. Suppose that, for the map T1 : S -*¦ $8
there is a constant 0 such that |7\[*i] — 7\[*2]ls = O.\xi — X2\x> f°r
a;1; *2 6 S. Then 70 satisfies ir^?)] - ro[a;2(?)]|D ^ 0A>! - x2\v.
According to Theorem 0.1, the sequence of successive approximations
will converge to a fixed point of To (under suitable conditions on S, xlf
and OK).
Use of Implicit Function and Fixed Point Theorems 407
In some situations, the equation L[y] = g may have solutions y satisfy-
satisfying \y\x> = K\g\% although y is not unique; cf., e.g., Theorem 0.3. In
this case, y need not depend linearly on g but it might be possible to form
convergent successive approximations in the following way: For a given
*!, let x2 = y be a solution of L[y] = 7\[«i(?)]- If xi, x2, ¦ ¦ ¦, *„_! have
been defined for n > 2, determine an xn from the equation L[xn — xn_1] =
TAxn-i] ~ 7\K-2] and the inequality \xn - x^ < K I^K^] -
^lfcrj^lls- This situation will not arise below.
When the inequality l^foW] — 7\[«2W]I<b = e \xi — X2\x> is not
available, Theorem 0.2 may still be applicable to assure the existence of
a fixed point of To.
PART I. PERIODIC SOLUTIONS
1. Linear Equations
In this section, unless otherwise specified, the components of the d-
dimensional vectors y, z are real- or complex-valued. Let p > 0 be fixed.
Consider an inhomogeneous system of linear equations
A-1)
V = A(t)y + g(t)
and the corresponding homogeneous system
A.2) y' = A(t)y,
where A(t) is a continuous d x d matrix and g(t) a continuous vector-
valued function for 0 ^ t ^ p. In addition, consider a set of boundary
conditions
A.3)
My(S>) - Ny(p) = 0,
where M, N are constant d x d matrices. For example, if M = N = /
and A(t), g(t) are periodic of period p, then a solution y(t) of A.1) or A.2)
satisfying A.3) is of period/;.
Lemma 1.1. Let A(t) be continuous for 0 ^ t ^ p and M, N constant
d X d matrices. Let Y(t) be a fundamental matrix for A.2). Then a
necessary and sufficient condition for A.2) to have a nontrivial (^ 0) solution
satisfying A.3) is that MY@) — NY(p) be singular. In fact, the number k,
0 ^ k ^ d, of linearly independent solutions of A.2), A.3) is the number of
linearly independent vectors c satisfying
A.4)
[MY@)~ NY(p)]c =
i.e., d-k = rank [MY@) - NY(p)].
This is clear since the general solution of A.2) is y = Y(t)c.
468 Ordinary Differential Equations
Exercise 1.1. Let A(t) be periodic of period p and
A.5) Y(t) = Z(t)eRt, where Z(t + p) = Z(t)
and R is a constant matrix; cf. the Floquet theory in § IV 6. Then A.2)
has a nontrivial (^ 0) solution of period p if and only if X = 1 is a
characteristic root of A.2); i.e., eRp — /is singular. In fact, the number
of linearly independent solutions of period p is the number of linearly
independent solutions c of
A.6)
[ 7@) - Y(j>)]c = 0, i.e.,
- I)c = 0.
For algebraic linear equations, the inhomogeneous system Cy = g has a
solution y for every g if and only if the only solution of Cy = 0 is y = 0.
The analogous situation is valid here.
Theorem 1.1. Let A(t) be continuous for 0 ^ t ^ p; M, N constant
d x d matrices such that the d X 2 d matrix (M, N) is of rank d. Then A.1)
has a solution y(t) satisfying A.3) for every continuous g(t) if and only if
A.2), A.3) has no nontrivial (^ 0) solution; in which case y(t) is unique and
there exists a constant K, independent ofg(t), such that
A.7)
\\y(t)\\
^ K P
Jo
W(s)|| ds for 0^^ p.
Proof. The general solution of A.1) is given by
A.8) 2/@ = Y(t){c + jj-\s)g(s) Jsj;
Corollary IV 2.1. This solution satisfies A.3) if and only if
A.9) [MY@) - NY(p)]c = NY(p) fV^s) ds.
Jo
Assume that A.2), A.3) has no nontrivial solution. Then, by Lemma
1.1, the matrix V = MY@) — NY{p) is nonsingular, thus A.9) has a
unique solution. Substituting this value of c in A.8) gives the unique
solution of A.1), A.3):
jj-\
1E)gE) ds +jj-\s)g(s) Jsj.
A.10) y(t) =
It is clear that there exists a constant K satisfying A.7) for 0 ^ t ^ p.
This proves one-half of Theorem 1.1 (and this part did not use the
assumption that rank (M, N) = d). The converse follows from Theorem
1.2.
Use of Implicit Function and Fixed Point Theorems 409
Exercise 1.2. What is the Green's function G(t, s) in the last part of
Theorem 1.1, i.e., what is the function G(t, s), 0 ^ s, t ^ p, such that
2/@
Jo
s)g(s) ds
is the unique solution A.10) of A.1), A.3)?
Consider the equations adjoint to A.1), A.2)
A.11) z' + A*(t)z + h(t) = 0,
A.12) z' + A*(t)z = 0,
where A* is the complex conjugate transpose of A; cf. § IV 7. Consider
also a set of boundary conditions
A.13) Pz@) - Qz{p) = 0,
where P, Q are constant d x d matrices. If y(t) is a solution of A.1) and
z(t) a solution A.11), the Green formula (IV 7.3) is
l\(s) • z(s) - j/E) • h(s)] ds = [y(t) •
Jo
A.14)
When do the boundary conditions A.3) and A.13) imply that
A.15) y(p)-*(p)-y@)-z@) = 0,
i.e., that the right side of A.14) is 0? Note that if M, Q are nonsingular,
then this is the case if and only if 0 = y{p) • Q^Pz^) — M^Nyip) • z@) =
(P*Q*~1 — Af-WJ/(p) • z@) = [M~\MP* — NQ*)Q*-*]y{p) • z@). In
this case, necessary and sufficient for A.3), A.13) to imply A.15) is that
A.16) MP* - NQ* = 0.
Lemma 1.2. LetM, Nbeconstantd X d matrices such that rank(Af, N) =
d. Then there exist d x d matrices P, Q satisfying rank (P, Q) = d,
A.16), and having the property that the relations A.3), A.13) imply
A.15). The pairs of vectors z@), z(p) satisfying A.13) are independent of
the choice ofP, Q.
Proof. Since rank (M, N) = d, there exist d X d matrices Mx, Nx such
that the Id x Id matrix
IM -N'
A.17) W=[
is nonsingular. Write the inverse of W as
or W *~J =
\Qf Q*J \P Q
so that A.16) holds and rank (P, Q) = d.
410 Ordinary Differential Equations
Let ylf y2, z1( z2 be J-dimensional vectors and r\ = (ylt y2), ? = (z1( z2)
be corresponding 2J-dimensional vectors. Then
A.18)
thus
• ?
A.19) My1- Ny2 = 0, PZl+Qz2 = 0 imply that rj • ? = 0.
The choices y1 = y@), y2 = y{p), z1 = z@), z2 = — z(p) show that A.3),
A.13) imply A.16). This completes the existence proof.
The formulation A.19) of the implication A.3), A.13) => A.16) makes
the last part of the lemma clear. For if rj = (j/1( y2) 5^ 0 satisfies My1 —
Ny2 = 0, then Py1 + Qy2 ^ 0. In fact, since rank (P, Q) = d, the set of
vectors ? = (z@), -z(p)) satisfying Pz@) - Qz(p) = 0 is the set of
vectors satisfying rj ¦ ? = 0 for all rj = (j/1( y2) such that My± — Ny2 = 0.
Since this set of vectors ? = (z@), — z(p)) is determined by M, N, the
proof of the theorem is complete.
Boundary conditions A13) satisfying the conditions of Lemma 1.2 will
be called the adjoint boundary conditions of A.3). Correspondingly, the
problems A.2)-A.3) and A.12)-A.13) will be called "adjoint problems."
(Note that the adjoint of the "periodic boundary conditions" y(p) = y@),
i.e., M = N = I, are equivalent to the "periodic conditions" z(p) = z@),
There is an analogue of the algebraic fact that if C is a d X d matrix,
then the number of linearly independent solutions of Cy = 0 and of the
"adjoint" equation C*z = 0 is the same:
Lemma 1.3. Let A(t) be continuous for 0 ^ t ^ p; M, N constant
d X d matrices such that rank (M, N) = d; and A.13) boundary conditions
adjoint to A.3). Then A.2)—A.3) am/A.12)-A.13) have the same number of
linearly independent solutions.
Proof. Since the relationship between A.2)-A.3) and A.12)-A.13) is
symmetric, it suffices to show that if A.12)-A.13) has k linearly independent
solutions, where 0 ^ k ^ d, then A.2)—A.3) has at least k linearly
independent solutions.
Let Y(t) be a fundamental matrix of A.2), then Y*~\t) is a fundamental
solution of A.12) by Lemma IV 7.1. In terms of A.17), define a constant
Id X Id matrix
(MY@) -NY(p)\
U=Wdiag[Y@), Y(p)] = [ );
A.20)
so that U is nonsingular and
U*-i = W*-i diag [7*-i
V
N^ip))
QY*~Hp)
Use of Implicit Function and Fixed Point Theorems 411
Thus, if c0 is a constant J-dimensional vector such that z(t) = y*-1^)^
is a solution of A.12)—A.13), then U*-1^, -c0) = (b, 0). Here b is a
J-dimensional vector, and if c0 varies over a set of k linearly independent
vectors, then b varies over a set of k linearly independent vectors, since
U*'1 is nonsingular. From A.20), it is easy to see that the equation
(c0, -Co) = U*{b, 0) gives
A.21)
so that
A.22)
c0 = Y*@)M *b = Y*(p)N*b,
[Y*@)M* - Y*(p)N*]b = 0.
Hence the matrix Y*@)M* — Y*(p)N* annihilates k linearly independent
vectors b; therefore, the same is true of its complex conjugate transpose
M Y@) — NY(p). In view of Lemma 1.1, this proves Lemma 1.3.
Remark. For the purpose of the next proof, note that the lemma just
proved implies that A.22) holds if and only if the vector c0 in A.21) is
such that the solution z = Y*~\t)c0 of A.12) satisfies A.13).
Another algebraic fact is that if C is a singular matrix, then Cy = g has
a solution y if and only ifg is orthogonal (i-e.,g • z = 0) to all solutions z
of the homogeneous "adjoint" system C*z = 0. Again an analogous
situation is valid here:
Theorem 1.2. Let A(t) be continuous for 0 ^ t ^ p, M and N constant
d x d matrices such that rank (M, N) = d, and let A.2)—A.3) and A.12)-
A.13) be adjoint problems. Suppose that A.2)—A.3) has exactly k linearly
independent solutions y^t),. . ., yk(t) and let z^t),.. . , zk(t) be linearly
independent solutions o/A.12)-A.13). Let g(t) be continuous for 0 ^ t ^ p.
Then A.1) has a solution yo(t) satisfying A.3) if and only if
A.23)
g(s) ¦ Zj(s) ds = 0 for j = 1, . . . , k.
+ • • • +
In this case, the solutions of A.1), A.3) are given by yo(t) +
a.kyk(t), where «1( . . . , a.k are arbitrary constants.
Proof. Note that, by the proof of Theorem 1.1, the problem A.1), A.3)
has a solution if and only if A.9) has a solution c. This is the case if and
only if
for all solutions b of A.22). In view of A.21), this is equivalent to the
condition that
0 = (\g(s
Jo
s) ¦ Y*-\s)Y*(p)N*b] ds
= r
Jo
¦ z(s) ds
412 Ordinary Differential Equations
for all solutions z = Y*-^)^ of A.12)-A.13), i.e., that A.23) holds.
This proves the theorem.
The next theorem is a rather particular result for the case that A(t), g(t)
are of period p.
Theorem 1.3. Let A(t) be continuous and of period p. Then, for a fixed
continuous g(t) of period p, A.1) has a solution of period p if and only if(\. 1)
has at least one bounded solution for t ^ 0.
Proof. The necessity of the existence of a bounded solution is clear.
In order to prove the converse, assume that A.1) has a solution y(t)
bounded for t ^ 0. Let Y(t) be the fundamental matrix of A.2) satisfying
7@) = /. Then A.1) has a solution of period/? if and only if the equation
c = Y(p)c + b, where
b = Y(p)
Jo
ds,
has a solution c; cf. A.9) in the proof of Theorem 1.1.
If c = 2/@) in A.8), then y{p) = Y(p)y@) + b holds for every solution
y(t) of A.1). Since y(t + p) is also a solution, yBp) = Y(p)y(p) + b =
Y2(p)y@) + Y(p)b + b, or more generally,
y(np) = Yn(p)y@)
/n-l
I Yk(p)
\*=o
Suppose, if possible, that [/ — Y(p)]c = b has no solution. Then [Y(p) —
I]* is singular and there exists a vector c0 such that [Y(p) — I]*c0 = 0
and b ¦ c0 ^ 0. Thus c0 = Y*(p)c0 and c0 = (Yk(p))*c0 for k = 0, 1,
Multiply the equation in the last formula line scalarly by c0 to obtain
y(np) • c0 = j/@) ¦ c0 + n(b • c0),
since Yk(p)y@) ¦ co = 2/@) • (Yk(p))*co. Asi'C0#0 and the sequence
y(p), yBp), ... is bounded, a contradiction results. This proves the
theorem.
2. Nonlinear Problems
This section deals with the existence of periodic solutions for non-
nonlinear systems. With very minor changes, the methods and results are
applicable to the situation when the requirement of "periodicity" is
replaced by boundary conditions of the type A.3). The results depend on
those of the last section for linear equations and, in particular, on the "a
priori bound" for certain solutions of A.1) given by A.7). The first two
theorems concern a nonlinear system of the form
B.1) V' = A(t)y+f(t,y)
in which y is a vector with real- or complex-valued components.
Use of Implicit Function and Fixed Point Theorems 413
Theorem 2.1. Let A(t) be continuous and periodic of period p and such
that A.2) has no nontrivial solution of period p. Let K be as in A.7) in
Theorem 1.1, where M = N = I. Let f(t, y) be continuous for all (t,y),
of period p in t for fixed y, and satisfy a Lipschitz condition of the form
B.2) Wf(t,y1)-f(t,y2)\\ ^eilyi-y.ll
for all t, 2/1,2/2 with a Lipschitz constant 6 so small that KBp < 1. Then B.1)
has a unique solution of period p.
Actually, it is not necessary that f(t, y) be defined for all y. If m =
max ||/0,0)||, it is sufficient to require that f(t, y) be defined for
II2/II ^ r, where
B.3)
1 -
< r.
Proof. Introduce the Banach space D of continuous periodic functions
g(t) of period p with the norm \g\ = max ||gO)ll- Thus convergence of
gi0)> ^20)i • . . in D is equivalent to the usual uniform convergence over
0 ^ t ^ p.
Let g(t) be a continuous function of period p satisfying ||gO)ll ^ r-
Thus by Theorem 1.1 the equation
B.4) y' -A(t)y=f(t,g(t))
has a unique solution y(t) of period/?. Define an operator To on the set of
all suchgO) by putting y(t) = T0[g]. Note that A.7), B.4) arid B.2) show
that if z@ = T0[h], then
B.5) \y-z\? KpB \g - h\; i.e., \T0[g] - T0[h]\ < Kp6 \g - h\,
where \y\ = max \\y(t)\\ for 0 ^ t ^ p. In addition, if m = max \\f(t, 0)||,
then |7*0[0]| ^ Kpm.
Thus Theorem 2.1 follows from Theorem 0.1, for yo(t) is a fixed point of
To, ro[2/o] = 2/o> if and only if 2/00) is a solution of B.1) of period/?; cf.
B.4) where y = T0[g].
In Theorem 2.1, we can omit assumption B.2) when \\f(t, y)\\ is "small,"
at the cost of losing "uniqueness."
Theorem 2.2. Let A(t), Kbe as in Theorem 2.1. Letf(t, y) be continuous
for all t and \\y\\ ^ r, of period p in t for fixed y, and satisfy
B.6) Kp 11/0,2/)|| ^ r for 0 ^ t ^ p, \\y\\ ^ r.
Then B.1) has at least one periodic solution of period p.
Proof. As in the last proof, define y(t) = T0[g] as the unique solution of
B.4) of period p, where g(t) is of period p and \g\ ^ r. In order to prove
the theorem, it suffices to show that To has a fixed point y0, T0\y0] = y0.
This will be proved by an appeal to Corollary 0.1 of Tychonov's theorem.
414 Ordinary Differential Equations
It follows from A.7) and B.6) that y = T0[g] satisfies \y\ ^ r. In other
words, if D is the same Banach space as in the last proof, then To maps
the sphere |g| < r of X> into itself. Also, A.7) gives
TJLg] ~ T0[h]\ ^ K \\fit, git)) -fit, hit))\\ dt.
Since/is continuous, it is clear that if \g — h\ = max \\g{t) — h{t)\\ -*¦ 0,
then T0[g] — T0[h] -*¦ 0. Thus To is a continuous map.
If y = T0[g], then \\y(t)\\ ^ r and B.4) show that there is a constant C,
independent of g, such that ||y'@ll = C. This implies that the set of
functions y(t) = T0[g] in the range of To is bounded and equicontinuous.
Hence, by Arzela's theorem, it has a compact closure in X> (i.e., any
sequence yx, y2, . . . has a uniformly convergent subsequence). Conse-
Consequently, Corollary 0.1 implies that 70 has a fixed point y0. Clearly
V = 2/o(O is a periodic solution of period p. This proves the theorem.
Remark. In the deduction of Corollary 0.1 from the Tychonov
Theorem 0.2, it is necessary to know that the convex closure of the
range &(T0) of To is compact. This is clear in the proof just completed,
for y(t) in the range of Tsatisfies the conditions: (i) y(t) is continuous of
period/?; (ii) \\y(t)\\ ^ r; and (iii) \\y(t) - y(s)\\ ^ C \t - s\. The convex
hull of &(T0) [i.e., the smallest convex set containing ,^G0)] is the set of
functions y(t) representable in the form hy^t) + • • • + ^«y»@» where
n = 1, 2, . . . ; Xi ^ 0 and Xx + ¦ ¦ ¦ + Xn = 1. It is clear that functions
in this set satisfy (i)-(iii). The closure of this set of functions under the
norm of D (i.e., under uniform convergence over 0 ^ ? </?) gives a set
of functions satisfying (i)-(iii). Thus the compactness of this set in D is
clear from Arzela's theorem. (A remark similar to this can be made for the
other applications of Corollary 0.1 in this chapter; see Theorem 4.2 and
Theorem 8.2.)
Consider now a system of nonlinear differential equations depending on
a parameter fi,
B.7) x' = Fit, x, p),
where F is continuous, of period p in t for fixed (*, fi), and x, F are real
J-dimensional vectors. Suppose that for fi = 0, B.7) has a periodic
solution x = go(t). Write y = x — go(t); then B.7) becomes
y' = F(t, y + git), fi) - F(t, go(t), 0).
If F has continuous partial derivatives with respect to x and A(t) =
dxF(t, go(t), 0), where d^F is the Jacobian matrix of F with respect to x,
then the last equation is of the form B.1), where
fit, y) = F(t, y + go(t), fi) - F(t,go(t), 0) - A(t)y
Use of Implicit Function and Fixed Point Theorems 415
and \\f{t, y)\\l\\y\\ -*0 as (y,fi)-+0 uniformly in t for 0 ^ t ?p. In
particular, when \/i\ is small, B.6) holds for small r > 0; in fact, B.2)
holds for small WyJ, \\y2\\ with arbitrarily small 8 and /(?, 0) = 0. It
follows from Theorem 2.1 that if A.2) has no nontrivial periodic solution
of period p, then B.7) has a unique solution x(t) = x(t, /i) of period p
for each small \fi\. The proof of Theorem 2.1 can also be used to show
that if F depends smoothly on fi, then x(t, fi) depends smoothly on fi.
All of these assertions can, however, be proved more directly by the use of
the classical implicit function theorem.
Theorem 2.3. Let x, F be real vectors. Let F(t, x, fi) be continuous for
all t, small \/i\, and x on some d-dimensional domain. Let F be of period p in
t for fixed (x, fi) and have continuous partial derivatives with respect to the
components ofx. Let B.7), where fi = 0, have a solution x = go(t) of period
p with the property that if Ait) = dxFit, goit), 0), then A.2) has no nontrivial
solution of period p. Then, for each small \/i\, B.7) has a unique solution
x = x{t, fi) of period p with initial point *@, fi) near goiO); x(t, fi) is a
continuous function of it, /i), and *(?, 0) = go(?)- If, in addition, F has a
continuous partial derivative with respect to fi, then x(t, fi) is of class C1.
It will be clear from the proof that if more smoothness is assumed for
F (e.g., F 6 Ck or F analytic), then *(?, fi) is correspondingly smoother (e.g.,
*(?, fi) 6 Ck or *(?, fi) analytic).
Proof. Let x = f (?, x0, /i) be the unique solution of B.7) satisfying the
initial condition *@) = x0. Then f (?, x0, /i) is continuous and has con-
continuous partial derivatives with respect to t and the components of x0;
see Corollary V 3.3. Also, if x0 is near to go(O), then f(?, x0, fi) exists on the
interval 0 ^ t ^ p; see Theorem V 2.1. The solution x = f(?, x0, /i) is
periodic of period p if and only if
B.8)
, x0, /i) — xo = 0.
Since f(?,go(O), 0) = goit), the equation B.8) is satisfied if(«0)/«) = (g0@),0).
Hence it can be solved for x0 = xo(jx) if the Jacobian matrix of the left
side, 3Xof(/>, x0, fi) - I, is nonsingular at (*„, fi) = (go(O), 0). The partial
derivatives of f (?, x0, fi) with respect to a component of x0, when (#0, /i) =
(#o(O), 0), is a solution of the equations of variation A.2); see Theorem
V 3.1. In fact, 7(?) = dxjit,g<$), 0) is a fundamental matrix for A.2)
satisfying K@) = /. Hence the assumption that A.2) has no periodic
solution is equivalent to the assumption that Yip) — I is nonsingular;
cf. Lemma 1.1, where M = N = I. Thus the implicit function theorem is
applicable to B.8) and gives a continuous function x0 = xo(jx). Corre-
Correspondingly, x = f(r, *oC")» A*) is a periodic solution of B.7) of period p
and the only such solution with initial point x0 near go(O). The other
assertions of Theorem 2.3 also follow from the implicit function theorem.
416 Ordinary Differential Equations
The question of the existence of periodic solutions when
det [Y(p)-I] = 0
has a vast literature and will not be pursued here.
Note that if Fin B.7) does not depend on t and go(t) ^ const., then the
conditions of Theorem 2.3 cannot be satisfied since x = go'(t) is a non-
trivial periodic solution of the equations of variation A.2). Here, however,
we have the following analogue.
Theorem 2.4. Let x, F be real vectors. Let F(x, fi) be continuous for
small \fi\ and for x on some d-dimensional domain and have continuous
partial derivatives with respect to the components of x. When fi = 0, let
B.9)
x' = F(x, fi)
have a solution x = go(t) ^ const, of period p0 > 0 such that if A(t) =
dxF(g0(t), 0), then exactly one of the characteristic roots of A.2) is 1 [i.e.,
eRp° has X = 1 as a simple eigenvalue; cf A.5) where p = p0]. Then, for
small \/i\, B.9) has a unique periodic solution x = x(t, /i) with a period p(ji),
depending on fi, such that x(t, fi) is neargo(t) and the periodp(p) is near p0;
furthermore x(t, fi), p(/j,) are continuous, x(t, 0) = go(t), andp@) = p0.
Remarks similar to those for Theorem 2.3 concerning the smoothness of
F and corresponding smoothness of x(t, /j), p(ju) hold.
The geometrical considerations in the proof to follow are clarified by
reference to Lemma IX 10.1, which shows that we obtain all solutions of
B.9) near x = go(t) by considering solutions with initial points x(fi) = x0
near to go(O) and x0 restricted to be on the hyperplane it normal to
•F(?o(O), 0) and passing through go(O).
Proof. Let x = f(?, x0, fi) be the unique solution of B.9) satisfying
x@) = x0. This solution is of period p if and only if B.8) holds. The
equation B.8) is satisfied when (p, x0, fi) = (p0, go(O), fi).
Since solutions of B.9) are uniquely determined by initial conditions and
go(t) =? const., it follows that F(go(t), 0) ^ 0 for all t. Suppose that the
coordinates in the a>space are chosen so that go(O) = 0 and F@, 0) =
@, . . . , 0, a), a^O, and let -n denote the hyperplane a^ = 0 through the
point go(O) = 0 normal to F@, 0). Consider x0 on this hyperplane,
x0 = (a;,,1, . . . , xft-1, 0). Then for small \/i\, the equation B.8) has a
unique solution for/?, x0, in terms of /i if the Jacobian matrix of f(?, x0, fi)
— x0 with respect to a:,,1, . . . , x^ and t is nonsingular at (t, x0, ju) =
(Po, 0, 0).
The matrix Y(t), in which the columns are the vectors d^jdx,,1, . . .,
dti/drf and f at (x0, fi) = @, 0), is a fundamental matrix for A.2) and
its last column is F(go(t), 0). At t = 0,
B.10)
7@) = diag [/,_!, a] = Z@),
Use of Implicit Function and Fixed Point Theorems 417
by A.5). Since A.2) has, up to constant factors, only the last column
go'(t) of Y(t) at (x0, fi) = @, 0) as a periodic solution of period p0, the
matrix Y(p0) — 7@) annihilates vectors c of the form c = @, . . ., 0, cd)
and no others.
The Jacobian matrix J of f(?, x0, fi) — x0 with respect to x^, . . ., x^-1,
and t at (?, x, fi) = (/?„, 0, 0) is
J = Y(p0) - diag [/,_!, 0],
and the last column of Y(p0) is F(g0@), 0) = @, . . . , 0, a), so that J =
[Y(p0) - 7@)] + diag [0,. . . , 0, a]. If J is singular, then there exists a
vector c = (c1, . . . , cd) ^ 0 such that Jc = 0; i.e.,
[Y(p0) - Y@)]c + @, . . . , 0, ac") = 0.
In view of B.10) and Z@) = Z(p0), this is the same as
Z@){(eR*° - I)c + @, . . . , 0, cd)} =0 or
(eRv" - I)c + @, . . . , 0, cd) = 0.
If cd = 0, then c = 0 for eRp« — I only annihilates vectors of the form
@, . . . , 0, cd). If cd 5* 0, then (eBp» - Ifc = 0. But this implies that
X = 1 is at least a double eigenvalue of eRp°. This contradiction shows that
J is nonsingular.
Hence the implicit function theorem is applicable to B.8) and gives the
desired functions x^(ji),. . ., xf,-1^), and p(ji). Correspondingly, if
xo(ji) = (VC")> • ¦ • , ^"V)' °)» then x(*> /") = f(?> *<>(>)> /") is a periodic
solution of B.9) and is the only periodic solution having an initial point
x0, with xod = 0, near to go(O) and a period near to p0. This proves
Theorem 2.4.
Exercise 2.1. Let dim a; = 2; F(t,x) continuous for all t and x,
periodic of period p in t for fixed x. Let the solution x = x(t, t0, x0) of
B.11) x' = F{t,x)
satisfying x(t0) = x0 be unique for all ?0, x0 and exist for t ^ ?0. Finally,
for some (?0, x0), let x(t, t0, x0) be bounded for t ^ ?0. Then B.11) has at
least one periodic solution of period/?. See Massera [1].
Exercise 2.2. Let a(?) = (^@, • • ¦, *d(t)), 0(t) = (jSHO, • ¦ ¦ , j8"@) be
piecewise continuously differentiable for §<t^p; a.\t) ^ fij(t) for
./ = 1,... , d; and a@) = a.(p), ?@) = p(p). Let
nt,y)={f\t,y),...,fd{t,y))
be continuous on an open set containing Cl° = {(?, y): a.'(t) ^ y' ^ fi'(t) for
0 ^ t ^ p) and let/(f, y) be uniformly Lipschitz continuous with respect
to ?/. Suppose finally that the functions u\t, y) = a.j'(t) —f'(t, y1, . . .,
y> "', oi>(t), y> • \ ...,/) and r'(r. y) = (i''(t) -f\t, y\ . . . , y*-\
418 Ordinary Differential Equations
y1+1,. . ., yd) do not change signs (e.g., u> ^ 0 or u1 ^ 0) and that uivi ^ 0
for all (t, y) e Q°. Then y' = f(t, y) has at least one solution y = y(t),
0 ^ t ^ p, such that (t, y(t)) e Q° and y@) = y{p). See Knobloch [1].
PART II. SECOND ORDER BOUNDARY VALUE PROBLEMS
3. Linear Problems
This part of the chapter concerns boundary value problems involving
a system of second order equations. Consider first a linear inhomogeneous
system of the form
C.1) x" = B(t)x + F(t)x + h{t)
and the corresponding homogeneous system
C.2) *" = B(t)x + F(t)x
for a J-dimensional vector x (with real- or complex-valued components).
The problem involves solutions satisfying boundary conditions
C.3) z@) = x0, x(p) = xp,
when p > 0, x0, xv are given. For the inhomogeneous equation C.1), the
conditions C.3) are not more general than
C.4) x@) = 0, x{p) = 0,
for if a; — [(xv — xo)t/p + x0] is introduced as a new dependent variable,
the equation C.1) goes over into another equation of the same form with
h(t) replaced by hit) + B(t)(xv - xo)t/p + B(t)x0 + /¦(?)(*„ - *„)//>•
Actually, the theory of the boundary value problem C.1), C.4) is
contained in § 1. In order to see this, write C.1) as a first order system
C-5) y' = A(t)y + g(t),
where y = (*, *') is a 2J-dimensional vector, g(t) = @, h(t)), and A(t) is a
Id X Id matrix
/ 0 / \
C.6) A(t) =
\B(t) F(t)J
The boundary conditions C.4) can be written as
C.7) My@) - Ny{p) = 0,
where M, N are the constant Id x 2d matrices
C.8)
M =
0 0
and N =
'0 0'
/ 0
Use of Implicit Function and Fixed Point Theorems 419
Note that
// 0 0 0\
rank (Af, N) = rank = 2d.
\0 0 / 0/
Instead of restricting M, Nto be of the type C.8), it is possible to choose
more general matrices; in this case, C.4) is replaced by conditions of the
form
Mnx@) + Mj2x'@) - Nnx(p) - Nj2x'{p) = 0 for j = 1, 2,
where Mjk, Njk are constant d X d matrices such that
M12 Nu N12\
(M,N) =
\M21 M22 N21 N2,
is of rank Id. For the sake of simplicity, only the choice C.8), i.e., only
the boundary conditions C.4), will be considered.
Lemma 1.1 implies the following:
Lemma 3.1. Let B(t), F(t) be continuous d X d matrices for 0 ^ t ^ p;
U(t) the d X d matrix solution of
C.9) U" = B(t)U + F(t)U', U@) = 0, U'@) = I.
Then C.2) has a nontrivial solution (^ 0) solution satisfying C.4) if and only
ifU{p) is singular. In fact, the number k,0 ^ k ^ d, of linearly independent
solutions of C.2), C.4) is the number of linearly independent vectors c
satisfying U(p)c = 0.
The corresponding corollary of Theorem 1.1 is
Theorem 3.1. Let B(t), F(t) be continuous for 0 ^ t ^ p. Then C.1) has
a solution x(t) satisfying C.4) for every h(t) continuous on [0,p] if and only
if C.2), C.4) has no nontrivial (^ 0) solution. In this case, x(t) is unique and
there exists a constant K such that
C.10)
\\<t)\\,
Exercise 3.1. Verify Theorem 3.1.
The homogeneous adjoint system for C.5) is y' = —A*(t)y which is not
equivalent to a second order system without additional assumptions on B
or F. The simplest assumption of this type is that F(t) is continuously
differentiable. In this case, the homogeneous adjoint system y' = — A*(t)y
is equivalent to
C.11)
z" = [B*(t) - F*'(t)]z - F*(t)z'
and the corresponding inhomogeneous system is
C.12) z" = [fl*@ - F*\t)]z - F*(t)z'
420 Ordinary Differential Equations
[Actually, the differentiability condition can be avoided by writing the
terms involving F* as (F*z)', and interpreting C.11), C.12) as first order
systems for the 2d-dimensional vector (—z — F*z, z).]
In order to obtain the corresponding Green's relation, multiply C.1)
scalarly by z, C.12) by x, subtract and integrate over [0,p] to obtain
C.13) f \h{t) ¦ z(t) - x(t) • f{t)] dt=[x'-z- x-z'-Fx- z\J>.
Jo
Thus, if x satisfies C.4) and z satisfies
C.14) z@) = 0, z{p) = 0
then
C.15)
[\hit) ¦ zit) - xit) -fit)] dt = 0,
Jo
so that C.4) and C.14) are adjoint boundary conditions.
Exercise 3.2. Verify that C.2), C.4) and C.11), C.14) are adjoint
boundary problems in the sense of § 1.
Lemma 3.2. Let Bit) be continuous and Fit) continuously differentiable
for 0 ^ t ^p. Then C.2), C.4) have the same number of linearly inde-
independent solutions as the adjoint problem C.11), C.14).
Finally, a corollary of Theorem 1.2 is
Theorem 3.2. Let Bit) be continuous and Fit) continuously differentiable
on [0, p] and such that C.2), C.4) has k, 1 ^ k ^ d, linearly independent
solutions. Let z-^t) zkit) be k linearly independent solutions o/C.11),
C.14). Let hit) be continuous on [0,p]. Then C.1), C.4) has a solution if
and only if
C.16)
fV0
Jo
for j=l,..., k.
The next uniqueness theorem has no analogue in § 1.
Theorem 3.3. Let Bit), Fit) be continuous d x d matrices on 0 ^ t ^ p
such that
C.17)
Re
- \Fit)F*it))x ¦ x] ^ 0
for all vectors x (i.e., let the Hermitian part of the matrix B — \FF* be non-
negative definite). Let git) be continuous for 0 _ t ^ p. Then
C.18)
x" = Bit)x + Fit)x + hit)
has at most one solution satisfying given boundary conditions x(Q) = x0,
x(p) = xp-
Use of Implicit Function and Fixed Point Theorems 421
Remark 1. Actually, Theorem 3.3 remains valid if C.17) is relaxed to
C.19) 2 Re [E(?) - \Fit)F*it))x ¦ x] > -(ff/p)« ||*||«
for all vectors x ^ 0; cf. Exercise 3.3.
Proof. Since the difference of two solutions of the given boundary value
problem is a solution of
C.20) x" = Bit)x + Fit)x', z@) = xip) = 0,
it suffices to show that the only solution of C.20) is x = 0.
Let xit) be a solution of C.20). Put r(t) = IKOII2- Then r' = 2 Re x • x
and r" = 2 Re (a; • x" + \\x'\\2), so that r" = 2 Re [iBit)x + F(t)x') • x +
|| a;'II2]. It is readily verified that
Re iBit)x + F(t)x') ¦ x + ||a;'||2 = \\x' + \F*x\\* + Re E* - \FF*x) ¦ x.
Thus
C.21) r" = 2 II*' + \F*x\\* + 2 Re [E - \FF*)x • *].
Hence C.17) implies that r" ^ 0. Since the last part of C.20) means that
r@) = rip) = 0, it follows that r(t) = 0 for 0 ^ t ^ p. This proves
Theorem 3.3.
Exercise 3.3. (a) Show that if there exists a continuous real-valued
function qit), 0 ^ t ^ p, such that the equation
r" + qit)r = 0
has no solution r{t) ^ 0 with two zeros on 0 _ t ^ p [e.g., </@ < i^lpJ]
and C.17) is relaxed to
C.22) 2 Re [E@ - \Fit)F*it))x ¦ x] = -qit) \\x\\*
for all vectors x, then the conclusion of Theorem 3.3 remains valid. F)
Let there exist a continuously differentiable d X d matrix Kit) on [0,p]
such that
C.23) Re [5 - K' + (|F - KH)i\F* - KH)]x -x^O
for all vectors x and 0 = t = p, where KH = ^K + A:*). Then the
conclusion of Theorem 3.3 is valid. [Note that C.23) reduces to C.17) if
Kit) = 0, so that ib) generalizes Theorem 3.3, but not part (a) of this
exercise.]
Remark 2. If F(t) has a continuous derivative, then C.20) implies that
x = 0 if and only if z = 0 is the only solution of
C.24) z' = [5*@ - F*'it)]z - F*it)z', z@) = zip) = 0;
cf. Lemma 3.2. Hence, the conclusion of Theorem 3.3 is valid if 5, Fin
thecriteriaC.l7),C.22),C.23)arereplacedby5* — F*', —F*, respectively.
422 Ordinary Differential Equations
4. Nonlinear Problems
Let x and/denote vectors with real-valued components. This section
deals with second order equations of the form
D.1) x"=f(t,x,x')
and the question of the existence of solutions satisfying the boundary
conditions
D.2) x@) = 0, x(p) = 0
or, for given x0 and xv,
D.3) z@) = x0, x(p) = xv.
The equation D.1) will be viewed as an "inhomogeneous form" of
D.4) x" = 0.
The problem D.2), D.4) has no nontrivial solution. Thus, by Theorem
3.1,'an equation
D.5) x" = h(t)
has a unique solution satisfying D.2). In fact, this solution is given by
D.6) x(t) = - -\(p - t) \\his) ds + t [\p - s)h(s) ds\
pL Jo Jt J
This can be verified by differentiating D.6) twice; cf. (XI 2.18). The
relation D.6) can be abbreviated to
D.7)
where
D.8)
according
x(t) = — G(t, s)h(s) ds,
Jo
G(t, s)=-(p- t)s or G(t, s) = - t(p - s)
P P
D.9)
0 ^ G(t, s) ^ ^ , f PG(t,
4 Jo
I'd
Jo '
^t^s^p. Thus
s = \t(p-
2
8
where Gt = dGjdt. Thus D.6) or D.7) and its differentiated form imply
2
D.10) \\x{t)\\ < V- max ||ft(s)ll» K@ll ^ - max ||ft(s)|| ,
~ 8 2
where the max refers to 0 ^ s ^ p.
1
Use of Implicit Function and Fixed Point Theorems 423
Theorem 4.1. Let f(t, x, x') be continuous for 0 ^ t ^ p and all (x, x')
and satisfy a Lipschitz condition with respect to x, x of the form
(A. 1 n II fft -r v '\ fft f -r 'Ml < fl ll-y <r II J_ fl ll-y ' <r 'II
14-'U \\J\h Xl> Xl ) JVt X2> X2 )\\ = 11*1 "~ X2\\ + "l iFl — X2 II
with Lipschitz constants 60, 61 so small that
D,2, «f! + ^<l.
Then D.1) has a unique solution satisfying D.2).
Remark 1. Instead of requiring /to be defined for 0 ^ t ^ p and all
(x,xl), it is sufficient to have/defined for 0 ^ t ^p, \\x\\ ^ R, \\x'\\ ^ 4/?//>,
where R satisfies either
D.13)
?**['-(?+?)]
if m = max \\f(t, 0, 0)|| for 0 < t < p, or merely
Mp2
D.14)
< R
if M = max \\f(t, x, x')\\ for ||x|| ^ /?, ||*'|| < 4R/p.
Proof. Let D be the Banach space of functions h(t), 0 ^ t ^ p, having
continuous first derivatives and the norm
D.15)
\h\ = max
,? max||fc'@ll).
4os«< /
Consider an h(t) in the sphere \h\ ^ R of X). Let x(t) be the unique
solution of
D.16)
x" =f{t,h{t\ h'(
satisfying *@) = x(p) = 0. Define an operator To on the sphere \h\ ^ r
of D by putting T0[h(t)] = x(t).
\fx0 = 70[0] and ||/(?,0,0)|| ^ m, then
D.17)
8
J
4
8
by the case h =f(t, 0, 0) of D.10). Thus the norm xo(t) = 70[0] e D
satisfies
D.18) |To[O]|^^.
424 Ordinary Differential Equations
Also, if *! = TM x2 = T0[ht], then, by D.10) and D.11),
2
(eo max Pi - A«II + °i max HAi'
2
-
8
max \\h, -
max HV -
If the last inequality is multiplied by p\4 and 0i(p2/8) max p/ — A2'|| is
written as @i//2)[(p/4) max p/ - A2'||], it follows that
D.19)
Thus the inequalities D.12), D.13) and D.18) show that Theorem 0.1 is
applicable and give Theorem 4.1.
Similarly, if \\f(t,x,x')\\ ^ M for ||*|| <j R, \\x'\\ ^ARjp, then the
derivation of D.17) shows that if \h\ ^ R, then x = T0[h] satisfies \x\ ^
A//>2/8. Thus if D.14) holds, To maps the sphere \h\ ^ R into itself and the
Remark following Theorem 0.1 is applicable in view of D.12). Hence the
proof of Theorem 4.1 and Remark 1 following it is complete.
Corollary 4.1. Let f(t, x, x') be continuous for 0 ^ t ^ p, \\x\\ ^ Ro,
||a;'|| <j R1 and satisfy D.11), D.12) and \\f(t, x, x')\\ <j M. Ut
D.20)
\\xo\\ <, R0, M-P
Z
8 I p
Then D.1) has a unique solution satisfying
D.21) *@) = 0 and x(p) = x0.
Exercise 4.1. (a) Prove Corollary 4.1. (b) In Corollary 4.1, let
\\f(t, x, x')\\ < M be relaxed to \\f(t, txjp, xo/p)\\ ^ m for 0 ^ t ^ /> and
J? be defined by replacing "<" by "=" in D.13). Show that the con-
conclusion of Corollary 4.1 remains valid if R + \\xj <; R0,4Rjp + \\xo\\/p ^
J?j replaces D.20).
Theorem 4.2. Let f(t, x, x) be continuous and bounded, say,
\\f(t,x,x')\\^m,
for 0 ^ t ^p and all (x, x'). Then D.1) has at least one solution x(t)
satisfying x@) = x(p) = 0 and
. __x ,, , ... . mp2 „ ,, .,, ^ mp
D.22) KOII ^ —— , II*(Oil = ~ ¦
It is sufficient to require that f(t,x, x') be defined only for ||*|| ^ w/>2/8,
||a;'|| ^ mp\2.
Proof. Let I) be the Banach space of continuously differentiable
functions h(t), 0 ^ t ^p, with norm |A| defined D.15). Consider h(t) in
I
Use of Implicit Function and Fixed Point Theorems 425
the sphere \h\ ^ mp2/% of X>. For such an h, put x = T0[h], where x(t) is
the unique solution of D.16) satisfying *@) = x(p) = 0. Then \\x(t)\\ ^
w/>2/8 and ||*'@ll ^ w/?/2, so that 7O maps the sphere \h\ < w/>2/8 into
itself.
If l/iil, \h2\ ^ w/>2/8 and ^ = T^], x2 = T0[h2], then D.7) and D.9)
imply that
4Jo
Since/is a continuous function, it follows that if \h-^ — h2\ ->-0, then
1*1 — x2\ —>-0. Thus Jo is continuous.
For any x(t) in the range of 70, i.e., a; = T0[h] for some A, D.16) implies
||*"@ll ^ w. It follows that the set of functions x(t) in the range of
To[h], \h\ ^ w/>2/8, are such that x(t), x'(t) are bounded and equicontinuous
since
Hence Arzela's theorem implies that the range of T0[h] has a compact
closure. Consequently, Tychonov's theorem is applicable and gives
Theorem 4.2.
Corollary 4.2. Let f(t, x, x') be continuous and satisfy ||/|| ^ M for
0 < t < T, ||a;|| ^ Ro, II*'|| < J?j. Le? /? anJ *0 ^a?«_^ 0 < p < T and
D.20). Then D.1) /iay a solution satisfying D.21). (/n particular, if0<
T < min ((8RJMIA, 2RJM), then there exists a d > 0 jmc/i that if
\\xo\\ ^ ^, then D.1) /ios a solution satisfying D.21) for p = T.)
Exercise 4.2. Prove Corollary 4.2.
Exercise 4.3. Let /(?, *, *') be continuous for 0 ^ ? ^ />, ||%|| ^ J?o,
and arbitrary *'. Let there exist positive constants a, b such that
\\f{t, x, Oil ^ a ll*'ll2 + * for 0 < t ^p, ||a;|| ^ J?o. Assume that
a, b, \\xo\\ are such that a(bp2 + 2 \\xo\\) < 1 and r* = (ap)-1^ — [1 —
a(bp2 + 2 llxoll)]'^} satisfies r*p + 3 ||*0|| ^ 4R0. Then the boundary
value problem D.1), D.2) has a solution.
Note that Corollaries 4.1 and 4.2 are similar except that in Corollary
4.1 there is the extra assumption that D.11) and D.12) hold; corre-
correspondingly, there is the extra assertion that the solution of D.1), D.21) is
unique. We can prove another type of uniqueness theorem.
Theorem 4.3. Let f(t, x, x') be continuous for 0 ^ t ^ p and for (x, x')
on some 2d-dimensional convex set. Let f(t, x, x') have continuous partial
derivatives with respect to the components of x and x . Let the Jacobian
matrices off with respect to x, x'
D.23) B (t, x, x') = dj(t, x, x\ F{t, x, x') = dx.f(t, x, x')
426 Ordinary Differential Equations
satisfy
D.24)
2(B-iFF*)z-z>--2\\z\\2
P
for all (constant) vectors z^O. Then D.1) has at most one solution
satisfying given boundary conditions x@) = x0, x(p) = xv.
By the use of Exercise 3.3(a), condition D.24) can be relaxed to
2(B-\F*F)z-z^ -q(t)\\z\\2
for all constant vectors z and (t, x, x'), where q(t) satisfies the conditions
of Exercise 3.3(a).
Proof. Suppose that there exist two solutions xx(t), x2(t). Put x(t) =
x2(t) — xx(t), so that
x" = f{t, x2(t), x2'(t)) - f(t, *i@> *i'@). *(°) = x(S = °-
This can be written as
x" = Bitty* + F&yc1, x@) = x{p) = 0,
where
D.25) Bi@ = f XB ds, F,(t) = F ds,
Jo Jo
and the argument of B, F in D.25) is
D.26) (t, A - s)x1(t) + sx2(t), A - s)x1'(t) + sx2'(t)).
This is a consequence of Lemma V 3.1.
For any constant vector z, an application of Schwarz's inequality to
the formula in D.25) for each component of Fx*(t)z gives
where the argument of F* is D.26). Hence,
[5,@ - iFiCO^i*@]« 'z ^ iB - iFF*1z ' z ds-
Jo
Thus by D.24)
2[Bi@ - iF&Wfmz ¦ z > - - ||z||
P
for all vectors ?#0. Consequently, Theorem 3.3 and Remark 1 following
it imply that x(t) = 0. This proves the theorem.
Exercise A A. Let f(t, x, x') be continuous for 0 ^ t ^ p and (x, x') on
some 2J-dimensional domain and satisfy a Lipschitz condition of the form
D.11), where
D.27) 2eo + R2<3-
Use of Implicit Function and Fixed Point Theorems 427
Then D.1) has at most one solution satisfying given boundary conditions
x@) = x0, x{p) = xv.
Exercise 4.5. Let/(?, x, x') be continuous for 0 ^ t ^ p and (x, x') on
a 2rf-dimensional domain. Let A.x = x2 — xx, A.x' = x2 — x{, A/ =
f(t, x2, x2) — f(t, xu x-f), where xlt x2, x^, x2 are independent variables
and assume that
Ax- A/+ |Aa;'|2> 0 if Ax jt 0, Ax ¦ Ax' = 0.
Then the boundary value problem x" = f(t, x, x'), x@) = x0, x(p) = xp
has at most one solution.
Exercise 4.6. (a) Let a; be a real variable. Let/(f, x, x') be continuous
and strictly increasing in x for fixed (t, x'). Then D.1) can have at most
one solution satisfying given boundary conditions x@) = x0, x(p) = x^.
(b) Show that (a) is false if "strictly increasing" is replaced by "non-
decreasing." (c) Show that if, in part (a), "strictly increasing" is replaced
by "nondecreasing" and, in addition, / satisfies a uniform Lipschitz
condition with respect to x', then the conclusion in (a) is valid. [For an
existence theorem under the conditions of part (c), see Exercise 5.4.]
Exercise 4.7 (Continuity Method). Let a; be a real variable. Let cn(t, x'),
fl(t, x') be real-valued, continuous functions for — oo < t, x' < oo with
the properties that (i) a, /? are periodic of period p > 0 in t for fixed x';
(ii) a > 0; (iii) |j8(f, x')\ -+ oo and |a(f, x')/P(t, x')\-*0 as \x'\ -+ oo
uniformly in t. (a) Show that
D.28) x" = zait, x') + fi(f, x')
has at most one solution of period p,
D.29) x@) - x[p) = 0, x'@) - x'[p) = 0.
(b) Show that if C = max |/3(f, O)|/oc(f, 0) and K is so large that Ca(t, x') ^
i \P(t, x')\ and \P(t, 0)| ^ |/5(?, x')|/4 when \x'\ ^ AT, then any periodic
solution x(t) of D.28) satisfies KOI ^ C, \x'(t)\ ^ AT. (c) Assume that
a, /? are of class C1. By showing that the set of 2-values on 0 < X ^ 1 for
WhlCh
x" =
, *') - /?(?, 0)
, 0)
has a periodic solution is open and closed on 0 ^ 2 ^ 1, prove that
D.28) has a unique periodic solution, (d) Show that the assumption in
(c) that a, p are of class C1 can be omitted.
Exercise 4.8 (Continuation). Let <x(?, a;, a;'), /?(?, x, x') be continuous
for — oo < t, x, x' < oo with the properties that (i) a, /5 are periodic of
period p > 0 in t for fixed (a;, a;'); (ii) a > 0; (iii) there is a constant C
such that \P(t, x, 0)| ^ Ca.(t, x, 0) for — oo < t, x < oo; (iv) |/3(f, a;, a/)| ->¦
a; and |a(^, x, x')/p(t, x, x')\ -*¦ 0 as |a;'| -»• oo uniformly on bounded
428 Ordinary Differential Equations
(t, *)-sets. Show that
x" = xa.it, x, *') + ^t, x, x')
has at least one periodic solution.
5. A Priori Bounds
The proofs for the existence theorems for solutions of boundary value
problems in the last section depended on finding bounds for the solution
and its derivative. This section deals with more a priori bounds and their
applications. The main problem to be considered is of the following type:
Given a rf-dimensional vector function xit) of class C2 on some interval
0 ^ t ^ p, a bound for ||*(OII, and some majorants for ||*"||, find a bound
for ||*'||. The following result holds for the case when a; is a real-valued
function:
Lemma 5.1. Let (pis), where 0 ^ s < oo, be a positive continuous func-
function satisfying
E.1)
f°° sds _
J wis) ~
oo.
Let R ^ 0 and r > 0. Then there exists a number M [depending only on
(pis), R, t] with the following property: If xit) is a real-valued function of
class C2 for 0 ^ t ^p, where p ^ r, satisfying
E.2)
^ R,
\x"\ ^ (pi\x'\),
then |*'| i
Proof. In view of E.1), there exists a number M such that
j2R/r 9?(S)
It will be shown that M has the desired property. [Instead of assumption
E.1), it would be sufficient to assume the existence of an M satisfying E.3).]
Let |*'@l assume its maximum value at a point t = a, 0 ^ a ^ p.
We can suppose that x\a) > 0, otherwise * is replaced by —*. If x\a) >
2R/t, then there exists a point t on 0 ^ t ^ p where *'@ ^ 2R/p ^ 2.R/t.
Otherwise *(/>) — *@) > 2R which contradicts |*| ^ i?. Assume x\a) >
2/?/t and let t = Z> be a point nearest t = a where *'@ = 2/?/t. For sake
of definiteness, let b > a. Thus 0 ^ 2R\t = *'(Z>) ^ *'@ ^ *'(<0 for
a ^ t ^ Z>.
If the second inequality in E.2) is multiplied by x\t) > 0, a quadrature
over a ^ t ^ b gives
' x'jt)x"jt) dt
(pix'it))
Use of Implicit Function and Fixed Point Theorems 429
Even though it is not assumed that x" j& 0, the formal change of variables
s = x\t) is permitted on the left and gives
2RIt (pis)
cf. Lemma I 4.1. From E.3), it is seen that x\a) ^ M. Thus it follows
that either x\a) ^ 2R/r or x\a) ^ M. In either case *'(a) ^ Af. Since
*'(a) = max |*'@l f°r 0 = t =P> the lemma follows.
Lemma 5.1 is false if * is a J-dimensional vector, d ^ 2, and absolute
values are replaced by norms in E.2). In order to see this, note that
(pis) = ys2 + C > 0, where y and C are constants, satisfies the condition
of Lemma 5.1. Let *(?) denote the binary vector *(?) = (cos nt, sin nt).
Thus ||*|| = 1, ||*'@ll = M, ll*"@ll = n2 = ||*'||2. Thus the inequalities
analogous to E.2),
E.4) ||*|| ^ R, ||*"|| ^ rtllz'U),
hold for R = 1, (pis) = s2 + 1. But there does not exist a number M such
that ||*'(Oil ^ Af for all choices of n. The main result for vector-valued
functions will be the next lemma.
Lemma 5.2. Let (pis), where 0 ^ s < oo, Z>e a positive continuous
function satisfying E.1). Let <x, K, R, r be non-negative constants. Then
there exists a constant M [depending only on (pis), a, R, r, K] with the
following property: If xit) is a vector-valued function of class C2 on
0 ^ t ^ p, where p ^ t, satisfying E.4) and
E.5) ||*|| ^ R, ||*"|| ^ a/-" + K, where r = ||*||2,
then ||*'|| ^ Af on 0 ^ t^p.
Proof. The first step of the proof is to show that E.5) alone implies the
existence of a bound for ||*'@ll on any interval [/u,p — fi], 0 < fx ^ %p.
Let 0 < /u < p and 0 ^ ? ^ /? — /u, then
E.6) *(r + /x)- xit) - ixx'it) = J (f + ^ - s)x"is) ds,
t + fx — s > 0, and E.5) imply that
fx ||*'@ll ^ 2R + 1 (? + ^ - s)(a/-"(s) + ^) rfs.
This inequality and the analogue of E.6) in which * is replaced by r give
hence
\\x\t)\\ ^ 2R + x[rit + /*)- rit) - /xr\t)] +
E.7) /* ||z'(f)ll ^ 2R(l + a) +
- zfir'O) for 0 ^ ? ^ ^ -
430 Ordinary Differential Equations
Similarly, for fx ^ t ^ p, the relation
x(t) - x(t - [x)~ fxx'(t) = - (t-t*- sKXs) ds
Jt-11
implies that
E.8) /x Ik'(Oil ^ 2RA + a) + \Kn* + afir'(t) for n^t^
The function M^ip) denned by
E.9)
/X
Mjjx) =
for
for g ^
is a continuous, nonincreasing function of fx > 0 (as can be seen by
calculating its derivative dMJdfi). If \p ^ 2[.RA + oc)/tf]w, the choice
[x = \p in E.7) gives
E.10) ||x'@ll ^ M-Skp) - a.r'(t) for 0 ^ ? ^ \p.
If |/7 ^ 2[.RA + a)/Kp, the choice ^ = 2[R(l + a)/tf]w gives E.10).
Similarly, E.8) implies that
E.11) ||x'@ll ^ M^I/7) + o/-'(f) for hp^t^p.
Adding E.10), E.11) for f = /»/2 shows that
E.12) ll*'(i/OII ^ MiQ/7).
The assumption E.4) and E.1O)-E.11) imply that
I "^ ' X I <r- II ~' II .^ »
E.13)
<P(\\z'\\)
where ± is required according as t > \p or ? ^ |/7. Let 0E) be denned by
... fs u du
E.14) O(s)= —-.
Jo w(u)
Then, by Lemma 14.1,
E.15)
[x' ¦ x" -*<-
J <p(\\x'\\
where the integral is taken over the ^-interval with endpoints t and p/2.
In view of E.13), the integral is majorized by
00 \r(t) - r{\p)\ ^
Hence
Use of Implicit Function and Fixed Point Theorems 431
In view of E.12) and the fact that O is an increasing function, ||x'@ll ^
M(p), where
M(p) = O
and O^1 is the function inverse to O. If p ^ t, then t e [0, p] is contained
in an interval of length t in [0, p\. Thus the considerations just completed
show that/7 can be replaced by t, and the lemma is proved with M{r) as
an admissible choice of M.
Exercise 5.1. Show that an analogue of Lemma 5.2 remains valid if
E.5) is replaced by
\\x\\ ^ R, \\x"\\ ^ p",
where p(t) is real-valued function of class C2 on 0 ^ t ^ p such that
\p(t)\ ^ Kv In this case, M depends only on cp(s), <x, R, r, and K±.
The choice <p(s) = ys2 + C in Lemma 5.1 gives the following:
Corollary 5.1. Let y, C, a, K, R, r be non-negative constants. Then
there exists a constant M [depending only on y, C, a, R, t, K] such that if
x(t) is of class C2 on 0 ^ t ^ p, where p ^ t, satisfying E.5) and
E.16) \\x\\^R, \\x"
then \\x'\\ ^ M for 0 ^ t ^p.
Remark 1. If y in E.16) satisfies
^ y \\x'\\2 + C,
< 1, then E.5) holds with
E.17)
2A - yR)
1 - yR
Thus assumption E.5) is redundant in Corollary 5.1 when yR < 1 (but
the example preceding Lemma 5.2 shows that E.5) cannot be omitted if
yR = 1). Also if a in E.5) satisfies 2xR < 1, then E.16) holds with
E.18)
2a
-2olR
and C =
K
so that E.16) is redundant in this case. Even if d = 1 (so that x(t) is
real-valued), condition E.16) cannot be omitted if 2olR > 1).
In order to verify the first part of Remark 1, note that
E.19) r" = 2(x-x" + \\x'\\2).
Hence E.16) shows that r" ^ 2[A - yR) \\x'\\2 - CR]. Another applica-
application of E.16) gives yr" ^ 2[A - y^)(||a;"|| - C) - CRy] = 2[A -
yR) \\x"\\ - C]. This is the same as E.5) with the choices E.17). The
proof of the remark concerning E.18) is similar.
Exercise 5.2. Show that if 2<zR > 1, then assumption E.5) cannot be
dropped in Corollary 5.1.
432 Ordinary Differential Equations
The following simple fact will be needed subsequently.
Lemma 5.3. Let fit, x, x') be a continuous function on a set
E.20) Eip, R) = {it, x,x): O^t <^p, \\x\\ < R, x' arbitrary},
and let f have one or more of the following properties:
E.21) x-f+ \\x'\\2 > 0 when x-x' = 0 and ||x|| > 0,
E.22) x •/+ \\x'\\2 > 0 when x ¦ x = 0 and ||x|| = R,
E.23)
E.24) imi <
Let M > 0. Then there exists a continuous bounded function g(t, x, x')
defined for O^t <,pand arbitrary (x, x') satisfying
E.25) g(t, x, x') =f(t, x, x') for 0 ^ t ^ p, \\x\\ ^ R, \\x'\\ ^ M
and having the corresponding set of properties among the following:
E.21') x-g+ \\x'\\2>0 when x-x' = 0 and ||z|| ^ 0,
E.22') x ¦ g + \\x'\\2 > 0 when x ¦ x = 0 and ||z|| ^ R,
E.23') llgll ^ cp(\\x'\\),
E.24') \\g\\^2a(x-g+\\x'V) + K.
Proof. We can obtain such a function g as follows: Let <5(s), where
0 ^ s < oo, be a real-valued continuous function satisfying E=1,
0<<5<l,E = 0 according as <5 =" M, M < s =" 2M, s > 2M. Put
g(f,*,aO=a(||*'||)/(f,*,aO on ?(P'^)'
g(?, x, x') = — git, , x'\ for ||x|| > R.
On Eip, R), the identity
x.g+ ||a/||. = 3(||a/||)(a;./+ llx'll2)-^ [1 - -5(lk'||)] Ik'H2
makes it clear that g has the desired properties on Eip, R). Furthermore
the validity of any of the relations E.21')-E.24') for ||x|| = R implies its
validity for ||x|| > R. This proves the lemma.
Note that inequalities of the type E.23), E.24) imply that solutions of
E.26) x"=fit,x,x')
satisfy E.4), E.5), respectively; cf. E.19).
Use of Implicit Function and Fixed Point Theorems 433
Theorem 5.1. Let f(t, x, x') be a continuous function on the set E(p, R)
in E.20) satisfying
E.27)
if x-x' =
and \\x\\ = R,
x-f+ Hz'P^
E.24) and E.23), where <p(s), 0 _ s < oo, is a positive continuous function
satisfying E.1). Let \\xo\\, \\xj ^ R. Then E.26) has at least one solution
satisfying x@) = x0, x(p) = xv.
It will be clear from the proof that assumption E.23) can be omitted
if 2<xR < 1. Furthermore, if/satisfies
E-28) ||/1| ^ y \\x'\\2 + C,
where y, C are non-negative constants and yR < 1, then both assumptions
E.23) and E.24) can be omitted.
If the vector x is 1-dimensional, Lemma 5.1 can be used in the proof
instead of Lemma 5.2. This gives the following:
Corollary 5.2. Let x be a real variable andf(t, x, x') be a real-valued
function in Theorem 5.1. Then the conclusion of Theorem 5.1 remains
valid if condition E.24) is omitted.
Note that, in this case, condition E.27) becomes simply f(t, +R, 0) ^ 0
and f(t, -R, 0) ^ 0 for 0 ^ t ^ p.
Proof of Theorem 5.1. The proof will be given first for the case that/
satisfies E.22) instead of E.27). Let M > 0 be a constant (with p = t)
supplied by Lemma 5.2. Let git, x, x) be a continuous bounded function
for 0 ^ t ^p and arbitrary (a;, x') satisfying E.25), E.22'), E.23'), and
E.24'). By Theorem 4.2, the boundary value problem
x" = git, x, x'), x@) = x0, and x(jj) = xv
has a solution xit). Condition E.22') means that r = ||a;(?)||2 satisfies
r" > 0 if r' = 0 and r = R2; cf. E.19). Hence /¦(?) does not have a
maximum at any point t, 0 < t < p, where /¦(?) ^ R2. Since /-@) =
Ikoll2, rip) = \\xj2 satisfy /-@), rip) = R2, it follows that /¦(*) ^ R2 (i.e.,
\\xit)\\ ^ R) for 0 ^ t^p. By virtue of x" =g and E.23'), E.24'),
Lemma 5.2 is applicable to x(?) and implies that \\x\t) || ^ Af forO ^ t ^p.
Consequently, E.25) shows that xit) is a solution of E.26). This proves
Theorem 5.1 provided that E.27) is strengthened to E.22). In order to
remove this proviso, note that if e > 0, the function fit, x, x') + ex
satisfies the conditions of Theorem 5.1 as well as E.22) if cp, K in E.23),
E.24) are replaced by cp + eR, K + eR, respectively. Hence
x"=fit,x,x') + ex
has a solution x = xfit) satisfying the boundary conditions. It is clear that
lk€@ll ^ R and that there exists a constant M (independent of e, 0 < e ^
1) such that |k€'(/)|| < M. Consequently, if N = max ||/(f, x, x')\\ + 1
434 Ordinary Differential Equations
for 0 ^ t ^p, \\x\\ ^ R, \\x'\\ ^ M, then \\x€\t)\\ ^ N. Thus the family
of functions x€(t), x('(t) for 0 ^ t ^ p are uniformly bounded and equi-
continuous. By Arzela's theorem, there is a sequence 1 > ex > e2 > • • •
such that en ->- 0 as n ->- oo, and x(t) = lim xe@ exists as e = en ->¦ 0 and
is a solution of E.26) satisfying x@) = x0, x(p) = a^. This completes the
proof of Theorem 5.1.
Exercise 5.3. Show that if E.27) in Theorem 5.1 is strengthened to
E.29) x •/+ \\x'\\2 ^ 0 when x • x' = 0,
then E.26) has a solution x(t) satisfying x@) = x0, x(p) = 0, and
E.30) r^O, /"' <0 if /"= ||a;||2.
Exercise 5.4. Let « be a real variable. Let A(?, m, m') be real-valued
and continuous for 0 ^ t ^ /> and all (m, m'), and satisfy the following
conditions: (i) h is a nondecreasing function of u for fixed (?, «'); 00 \h\ ^
<P(|m'|) where (p(s) is a positive, continuous, nondecreasing function for
5^0 satisfying E.1); (iii) u" = h(t,u,u) has at least one solution
uo(t) which exists on 0 ^ t ^ p [e.g., (ii) and (iii) hold if \h\ ¦?. a. \u'\ + K
for constants a, K ]. Let u0, uv be arbitrary numbers. Then u" — h (t, u, u')
has at least one solution u(t) satisfying m@) = u0, u(p) = uv. [For a related
uniqueness assertion, see Exercise 4.5(c).]
Theorem 5.2. Letf(t, x, x') be continuous in
E.31) E{R) = {(t, x,x'): 0 < t < oo, Hxll < R, x' arbitrary}.
For every p > 0, let/satisfy the conditions of Theorem 5.1 on E(p, R) in
E.20), where <p(s) and the constants ol, K in E.23), E.24) can depend on p.
Let \\xo\\ ^ R. Then E.26) has a solution x(t) which satisfies x@) = x0
and exists for t ^ 0.
Exercise 5.5. (a) Prove Theorem 5.2. (b) Show that if, in addition,
E.27) is strengthened to E.29) in Theorem 5.2, then the solution x(t) can
be chosen so that E.30) holds, (c) Furthermore, if E.29) is strengthened to
x-f+ IkT ^ 0, then r ^ 0, r' ^ 0, r" ^ 0 for t > 0. (d) If x is 1-
dimensional, show that condition E.24) can be omitted from Theorem 5.2
and parts (b) and (c) of this exercise.
Exercise 5.6. Let/(f, x, x') be continuous on the set E(R) in E.31).
For every m, 0 < m < R, let there exist a continuous function h(t) =
h(t, m) for large t such that fA(f) dt = oo and a; •/(?, a;, a;') ^ A@ ^ 0
for large ?, 0 < m ^ ||a;|| ^ J?, a;' arbitrary. Let x(t) be a solution of
E.26) for large t. Then x(t) ->¦ 0 as ? ->¦ oo.
Exercise 5.7. Let/(?, a;, a;') be continuous on ?(.R) in E.31) and have
continuous partial derivatives with respect to the components of x,x';
let the Jacobian matrices D.23) satisfy \{B + B*) - \FF* > 0; cf. C.17).
Use of Implicit Function and Fixed Point Theorems 435
Let ||a;0|| ^ R. Then E.26) has at most one solution satisfying a;@) = x0
and ||a#)ll ^ R for t ^ 0.
Remark 2. The main role of the assumptions involving E.23) and/or
E.24) in Theorems 5.1, 5.2 is to assure that the following holds:
Assumption (Av). There exists a constant M = M(p) with the property
that if x(t) is a solution of x" = f(t, x, x') for 0 ^ t ^ p satisfying
\\x(t)\\ ^ R, then \\x'(t)\\ ^ M for 0 ^ t <p.
Exercise 5.8. Show that if E.27) is replaced by E.22) and assumptions
involving E.23) and/or E.24) are replaced by the assumption (A,,), then
Theorem 5.1, Corollary 5.2, and Theorem 5.2 remain valid. (In the case
of Theorem 5.2, it is, of course, assumed that (Av) holds for all large
p>0.)
Exercise 5.9. Let f(t, x, x') be continuous on E(R) in E.31) and
satisfy assumption (Av) for all p ^ p0 > 0. Suppose that, for each x0 in
Ikoll = R, E.26) has exactly one solution x(t) — x(t, x0) satisfying a;@) =
x0 and existing for t > 0 (cf., e.g., Theorem 5.2 and Exercise 5.7.) (a) Show
that x(t, x0) is a continuous function of (t, x0) for t ^ 0, ||a;0|| ^ R.
(b) Suppose, in addition, that f(t, x, x') is periodic of period p0 in t for
fixed (x, x'). Then E.26) has at least one solution x(t) of period p0.
PART III. GENERAL THEORY
6. Basic Facts
The main objects of study in this part of the chapter will be a linear
inhomogeneous system of differential equations
F-1) y' = A(t)y+g(t),
the corresponding homogeneous system
F-2) y' = A (t)y,
and a related nonlinear system
F-3) y' = A(tyy+f(t,y).
Let / denote a fixed ^-interval /: 0 ^ t < a> (< oo). The symbols
x> y>fg> ¦ ¦ ¦ denote elements of a rf-dimensional Banach space Y over
the real or complex number field with norms ||a;||, ||y||, ||/||, \\g\\,
(Here ||a;|| is not necessarily the Euclidean norm.) In F.1), g = g(t) is a
locally integrable function on / (i.e., integrable on every closed, bounded
subinterval of /). A (t) is an endomorphism of Y for (almost all) fixed t
and is locally integrable on /. Thus if a fixed coordinate system is chosen
on V, A (t) is a locally integrable d x d matrix function on /.
436 Ordinary Differential Equations
When y(t) is a solution of F.1) on the interval [0, a] <= J, the fundamen-
fundamental inequality
F.4)
for O^t,t'^a
follows from Lemma IV 4.1. If this relation is integrated with respect to
t' over [0, a], we obtain
F.5) ||2/@II ^ (- f II2/0)II ds + (a\\g(s)\\ ds) exp f M(s)|| ds
\aJo Jo ) Jo
for 0 ^ t ^ a.
Let L = Lj denote the space of real-valued functions <p@ on / with
the topology of convergence in the mean L1 on compact intervals of /.
Thus L is a Frechet (= complete, linear metric) space. For example, the
following metric, which will not be used below, can be introduced on
L: let 0 = t0 < tx < t2 < . . . , tn -»¦ co as n -»¦ co, and let the distance
between <p, f e L be
where
Correspondingly, let C = Cj denote the space of continuous, real-
valued functions <p@ on J with the topology of uniform convergence on
compact interval of /. Thus C is also a Frechet space. A metric on C,
e.g., is
» iM > Where m(n) = max \<p(t) - V<OI •
The symbols V = L/, 1 ^ p ^ co, denote the usual Banach spaces of
real-valued functions (fit) on /: 0 ^ t < co (^ co) with the norm
if 1 ^ p < co,
1^1^ = ess sup |99(OI if p = co.
j
Lo00 is the subspace of Lw consisting of functions 99@ satisfying 99@ -»¦ 0
as t _>. w. For other Banach spaces B of real-valued, measurable functions
99@ in /, the notation \q>\B will be used for the norm of 99@ in B.
Remark. Strictly speaking, the spaces L, L00, Lox,... are not spaces
of "real-valued functions" but rather spaces of "equivalence classes of
real-valued functions," where two functions are in the same equivalence
class if they are equal except on a set of Lebesgue measure zero. Since no
confusion will arise, however, over this "abuse of language," the
abbreviated terminology will be used. In this terminology, the meaning
of a "continuous function in L" or the "intersection L C\ C" is clear.
Use of Implicit Function and Fixed Point Theorems 437
L(Y), LV(Y), B(Y), . . . will represent the space of measurable vector-
valued functions y(t) on /: 0 ^ t < co (^ 00) with values in Y such that
9>@ = \\y(t)\\ is in L, V,B, With V or B, the norm \(p\v or \<p\B
will be abbreviated to \y\v or \y\B.
A Banach space D will be said to be stronger than L(Y) when (i) D is
contained in L( Y) algebraically and (ii) for every a, 0 < a < co, there is a
number a = aj,(a) such that y(t) e D implies
F.6)
f
Jo
II 2/@II <*f ^ a |2/|j,, where a = 04
[It is easily seen from the Open Mapping Theorem 0.3 that condition (ii) is
equivalent to: "convergence in D implies convergence in L(F)."]
If D is a Banach space stronger than L( 7), a D-solution 2/@ of F.1) or
F.2) means a solution 2/@ e D. Let Fj, denote the set of initial points
2/@) g 7 of D-solutions 2/@ of F.2). Then Fj, is a subspace of Y. Let 1^
be a subspace of 7complementary to Yj,; i.e., 1^ is a subspace of Y such
that Y = Yj, © 7X is the direct sum of Yz and Fl5 so that every element
y g Y has a unique representation y = y0 + yx with 2/0 G ^d> 2/i g ^1
(e.g., if Y is a Euclidean space, Yx can be, but need not be, the subspace
of Y orthogonal to Yj,). Let Po be the projection of Y onto Yj,
annihilating Yx; thus if 2/ = y0 + 2/1 withy0 e Yj,, 2/1 e ^i, then P,^ = 2/0-
Lemma 6.1. Let A(t) be locally integrable on J and let X) be a Banach
space stronger than L(Y). Then there exist constants Co, Q such that if
2/@ is a ^-solution of F.2), then
F-7) |2/b ^ Co ||2/@)|| and \\y@)\\ ^ C, \y\z.
Proof. Yj, is a subspace of the finite dimensional space Y. In addition,
there is a one-to-one, linear correspondence between solutions 2/@ of
F.2) and their initial points 2/@). Thus the set of D-solutions of F.2) is a
finite dimensional subspace of D which is in one-to-one, linear corre-
correspondence with Yj,. It is a well known and easily verified fact that if two
finite-dimensional, normed linear spaces can be put into one-to-one
correspondence, then the norm of an element of one space is majorized by
a constant times the norm of the corresponding element of the other
space. [For example, an admissible choice of Cx is a
[
Jo
\\A(s)\\ds
for any a, 0 < a < co. This follows from F.6) and the choices t = 0,
/>(*) = 0 in F.5).]
Let 93, D be Banach spaces stronger than L(X). Define an operator
T = Tvj> from X) to 23 as follows: The domain 2{T) <= X) of Tis the set
of functions y(t), t ej, which are absolutely continuous (on compact
438 Ordinary Differential Equations
subintervals of/), y{t) g T>, and y'(t) — A(t) y(t) g 23. For such a function
y{t), Ty is defined to be y'(t) — A(t) y(t). In other words, Ty = g, where
g(t) g 23 is given by F.1).
Lemma 6.2. Let A(t) be locally integrable on J and let 23, D be Banach
spaces stronger than L{Y). Then T = TSI) is a closed operator; that is the
graph of T, <g(T)={(y(t),g(t)):y(t)eS>(T),g = Ty}, is a closed set of
the Banach space D x 23.
Proof. In order to prove this, it must be shown that if yx{i), y2(t), ¦ ¦.
are elements of 9(T), gn = Tyn, y(t) = lim yn(t) exists in $ and g(t) =
lim gn(t) exists in 23, then y(t) e 9)(T) and g(t) = Ty.
The basic inequality F.5) combined with F.6) and the analogue of
F.6) for the space 23 give
A
^ - aD
[a
\yn -
\g
1 f
n - sJ* exp
I Jo
a
\\A(s)\\ ds.
/.
[a
Hence y(t) is the uniform limit of yi(t), y2(t),... on any interval [0, a] c
The differential equation F.1) is equivalent to the integral equation
2/@ = 2/0*) + \A{s)y(s) ds + \g(s) ds.
J a J a
Since the convergence of gx, g2, ... in 23 implies its convergence in L(Y), it
follows that F.1) holds where y = lim yjt) in $), g = lim gn(t) in 23.
Finally, y g D, g g 23 show that y g 2{T). This proves Lemma 6.2.
The pair of Banach spaces B3, D) is said to be admissible for F.1) or for
A(t) if each is stronger than L( Y), and, for every g(t) e 23, the differential
equation F.1) has aD-solution. In other words, the map T = T^: @(T)->
23 is onto, i.e., the range of T is 23. (For example, if J:0 < t < oo,
A(t) is continuous of period p, and 23 = 1) is the Banach space of con-
continuous functions y(t) of period p with norm \y\% = sup ||?/@ll, then
B3, $)) is admissible for F.1) if and only if F.2) has no nontrivial solution
of period p; see Theorem 1.1.)
Lemma 6.3. Let A(t) be locally integrable on J, let B3, D) be admissible
for F.1), and let y0 e Y^. Then, ifg(t) g23, F.1) has a unique l)-solution
y(t) such that Poy(fy = 2/o- Furthermore, there exist positive constants
Co and K, independent of g{t), satisfying
F.8) \yh^C0\\y0\\+K\g\v.
Proof. Consider first the case that y0 = 0, so that we seek D-solutions
y(t) with ?/@) g Yx. For any ge23, F.1) has a solution y(t) g D, by
assumption. Let y@) = y0 + ylt where y0 = P02/@) e ^s. ?/i e ^i- Let
2/0(() be the solution of the homogeneous equation F.2) such that yo(O) =
y0, so that yo(t) g D. Then y^t) = y(t) — yo(t) g D is a solution of F.1)
and
Use of Implicit Function and Fixed Point Theorems 439
It is clear that y^t) is a unique D-solution of F.2) with initial point
in Yv Thus there is a one-to-one linear correspondence between g g 23
and D-solutions y^t) of F.2) with y^O) g Yv The proof of Lemma 6.2
shows that if 7\ is the restriction of T = T^ with domain consisting of
elements y(t) e @(T) satisfying y@) g Yx, then 7\ is closed. Thus 7\ is
a closed, linear, one-to-one operator which maps its domain in D onto 23.
By the Open Mapping Theorem 0.3, there is a constant K such that if
T^y = g, then |j/|j, ^ AT |g|<g. This proves the theorem for y0 = 0.
If y0 ,* 0, let ^(f) be the unique D-solution of F.2) satisfying y^O) e 7^
Let j/0(() be the unique J)-solution of the homogeneous equation F.2)
satisfying yo(O) = y0. Then y(t) = yo(t) + y^t) is a D-solution of F.1),
A»/@) = yo, and |y|s ^ |yo(Ob + l^iCOIs- By the part of the lemma
already proved, MOIs ^ ^Iglse and> by Lemma 6.1, |yo(Ols ^ Q ||yo||.
This completes the proof of Lemma 6.3.
7. Green's Functions
Let hOa(t) be the characteristic function of the interval 0 ^ ( ^ a, so
that /jOa(O = 1 or 0 according as 0 ^ t ^ a does or does not hold.
Similarly, let ha(t) be the characteristic function of the half-line ( ^ a, so
that /ja(() = 1 or 0 according as t ^ a or f < a.
A Banach space 23 of functions on J: 0 ¦< t < co (^ co) will be called
lean at t = co if y(t) g 23 and 0 < a < co imply that ^@^@. K(t)vit) G
23; IV^Is> l^aVlsB = I Vis! and l^aVls — ° as a-> co. Since Aa(()f(t) =
y>(t) — hOa(t)y>(t) on /, the property "lean at ( = co" implies that the
set of functions hOa(t)ii>(t) of 23 vanishing outside of compact intervals
[0, a] c /is dense in 23.
Let D be a Banach space stronger than L( Y). As above, let Y^ = Yltl be
a subspace of Y complementary to Y^. Let Po = Pod be the projection
of Y onto Fj, annihilating Ylf and Px = I — Po the projection of Y onto
Yl annihilating Yj,. In terms of a fixed basis on Y, Po and Pi are
representable as matrices.
Let U(t) be the fundamental matrix for F.1) on 0 ^ ( < co satisfying
/ /@) = /. For 0 ^ s, t < co, define a (matrix) function G(t, s) by
G.1)
G(t, s) =
C((, 5) = -
for 0 < s ^ (,
for 0 ^ ( < s.
lor a fixed t, G(t, s) is continuous on 0 < s < co, except at s = t, where
it has left and right limits, U{t)P0U~\t) and -U(j)PJJ-\i).
Theorem 7.1. Let A(t) be locally integrable on J. Suppose that 23, 1)
<//•(• Banach spaces stronger than L(Y); that 23 is lean at co; and that
440 Ordinary Differential Equations
X has the property that ify(t), 2/i@ are continuous functions from J to Y
and y(t) — 2/i@ = 0 near t = a> (i.e., 2/1 — y2 = 0 except on an interval
[0, a] c /), then y(t) e X) implies that y-Jt) e X. Then E8, X) is admissible
for F.1) if and only if, for every g(t) e 58,
G.2)
2,@ = f aG(t, s)g(s) ds = lim [ G(t, s)g(s) ds
JO a~*a> JO
exists in X>. In this case, the limit is uniform on compact intervals of J and
is the unique ^-solution of F.1) with y@) e Y±.
Proof. "Only if". Let g(t) e 58, ga(t) = hOa(t)g(t). Then G.2) becomes
G.3)
ya(i) = \aG(t,
Jo
s)ga(s) ds = pG(f, s)g(s) ds,
J
where the integral exists as a Lebesgue integral for every fixed t, since
G(t, s) is bounded for 0 < s ^ a and ga(s) is integrable over /. In view of
the first part of G.1), the contribution of 0 ^ s ^ t to G.3) is
(s) ds.
u(t)p0 Cu-^gM ds = u(t) f'u-XsigjLs) ds - u^p, fV-1(s)ga
Jo Jo Jo
Hence, by the second part of G.1), G.3) is
G.4) ya(t) = l/(») ('iTXsteM ds + U(t)ya@),
Jo
ya@)=-P1\ U~\s)ga(s)ds.
Jo
It follows from G.4) and Corollary IV 2.1 that ya(t) is a solution of F.1)
when g(t) is replaced by ga{t).
An analogue of the derivation of G.4) gives
where
G.5)
ya(t) = -U(t)
ds + U(t)P
Hence
0 ("irXs)g
Jo
U-Xa)ya(a) = Po \'irXs)ga(s) ds e Yx.
Jo
ds.
Thus for a ^ t < co, ya(t) is identical with the solution U(t) U~1(a)ya(d)
of the homogeneous equation F.2). Since the initial point of the latter
solution is in Y%, the property assumed for X) implies that ya(t) e X).
Since ya@) e Yx by G.5), it follows that ya@ is the unique solution of
F.1), where g = ga(t), satisfying ya@) g Y±. Hence, by Lemma 6.3,
Use of Implicit Function and Fixed Point Theorems 441
Let 0 < a < b < w. Then, since 58 is lean at t = a>,
\ya-y<>h^K\ga-gb\$^2K\hag\v^0 as a^co
Thus y = lim ya(t) exists in X as a -*¦ a>. Also g = lim ga(t) in 58. Since
T = TSI) in Lemma 6.2 is closed, y(t) is a X-solution of F.1). The proof
of Lemma 6.3 shows that?/ = \imya(t) uniformly on compact intervals
of /. Hence, 2/@) = \imya@) e YX. This proves "only if" in Theorem
7.1. The "if" part is easy.
Corollary 7.1. Let a> = oo; B and D be Banach spaces of class $~#;
B' be the space associate to B; cf. § XIII 9. For the admissibility of(B(Y),
D(Y)),(i) it is necessary that \\G(t, -)|| e B' for fixed t—thus the integrals
in G.2) are Lebesgue integrals; (ii) when B is lean at oo, it is necessary and
sufficient that G.2) define a bounded operator g -»- y from B(Y) to D(Y);
(iii) it is sufficient that r(i) eD where r(t) = | \\G(t, -)|| U-; (iv) when
D = L°°, it is necessary and sufficient that r(t) eL00'.
Exercise 7.1. Verify this corollary.
8. Nonlinear Equations
Lemmas 6.1-6.3 will be used to study the nonlinear equation
(8.1) y' = A(t)y+f(t,y).
Let 58, X be Banach spaces stronger than L( Y) and 2p the closed ball
S p] in 3).
Theorem 8.1. Let J; 0 ^ t < w (^oo); A(t) a locally, integrable
d X d matrix function on J, and E8, X) admissible for F.1). Let f(t, y(t))
be an element of 58 for every y(t) e 2P and satisfy
(8.2)
\f(t,y,(t))-f(u
for all y1(t),y2(t)e'Ll>and some constant B; r = \f(t, 0)|s; ?/„ e Yj,.
Suppose that if Co, K are the constants in Lemma 6.3, then B, r, ||2/0|| are
so small that
(8.3) C0II2/0II + Kr ^ />A - 0tf) and 0^ < 1.
77;en (8.1) has a unique solution y(t) eEp satisfying
(8.4) Po^O) = 2/o-
It will be clear from the proof that the first part of (8.3) can be replaced
by the assumption
<K-5) Co ||.yo|| + K I J{t, y(t)\* ^ P for all y(t) e Sp.
442 Ordinary Differential Equations
In fact, the role of the assumption in (8.3) is to assure (8.5). In (8.4), Po
is the projection of Y onto Y% annihilating a fixed subspace Ylt where
Y=YX® IV
Proof. Theorem 8.1 is an immediate consequence of Theorem 0.1 and
Lemma 6.3. Since f(t, x(t)) e 23 for any x(t) e Zp, Lemma 6.3 and the
assumption that B3, D) is admissible imply that
(8.6) y' = A{t)y+f(t,x(t))
has a unique J)-solution^(f)satisfying(8.4) andF.8), whereg(t) = f(t, x(t)).
Define the operator To from 2p into D by y(t) = T0[x(t)]. In particular,
ifm = |ro[0]|jj,then
(8.7)
m ^ Co \\yo\\ + Kr,
where r = |/(f,O)|e.
J, y2 = T0[x2], it follows that 2/i@ —
If x-.it), x2(t)e~Lp and ^ = To
y2(t) is the unique D-solution of
y' = A(t)y + f(t, Xl(t)) -f(t, x2(t))
satisfying P<&@) = 0. Hence, by Lemma 6.3 and by (8.2),
(8.8) I ft-yil s^fl* l*i-*.!»•
Consequently, Theorem 0.1 is applicable, and so To has a unique fixed
point 2/@ g Zp. This proves Theorem 8.1.
The statement of the next theorem involves the space C( Y) of continuous
functions y(t) from J to Y with the topology of uniform convergence on
compact intervals in J. The theorem will also involve an assumption
concerning the continuity of the map Tx\y{t)] =f(t, 2/@) from the closure
of the subset Sp O C(Y) of C(Y) into 23. This condition is rather natural
in dealing with Banach spaces 23, 35 of continuous functions on J with
norms which imply uniform convergence on /. This is the case in Parts
I and II, where J is replaced by a closed bounded interval 0 ^ t ^ p.
This continuity condition will also be satisfied under different circumstances
in Corollary 8.1.
Theorem 8.2. Let A{t) be locally integrable on J; 23, D Banach spaces
stronger that L{Y); 2p the closed ball of radius p in D; and S the closure
o/2p O C(Y) in C(Y). Let A(t) andf(t, y) satisfy (i) B3, D) is admissible
for F.1); (ii) y(t)->f(t, y{t)) is a continuous map of the subset S of the
space C( 7) into 23; (iii) there exists an r > 0 such that
(8.9) \f(t,y(t))\%^r for y(t)eS;
and (iv) there exists a function X(t) e L such that
(8.10) \\f(t,y(t))\\^KO for teJ, y(t)eS.
Use of Implicit Function and Fixed Point Theorems 443
Let Co, K be the constants of Lemma 6.3 and let y0 e Yj,. Let r, \\yo\\ be so
small that
(8.11) CJyJ + Kr ^ p.
Then (8.1) has at least one solution y(t) e 2p satisfying Poy@) = y0.
Proof. As in the last proof, define an operator To ofS into J) by putting
y = T0[x], where x(t) e S and y(t) is the unique D-solution of (8.6)
satisfying (8.4). Thus, by Lemma 6.3,
Ms ^ Co \\yo\\ +K\f(t, x(t))\B ^ Co \\yo\\ + Kr.
Hence assumption (8.11) implies that To maps S into itself, in fact, into
2p O C(Y) c S.
Note that the basic inequality F.5) implies that
WOII ^ (- \"\\y(s)\\ ds + (a\\g(s)\\ ds) exppME)|| ds
\a Jo Jo i Jo
for 0 ^ t ^ a if g(t) =f(t, x(t)). Since D is stronger than L(Y), F.6)
holds. Also there is a similar inequality for elements g e 23 with a suitable
constant as (a). Hence, for 0 ^ ? ^ a,
(8.12) WOII < (-(^(fl) 12/lj, + ae(a) |g|J expflU^II ds.
\a I Jo
It will first be verified that To: S —>- S is continuous where .S is considered
to be a subset of C(Y). Let a;/f) e S, g}{t) =f(t, x^t)), y}(t) = T^xfc)]
for j = 1, 2, then 2/i@ — 2/2@ is the unique i-solution of F.1), where
g = gi — g2, satisfying Pobi(°) — ^a@)] = 0. Hence Lemma 6.3 implies
that
Ift-Skis ^-^l^i "gals-
Also, (8.12) holds if y = yx — y2 and g = gx — g2. Thus, for 0 ^ t ^ a
\gl -
exp f°
Jo
ds.
Since, by assumption (ii), xx(t)-*¦ x2(t) in C(Y) implies gi~>-g2 in 23. it
follows that 2/i@ -*¦ 2/2@ uniformly on intervals [0, a] of J; i.e., yx(t) -»¦
!/2(t) in C(Y). This proves the continuity of To: S->-S.
It will now be shown that the image T0S of S has a compact closure in
C(Y). It follows from (8.12), where g(t) =f(t,x(t)) and 2/@ = ^oKOl
that, forO <t ^ a,
IIKOII ^ (-*?(«)/> + ae(fl)'1] exp VWII d*.
\a ) Jo
Thus the set of functions 2/@ e T0S are uniformly bounded on every
interval [0, a] of J. If c(a) is the number on the right of the last inequality,
| du
for 0 ^ s ^ t ^ a.
444 Ordinary Differential Equations
then (8.6) and (8.10) show that
\\y(t) - y(s)\\ ^c(a)(t\\
J s
Therefore, the functions y(t) in the image T0S of S are equicontinuous on
every interval [0, a] <= /. Consequently, Arzela's theorem shows that
T0S has a compact closure in C(Y). Since S is convex and closed in
C(Y), it follows from Corollary 0.1 that To has a fixed point y(t) g S.
Thus Theorem 8.2 is a consequence of the fact that y(t) = T0[y(t)] e
It is convenient to have conditions on SB, 5), fit, y), 2@ which imply
(ii), (iii), (iv) in Theorem 8.2.
Assumption (Ho) on SB = B(X): Let SB = B(X) (cf § 6), where X is a
subspace of Y and B is a Banach space of real-valued functions on J such
that (/') B is stronger than L; (ii) B is lean at t = a> (cf. § 7); (///) B contains
the characteristic function hOa(t) of the intervals [0, a] <= /; and (iv) if
(p^t) g B and <p2(t) is a measurable function on J such that \<p2(t)\ ^ l99i(OI>
then <p2(t) e B and \(p2\<g, < {tp^.
It is important to have SB = B(X) rather than SB = B( Y) for applications
to higher order equations. If such equations are written as systems of
differential equations of the first order, the "inhomogeneous term/(f, y)"
will generally belong to a subspace A'of Y; e.g.,f(t, y) might be of the
form (h, 0, . . ., 0).
Examples of spaces B satisfying the conditions in (Ho) are B = Lp,
1 ^ p < oo, and B = Lo"" (but not B = Z,00). Other such spaces B can
be obtained as follows: Let y(t) > 0 be a measurable function such that
xp(t) and l/f(t) are bounded on every interval 0 5f t ^ a «oS). Denote
by B = ?™0 the space of functions q>(t) on / such that (p(t)jxp(t) e L^
with the norm \<p\% = \q>ly>\oa- The space B = ?™0 satisfies conditions
(i)-(iv). For this space, 2(f) g B holds if
(8.13)
0 ^ 2@ ^ rp(t)
and ^
xp(t)
as t
Assumption (Hj) on f(t, y): Letf(t, y) be continuous on the product set
of J and the ball \y\ ^ p in Y, let f have values in X, and let there exist a
function l(t) e L such that
(8.14) \\f(t,y)\\ ^2(?) for teJ, \\y\\ ? p.
Corollary 8.1. Let A(t) be locally integrable on J, (SB, D) admissible for
F.1), SB satisfies (Ho), D = LCO(Y) [or D = L0CO(Y)], f(t, y) satisfies (Hx)
and X(t) g B with r = \X\B. Let y0 e Y^. Then, if (8.11) holds, (8.1) has
Use of Implicit Function and Fixed Point Theorems 445
at least one solution y(t) on 0 ^ t < a> satisfying P$@) = y0, \\y(t)\\ ^ p
[and y(t) —»- 0 as t —»- co].
Exercise 8.1. Verify Corollary 8.1.
Exercise 8.2. Let Y be expressed as a direct sum Fj, © Fx; let Po
be the projection of Y onto Yj, annihilating Y1; and P^ = I — Po the
projection of 7 onto Y1 annihilating Fj,. Let ^4(?) be locally integrable
on J: 0 ^ ? < oo. Define G(t, s) by G.1) and suppose that there exist
constants N, v > 0 such that \\G(t, s)\\ ^ Ne~^l~A for 5, ? ^ 0. Let
f(t, y) be continuous for 0 ^ ? < oo, ||^|| ^ p, and let ||/(?, y)|| ^ /-. Let
2/0 g 7j,. Show that if ||yo|| and r > 0 are sufficiently small, then (8.1) has
a solution ?/(?) for 0 ^ ? < oo satisfying \\y(t)\\ ^ p and P$@) = ?/0.
(For necessary and sufficient conditions assuring these assumptions on G,
see Theorems XIII 2.1 and XIII 6.4.)
9. Asymptotic Integration
In this section, let/be the half-line/: 0 ^ t < oo (so that w = oo). As
a corollary of Theorem 8.2, we have:
Theorem 9.1. Lef A(t) be continuous on J: 0 ^ ? < oo. Let f(t, y) be
continuous for t ^ 0, ||y|| < p, satisfy
(9.1) ||/(f, S0|| ^ 2@ for r^0, II^H ?P,
and have values in a subspace X of Y. Assume either (i) that X(t) g L1 and
that (L\X), D), where D = Lc0(y) [or D = Lo°°(y)], w admissible for
(9.2) y'
or (ii) that there exists a measurable function rp(t) > 0 on J such that
y>(t) and \jf(t) are locally bounded, that
(9.3)
0 <
and that for every g(t) g L(X),for which
,9.4)
< f(t) and -^ -+ 0 as t — oo,
oo,
¦0
as
(9.2) has a l)-solution. Then ift0 is sufficiently large, the system
(9.5) y' = A(t)y+f(t,y)
has a solution for t ^ t0 such that \\y(t)\\ ^ p [and y(t) -*¦ 0 as t -*¦ oo].
Remark 1. Assumption (ii) merely means that (L™0(X), D) is admis-
admissible for (9.2). Actually, assumption (i) is a special case of (ii) but is isolated
for convenience. For a discussion of conditions necessary and sufficient
446 Ordinary Differential Equations
for (L\X), LX(X)) or (Ll(X), LOX(X)) to be admissible for (9.2), where
X = Y, see Theorem XIII 6.3.
Remark 2. Let U(t) be the fundamental solution for
(9.6) y' = A{t)y
satisfying ?/@) = /. Let y0 e Y^. Then if \\yo\\ is sufficiently small and
?0 > 0 is sufficiently large, the solution y{t) in Theorem 9.1 can be chosen
so as to satisfy
\){Q = y0.
Let Co, K be the constants of Lemma 6.3 associated with the admissibility
of the appropriate pairs of spaces {L\X), D) or (L™0(X), t>). According
as (i) or (ii) is assumed, the conditions of smallness on \\yo\\ and largeness
of t0 are
Co
?P or Co
y(t)
Proof. Let SB = L\X) or SB = L™0(X) according as (i) or (ii) is
assumed. Then Theorem 9.1 is a consequence of Corollary 8.1 obtained
by replacing f(t, y), X(t) by the functions ha(t)f(t, y), ha(t)X(t), where
a = t0 and ha{t) is 1 or 0 according as t ^ a or t < a.
Exercise 9.1. The following type of question often arises: Let y^t) be
a solution of the homogeneous linear system (9.6). When does (9.5) have
a solution y(t) for large t such that y — yx—>-0 as t —>-oo? Deduce
sufficient conditions from Theorem 9.1.
As an application of Theorem 9.1, consider a second order equation
(9.7) u" = h(t, u, u')
for a real-valued function u. Assume that h(t, u, u') is continuous for
t ^ 0 and arbitrary (u, u'). Let a, /3 be constants and consider the question
whether (9.7) has a solution for large t satisfying
(9.8) u(t) — a.t — P ->- 0 and u'(t) — a. ->- 0 as ? -» oo
Introduce the change of variables m —*¦ v, where
(9.9) u = xt + 0 + v,
then (9.7) becomes
(9.10) v" = h(t, oit + P + v, a + v')
and (9.8) is v, v' ->¦ 0 as t ->¦ oo. Theorem 9.1 implies the following:
Corollary 9.1 Let h{t, u, u) be continuous for t ^ 0 and arbitrary (u, u)
such that
for \u\, \u'\<P,
Use of Implicit Function and Fixed Point Theorems 447
where X(t) is a function satisfying
0
tX(t) dt < oo.
Then (9.7) has a solution u{t) for large t satisfying (9.8).
Exercise 9.2. (a) Verify Corollary 9.1. (b) Apply it to the case that
h = f(t)g{u), where a 5^ 0 or a = 0. (c) Generalize it by replacing (9.7)
by k«» = *(f, «,«',..., m"*1).
Actually Corollary 9.1 is a special case of Theorem X 13.1, but Theorem
X 13.1 can itself be deduced from Theorem 9.1; cf. Exercise 9.3 below.
Many problems involving asymptotic integrations can be solved by the
use of Theorem 9.1. Often these problems can be put into the following
form: Let Q(t) be a continuously differentiate matrix for ? > 0. Does
the nonlinear system (9.5) have a solution y(t) such that if
(9.11)
V = Q(t>,
then c = lim x(t) exists as r ->¦ 00 ? The differential equation for x(t) is
(9.12) x' = Q-\t)[A(t)Q{t) - Q'(t)]x + Q~\t)f(t, Q(t)x).
The change of variables
(9.13)
transforms (9.12) into
2 = x — c
(9.14)
where
(9.15)
z' = Q~\AQ- Q')z+g(t,z,c),
g(t, z, c) = Q~\AQ - Q')c + Q~Y(t, Qz + Qc).
The problem is thus reduced to the question: Does (9.14) have a solution
z{t) for large t such that z(t)-^0 as (^go? Clearly, Theorem 9.1 is
adapted to answer such questions.
We should point out that if the answer is affirmative, then (9.11) and
the conclusion x(t) — c->-0as?->-oo need not be very informative unless
estimates for \\x(t) — c\\ are obtained [e.g., if Q(t) is the 2x2 matrix
Q@ = (fc*@). where qkl = (- l)V, qk2 = el for k = 1, 2 and c = A, 0),
then we can only deduce y(t) = o{el), but not an asymptotic formula of
the type y(t) = (- 1 + o(l), 1 + o(l))e~' as t -* 00.]
Exercise 9.3. Follow the procedure just mentioned and deduce
Theorem X 13.1 by using Theorem 9.1 (instead of Lemma X 4.3).
Notes
introduction. The use of fixed point theorems in function spaces was initiated by
Birkhoff and Kellogg [1]. For Theorem 0.2, see Tychonov [1]. For Schauder's fixed
448 Ordinary Differential Equations
point theorem, see Schauder [1]. For the remark at the end of the Introduction, see
Graves [1]. As mentioned in the text, Theorem 0.3 is a result of Banach [1].
section 1. Results analogous to those of this section but dealing with one equation
of the second order, e.g., go back to Sturm. Boundary value problems for systems of
second order equations were considered by Mason [1]. The results of this section
(except for Theorem 1.3) are due to Bounitzky [1]; the treatment in the text follows
Bliss [1]. These results are merely the introduction to the subject which is usually
concerned with eigenfunction expansions; see Bliss [1] for older references to Hilde-
brandt, Birkhoff, Langer, and others. For an excellent recent treatment for the singular,
self-adjoint problem; see Brauer [2]. Theorem 1.3 is given by Massera [1], who at-
attributes the proof in the text to Bohnenblust.
section 2. Theorems 2.1 and 2.2 are similar to Theorems 4.1 and 4.2, respectively.
Exercise 2.1 is a result of Massera [1] and generalizes a theorem of Levinson [2]; its
proof depends on a B-dimensional) fixed point theorem of Brouwer. Exercise 2.2 is a
result of Knobloch [1], who uses a variant of Brouwer's fixed point theorem due to
Miranda [1]; cf. Conti and Sansone [1, pp. 438-444].
Theorems 2.3 and 2.4 are due to Poincare [5, I, chap. 3 and 4]; see Picard [2, III,
chap. 8]. Problems concerning "degenerate" cases of Theorems 2.3 and 2.4 when the
Jacobians in the proofs vanish were also treated by Poincare and since then by many
others, including Lyapunov. For some more recent work and older references, see
E. Holder [1], Friedrichs [1], and J. Hale [1]; for the problem in a very general setting,
see D. C. Lewis [4].
section 3. The scalar case of Theorem 3.3 is a result of Picard [4]; the extension to
systems is in Hartman and Wintner [22]. In the scalar case, C.17) can be relaxed to the
condition Re B(t)x ¦ x ^ 0, Rosenblatt [2]; see also Exercise 4.5(c). The uniqueness
criterion in Exercise 3.3F), among others, is given by Hartman and Wintner [22].
Sturm types of comparison theorems for self-adjoint systems have been given by Morse
[1].
section 4. Theorem 4.1 and its proof are due to Picard [4, pp. 2-7]. For related
results in the scalar case, see Nagumo [2], [4], references in Hartman and Wintner [8]
and Lees [1] to Rosenblatt, Cinquini, Zwirner, and others. Theorem 4.2 is a result of
Scroza-Dragoni [1]. The uniqueness Theorem 4.3 is due to Hartman [19]. For Exercise
4.6F), see Hartman and Wintner [8]; for part (c), with the additional condition that
/ has a continuous partial derivative dfjdx g; 0, see Rosenblatt [2]. For Exercises 4.7
and 4.8, see Nirenberg [1].
section 5. Lemma 5.1 and Corollary 5.2 are results of Nagumo [2]. The example
following Lemma 5.1 is due to Heinz [1], The other theorems of this section are
contained in Hartman [19]. Exercise 5.4 is a generalization of a result of Lees [1] who
gives a very different proof from that in the Hints. For the scalar case in Exercise
5.5(<f), see Hartman and Wintner [8]; this result was first proved by A. Kneser [2] (see
Mambriani [1]) for the case when/does not depend on x'. For related results, see
Exercises XIV 2.8 and 2.9. A generalization of Exercise 5.9 involving almost periodic
functions is given in Hartman [19] and is based on a paper of Amerio [1].
section 6. Part III is an outgrowth of a paper of Perron [12], whose results were
carried farther by Persidskil [1], Malkin [1], Krein [1], Bellman [2], Kucer [1], and
Maizel' [1]. Except for Kucer, these authors deal, for the most part, with the case
SB = ico(l'), D = LX(Y). (For a statement concerning the results of these earlier
papers, see Massera and Schaffer [1,1].) The results of this section are due to Massera
and Schaffer [1] who deal with the more general situation when the space Y need not be
finite-dimensional.
Use of Implicit Function and Fixed Point Theorems 449
section 7. For the notion of "lean at w," see Schaffer [2, VI]. The Green's functions
G of this section occur in Massera and Schaffer [1, I and IV]. Theorem 7.1 and
Corollary 7.1 may be new.
section 8. Theorem 8.1 is a result of Corduneanu [1]. Theorem 8.2 is a corrected
version of a similar result of Cordunean [1] (see Hartman and Onuchic [1])- also
Massera [8] For Corollary 8.1, see Hartman and Onuchic [9]. For Exercise 8.2, see
Massera and Schaffer [1, I or IV].
section 9. This application of the results of § 8 is given by Hartman and Onuchic
[1]. For Corollary 9.1, see Hale and Onuchic [1].
Chapter XIII
Dichotomies for Solutions of Linear Equations
For t ^ 0, consider an inhomogeneous linear system of differential
equations
@.1) y - A{t)y = g(t)
and the corresponding homogeneous system
@.2) y - A(t)y = 0,
or, more generally, an inhomogeneous, linear system of equations of
(m + l)st order
m
@.3) u + 2.^kV)u —JKO
and the corresponding homogeneous system
@-4)
Am-i
Suppose that 23, t) are Banach spaces of vector-valued functions and
that @.1) [or @.3)] has a solution y(t) e D [or u(t) e t>] for every g(t) e 23
[or/@ e 23]; i.e., that B3, D) is admissible in the sense of § XII6. Then,
under suitable conditions on the coefficients and the spaces 23 and t>,
this implies a [an exponential] dichotomy for the solutions of the homo-
homogeneous equations, roughly in the sense that some of the solutions are
small [or exponentially small] and that others are large [or exponentially
large] as t ->¦ oo. This type of assertion and its converse will be the subject
of this chapter.
In particular, for @.2), we shall obtain conditions necessary and/or
sufficient in order that there exist Green's matrices G{t, s) defined as in
§ XII 7, satisfying
\\G(t,s)\\ ^K or \\G(t,s)\\ ^ Ke-vlt-sl
for s, t ^ 0, where K, v > 0 are constants. The main results for @.2) are
given in § 6; corresponding results for @.4) are given in § 7. The main
450
Dichotomies for Solutions of Linear Equations 451
tools will be the Open Mapping Theorem XII 0.3 (in fact, analogues of
the lemmas of § XII 6) and the following basic inequality for solutions of
@.1): Let y be a point of the (real or complex) vector space Y with norm
||y|| and let \\A(t)\\ = sup \\A(t)y\\ for \\y\\ = 1, then solutions of @.1)
satisfy
@-5) ||2/@ll ^ [\\y(s)\\ + I j\\g(r)\\ dr |) exp | J'mOOII dr
for arbitrary s, t. When 7 is the Euclidean space, this can be strengthened
to
@.6)
\\g(r)\\dr
exp
dr
where fx{t) = sup |Re A(t)y • y\ for \\y\\ = 1; see § IV 4.
It will be convenient to write @.1), @.2) as equations Ty = g, Ty = 0,
where T is the operator Ty = y' — A(t)y. In order to avoid a special
treatment of @.3), this equation will be written as @.1), where y = (u, u',
. . ., «(m)). It will be advantageous, however, to introduce a "projection"
operator P, which in the general case of @.1) is the identity Py = y but
in the case @.3) of @.1) is Py = u.
This chapter will be divided into two parts: Part I deals with analogues
of @.1), @.3) and Part II with the analogues of the adjoint equations.
PART I. GENERAL THEORY
1. Notations and Definitions
(i) Below 2/, 2, ... [or u, v,...] are elements of a finite dimensional real
or complex Banach space Y [or U] with given norms \\y\\, \\z\\,.. . [or
||«||, \\v\\, .. .]. It will not be assumed that these spaces are Euclidean.
For example, in dealing with a product space X x Y, it is often more
convenient to use the norm \\(x, y)\\ = max (||x||, \\y\\). It is also more
convenient to work with the angular distance between two nonzero ele-
elements y,ze Y defined by
2/ 2
A.1)
y[y, 2] =
than to assume that Y is Euclidean and to deal, e.g., with the Euclidean
angle between y, z or with |sin (y, z)\. (If 7 is Euclidean, then y in A.1) is
2|sinJB/,2)|.)
Note that \y\ • ||z|| y\y, z] is the norm of y \\z\\ — \\y\\ z and hence is
IK?/ — z) \\z\\ — (\\y\\ — l|2||J|| ^ 2 ||2|| • ||2/ — z\\. Interchanging y and z
shows that
A.2)
>'[?/, z] max
452 Ordinary Differential Equations
If A" is a linear manifold in 7 and y e 7, let d(X, y) = dist (X, y) =
inf ||x — y\\ for x e A\ In particular,
A.3) II2/II ^ d(z, */)¦
The condition
A.4)
and A?
will frequently occur, where A" is a subspace (i.e., a closed linear manifold).
Note that the inequality A.4) can hold for y e X only if y = 0. If j/ ^ 0
and X is an admissible number in A.4), then I/A can be interpreted as a
"rough measure of the angle between y and the linear manifold X." This
is clear if A.4) is written as I/A < d(X, y \\y\\-1) = inf \\x - y [|2/||—x|| for
xeX.
Y* denotes the space dual to 7 and {y, y*} is the corresponding pairing
(i.e., "scalar product") of y e Y, y* e 7*.
(ii) It will be supposed that a coordinate system in Y [or I/] is fixed.
Thus an element y e Y can be represented as y = (j/1,.. ., «/*), where
d = dim 7, and a linear operator from 7 to 7 is a d X d matrix A with
the norm \\A\\ = sup \\Ay\\ for \\y\\ = 1. (This is only for the purpose of
making the theorems of Chapter IV, as stated, available here.)
(iii) Let J denote the closed half-line 0 ^ t < oo and /' a bounded
subinterval of J. The characteristic function of J' will be denoted by
hj.(t), so that hj.{t) = 0 or 1 according as t $J' or (e/. Correspond-
Correspondingly, hs{t) is the characteristic function of the half-line s ^ t < oo and
hse(t) the characteristic function of J' = [s, s + e].
<pse@ will always denote a non-negative, integrable function on J with
support on [s, s + e] [i.e., vanishing for t < s or (> j + e, so that
(iv) Let &~ denote the set of normed spaces O whose elements are
(equivalence classes modulo null sets of) real-valued measurable functions
<p{t) on J satisfying the following conditions: (a) O ^ {0}; (b) the elements
y{t) of O are locally integrable and for every bounded /' there exists a
number a = a(/, O) such that
for all
j-
the least number a satisfying this relation will be denoted by \hj.\^ , so
that
A.5) f ItfQl dr < |?U IM»< f°r a11 9e(I)
Jr
cf. § 9. (c) if 9? e O and y> is a real-valued measurable function on / such
that 1^@1 ^ W0l» then ^ e O and 1^1® ^ I^U; (d) if ^ eO, j > 0, and
= 0 or y(t) = y{t — s) according as 0 ^ t < s or ? ^ j, then
Dichotomies for Solutions of Linear Equations 453
tp eO and |-yj|o = l^; (e) the characteristic functions hj,(t) of bounded
intervals /' are elements of O.
Unless the contrary is implied, B and D below denote Banach spaces
in^~. It is clear that all of the spaces Lp on/, 1 ^/>^ ooarein^". Also,
the subspace Lox of I™, consisting of functions <p(t) e L" satisfying
<p(t) -»¦ 0 as ? ->¦ oo, is in F.
When O = L11, 1 ^ p ^ oo, the norm \<p\Lv will be abbreviated to
(v) M will denote the Banach space of (equivalence classes modulo null
sets of) locally integrable functions <p(t) on / with the norm
A.6)
t+i
\<p\M= sup \<p(s)\ ds.
(SO Jt
Clearly, MeJ.
(vi) If O e 3", O „ denotes the linear manifold of functions (fit) e O
with compact support, i.e., functions <p(t) e O vanishing for large t. If,
in addition, O is a Banach space (i.e., complete), then 0^ is the completion
(closure) of O^ in O.
(vii) If O e &~ and Y is a finite-dimensional Banach space (over the
real or complex numbers), O(F) will denote the normed vector space of
(equivalence classes modulo null sets of) measurable functions y(t) from
J\.o Y [i.e., functions y(t) with components which are measurable functions]
such that <f(t) = ||2/@II is m ^ with the norm \y(t)\(S>(Y) defined to be
l^l^. ¦ For brevity, the norm of y(t) e 0G) will be denoted by Ij/I^.
It is easy to see that if O is Banach space, then so is 0G).
(viii) Let L denote the space of (equivalence classes modulo null sets of)
real-valued measurable functions y{t) on / with the topology of con-
convergence in the mean L1 on bounded intervals. Correspondingly, L(Y) is
the space of locally integrable functions y(t) from / to 7 with the topology
of convergence in the mean L1 on bounded intervals.
Condition (b) in (iv), cf. A.5), on spaces OeJ means that O is stronger
than L, so that convergence in O implies convergence in L; see § XII 6.
(ix) A space O e &~ is called quasi-full if it has the property that q>(t) e L,
<p(t) $ O implies that either hOA(t)(p(t) <? O for some A > 0 or that
K'o^ld) -* oo as A -»- oo. Clearly, the spaces O = Lp for 1 ^ p ^ oo are
quasi-full.
(x) Dichotomies. Let 7, W be Banach spaces and ^V a linear manifold
of functions y = y(t) from / to 7. With each y(t) e ^f", let there be
associated a non-negative function pv(t) on / and an element y[0] of fF.
We shall assume that the map Q from./f to W given by Qy(t) = y[0] is
linear and one-to-one; .y[0] will be called the "initial value" of y(t). Let
WQ be a linear manifold in the range of Q.
454 Ordinary Differential Equations
When Wo is a subspace (i.e., closed linear manifold), it is said to induce
a partial dichotomy for (yV, py, y[0]) if there exist a positive constant Mo
and a non-negative number 0° such that
(a) if y(t) e ./F with y[0] e fF0, then
A.7) pjit) < MoPJs) if 0°^*^;
F) if 2@ e ./f with ||z[0]|| ^ A d(fF0, z[0]) and X > 1, then
A.8) pAO ? XMoP,(s) if 5^0°, O^f^j.
A subspace PF0 is said to induce a tota/ dichotomy for (./f, 2/[0]) if it
induces a partial dichotomy for (./F, p,,, y[0]), where pv{t) = ||«/@ll» and m
addition the following holds:
(c) there exists a constant y0 > 0 such that if 2/@> 2@ are as in (a) and
(b), respectively, then
A.9) Xy[y{t\ 2@] ^ yo > 0 when t ^ 0°
and y{t) * 0, z(f) 5* 0.
A subspace Wo is said to induce an exponential dichotomy for (yV,
Pv> 2/[0]) if there exist a non-negative number 0°, positive numbers Mlt
v, v and, for every X > 1, a positive number M/ = A/i'(^) such that
(a) if 2/@ e jV with j/[0] e fF0, then
A.10) Pv(t) ^ Mie-"{t-s)pv(s) for 0° ^ s ^ t;
(b) if 2@ e JT with ||z[0]|| < A d(W0, z[Q]) and A > 1, then
A.11) pz{t)^M^ev{t~s)PXs) for f^ 0°, 0 < s < t.
A subspace Wo is said to induce a tota/ exponential dichotomy for
(,/F, j/[0]) if it induces an exponential dichotomy for (Jf, pv, y[0]), where
Pv@ = II 2/@II > and condition (c) of a total dichotomy holds (i.e., Wo
induces a total dichotomy for (Jf, y[Q]) and an exponential dichotomy for
(yV, PV, y[OJ) with Py(t) = ||2/@II).
A manifold Wo (not necessarily closed) is said to induce an individual
partial [or exponential] dichotomy for (yV, py, j/fO]) if
(a) for every y(t) e ^V with j/[0] e fF0, there exist constants 0° ^ 0,
M0>0 [or Mlt v>0] depending on y(t) such that A.7) [or A.10)]
holds;
(b) for every 2@ e ^ with 2@) ? fF0, there exist contants 0° ^ 0,
Mo' > 0 [or Mj', v > 0] depending on z(t) such that
A.12) p2@ <; Afo'p,(j) if j ^ 0° and 0 < t < s
[or A.11)] holds.
If no confusion results {^V, pv, y[0]) will be shortened to (yV, pv); also,
if py(t) = HjKOII.^ will be written in place of {Jf, pv).
Dichotomies for Solutions of Linear Equations
2. Preliminary Lemmas
455
The notation Y, JT, W, Wo, Q, pv{t) is the same in this section as in
paragraph (x) of the last section. It will sometimes be assumed that
pv{t\ for fixed t, is a "semi-norm,"
B.1) PJf) ^ |c| Pv(t), Py_?t) ^ Pv{t) + Pz(t) for 2/@,2@ eJT,
c an arbitrary constant; and/or that there exist 01 ^ 0 and Ko > 0 such
that
B.2) p
B.3) ||;
whenever
B.4)
B.5) 2@ 6 JT
^ XKoPz(t)
for t^
for t^
with y[0]eW0,
with ||2[0]|| ^Xd(Wo,z[0]),
respectively.
Note that if Wo is a subspace, a sufficient condition for Wo to induce a
partial dichotomy for (yV, pv(t)) is that there exist 0° ^ 0, Mo > 0 such
that B.4), B.5) imply that
B.6) max(pJir),pJit))<XMopv_Jis) for 0 < r ^ s ^ t, s ^ 0°.
In fact, conditions (a), (b) follow from the cases 2 = 0, arbitrary X > 1,
and the case y = 0 of B.6).
The first lemma will be useful and will illustrate the meaning of "total
dichotomy."
Lemma 2.1. Let Wo be a subspace in the range of Q. Let y(t), z(t)
denote arbitrary elements of^V satisfying B.4), B.5), respectively. If there
exist 0° = 0, Mo > 0 such that
B.7) max (||2(OH,||2/(OII)^^MO ||2E)-2/E)|| for 0 ^ r ^ s < t
and s ^ 0°, then Wo induces a total dichotomy for ^V {with the corresponding
0°, Mo, yQ = 1/MO). Conversely, if Wo induces a total dichotomy for jV,
then
B.8) max Q \\z{r)\\, ||2/@ll) ^ 2XM0y^ \\z(s) - y(s)\\ for 0 ^ r ^ s < t
and s > 0°.
Remark 1. The factor \jX of ||z(OH on the left of B.8) can be removed
under some additional conditions: If Wo induces a total dichotomy for
. ¦'', and there exist 0l ^ 0, Ko > 0 such that py(t) = ||2/@ II satisfies B.2),
456 Ordinary Differential Equations
B.3) whenever B.4), B.5) hold, then there exists a constant Mo' > 0 such
that
||z(r)|| < XM0' \\z(s) - y(s)\\ for max @°, 01) < r ^ s.
Proof of Lemma 2.1. Assume B.7). The cases z = 0, A > 1 arbitrary,
and 2/ = 0 give conditions (a), (b) of a partial dichotomy. Replacing
y(t),z(t)e^V by the elements 2/@/112/(^I1, z@/||z(j)|| G./T for a fixed
j ^ 0° for which 2/(s) 5^ 0, z(s) ?? 0 gives A.9) when r = s or t = s with
y0 = i/m0.
Conversely, assume that Wo induces a total dichotomy for jV. Then
A.9) and A.2) show that
y0 max
< 2X \\y(s) - z(s)
if s ^ 0° and z(s) 5* 0, j/(j) 5* 0. Conditions (a), F) of a total dichotomy
give B.8) if s ^ 0° and z(j) 5* 0, j/(j) 5* 0.
The proof of the last part of this lemma can obviously be modified to
give
Corollary 2.1. Let Wo be a subspace of Q inducing a total exponential
dichotomy for ^V. Let y(t), z(t) be as in Lemma 2.1. Then, for 0 ^ r ^
s ^ t and s ^ 0°,
B.9) y0 max
||, AffV
- y(s)\\,
where v, v , M/ = Mi(X) > 0, Mlf and yQ occur in the definition of total
exponential dichotomy.
Instead of proving Remark 1 following Lemma 2,1, the following more
general assertion will be proved.
Lemma 2.2. Let Wo be a subspace of W with the property that there exist
0° ^ 0, Mo > 0, Xo > 1 such that B.4), B.5) with X = Xo imply B.6) with
X = Xo. Assume that py(t) satisfies B.1) and that there exist 01 ^ 0,
Ko > 0 such that B.4), B.5) imply B.2), B.3). Then there exists an Mo' > 0
such that B.4), B.5) imply
B.10)
for 0O < r < s
where 0O = max @°, 01).
Proof. The definition of d(W0,z[0]) shows that there exist elements
y° e W such that \\y° — z[0]|| is arbitrarily near to rf(^o, z[0]). Since
Ao > 1, 2/0 can be chosen so that z° = y° — z[0] satisfies ||z°|| <
XQd(WQ, z[0]) = Xod(Wo,z°). As Wo is in the range of Q, there is a
2/°@ g ^T such that ^>[0] = y° and z°@ = y°(t) - z(t) g .yT with z°[0] =
z°. Since ||z[0]|| < Xd(W0, z[0]) = Ad(fF0, z°) ^ A ||z°[0]||, it follows that
B.11)
^ ||z0t0]||
^ (X
< 2X
Dichotomies for Solutions of Linear Equations 457
From pz{r) ^ pz"ir) + P^W and the case z = 0 and X = Xo of B.6),
Mr) ^ AoMop!,o@o). The inequality B.2) implies that Py0F0) ^
Ko \\y°[0]\\, so that Pt(f) < p2o(r) + M0K0X0 ||2/°[0]||. By B.3) applied to
z = z°(t) with X = Xo,
Hence the last two displays and X > 1 give
B.12) Pz(r) ^ X(l
Thus B.10) with Mo' = A + 2M0K02X02)X0M0 follows from B.6), where
(X, z, y) are replaced by (Ao, z°, 2/0 + y).
The following lemma, which is of interest in itself, will be used several
times. (It is false if the assumption that W is finite dimensional is omitted.)
Lemma 2.3. Let W be finite dimensional, Wo a subspace of W in the
range of Q with the property that there exist 0° ^ 0, Mo > 0 such that
B.4), B.5) imply B.6). Assume also that py(t) is a continuous function of t
for each y(t) g jV, that B.1) holds and that there exist 01 ^ 0, Ko > Osuch
that B.4), B.5) imply B.2), B.3). Let X be the set of initial values 2/[0] of
elements y(t)e<A/" satisfying py(t)^>-0 as t-*-ao. Then X <=¦ WO is a
subspace of W and there exists a constant Mo' > 0 such that the conditions
B.13)
imply that
y(t)ejV with y[0]<=X,
z{t)ejr with ||z[0]|| ^ Xd(X,z[0]), X
B.14) max (Pz(r), Py{t)) ^ XM0'py^z(s) for 0O < r < s^ t,
where 0O = max @°, 01).
This clearly has the following corollary:
Corollary 2.2. Let W be finite dimensional and Wo a subspace of W
which induces a total dichotomy for jV'. Assume that \\y(t)\\ is a continuous
function of t for y(t) g jV and that B.4), B.5) imply B.2), B.3)/or py(t) =
||y@ll and 01 = 0 (e.g., ify[0] = 2/@)). Let X be the set of initial values
!/[0] of elements y(t) g Jf satisfying \y(t)\ ->¦ 0 as t ->¦ 00. Then X ^ Wo
induces a total dichotomy for ^V.
Proof of Lemma 2.3. If the norm ||w|| in PFis replaced by an equivalent
norm ||w||0 (i.e., if cx ||w|| ^ ||w||0 ^ ca ||w|| for constants clf ca > 0), then
assumption and assertion of this lemma remain unchanged. Thus,
without loss of generality, we can suppose that W is a Euclidean space.
458 Ordinary Differential Equations
Let Wx be the subspace of W orthogonal to WQ. Then, if y(t) e JT with
y[0] e Wo and z(f) e ^f with z[0] e Wx, B.6) implies that
B.15) mnx(PMPv(t))^MoPy_z(s) for 0 < r ^ 5 < r, ^ 0°
since B.6) holds for all X > 1.
It is clear from B.6) that X «= f*V Let A be the subspace of fF0
orthogonal to X. First we will show that there exists a constant a > 0
such that if x\t) e .yK" with ^[0] e A, then
B.16) fc,i@ ^ oc/vO) if 0o ^ J, ' < «•
To this end, it is sufficient to show the existence of a constant ax > 0
satisfying
B.17) \\xl[0]\\ ^ xlPai(s) for * ^ e«-
For then B.16) follows from B.2) and B.17) with a = olJ^. In order to
verify the existence of an a1; note that, by B.2), y[0] = 0 gives py(t) = 0
for t ^ 0O. In particular pxi(t) is uniquely determined by a^fO]. Let
fta^O]) = inf p^ilKOII for t ^ 0O. It is clear that jS^O]) > 0 unless
xi[0] = 0 (otherwise pxi(t)-*O, f->- oo, but 0 5* ^[O] e A). Since B.2)
shows that convergence of xx[0] in W implies the convergence of pxi(t) in
the norm "sup pjt) for t ^ 0O," it follows that /S^fO]) is a continuous
function of a^fO]. Thus if X1 5* {0}, jS^O]) has a positive minimum l/o^
on the sphere ||^Ip]|| = 1. This gives B.17) if X1 * {0} while B.17) is
trivial if X1 = {0}.
It will next be verified that if x(t) e Jf with x[0] e X, x\t) e ^f with
x![0] e X\ then
B.18) max (^(y), Px(s)) ^ 3aMop^i_^) for 5 ^ fl0.
Suppose that B.18) is false. Then there exist x(t), x\t) as specified and an
s ^ d0 such that
B.19) max (pxi(s), px(s)) > 3a.Mopxi-x(s).
Since y = x\t) - x(t) e Jf with 2/[0] e fF0, it follows that Mopxi_x(s) ^
Pa,i_i0for ? ^ 5 ^ 6O. ByB.1),B.16),andthefactthatpa,@^0as?->oo,
it follows that, for large t,
max (fids), Px(s)) > 3«.M0(Pxi(t) - Px(t)) ^ 3pjs),
where the last inequality is a consequence of B.16). Since the last two
formula lines hold, max (pxi(s), px(s)) = px(s), and so px(s) ^ 2pxi(s).
Thus by B.1) Pxi_x(s) ^ 2pB(j)/3. Hence B.19) implies Px(s) > 2«.MoPx(s),
which is impossible since the constants a, Mo in B.15), B.16) must satisfy
a ^ 1, Mo ^ 1. Hence B.18) holds. An immediate consequence of B.18)
Dichotomies for Solutions of Linear Equations 459
is that
B.20) max (Pxl(r), Px(t)) ^ 3otW,»p,^) if d0 <: r <: s < t.
The subspace A of W orthogonal to X is the direct sum of X1 and Wx.
It will now be shown that if x{i) e Jf with x[0] e X, x°(t) e ^f with
z°[0] e AT0, then
B.21) max (Px<>{r),Px(t)) ^ MlP^J.s) for 0O ^ r < 5 < t
if Mi = 6aaM03. To this end, let x°[0] = xx[0] + x^O] be the decomposi-
decomposition of z°[0] into orthogonal components xx[0] e Wx, xx[0] e A. Since
xx[0] 6TC fF0 is in the range of Q, there is an x\t) e ^f with Qxx@ =
*![()]. Let xx@ = x°(t) - x\t) e ^f, so that Qx^t) = x^O]. By B.15)
applied to 2 = — xx, y = x\t) — x{t),
max (pXl(r), pxi_x(s)) < Mopx«_x(s) for 60 ^ r ^ s.
Hence B.20) gives
max (pXi(r), Pxl(r), Px(t)) <: 3z*Mo3Pxo_x(s) for 80 <: r <: s ^ t
and B.21) follows with Mx = 6aaM03 since pxo ^ pxi + pXi by B.1).
Lemma 2.3 now follows from Lemma 2.2 (or its proof), where Ao = 1 is
permitted here since orthogonal decomposition can be used in the
Euclidean space W.
The proof of the existence of exponential dichotomies below will
generally depend on proving first the existence of a dichotomy and then
the applicability of the following:
Lemma 2.4. Let a(t) be non-negative for a ^ t < 00 with the properties
that there exist positive constants 6 < 1, Mo, 6 such that a(t) ^ M0a(s)for
a ^s ^t<^s + 8 and a(t + 8) < 6o(t) for t ^ a. Then a{t) ^
MQd~1e-'">t-s)o(s)for a <| 5 ^ t < 00, where v = -8~l log 6 > 0.
If, in this lemma, the assumption a{t + 8) ^ 0cr(f) holds only for
* ^ A (^ a), then the main inequality in this assertion is valid for
b ^s ^t < co. It can, however, be replaced by a(t) ^ K'6-1e-vii->)o(s)
valid for a ^ 5 < t < 00 if K' = Mome"(i'-a> and m = 1 + 6 - a.
Proof of Lemma 2.4. Clearly, tf(s + n<3) ^ 0M.s) for 5 ^ a and
n = 0, 1,.... Hence,
s + n8^t <s + (n+l)8 implies
^ (Afo0-1Hn+1(T(.y).
Since e-vit-a) > e-"(n+i)« = en+\ the assertion follows.
Applications of Lemma 2.4, in proving the existence of exponential
dichotomies, generally lead to an exponent v in A.11) which depends on
X > I. In order to get a v independent of A, the following will be used.
It is derived by the arguments used in the proof of Lemma 2.2.
460 Ordinary Differential Equations
Lemma 2.5. Let Wa be a subspace of W in the range of Q. Let
there exist 0° ^ 0, Mo' > 0, v > 0 such that if z(t) e Jf with ||z[0]|| <
XQ d{W0, z[0])for a fixed Xo > 1, then
B.22) Pl{t) ^ M0'ev'«-s)p?s) for 0° < s ^ t.
Assume that pv(t) satisfies B.1) and that there exist 01 ^ 0, Ko > 0, A ^ 0
such that B.4), B.5) imply B.2), B.3), and
B.23) ft(f) ^ A*oP_('+ A) for t ^ 8\
Finally, suppose that condition (b) for a partial dichotomy holds; cf A.8).
Then condition (b) of an exponential dichotomy holds with the given v {for
allX> 1); cf A.11).
Proof. Let B.5) hold and let z(t) = y°(t) — z°(t) be the decomposition
used in the proof of Lemma 2.2, so that B.11) holds. Then B.2) and B.3)
with z — z°, X = XQ give
Pyo(t) < 2Ko2XoXPzo(t) for t > 01.
Since z(t) = 2/°@ — z°(t), B.1) implies that
B.24) Pz{t) ^ A + IKfX^p.tt) for t ^ 01.
Applying B.22) for z = 2°, with s and ? interchanged, gives
MQ'pAt) ^ e-v'<s-"p*°(s) for 6° ^ ^ s,
so that if s is replaced by s — A,
M0'Pl°(t) ^ ev'V*'<8-"p*0(s - A) for 0° ^ * ^ s - A.
By B.23), with 2, f, 2/, A replaced by z°, s - A, 2/0, Xo,
Pz0(s - A) ^ Ao*:oft(j) for s - A ^ 01.
Thus B.24) and the last two inequalities give
Pz{t) ^ KVv'(a-"p.(s) for 01 ^ ^ s - A
if AT' = A + 2Ko2XoX)XoKoev'AIMo' > 0. Finally, A.8) in condition F)
for a partial dichotomy shows that
p,@ ^ AK'Moev' Vv'(s-"p,(s) if 01 ^ ^ s,
and j ^ 0° + A. Another use of condition (b) of a partial dichotomy
allows the removal of the restriction ? > 01 by a suitable alteration of the
factor XK'Moev^. This proves Lemma 2.5.
The preceding lemmas and their proofs can be used to obtain a charac-
characterization of "total dichotomy" or "total exponential dichotomy" for the
linear manifold ./T of solutions of the homogeneous equation @.2), where
7 is a finite dimensional (Banach) space. Let U(t) be the fundamental
Dichotomies for Solutions of Linear Equations 461
matrix of @.2) satisfying t/@) = /. Let WQ, Wx be complementary
subspaces of Y (i.e., Y = Wo © Wx), and let Po [or Px] be the projection
of Y onto Wo [or Wx] annihilating Wx [or Wo]. Define a Green's matrix
G(t, s) by
B.25) G(?, s) = UiOPtU-H?) or G(/, 5) = - UiQP^is)
according as 0 < s ^ f < 00 or 0 ^ f < s < 00; cf. (XII 7.1).
Theorem 2.1 Le? A(t) be a matrix of locally integrable (real- or
complex-valued) functions on t ^ Q,jV the set of solutions of @.2), W = Y,
and 2/[0] = 2/@) e Y = W. A subspace Wo of Y induces a total dichotomy
[or total exponential dichotomy] for (yV, ||2/(OL 2/@)) '/ and only if, for
one andjor every subspace Wx of Y complementary to Wo, the norm of the
Green's matrix B.25) satisfies
B.26) || G(t, s) || ^ K for i,(^0
for some constant K = K(WX) [or
B.27) \\G(t, s)\\ ^ Ke~vlt-Sl for s, t^ 0
/or some constants K = A"^), v = v(^) > 0].
Proof ("Onlyif"). Let ff0 induce a total dichotomy for (Jf, || 2/@ II, 2/@))
and Wx be a subspace of Y complementary to Wo. There exists a Ao > 1
such that if z@) e Wu then ||z@)|| ^ X0d(W0,z@)). Hence Lemma 2.1
implies that if 2/@, 20) are solutions of @.2) and 2/@) e WQ, z@) e fF1; then
B.7) holds with X = Ao.
Let ce 7 be arbitrary. Then y(t) = [/@Pot/(J)c, for a fixed j, is a
solution of @.2) and 2/@) = At/^ e fT0, y(s) = t/(j)P0[/^(^c. Also
2@ = — [/@Pit/^1(^)c, for a fixed s, is a solution of @.2) and 2@) =
-At/^ e Wx, z(s) = -t/^PitZ-H^c. Thus B.7) and Po + Px = I
give
B.28) max (|| U(r)PxU~\s)c\\, \\ U{t)P0U-\s)c\\) ^ K \\c\\
for 0 ^ r ^ s ^ Mf K = A0M0. In view of B.25), this is equivalent to
B.26).
Similarly, if Wo induces a total exponential dichotomy for^, then a use
of B.9) instead of B.7) leads to B.27) with A", v in B.27) given, e.g., by
2X0y^1maii.(M1,llM1'(X0)),mm(v,v'(X0)), respectively, in terms of the
constants in B.9).
Exercise 2.1. Prove the "if" portion of Theorem 2.1.
3. The Operator T
The general theory will be presented in a somewhat abstract form which
can then be applied to @.1) or @.3) or other situations. In what follows,
462 Ordinary Differential Equations
B and D are Banach spaces in &~. The results of this section are analogues
of the lemmas of § XII 6.
Let Y, F be finite dimensional Banach spaces. The objects of study
will be a linear operator Thorn L(Y) to L{F),
C-1) Ty(t)=f(t),
and the elements y = y(t) of its null space Jf(T)
C.2)
Ty(t) = 0.
The domain of definition of T will be denoted by S)(T).
Let U be a finite dimensional Banach space and W another Banach
space. (It will not be assumed that W\s finite dimensional although in the
applications below this will be the case. In applications, e.g., to difference-
differential equations, this need not be the case.) P will denote an operator
from L(Y) to L(U) and Q an operator from L(Y) to W with the same
domain of definition as T, @(P) = 2{Q) = S){T). The element w =
Qy(t) e W will be called the "initial value" of y(t) and denoted by y[0].
There will be no confusion even though || . . . || denotes norm in either
Y, F, U, or W.
Remark. It will be convenient to illustrate various statements in the
general theory from time to time by references to @.1). In such references,
it is always assumed that A(t), g(t) are defined on / and are integrable
over all finite intervals J' c J and that y{t) is an absolutely continuous
solution @.1). In this case, the spaces Y, F, U, W are taken to be identical;
Ty(t) is defined by Ty = y'(t) - A(t)y(t) and 3>{T) is the set of functions
y(t) from J to 7which are absolutely continuous (on every/'); Py{t) =
y(t) is the identity operator; and Qy(t) = 2/@). In applying the general
theory to @.3), where u can be a vector, it is supposed that @.3) is written
as a system @.1) for y = (u,ur, . . ., «(m)), but Py(t) = u(i), Qy(t) = y@).
Definition. PD-Solutions and WD. Let Z)eJ bp a Banach space.
2/@ is called a PZ)-solution of C.1) for a given/@ e L{F) if C.1) holds
and Py(i) e D(U). WD = WD(P) denotes the linear manifold in W
consisting of initial values w = 2/[0] of PZ)-solutions of C.2).
Definition. P-Admissibility. The pair (B, D) of Banach spaces in &"
is called P-admissible for C.1) if, for every f(t) e B(F), C.1) has at least
one PZ)-solution y(t).
Various assumptions (Ao), (Ax),... or (Bj), (B2), . . . concerning T will
be made from time to time. These and some of their consequences will
now be discussed.
(Ao) If 2/@ e 3>{T), then u(t) = Py(t) is (essentially) bounded on
every bounded interval J' of J.
Dichotomies for Solutions of Linear Equations 463
This assumption implies that the answer to the question whether or not
2/@ is a PZ)-solution of C.1) depends only on the behavior of u(t) = Py(t)
for large t; cf. conditions (c), (e) for O = D ej in § 1 (iv).
(Aj) Uniqueness for Q. (a) If C.2) holds and y[0] = 0, then y(t) = 0.
(b) There exist positive constants a, Ct such that C.1) implies that
C.3)
\\y[0]\\ ^
II 2/@ II dt + 11/@11 dt
This assumption is, of course, suggested by the inequality @.5).
(A/) Same as (Ax) except that C.3) is replaced by
C.3')
\\Py{t)\\ dt + 11/@11 dt
(A2) Normality for P. If C.1) holds, then y(t) is uniquely determined
by Py(t) and/@; furthermore, the linear map from L(U) x L(F) to
L( Y) defined by (Py(t), Ty(t)) -> y(t) is continuous in the following sense:
ifyn(t)e 2{T) and the two limits u(t) = \im Pyn(t) in L(U),f(t) = limTyn(t)
in L(F) exist, then 2/@ = lim 2/«@ in L( Y) exists and 2/@ e 3>(T),
Py(t) = u(t),Ty(t)=f(t).
The main role of (A2) is the following (cf. Lemma XII 6.2):
Lemma 3.1. Assume (A2). The operator T0from D(U) to B(F) defined
by T0[Py(t)] = Ty(t), with a domain 3>(T^ consisting of those elements
u = Py{t) in the range of P for which u = Py(t) e D(U) and Ty(t) e B(F),
is closed. Also, for every t > 0, there exists a constant C2 = C2@ such
that
C.4)
P||
Jo
ds <: C2(t)[\Py\D + \f\B].
Proof. In order to prove that To is closed, let («[,/,), (u2,f2),... be a
convergent sequence of elements in the graph of To, where un = Pyn{t),
fn = Tyjj) and yn(t) e 2>(T). Thus u = \imun(t) in D(U) and / =
lim/n@ in B(F) exist. Since convergence in B or D implies convergence
in L [cf. condition A.5) on O = B, D e^"], (A^ implies that y(t) =
lim yn(t) exists in L( Y) and y(t) e ®(T), Py = u,Ty = /. Hence u e @(T0)
and 70« =/; thus To is closed.
Let rx be the map from D(U) x B(F) to the space L\ot](Y) of 7-
valued functions which are integrable over the interval [0, t], where the
domain S>(Tj) is the graph of To, and T1 is defined by T^Py, Ty) = y.
Thus 7\ is defined on a subspace of Z>(t/) X B(F). (A2) implies that
T1 is continuous, hence bounded. The inequality C.4) is equivalent to the
boundedness of 7\.
Although the trivial space B = {0} is not in F, this choice of B is
permitted in Lemma 3.1. The fact that TB is closed gives
464 Ordinary Differential Equations
Corollary 3.1. Assume (A2). The set of u = Py(t), where 2/@ varies
over the PD-solutions of C.2), is a subspace {i.e., closed linear manifold) of
D(U).
(A3) (B, D) is P-admissible.
Lemma 3.2. (A0 [or (A/)], (A2), and(A3) imply that there exist constants
C3, C30 such that iff(t) e B(F), then C.1) has a PD-solution 2/@ satisfying
C.50
C.5,)
^ C30 \f\B.
Proof. Let To be the operator from D(U) to B(F) occurring in Lemma
3.1. Thus To is closed by Lemma 3.1 and onto by (A3). Hence the
existence of C3 follows from the Open Mapping Theorem XII 0.3.
If (Ax) is assumed, then C.3), C.4) with t = a, and C.50 give
\f\
Thus, by A.5), C.5,) holds with C30 = Q^C^XQ + 1)+ |*0o|B,}.
If (A/) holds, it is similarly seen that A.5), C.3'), and C.50 imply C.52)
with C30 = Qfa |/!0JD-C3 + |/!«JB,}.
(A0 WD is closed.
This assumption is of course trivial if W, hence WD, is finite dimensional.
Lemma 3.3. Assume (A0 [or (A/)] and (A2). Le? Wm be a subspace of
W contained in WD [e.g., i/(A4) holds, let Wm = WD]. Then there exists
a constant C4 = Q(fFD0) such that if y{t) e^V(T) and y[0] e Wm, then
C-6) |P2/ID^Q||2/[O]||.
Proof. Let Qo be the operator from D{U) to Wm defined by
Qo[Py(t)] = 2/[0], where ^(g0) is the set of u = Py(t) such that 2/@ is a
PD-solution of C.2) with y[0] e «V ?H is closed by C.3) [or C.3')] and
(A0, one-to-one by (Aja), and onto since WD0 c WD. Thus the open
mapping theorem is applicable to Qo and implies Lemma 3.3.
Lemma 3.4. Assume (A0 [or (A/)], (A2), (A3), and (A4). Then there
exists a constant C5 with the property that if X ^ 1, Ty = f, Py e D(U),
fe B(F), and \\y[0]\\ ^Xd(WD, y[0]), then
C.7) \Py\D < XC, \f\B, ||2/[0]|| ^ XC3I) \f\B,
where C30 is the same as in C.5). (In particular, X = 1 is permitted in
C.7) ify[0] = 0.)
Proof. Let y = yo(t) be a PD-solution of C.1) supplied by Lemma 3.3,
so that \Pyo\D < C3 \f\B, \\yMW ^ Qo l/b- Since 2/@ - y?t) is a
PD-solution of C.2), C.6) gives
\Py - Py»\D ^ Q \\y[0] -
Dichotomies for Solutions of Linear Equations 465
As w = y[0] — 2/0[0] e WD, we have
ll2/«[0]|| = ||2/[0] - w|| ^ d(WD,y[0]) ^ A ||y[0]||.
Thus the second inequality in C.7) holds. In addition,
l^2/lD^l^2/alD+Q(l + A)||2/o[O]||.
Thus X ^ 1 shows that the first inequality in C.7) holds with C5 =
C3 + 2QC30.
4. Slices of \\Py(t)\\
Recall that <psc(t) always denotes a non-negative, integrable function on
/ with support on [s, s + e] and that | . . . |x refers to the L1 norm on /.
(B^) Let 0O > 0, e > 0 and K > 0 be fixed. For every pair of
solutions 2/@. 2@ of C.2), with y(t) — z@ ^ 0, and for any given function
<pse(t) as in § l(iii), with s ^ 0O, let there exist a function yt(t) e ^"G) with
the properties that (i) yM = (const.) z[0]; (ii) ||Pz@ll ^ ^ ||P2/1@ll/l<?'sel1
for 0 < t? s; (iii) ||P2/@ll ? K \\Py1(t)\\l\<P.t\i for t^s+e; (iv) there
exists a constant ^T' (depending on y, z, <psc) such that ||P2/i@ll ^ -^'11-^2/@11
for ? > s + e; finally, (v) ||TVl(t)\\ <> K<pSi{t) \\Py{t) - Pz(t)\\ for t ^ 0.
Also, if 2/@ ^ 0 is a solution of C.2), then l/||P2/@ll is integrable over
any closed interval /' of /.
Remark 1. This assumption will be used only in the particular case
D.1) <pse(t) ^ .
^ \\Py(t) - Pz(t)\\
For D.1), condition (v) becomes
D.2) IITj^flU < Khse(t)
and, by Holder's inequality,
D.3)
\Py(t) - Pz(i)\\ dt.
Remark 2. Note that assumption (B^) holds for all 0O > 0, e > 0 with
K = 1 if T is the operator associated with @.1) as in the Remark in § 3;
thus 72/@ = y\t) - A @2/@, Py@ = 2/@ and 2/[0] = 2/@). In fact, let
2/i@
= 2/@ ?>«(
Jo
+
Jt
<pJir) dr.
Then Tyx = yx' - A {t)Vl = ^f@[2/@ - «(f)]andy,@ =
^ |?g, s(/) for 0 ^ ?^ j.
466 Ordinary Differential Equations
Theorem 4.1. Assume (Ao), (Ax) [or (Ax')], (A2), (A3), (A4), andB^e)
for a fixed e. Let y(t)eJ^(T), y[0] e WD, and z(?) eJ^r), \\z[0]\\ <;
X d(WD, \\z[0]\\)for aX>l,ands^ 6o. Then
D.4) max (\hOsPz\D, \hs+cPy\D) ^ XK2C,e-2 \hOe\B fS+VB/ - *)ll dt,
Js
where K and C5 are the constants in (B1e) and Lemma 3.4, respectively.
In particular, if A ^ e is fixed and either
D.5) Pv@ = \htAPy\D or
fnen W^, induces a partial dichotomy for (Jf(T), py{t)) with d° = 60 and
D.6) Mo = 1 + 2K2C5I\e), where T(e) = e |nO?|B |n0?li>'-
In the applications of Theorem 4.1, it will be important that Mo depends
on e, but not on A. The inequalities D,10), D.12) in the following proof
will be used in the proofs of Theorems 4.2 and 4.3.
Proof. Apply (Bxe) with the choice D.1) of (pse(t), so that (v) implies
D.2). Assumption (Ao) and (iv) in (Bxe) show that Py1eD(U). Since
ll2/i[0]|| ^ Xd(WD, 112/^OHI) by (i) and the condition on z[0], Lemma 3.4
and D.2) give IP^I^ < XKCS \hOe\B. It follows from (ii) and (iii) in (Bxe)
that
max (\hOsPz\D, \hs+cPy\n) ^
D.7)
Hence D.4) is a consequence of D.3).
In order to prove the assertions concerning D.5), note that \htAPy\D ^
\hs+€Py\D for t^ s + e, A > 0, and \hrAPz\D ^ \hOsPz\D if r + A ^ s.
Thus, for any A > 0 and j ^ d0, the inequality D.4) implies that, for
r + A < j, s + e ^t,
\"* - z)\\ dr.
- z)\D.
D.8) max (\hrAPz\D, \htAPy\D) ^ XK2C&e~2 \hOc\B
The relations A.5) and A ^ e show that
D.9) !S+e\\P(y - z)\\ dt ^ \hOe\n. \hseP(y - z)\D ^ \hOe\D,
Js
Thus, for A ^ e, s ^ 0O, r > 0, r + A ^ s, s + e ^ t,
D.10) max A^1^ \htAPy\D) ^ XK2C5Y(e) \hsAP(y - z)\D,
where F(e) is denned in D.6).
The case z = 0 and X = 1 of D.10) combined with
D.11) \htAPy\D ^ \hsAPy\D + \hs+AAPy\D for j^/^j+A
gives condition (a) [i.e., A.7)] for a partial dichotomy for py(t) = \htAPy\D
Dichotomies for Solutions of Linear Equations 467
with 6° = 60 and Mo given by D.6), even with the factor 2 omitted from
the second term. Similarly, the choice y = 0 in D.10) combined with an
analogue of D.11) gives condition (b) [i.e., A.8)] with 6° = 60 and the
same Mo.
In order to deal with the second function in D.5), apply an analogue of
D.9) to the left side of D.8) with A = e to obtain
D.12) max (J^ll-Pzll dr,^C\\Py\\ drj ^ XK2CiT(e)^+e\\P(y - z)\\ dr
if r + e ^ s, t ^ s + e, s > d0. For A ^ e, let k ^ 1 be an integer such
that ke ^ A < (k + l)e. Then it follows, by replacing tby t + je and s by
s + je fory = 0, . . ., k — 1 and adding, that
In addition,
r
\\Py\\ dr ^ XK2C5
fs+ke
||
Js
\\P(y - s)ll dr.
rt + A fs + e
\\Py\\dr <, XK2C5V(e)\ \\P(y - z)\\ dr.
Jt+kc Js
If the upper limits of integration on the right of the last two inequalities
are replaced by s + A, we obtain the second of the inequalities contained
in
r+A
\
\\Py\\drj ^ 2XK2C5
fs+A
) \\P(y-
Js
z)\\ dr
for r + A ^ s, t ^ s + e, s ^ 60. The other inequality, involving the
first integral, is obtained similarly. Combining D.13) with inequalities of
the type
\\Py\\ dr ^ r+A\\Py\\
J s
fs+2A
u \\Py\\
Js+A
dr for s ^ ; ^ 5 + A
leads to a partial dichotomy for the second function in D.5) with 0° = 60
and Mo given by D.6). This proves Theorem 4.1.
Corollary 4.1. In addition to the assumptions of Theorem 4.1, assume
that D is quasi-full {cf. (ix) in§ 1) and that z(t) is a non-PD-solution of C.2),
then
J( ||Pz(r)||dr->oo as ^ oo.
This follows from D.4) with y(t) = 0 and the definition of a quasi-full
space.
Corollary 4.2. In addition to the assumptions of Theorem 4.1, assume
that W is finite dimensional, that py(t) is defined in D.5) and, in the case of
the first choice, that py(t) is a continuous function oft. Let Wo be the set of
468 Ordinary Differential Equations
initial values y[0] of y(t)^Jf(T) satisfying py(t)-+O as t -*¦ oo. Then
Wo c WD and Wo induces a partial dichotomy for (^(T), py).
Exercise 4.1. Verify this corollary which follows from Lemma 2.3
combined with Theorem 4.1 and its proof.
Theorem 4.2. Let the assumptions of Theorem 4.1 hold with B = L1 and
D = L°° or D = Lox; in addition, let (B^) hold for all e, 0 < e ^ e0, with
60, K independent ofe. Let PJf be the set of functions Py(t) with y{t) eJf
and let the "initial value" y[0] be assigned to Py(t). Assume that \\Py(t) \\ is a
continuous function of t for y(t) eJf. Then WD induces a total dichotomy
forPJf.
Proof. By the proof of Theorem 4.1, D.12) holds for r + e < s,
t^s+ e, s^60. Since B = L1 and D = L°° (or Lox), T(e) in D.6) is
e-2e- e = 1; cf. A.5). Hence
max
Letting
Cr+e
ll^ll dr, e-1
<: AK2C
Cs+t
-1 \\P(y -
v S
dr.
¦ 0 gives
max(||Pz(r)||, \\Py(t)\\) < XK*C5 \\P(y - z)(s)
for 0 < r ^ s ^ t, s ^ 0°. In view of Lemma 2.1, this proves Theorem 4.2.
Theorem 4.3. Assume (Ao), (A/), (A2), (A3), (A4), andBx(e) for all
e ^ e0 > 0 and with 60, K independent of e. Assume also that either
D.14) A-^oaIb-^O or A hUD, -> 0 as A->oo.
Then WD induces an exponential dichotomy for the functions D.5) for
every fixed A ^ e0, with d° = d0 and constant M^{X, A) depending on A,
but Mi, v, v' > 0 independent o/A.
Note that if (B, D) = (Lp, Lq), where \^p,q^ oo, then D.14) holds
when(>, q) ^ A, oo).
Proof. Theorem 4.3 will be deduced from Lemmas 2.4, 2.5, and
Theorem 4.1. In the proof consider only the first function py{t) = \htAPy\D
in D.5); the proof for the other function is similar. The condition D.14)
is equivalent to
D.15) r(e) = 6-2|VIBIVlD'-0 as e—oo.
In order to see this, note that if k ^ 1 is an integer such that 0 < ke ^
f] < (k + l)e, then \hOrj\D, ^ (k + 1) \hOe\D'. This is a consequence of the
fact that \hr\D, is the "best" constant in A.5). Hence rf1 \hOn\D, ^
(k + l)^-^ \hOe\D. ^ 2c-1 |/!0?Id' for all r\ ^ e and so the second
function of A in D.14) is bounded for e >. e0. Also lim inf e-1 |/iOelz>- = 0
as e -*- oo implies that e |^Oelz>- -*¦ 0 as e -> oo. Similar remarks apply
to the first function in D.14). This makes it clear that D.14) and D.15)
are equivalent.
Dichotomies for Solutions of Linear Equations 469
Let e (^ e0) be fixed so large that
D.16) 0
Let y(t)e.A/~(T), y[O]e WD, so that A.7) in condition (a) of a partial
dichotomy is applicable for A ^ e0. By the case z = 0, X = 1 of D.10),
it follows that
D.17) \hlAPy\D^e\hsAPy\D if t^s+e
andA^e. Thus, for fixed A ^ e, condition (a) [cf. A.10)] for an exponen-
exponential dichotomy with 6° = 60, Mx = Mod-1, and v = — e-1 log d follows
from Lemma 2.4 applied to a(t) = \htAPy\D. (Hence Mx and v are
independent of A ^ e.) If e0 ^ A < e, let k ^ 1 be an integer such that
(k — 1)A < e ^ ?A; thus k < 1 + e/e0. Then, by what has been proved,
for 60 ^ s ^ t. By A.7) in a partial dichotomy,
These two inequalities give condition (a) of an exponential dichotomy
with 0° = 0O, Mj = kMod-1, and v = - e-1 log d.
In order to obtain condition (b) [cf. A.11)], consider first a fixed Xo > 1.
Let e = e(A0) be so large that 6 = A0/<:2C5r(e) < 1. Let z(t) eJ/~(T),
IIZ[O]|| ^ Xo d(WD, z[0]). Then A.8) in condition (b) of a partial dichotomy
is applicable for A ^ e. The case y = 0, X = Xo, e = e(X0) of D.10) gives
if
A<5 and A > e.
\KJz\v ^ 0 \hsAPz\D
An application of Lemma 2.4 to a{t) = l/\htAPz\D gives condition (b) for
an exponential dichotomy for X = Xo with 6° = 60, Af/ = X0M06-\ and
v' = -Alog0.
When A ^ e, the corresponding condition (b) for all X > 1 with the
same v' will be deduced from Lemma 2.5. In fact, condition B.22) has
just been verified. B.1) is clear and B.2) follows from C.6) with Ko = C4,
01 = 0. Condition B.3) follows from C.3') applied to y = z,f= 0 which,
together with A.5), gives
l|z[0]|| ^
dt ^ C.a-1 \h0aPz\D \hOa\D,
In fact, since A.8) implies that \hOaPz\D < XkM0 \htAPz\D for t ^ max (a, 60)
if a ^ A:A, B.3) follows with Ko = CjCt'1 \hOa\D, kM0, 61 = max (a, 60).
Finally, B.23) follows from D.10) with Ko = K*Csr(e). Consequently,
condition (b) of an exponential dichotomy holds for A ^ e with Mt' =
M/iX, A), v' = v'(A).
470 Ordinary Differential Equations
As in case (a), it can be shown that (b) holds for e0 ^ A < e with a
suitable Mi(X, A) and v = v'(e) independent of A. Finally, an analogous
argument shows that it is possible to choose v = v'(e) for all A ^ e0.
This proves Theorem 4.3.
Theorem 4.4. Let the conditions of Theorem 4.3 hold; in addition, let
there exist a subspace Wo of W which induces a partial dichotomy for
(J/~, \\Py(t)\\). Then Wo = WD and WD induces an exponential dichotomy
for {JT, \\Py(t)\\).
Proof. In view of Theorem 4.3, WD induces an exponential dichotomy
ft+A
for {jV, Py(t)), where Pv(t) = ||JMt)|| dr for a suitable A > 0. This
makes it clear that if Tz(t) = 0 and z[0] ^ WD, then pz(t) is not bounded as
t -> oo, and so z[0] <? Wo. Thus Wo c WD. Also, if z[0] <? Wo, then
"ftXO -* 0 as t -> oo" does not hold, and so z[0] <? WD. Thus ffB <= JF0
and, consequently, Wo = W^.
Using the fact that WD = Wo induces a partial dichotomy for (jV,
11-^2/@11) and an exponential dichotomy for(^T, py(tj), it is easy to see that
it induces an exponential dichotomy for (J/~, \Py(f)\). Details will be
left to the reader.
Theorem 4.1 and 4.4 are immediately applicable to the operator T
associated with @.1); the results will be given in § 6. These theorems are
not applicable to operators associated with @.3) without some boundedness
conditions on the coefficients Pk(t). The difficulty arises from the fact that,
in general, condition (B^) does not hold. The next section leads to
theorems applicable to @.3) as well as @.1).
5. Estimates for \\y(t)\\
In this section, the role of (B^) will be played by the following
condition:
CB2e) Let e > 0, 0O > 0, K% > 0 be fixed. For every s ^ 0O and every
pair y(t), z(t) e^V(T), where y[0] e WD, there exists a <pse(t) as in § l(iii) and
ayi@ e 3( T) such that (i) yi[0] = (const.) z[0]; (ii) \\Pz(t)\\ ^ K2\\Pyi(t)\\
for 0 ^ t? s; (iii) \\Py(t)\\ ^ K, \\PyMl for s + e < t < oo; (iv)
Pyi(t) eD{U); (v) TVl(t) satisfies
E.1) 117^@11 ^<PsXt)\\y(t)-z(t)\\;
(vi) <pS€(t) g B and there exists a number 6(e) satisfying
E.2) \<P»\B?l>(*) for s^B0.
Note that E.1) implies that Tyx(t) = 0 unless s ^ t < s + e.
(d
(B38) Let 6 ^ 0 be fixed. There exists a constant
such that
Dichotomies for Solutions of Linear Equations 471
(i) if y(t)^r(T) and y[0] e WD, then y(t) e D(Y) and \hs+dy\D <
Ks(d)\hsPy\D for s^0; (ii) if y(t)eJT(T), then hOs(t)Py(t)e D(U),
hOs(t)y(t) eD(Y) and \Ks-6y\D ^ Ks(d) \hOsPy\D for s ^ <J.
It is clear that if (B3(S) holds for d = d0, then it holds for all d ^ d0.
For convenient reference, the following variant of (B3E), which will be
needed in Part II of this chapter, is stated here.
(B308) Let d ^ 0 be fixed. There exists a constant K30(d) such that if
f(t) g BJF) [cf. (vi) in § 1], f(t) = 0 on an interval [s - 8, s + A + 8]
for some s ^ 8, and y(t) is a PD-solution of Ty = f then y(t) e D(Y) and
\hsAy\D ^ Kso(8) \Py\D.
(B4A) Let A > 0 be fixed. The solutions y(t) of Ty = 0 are continuous
functions of t (from J to Y) and there exists a constant ^T4(A) such that if
y(t) eJT(T), then ||y@ll ^ A^(A) ||y(j)|| if |j - t\ ^ A.
It is clear that if (B4A) holds for some A > 0, then it holds for all A > 0.
Remark. If A (t) satisfies
mi
E.3) \\A(s)\\ ds < const. for t ^ 0
or if Y is Euclidean and
E.4)
mi
Jt \\
sup I Re A(s)y -y\ds ^ const.
HvlUi
for t > 0,
then the operator T belonging to @.1), as described in the Remark of § 3,
satisfies (B4A). This is clear from @.5) or @.6).
Theorem 5.1. Let e, 8 > 0 be fixed. Assume (Aj)-(A4); (B2e) with b(e)
independent of y, z; (B38); and(B4&). Then WD induces a total dichotomy
for ^V = jV(T) with 0° = 0. If, in addition, D is quasi-full and z(t) is a
non-PD-solution of Tz = 0, then ||z(f)|| -> oo as t -* oo.
If the assumption that b(e) can be taken independent of y, z is omitted,
then we obtain an individual partial dichotomy instead of a total
dichotomy.
Proof. In view of E.1) and (B4e),
- 2(/!)|| for s^h^s + e
and s^60. Hence \TVl\B ^ Kt(e) \<pSe\B \\y{h) - z(tl)\\ for s ^ t, <
s + e. If ||z[0]|| ^ Xd(WD,z[tt\), then, since Py1eD(U), Lemma 3.4
gives
\PVi\d ^ *C5K4(e) \VJB \\y(tl) - z(tl)\\ ,
thus (ii) and (iii) in (B,e) imply that
, \h,uPy\») ? XK%ChK^)\(p.,\,} \\y(tl) - zOM .
472 Ordinary Differential Equations
By (B,<J),
max(\ho^dz\D, \hs+c+iy\D) ^ XKJftCJCk) \<pu\B \W(h) - zih)\\ ¦
The inequality in (B4e) shows that, for 0 < t ^ t0 ^ t + e,
and that a similar inequality holds for z. Consequently,
,XK(e,8)\<psc\B
\K\d
\\y(h) -
where 0 ^ r < s - e - 8, fZ s + e + 8, s ^ 0O, and K(e, 8) =
K^SjC^ie). Finally, applications of (B4e + 8) and (B40O) give
E.5) max (||z(r)||, \\y(t)\\) ^ XM0 \\y(s) - z(s)\\
for O^r^s^t if Mo= XK{€, <JN(e)Ai(e + <J)*4@o)/l*oJfl- Hence
Theorem 5.1 follows from Lemma 2.1.
Theorem 5.2. Assume (A^HAJ; (B2e)/or all e > e0 > 0 with 0O, K2
independent of e, |95,e|B independent of y, z, and
E.6)
\<P
sab\
A\D
0
as A-
oo
uniformly for large s; (B38); and (B4A). Then WD induces a total
exponential dichotomy for Jf = jV(T) with 0° = 0.
Proof. Let z = 0 in (B^). Then E.1), t ^ s, Theorem 5.1 imply
\\Tiy(t)\\ ^ Mo<pse(t) \\y(s)\\. Arguing as in the last proof, it follows that
\hs+e+dy\D ^ CbKl$)M0 \<pse\B \\y(s)\\. Also, by condition A.7) of a total
dichotomy, Mo \hTey\D ^ \\y(r + e)|| • \hOe\D. Hence if / > 5 + 2e + <J,
In view of E.6), e and s0 can be chosen so large that
c5x3^)M02 lyJ* < 0 < t for 5>5o;
l^otlzi
so that ||j/(OI| ^ 0 ||2/(s)|| for 5 ^ 50, f > 5 + 2e + 8. An application of
Lemma 2.4 and the remark following it to o(t) = \\y(t)\\ give condition (a)
for an exponential dichotomy with 01 = 0; cf. A.10).
Let Xo > 1 and z(t) e^f (J), ||z[0]|| ^ Xo d(WD, z[0]). Choosing y = 0
in (B2e) and arguing as before shows that if e = e(X0), s = s(X0) are so
large that
^WM«^<1 for 5^50,
I'lotlo
Dichotomies for Solutions of Linear Equations 473
then ||z(r)|| ^ 6 \\z(s)\\ if r < s - e - 8, s ^ s0. Applying Lemma 2.4
to a(t)= l/||z(OII gives condition (b) of an exponential dichotomy for
X = A,,with 0° = 0; cf. A.11).
Condition (b) for all X > 1, with v' independent of X, can now be
deduced from Lemma 2.5, where Pv(t) = \\y(t)\\. In fact, B.22) has just
been verified and B.1) is clear. In order to deduce B.2), note that C.6)
and (B3E) imply that
\htAy\D < K?8) \Py\D < K3(8)Ct \\y[0}\\
for any t^8, A > 0. By condition (a) of a partial dichotomy,
Mo \htAy\D ^ \\y(t)\\ \hOA\D. This gives B.2) with Ko = K^C^M^h^o
and 01 = 8. To obtain B.3), begin with C.3) applied to y = z, f = 0.
Then, by condition (b) of a partial dichotomy,
||z[0]|| ^ XC.Mo \\z(t)\\ for t ^ a,
which is B.3) with Ko = QMo and 01 = a. Finally, B.23) is implied by
E.5). Hence Lemma 2.5 implies condition A.11) of an exponential
dichotomy for yT( 7).
We can obtain results analogous to Theorems 5.1-5.2 under a condition
somewhat weaker than (B4A):
(B6A) Let A > 0 be fixed. The solutions y(t) of Ty = 0 are continuous
functions of t (from J to Y) and there exists a number isT6(A) such that if
j/@ ?JT{T), then
E.7) \\y(t)\\ ^ Js:6(AX||j/E)|| + \\y(s + <J)||) if s ^ t ^ s + 8, 8 ^ A.
Condition (B6A) is useful for applications to second order equations
and is suggested by Exercises XI 8.6 and 8.8. For applications, see
Exercises 7.1, 13.1, and 13.2, below.
Choosing 5 = t — \8 in E.7) and integrating with respect to 8 over an
interval [0, 8] gives
rt+d/2
E.8) ||j/@ll^2X4(A)r1 \\y(r)\\dr if t^ \8 > 0, 8 < A.
Jt-d/2
Introduce the Banach space 7<2) = Y x Y with the norm of r\ =
B/i, 2/2) e J™ defined by IMI = max (||yi||, ||j/2||).
Theorem 5.3. Let e, A > 0 be fixed. Assume (Aj)-(A4); (B2e) with
b(e) independent of y, z; (B3|e); and (B5A) with A > 2e. For 8 > 0, /ef
^4 *e f/ie manifold of functions r\ = ^(f) = (j/(;), y(t + 8)) from J to 7<2),
where y(t) e.,V~(T), and let rjd[O] = y[0]. If e <L 8 <^ \, then WD induces
a partial dichotomy for {jV\, \r?(f)\\, ^[0]) with 0° = max @O, e) and
Mo = M0(t, 8). If 3« < 8 ^ A, then Wv induces a total dichotomy for
.A'~d with 0° = max @O - «, 0) and y0 = yo(e, 8). If, in addition, D is
474 Ordinary Differential Equations
quasi-full, e < 6 ^ A, and y{t) is a non-PD-solution of Ty = 0, then
\\rf(t)\\ -> oo as t-+ oo.
Exercise 5.1. Prove Theorem 5.3.
Theorem 5.4. Lef e0 > 0, Ao ^ 2e0 an</ e0 ^ E ^ Ao- Assume (Aj)-
(A4); (B2e) for e ^ e0 wif/j d0, K2 independent of e, \<pS€\b independent of
y,z, and E.6) holds uniformly for large s; (B3|e0); am/(B5A0). Then WD
induces a total exponential dichotomy for the manifold jVd, defined in
Theorem 5.3, with 6° = max F0, e0).
Exercise 5.2. Prove Theorem 5.4.
6. Applications to First Order Systems
As was pointed out in §§ 3-4, the assumptions (A0)-(A4) and (B^) are
satisfied by the operator T associated with @.1) in the Remark of § 3,
with Py = y, Qy = 2/@). Hence Theorems 4.1-4.3 imply
Theorem 6.1. Let A{t) be a matrix of locally integrable (real- or complex-
valued) functions for t ^ 0. Let B, D be Banach spaces in O and let (B, D)
be admissible for @.1) in the sense that for every g(t) e B(Y), @.1) has a
solution y(t) e D(Y). Let jV denote the set of solutions y(t) of @.2) and YD
be the set of initial conditions 2/@) belonging to solutions y(t) bjV C\ D( Y).
(i) Then YD induces a partial dichotomy for (jV , pv(t),y(O)), where
F.1)
Pv(t) = \htA.y\D or
and A > 0 fixed arbitrarily, (ii) If, in addition, D is quasi-full and z(t) e
JT, but z(t) $ D(Y), then, for every A > 0,
[t+A
Ht)
dr-*¦ co as t-*co.
(iii) If D.14) holds, then YD induces an exponential dichotomy for (jV , pv(t),
2/@)), where the exponents v, v can be chosen independent of A > 0.
(iv) IfB = L1andD = La>orD = L0a>, then YD induces a total dichotomy
for.yf\
Exercise 6.1. State the consequences of Corollary 4.2 for parts (i)
and (iv) of Theorem 6.1.
The next theorem, except for condition (c) of a total dichotomy [cf.
A.9)], is easily deduced from Theorem 6.1 by virtue of the Remark
concerning E.3), E.4). The entire theorem, however, will be deduced
from Theorems 5.1 and 5.2.
Theorem 6.2. Let A(t) be a matrix of locally integrable, real- or complex-
valued functions satisfying E.3) or E.4). Let B, D be Banach spaces in O
and (B, D) admissible for @.1), andjV, YD as in Theorem 6.1. Then YD
Dichotomies for Solutions of Linear Equations 475
induces a total dichotomy forjV. If, in addition, D is quasi-full and z(t) e^ir,
but z(t)$D(Y), then
F.2) ||z(OI|-*oo as t-*co.
Finally, if
F.3) A^IVIb-^O or \hOA\D^co as A -* oo,
then YD induces a total exponential dichotomy for jV'.
Proof. It suffices to verify the conditions of Theorem 5.1 and/or 5.2
for the operator T, with Py = y and y[Q] = 2/@), associated with @.1).
Conditions (Ai)-(A4) have already been verified. As pointed out before,
(B^) holds for all 60 > 0, e > 0 with K = 1 and arbitrary q>S€(t). Choosing
<PseU) = ?~lhS€(t) in Remark 2 preceding Theorem 4.1 shows that (B2e)
holds for all d0 > 0, e > 0 with K2 = 1 (since K = 1 and ^^ = 1) and
l9Vta = ^\K\b
independent of s and 2/@. z@- Condition (B3d) is trivial since y = Py.
The Remark preceding Theorem 5.1 shows that E.3) or E.4) implies
condition (B4A). Thus Theorem 5.1 is applicable and gives the statements
concerning a total dichotomy and F.2).
In order to apply Theorem 5.2, the condition E.6), which is
J^Ab_^o as A^oo,
must be verified. But is readily seen that this is equivalent to F.3); cf.,
e.g., the beginning of the proof of Theorem 4.2. Hence the proof of
Theorem 6.2 is complete.
The next theorem is the main result on total dichotomies for solutions
of @.2) and admissibility for @.1). For the sake of brevity, let Yp and
Yx0 denote the subspace YD of Y, when D = Lp and D = Lo°°, respec-
respectively; i.e., Yp and Yx0 denote the set of initial points y@) e Y of
solutions 2/@ of @.2) of class W, 1 ^ p ^ oo, and Lo°°, respectively.
Theorem 6.3. Let A(t) be a matrix of locally integrable (real- or complex-
valued) functions of t ^ 0 and^V the set of solutions of @.2). Then there
exists a subspace Wo of Y = W which induces a total dichotomy for
(Jf, ||2/@ll, 2/@)) if and only if (L\ L°°) and/or (L\ Lo°°) is admissible for
@.1). In this case, Yx0 <= W0andboth Yx and Yx0 induce total dichotomies
for(J^,\\y(t)\\,y@)).
Exercise 6.2. The proof of Theorem 6.3 will depend on Lemma 2.3.
Without using this lemma, prove the parts of Theorem 6.3 which do not
involve Lo™ (an analogous result is applicable even if Y = W is not
finite dimensional and Lemma 2.3 is not applicable).
476 Ordinary Differential Equations
Proof. If (L1, Lo°°), hence (L1, L°°), is admissible for @.1), then Yx and
Yx0 induce total dichotomies foryT by (iv) in Theorem 6.1.
Conversely, if there exists a subspace Wo of Y which induces a total
dichotomy for^T, then 7^,0 <= JF0 and 7^,, induces a total dichotomy for
yT by Corollary 2.2. Thus, in order to complete the proof, it is sufficient
to show that if Yx0 induces a total dichotomy for jV, then (L1, L°°) or
(L1, Lo°°) is admissible for @.1).
The proof of this part will use the easier ("only if") portion of Theorem
2.1. Let Z be a subspace of 7 complementary to 7^, so that 7 is the
direct sum 7^,,, ©Z (e.g., if 7 is Euclidean, let Z be the orthogonal
complement of 7^ in 7). Let Po be the projection of 7 on 7^ annihilat-
annihilating Z, and P1 = 1 — Po the projection of 7 on Z annihilating 7^,,. Let
U(t) be the fundamental matrix of @.2) satisfying C/@) = /, and G(t, s)
the matrix function
F.4)
G(t, s) = U(t)P0U-\s) for 0 ^ s ^ t,
G(t, s) = -
for 0 < t < s.
Then, by Theorem 2.1, there exists a constant K such that
F.5) \\G(t,s)\\ ^K for s, t ^ 0.
Let g(t) be an arbitrary element of L\Y). In order to show that @.1)
has an L0°°-solution, put
F.6)
y(i) = G(t, s)g(s) ds.
The integral is absolutely convergent [in view of F.5) and g(t) g L\Y)]'
and defines a solution of @.1). Also \\y(t)\\ ^ Klgh; in particular,
It will be verified that y{t) in F.6) is in LOX(Y). By F.4) and F.5),
lly@ll ^ r\\hot(s)U(t)PoU-\s)g(s)\\ ds+ K (X\\g(s)\\ ds.
Jo Jt
The last integral tends to 0 as t -> oo. For fixed j, the solution
UUJPoU-K^gis) of @.2) tends to 0 as (^-oo, since its initial value
P0U-\s)g(s) e Yx0. Furthermore,
^ K \\g(s)\\ for j ^ 0 and t ^ 0.
Thus Lebesgue's term-by-term integration theorem (with majorized
convergence) shows that the first integral tends to 0 as t -> oo.
Consequently, y(t) g LoocG). This proves the theorem.
The main result on total exponential dichotomies is the following
analogue of Theorem 6.3.
Dichotomies for Solutions of Linear Equations 477
Theorem 6.4. Let A(t) be a matrix of locally integrable (real- or complex-
valued) functions oft^.0 andJf the set of solutions of @.2). Then there
exists a subspace JV0 of 7 which induces a total exponential dichotomy for
(•#", W0II. 2/@)) '/ and only if (L», L") for (p, q) = A, oo) and for some
(p,q) 9* A, oo), 1 ^ p ^ q ^ oo, is admissible for @.1). In this case,
Wo= Yp= Ya0, and (L\ Lox) and (V, U) are admissible for all (p, q),
where 1 ^ p ^ q ^ oo.
Proof. If (L\LX) and (V, L") for some (p,q) ^ A, oo) and 1 ? p,
q ^ oo are admissible for @.1), then Yx = YQ = Ya0 induces a total
exponential dichotomy for«/f by (iv) in Theorem 6.1 and by Theorem 4.4.
Conversely, let a subspace Wo of 7induce a total exponential dichotomy
for^T. Let W1 be a subspace of 7 complementary to Wo and let G(t, s)
be the Green's matrix F.4) defined in terms of projections Po, P1 of 7
onto Wo, Wx annihilating Wx, Wo, respectively. Then, by Theorem 2.1,
F-7) || G(t, 5) || < Ke-vl("sl for s, t^ 0
and some constants K, v > 0. Let g(t) e LV(Y), where 1 ^ p ^ oo. Then
the integral in F.6) is absolutely convergent and defines a solution of @.1).
Thus it remains to show that y(t) e L"(Y) forp ^ q < oo.
Consider first the case that/? =1. As in the last proof, it is easy to see
that t/@ g Lo°°( 7). Also y(t) e L\ 7) for
F.8)
t fY»l«-I ||^)|| rfs ^ f "||g(s)|| rf5 f" c-*
JO Jo J-oo
oo.
Hence y@ e ?,"G) for 1 ^ ^ ^ oo. Also, if ^ = oo, then y(t) g L°°G)
for
Jo
J — oo
e~vlsl
Consider 1 < ^ < q < oo. Let a, /S > 0, a + /S = 1, and <p@ = llg@ll
Then repeated applications of Holder's inequality show that \\y(t)\\" is at
most
K"
Y f+°°
o
e-v|'-'l/'('ap3>
Hence (J||t/(OII9
<s finite and is at most
478 Ordinary Differential Equations
cf. F.8). Consequently, y(t) g L"(Y) forp < q < oo. Since the majorant
remains finite as q-*p or q -> oo, it follows that y{t)sL"(Y) for
p ^q ^ oo. This proves the theorem.
Since F.7) is a necessary condition for Wo to induce a total exponential
dichotomy, it is possible to conclude in this case that many pairs (B, D)
are admissible. Such a result is given by the following:
Exercise 6.3. Let A(t) be a matrix of locally integrable functions for
t ^ 0 and Jf the set of solutions of @.2). Let B, D be Banach spaces in
3T such that
f(t) g D=>ip(t + s)e D for every fixed s ^ 0,
F.9)
9?@ e -
rt+i
ds e Z).
Let there exist a subspace Wo of Y which induces a total exponential
dichotomy for^T. Then (B, D) is admissible for @.1).
7. Applications to Higher Order Systems
In dealing with @.3), it will be assumed that U is a finite dimensional
vector space with elements u and norm ||w||. V is the corresponding
Banach space of linear operators Po of U into itself with the norm ||PJ =
sup \\Pou\\ for ||k|| = 1. Po can, of course, be considered to be a matrix if a
coordinate system is fixed on U.
A function u(t) from a ^-interval to U will be called an (m + \)st integral,
m ^ 1, if u(t) has continuous derivatives, u = u<0), u'(t),. .., u(m)(t) such
that u(m)(t) is absolutely continuous [with a derivative u(m+1)(t) almost
everywhere].
In this section, let Y = Uim+1) = U X ¦ • • X U and if y = («<0), . . . ,
«<m>) G Y, put
G.1) ||y|| = max(||«<o>||,...,||«<»>||).
If/@ g L(U) and Pk(t) g L@) for k = 0,. . ., m, then u(t) is a solution
of @.3) if it is an (m + l)st integral and y(t) = (um(t),. .., «<m»@)
satisfies the first order system @.1) corresponding to g(t) = @, ..., 0, f(t)),
and the m + 1 equations: u{i)l = uu+1) fory = 0, . . . , m — 1 and @.3).
It is clear that in this identification of @.3) with @.1), G.1) implies that
the norm of A (t) g Y satisfies
G.2)
^ max
i,
Correspondingly, the inequality @.5) is applicable.
Dichotomies for Solutions of Linear Equations 479
In dealing with @.3), Tis an operator from L(Y) to L(U). The domain
X)(T) of T is the set of y = (u(t), u\t) «<m)@)> where u(t) is an
(m + l)st integral for ; ^ 0, and «',..., u(m) are its derivatives. Also,
f=Tyis defined by @.3), Py(t) = u(t), and y[0] = y@); thus Y = W,
F= U; cf. § 3. Finally, Jf =^{T) is the set of y(t) where u(t) = Py(t)
is a solution of the homogeneous equation @.4); i.e., Jf is the set of
solutions y(t) of @.2).
The following preliminary lemma has nothing to do with differential
equations.
Lemma 7.1. Let m be a positive integer. Then there exists a constant
cm > 0 with the property that if u(t) g U is an (m + l)st integral on an
interval r^f^r + A, A>0, then its derivatives satisfy
G.3) \\uik)(t)\\ ^
2!
3=0
Cr+A
l dr
for k = 0, 1, . . . , m.
Proof. It is sufficient to verify the corresponding inequality
G.4)
m
fc2\<p(r
fr+A
k |
Jt
| dr,
when (f{t) is a real-valued function on [t, t + A] which is an (m + l)st
integral. If t ^ ^ ^ t + A, let w* g U* be chosen so that ||h*|| = 1 and
the "scalar" product (um(t1),u*)= ||«w0i)ll- If y(t) = Re(u(t),u*),
then G.4) at t = tx implies G.3) at t = tx.
In order to prove G.4), let (p(t) = p(t) + f(t), where p(t) is the poly-
polynomial of degree m satisfying p(t) = (p(t) for ; = t + jA/m,j = 0,. . ., m;
i.e.,
. t2 •
J — K
It is clear that there is a constant cm > 0 such that
3=0
) IT
G.5)
+ jA/m)| for r ^ t ^ t +
and A: = 0, .. ., m. Since y(t) = (p(t) — p(t) vanishes at the m + 1
points t = t + jA/m,J = 0,. . . ,m, it follows that there is a f-value
t' = tk' such that y>m(t') = 0. Hence, for t ^ t^ r + A,
G.6)
Integrating over [t, t + A] gives
^ r
dr.
/V+A /*r+
J |y<wl«/r^A
480 Ordinary Differential Equations
so that, by induction, G.6) implies that
G.7) | vw@l ^ A-* T+A | v(m+1)l dr.
Since p(t) is a polynomial of degree m, y>im+1>(t) = <p{m+1)(t). Hence
(p = p + rp and G.5) and G.7) give G.4).
Lemma 7.2. Let Pk(t) g L(U), k = 0,. .., m, and let a = a(*0) be a
positive integer satisfying
m fs+1
G.8) 2 II^OOII dr < ?a for 0 ^ s S /0.
jt=o Js
Let f(t) g L(C/) and u = u(t) a solution of @.3). Then, for 0 ^ s ^ t0,
s ^t ^s + l,andk = 0,...,m,
ma l*s+l
G.9) ||««>@|| ^ C*2 ||«(s + j/#»a)|| + C2 ||/(r)|| dr,
3=0 Js
where C1 = 2eacma.m and C2 = 3ea.
Remark. Note that f0 > 0 can be chosen arbitrarily and a (hence
C\ C2) independent of t0 if Pk(t) g M@), k = 0,. . ., m and
G.io) 2 2Ip*Im^«;
cf. A.6) and G.8).
Proof. Let s ^ 0. Then G.8) implies that there exists an integer
i = i(s), 0 ^ i < a, such that
m /*r+A
G.H) 2 II^(OII dr ^ * for r = 5 + fa, A = I/a.
Jfc=0 Jr
By @.3), ||w<m+1>@ll ^ S ||Pfc@ll • 11«**>C011 + 11/@11- ^the relations G.3)
are inserted into this inequality, an integration of the resulting inequality
over [t, t + A] gives, by G.11),
/V+A m Pt+A
G.12) \\uim+1)\\ dt ^ cm«mJ,\\u(t + j/moOH + 2 ||/|| dt.
Jt 3=0 Jt
Thus, if y = (u, «A),. . ., «(m))> G.3) implies that
ma Ps+1
\\y(r)\\ ^ 2cmam2 ll«(* + ;/««)ll + 2 11/11 dr.
Therefore, G.9) follows from G.2), G.8), and the corresponding inequality
@.5) with s replaced by t.
Corollary 7.1. Under the conditions of Lemma 7.2,
G.13)
rt+i /*(+
) \\u(s)\\ ds + C2
Jt-i Jt-i
for I ^ t ^ t0 — \ and k = 0, . . ., m.
This follows by integrating G.9) with respect to s for t — 1 ^ s < ;.
Dichotomies for Solutions of Linear Equations 481
Corollary 7.2. Assume the conditions of Lemma 7.2. Letf1(t),f2(t),. . .
be functions in L(U) and u = un(t) a solution of @.3) whenf = fn. Suppose
that f= Urn/,,, u = lim u exist in L(U). Then the function u = u(t)
(up to an equivalence modulo a null set) is a solution of @.3) and u{tf\t) ->
uik)(t), n -> oo, uniformly on bounded subsets of J for k = 0, . . . , m.
This follows by replacing «,/ in G.13) by uv — uQ,fp — fq for p, q =
1,2,....
Theorem 7.1. Let pk(t) e M@) for k = 0,. . ., m and
G.14) hsl(t)Pk(t) e B(U), sup \h$1Pk\B < oo for k = 1, . . ., m.
Suppose that (B, D) is admissible for @.3); i.e., that for every f(t) sB(U),
@.3) has a solution u(t) e D(U). Let YD be the set of initial conditions
2/@) = («@), h'@), . . ., «<m>@)) o/ solutions u(t) gD(U) of @.4). 7feen
YD induces a total dichotomy for Jf (with d° = 0). If, in addition, D is
quasi-full and u(t) ? D(U) is a solution 0/@.4), then \\y(t)\\ —*¦ oo as t -> oo.
The conditions G.14) on Pfc@ are satisfied if, e.g., Pk(t) e Lco(f7).
Note that G.14) is not required for Jt = 0. The condition that (B, D) is
admissible for @.3) is the condition that (B, D) be P-admissible for the
operator T described at the beginning of this section. It does not corre-
correspond to the admissibility of (B, D) for the corresponding linear system
@.1) for we do not consider arbitrary g e B(Y) but only g of the form
g = @,...,0,f),feB(U).
Proof. This theorem will be proved by verifying the applicability of
Theorem 5.1. (Ax) follows from @.5) in view of the identification of @.1)
and @.3); (A2) follows from Corollary 7.2; (A3) is an explicit assumption
of the theorem; (A4) is trivial since W = Y is finite dimensional.
In order to verify (B2e), let e > 1 and u(t), v(t) solutions of @.4). Let
f(t) be a non-negative function on — oo < ; < oo of class Cm which
vanishes except on [0, 1] and satisfies
G.15) \fdt = l.
Jo
Put y>Sf(t) = ryr'(( — s)) and
G.16) u,(t) = u(t) f V«(r) dr + v(t) f"y>K(r) dr.
Jo Jt
Then for k = 1, . . . , m + 1,
G.17) u[k)(t) = uik)(t) \\udr + v{k)(t) rVs€dr
Jo Jt
482 Ordinary Differential Equations
where Ckj is the binomial coefficient k\ //! (k —j)\ It follows that u = ux
is a solution of @.3), where/is
G.18)
m+l fc-1
= 2 l
k=l 3=0
= 1 when
and Pm+i(t) = / is the identity operator.
Since fS€{t) = 0 for t ^ s or t ^ s + e, it follows that yx(t) = z(t) for
O^t^s and j/^0 = 2/@ for f ^ J + e if y = (u,..., u<m>), z =
(v,..., t;<m>)> j/i = («! «!""»)¦ Thus, if s > 0, j/^O) = z@) and Py1 =
ux e D(U). There is a constant c > 0 such that 1^@1 = chse(t)/e for
A: = 0, . . ., m and s, t ^ 0 and e ^ 1. Thus if Tyx = /i,
m+l
G.19) ||ryi|| < 2»+Vr1M0 2 11^@11 • II 2/@ - K0II-
fc=i
Thus (B2e) with e ^ 1 holds for arbitrary d0 > 0 and
G.20) %f@ = 2m+1Ce-\X0 (l + 2 H^(Oll) •
\ k=l I
By virtue of G.14), yS€(t) e B and
/ m
G.21) |^|B ^ 2m+1c€-1 |VIB + 2 \KP*
\ k=l
Lemma 7.2 with/= 0, and the Remark following it show that (B3E)
holds for 8 ^ 1 if Ka(8) = CJA + ma). Finally, (B4A) is a consequence
of @.5) and G.2), since P0(t),.. ., Pm(t) e M(U). Thus Theorem 7.1
follows from Theorem 5.1.
Theorems 6.3 and 7.1 have a curious consequence. Let @.2) be the
first order system obtained in the standard way [as before G.2)] from
@.4). Then we can consider @.1) without the restriction that g is of the
form @ 0./) and we obtain from Theorem 6.3:
Corollary 7.3. Let the conditions of Theorem 1.1 hold. Then (L1, L°°)
and/or (L1, Lq00) is admissible for @.3) if and only if it is admissible for
@.1).
For example, if Pk(t) e LX(C), k = 0, 1,. . . , m, and if some pair
(B, D) is admissible for @.3), then (L1, Lo°°) is admissible for @.1) [and
for @.3)].
Theorem 7.2. Let the assumptions of Theorem 7.1 hold. In addition,
assume either the condition
G.22)
or the condition
G.23) A
\K\\d
k=l
; k\B
as A -> oo
0 uniformly for large 5
as A —>¦ oo. Then YD induces a total exponential dichotomy for Jf {with
6° = 0).
Dichotomies for Solutions of Linear Equations 483
When I/IqaIb/A -» 0 as A -+ oo and Pk(t) e LX(U) for k = 1,. . ., m,
then G.23) holds, since |/i0AIb ^ A + A) |Vb-
Proof. This is a consequence of Theorem 5.2. It suffices to verify the
analogue of condition E.6). By G.21), E.6) follows either from G.14),
G.22) or from G.23).
Exercise 1.1. This exercise concerns real second order equations
G.24) u"+p(t)u' + q(t)u=f(t),
G.25) u" + p(t)u' + q(t)u = 0,
for t ^ 0. Let U be the (Banach) space of real numbers; W the product
space Um = U x U with norm ||ve|| = max flw1!, |w2|) if w = (w\ h>2);
and y the product space Uw with norm ||y|| = max Qy1], \y*\, \y3\, |j/4|) if
V = B/\ 2/2. 2/3> 2/4)- For 8 > 0, let ^Vd denote the linear manifold of
functions y\t) = (u(t), u\t), u(t + 8), u'(t + 8)) e Y for fZ. 0 where
m@ is a solution of G.25). Let the "initial value" of y%t) be /[0] = («@),
m'@)) ? W. Assume that/>(;) is a real-valued function in J: t ^ 0 which
satisfies
G.26)
M0/K0
sup \hsl(t)p(t)\B < oo.
8 = 0
Let ^@ be a real-valued element of L for which there exists a constant C
satisfying the one-sided inequality
G.27)
q(r) dr^
for 0<s<f^
Finally, suppose that (B, D) is admissible for G.24); i.e., if/(f) e i?, then
G.24) has a solution h(*) e D. Let JFp be the 0-, 1-, or 2-dimensional
manifold of initial conditions (w@), u'@)) of D- solutions of G.25). (a) Then,
for sufficiently small 8 > 0, Wp induces a total dichotomy for J/ d. (b) If,
in addition, D is a quasi-full and h(*) ^ D is a solution of G.25), then
HsAOII —*¦ oo as ; —»- oo. (c) If, in addition to the conditions of (a), either
G.22) or
G.28) A-'flJUa + IMOKOIs) -¦ 0 uniformly for large 5
as A —*¦ oo, then Wp induces a total exponential dichotomy for jVd if
<5 > 0 is sufficiently small.
8. P(B, Z))-Manifolds
The main role of assumptions (A3), (A4) is in the proof of Lemma 3.2.
An analogous lemma follows from a similar pair of assumptions (A6), (A7).
The notation in this section is the same as in §§ 3-4.
(A5) If Ty(t) =f(l) and/(/) = 0 for large /, then there exists a unique
solution yx(t) of Ty = 0 such that/,„(/) =« »/(/) for large /.
484 Ordinary Differential Equations
Definition. P(B, D)-Manifold. Assume (A5). Let B, D be Banach
spaces in 3~ and Bx be defined as in (vi) of § 1. A submanifold X of WD
is called a P(B, D)-manifold if there exists a constant C3 with the property
that, for/@ g BJF), there exists a PD-solution y(t) of Ty =/such that
\Py\o ^ Cs\f\B, and yjt) [defined by (A5)] satisfies j/JO] e X. [When
(Aq) is assumed, then even without the assumption yx[0] g X, it follows
that yJO] e WD.]
Exercise 8.1. Assume (A5). Show that there exists a PE, D)-manifold
if and only if (Bx, D) is P-admissible.
(A6) X is a PE, D)-manifold.
Lemma 8.1. Assume (Ax) [or (Aj')], (A2), (A5), and (A6). Le*/, y, yx
be as in the definition of a P(B, D)-manifold. Then there exists a constant
C30 such that \\y[0]\\ <: CM\f\B.
This is a consequence of C.3) and C.4) [or C.3')], C.5!) and A.5).
(A7) X is a P(B, D)-subspace, i.e., a closed P(B, D)-manifold.
Lemma 8.2. Assume (Aj) [or (Aj')], (A2), (A5) and (A7). Let X > 1,
feBJF), 2/@ any PD-solution of Ty = f with yJ0]eX, \\y[0]\\ ^
Arf(X, j/[0]). 77!en f/jere e.ra; constants C5, C30 .wc/i */w* C.7) /joWj (with
X = 1 permitted ify[0] = 0).
The proof is identical to that of Lemma 3.4. If, in addition, to (A^ [or
(A/)]-^,), (Ao) and (A5) are assumed, then Lemma 3.4 is contained in
Lemma 8.2, for X = WD is a P(B, D)-subspace in this case.
Remark. In view of the uses of Lemmas 3.3 and 3.4, it follows that
(A3), (A4) can be replaced by assumptions (A5), (A6), (A7) in the theorems
of §§ 4-5 (and their applications in §§ 6-7) if conclusions of the type " WD
induces a ... dichotomy" are replaced by "X induces a ... dichotomy."
When (A4) holds (e.g., if W is finite dimensional), then (A5) and (A6)
imply, by Exercise 8.1, that (A4) holds with B replaced by Bx. The point
in the Remark above, however, is that a subspace X [which can be smaller
than WD] can replace WD in the conclusions of the theorems of §§ 4-5.
PART II. ADJOINT EQUATIONS
9. Associate Spaces
In this part of the chapter, the concept of an associate space will be
needed. Let ^"# denote the set of normed spaces $eJ satisfying the
additional condition: (d#) if<p(t) g <D, s ^ 0, andyi(t) = y(t + s)for t > 0,
then y>(t) g <D.
Hypothesis. In the remainder of this chapter, it is assumed that B, D
are Banach spaces in &~#.
Let O be a Banach space in 3~#. Let O' be the set of (equivalence classes
modulo null sets of) real-valued measurable functions y>(t), t ^ 0, such
Dichotomies for Solutions of Linear Equations 485
that for all f(t) g O, cp(t)y>(t) g L1. It is easy to see that there exists a
constant a = <x.(ip) satisfying
(9.0)
<p(t)y>(t) dt
Jo
for all
Otherwise, there are elements y,6$ such that I^Jj, ^ 2~n, cpn(t)y>(t) ^ 0,
and yn(i)f(t) dt > 1. Then <p(t) = 2 ^@ g <D, but, by Lebesgue's
Jo
theorem on monotone convergence,
f x<p(t)f(t) dt = 2 f "v»@v@ ^ Z i = oo.
Jo Jo
Let the least constant a satisfying (9.0) be denoted by |yU', so that
^Molvl*' fora11 ^e0-
for all
Lemma 9.1. Let O g^"# be a Banach space. Then O', with the norm
IvU' for V G ^> *s a Banach space in J~ (in fact, in $~#) and is quasi-full.
Exercise 9.1. Prove this lemma. The assumption O g 3~#, rather than
OeJ, assures that O' e J".
It is clear that O' is isomorphic and isometric to a subspace of the dual
space O* of O.
Exercise 9.2. Let 1 ^p ^ 00, \\p + \/q = 1. Show that the associate
space of Lp is L9.
Lemma 9.2. Lef O g^"# Z>e a Banach space and Ya (finite-dimensional)
Banach space with Y* its dual space. Let y*(t)?L(Y*). Then y*(t)e
O'(F*) if and only if there exists a constant a. = a(j/*) such that for every
y(t)€^(Y){y(t)y*(t))€L\ad
This is clearly equivalent to
(9.1)
, y*(t)) dt
(9.2)
In this case, the least constant a. satisfying (9.2) is |j/*U'(y») = |j/*U-
Recall that (y, y*) denotes the pairing ("scalar product") of elements
y g Y, y* g Y*. In the case under consideration (when Y is finite dimen-
dimensional), the nontrivial part of this lemma can be reduced to the 1-dimen-
sional case by the introduction of suitable bases in Y, Y* and examining
the components of y*(t).
Exercise 9.3. Prove Lemma 9.2.
486 Ordinary Differential Equations
10. The Operator T
This part of the chapter concerns an operator T as in § 3, an associate
operator
A0.1) T'y*{t) =f*(t),
and the corresponding null space .#"G").
A0.2) T'y*(t) = 0.
In dealing with the pair T and 7", the following will be assumed:
(Co) T is a linear operator from L(Y) to L(U), i.e., F = U; the ele-
elements y(t) e 3){T) are continuous (so as to avoid ambiguities on null sets);
and y[0] = 2/@), so W = Y. Correspondingly, 7" is a linear operator
from L(Y*) to L(t/*); the elements j/*@ ? 3>(T) are continuous with
j/*[0] = y*@); and «*@ = P'y*(t) is a linear operator from L(Y*) to
L(?/*) with SUP1) = S>G").
(Cj) For each t ^ 0, there exists a bounded bilinear form Vt(y, y*) on
7 X Y* such that for j/(?) ? 3>(T), y*(t) e S>(T'), u(t) = Py(t), u*(t) =
P'y*(t), and for 0 < a < t < oo, the following "Green's relation" holds:
A0
.3) P {Ty, u*) dt - P <«, T'j/*> d/ =
, 2/*@)]/-
For t ^ 0, /?(/) denotes a number satisfying
A0.4) | Vt{y, y*)\ ^ P(t) \\y\\ ¦ \\y*\\ for all y ? Y, y* e Y*.
Note that A0.3) implies that Vt(y(t), y*(t)) is constant on any interval
where Ty = 0, T'y* = 0. In particular, Vt(y(t), y*(t)) is constant on
J for y(t) ? J^(T), y*(t) ? ^G").
Definition. If X is a manifold in y, let Xv be the subspace of Y*
denned by Xv = {y*: V0(y, y*) = 0 for all j/ ? X}. Correspondingly, if
X* is a manifold in y*, let X*v denote the subspace of Y denned by
X*v = {y: V0(y, y*) = 0 for all y* e X*}.
Assumption (AJ* or (Bne)* will mean the analogue of (AJ or (Bne)
with T, Y, F = U, P, B, D, WD = YD, constants Cs,... replaced by
7", y*, ?/*, P', D', B', Y%., constants C}*, ..., respectively, where
B', D' are the associate spaces of B, D. Note the replacement of the
ordered pair (B, D) by (Dr, B').
11. Individual Dichotomies
The next theorem involves the following notion: A set S of real-valued
functions <p{t) ? L will be called small at oo if every cp(t) ? S is small at
oo in the sense that
liminf \w(r)\ dr = 0.
Dichotomies for Solutions of Linear Equations 487
Theorem 11.1. Assume (AJ [or (ADHA4); (B3O<5); (B3<5)*, (B4A)*;
and (Co), (Cj) with a /?(/) ? M in A0.4). Suppose also that /?(*) or B' or D
is small at oo. Then Y%,, induces an individual partial dichotomy for
C#X7") lly*@ll)
It will be clear from the proof that 60, Mo in A.7) of condition (a) of
an individual partial dichotomy can be chosen independent of y*(t).
The assumption that E(t) or B' or D is small at oo will be used only in
the derivation of the condition (a) of an individual partial dichotomy
involving y*(t) ? yTG") with P'y* ? B'. At the cost of allowing Mo to
depend on y* in condition (a) and of making the additional hypothesis
(A5) of § 8, the condition of "smallness at oo" will be eliminated in Theorem
11.3.
In view of Theorem 12.3, it follows that the main point of the theorems
of this section is that no assumption of the type (B2e)* occurs. The
importance of this can be seen by noting that assumption G.14) occurs in
Theorem 7.1 only to insure (B2e) for the operator T there.
Proof of Theorem 11.1.
Condition (a). The condition /?(/) ? M implies that, for 6 > 0,
A1-1)
rt+3
&,(<$) = sup (i
Jt
/?(/¦) dr < oo for t ^ 0.
The condition that /9@ or B' or D is small at oo will be used as follows :
if three functions ?(/), \\y(t)\\, l|y*@ll are in M[e.g., ?(/) ? M, y(t) e D(Y),
y*(t)eB'(Y*); cf. (9.1)] and at least one is small at oo, then
A1.2)
lim inf /?(T) \\y(r)\\ • ||j,*(T)|| = 0.
In order to see this, note that if cp(t) > 0 is integrable on an interval
[t, t + 1 ], the measure of the set of j-values where (p(s) > 3 <p{r) dr is
less than 1/3. If this remark is applied to cp = /?, \\y\\, \\y*\\, it follows that
for any t ^ 0, there is a common /-value r e [t, t + 1 ] satisfying
rt+x
A1.3) ?<T)^3J fi^dr for tp = 0, \\y\\, \\y*\\.
By the assumption cp e M, the integral on the right is bounded for t ^ 0
and, for at least one of the functions, <p(t) -»¦ 0 for suitable choices of
t = tn-+ oo as n ->• oo. Hence A1.2) follows.
Let y*(f) ? jV(T') with P'y* e B'(U*). Let s ^ 0 and f(t) any element
of B(U) with compact support on t ^ s + 3<5. By Lemma 3.2, Ty = /
has a ^/)-solution y(/) satisfying C.5), hence \Py\n ^ Cs|/|w. By
488 Ordinary Differential Equations
(Bso.5) and (B32<5)*, it is seen that y(t)eD(Y), y*{t)e D'(Y*). Since
T'y* = 0 and Ty = /, the Green relation A0.3) gives
T<f,u*)dr=[Vt(y(t\y*(t))]J,
f
where u*{t) = P'y*{t). The left side is independent of a ^ s + 3<5 and
of large t [since fit) = 0 for t ^ s + 36 and large t]. Thus A0.4) and
A1.4)
s+33
(f,u*)dr
The argument leading to A1.2) shows that for a suitable choice of a,
s ^ a ^ s + 3<5,
The first factor on the right is at most i36)-ipoC6) by A1.1); the second
fact is at most i36)~x \ho3d\D, \hs3ey\D by A.5), hence, at most
by (B3O<5). Since \Py\D ^ C3 \f\B, the right side of A1.4) can be replaced
by K' \f\B \\y*{a)\\, where K' = K'(S) is a constant and a is some point
of j ^ a ^ s + 36.
Since fit) is an arbitrary element of B(U) with compact support on
t ^ s + 36, Lemma 9.2 implies that
A1.5) Ih^u*^, ^ K'sup \\y*(a)\\ for s ^ a ^ s + 3d;
or by (B,d)*,
A1.6) \hs+iSy*\B,^K'K3*(d)sup\\y*(a)\\ for s<o?s + 3d.
Arguments involving (B4A)* similar to those used in the proof of Theorem
5.1 give condition (a) of a partial dichotomy for (^V(T'), \\y*(t)\\).
Condition (b). Let z*(t) e jV{T'\ P'z* = v*,andv*$ B'( Y*). Let fit)
be any element of -S([/) with support on [0, s] and yit) a solution of Ty =f
supplied by Lemma 3.2. Then t ^ s, Green's relation, and A0.4) give
(/, v*) dr
||z*@)|
\\yir)\\ •
The use of C.5) and the arguments in the derivation of A1.5) show that
A1.7) \hOsv*\B, ^ C30i8@) ||z*@)|| + K' sup ||z*(t)||
for s + 6 ^ t < s + 3d,
where K' = K\S) is the same as in A1.5).
Dichotomies for Solutions of Linear Equations 489
Since v* <? B'i Y*), it follows from the fact that B' is quasi-full (Lemma
9.1) that \hOsv*\B,—>- oo as s-*- oo. Thus there exists an s0 depending on
the solution z*{t) such that \hOsv*\B, ^ 2C3o/S(O) ||z*@)|| for s > s0. Then
by A1.7)
\Kjo*\b- ^ 2A" sup ||z*(t)|| for s + 6 ^ t < s + 3d, s > s0.
Using (B»d)*,
A1.8) |/2Os2*|^ ^ 2AX,*(E) sup ||z*(t)||
for s + 26 ^ t < s + 56, s ^ sg + 6.
The proof can be completed by the arguments in the proof of Theorem 5.1.
Theorem 11.2. Let the conditions of Theorem 6.1 hold. In addition,
assume either
A1.9)
¦ 0 or \h
0ts.\B"
oo
as A —> oo.
Then Y%, induces an individual exponential dichotomy for (.#"G"),
||2/*(/)||), where 6°, Mv v do not depend on y*it) in condition (a).
Exercise 11.1. Prove this theorem.
The elimination of the "smallness at oo" condition will depend on the
following lemma, which involves assumption (A5) of § 8. In this lemma,
the notation of (A5) is used; i.e., if Ty =/and Tyr —fu where/,fr vanish
for large t, then y^, ylx, are the corresponding solutions suppplied by (A5).
Lemma 11.1. Assume (Ai)-(A5) and (CoHCj). Let y*it) be a P'B'-
solution of T'y* = 0. Then there exist constants C3 and C30, depending on
y*iO), such that for any fit) e BMiU), Ty = f has a PD-solution yit)
satisfying C.5) and VoiyxiG), y*iO)) = 0; i.e., if{y*iO)} is the l-dimensional
manifold in Y* spanned by y*i<3), then YD n {y*iO)}v is a PD-manifold
for C.1).
Proof. We can suppose that there is a PD-solution yoit) of Ty = 0
satisfying a = VoiyoiO), 2/*@)) ^ 0. For if not, Lemma 11.1 follows from
Lemma 3.2. Let yit) be the solution of Ty =/supplied by Lemma 3.2
and put
A1.10)
= yit) - Voiy
Then y^t) is a PD-solution of Tyr =/satisfying K0(yloo@), 2/*@)) = 0.
By Green's relation and T'y* = 0,
A1.11) f(/, u*) dt = [Vtiyiit), y*it))V
Jo
if u* = P'y*. Since/vanishes for large t, (A5) implies that y^t) = ylx,i0
for large /, so that Kt(y,(/), y*(t)) = Kt(yl00(/), y*(t)) for large /. But the
490 Ordinary Differential Equations
latter expression does not depend on t (since Tyl00 = 0, T'y* = 0) and
is therefore V0(ylm@), y*@)) = 0. Thus A1.11) shows that
A1.12)
Thus, by A1.10),
(f, u*)dt = -vo(yi(% y*
{f,u*)dt
W0(y@), y*@))\.
The right side is at most |/|^(|P'y*|^ + |S@)C30 ||y*@)||), by A0.4) and
Lemma 3.2. Hence A1.10) shows that
\Pyi\D ^ \py\D + \f\B{\P'y*\B' + mc30 ll2/*@)ll)a-1 \Pyo\D.
By C.5i) in Lemma 3.2, \Py\D ^ Ca\f\B. Thus the analogue of C.5J
holds with y replaced by y1 and C3 replaced by the constant C3 +
(\P'y*\B' + ?@)C30 ||2/*@)||)a-1 \Pyo\D. The analogue of C.5.) follows as
in Lemma 8.1.
Theorem 11.3. /« Theorems 11.1 or 11.2, replace the assumption that
"P(t) or B' or D is small at oo" by the assumption (A5). Then the conclusions
of these theorems remain valid.
Exercise 11.2. Prove this theorem. In view of the remarks following
Theorem 11.1, only condition (a) need be considered. The proof of
condition (a) here is similar to (but simpler than) the proof of the corre-
corresponding condition in Theorem 11.1 or 11.2. The solution of Ty = f
supplied by Lemma 11.1 is used in place of that given by Lemma 3.2.
12. P'-Admissible Spaces for T
The object of this section is to show that, under suitable conditions, if
(B, D) is P-admissible for T, then (Dj, B') or (D', B') is P'-admissible for
T'.
Lemma 12.1 Assume (A1)-(A2), (A5)-(A6); (A5)*; and (C0)-(Ci).
[In particular, X is the P(B, D)-manifold in (A6).] Iff *{t) e Dj(U*) and
y*(t) is a solution of T'y* =/* with yM*@) e Xy, then y*{t) is a P'B'-
solution and
A2.1) \P'y*\B- ^ C3 \f*\D, + P@)C30 ll2/*@)||,
where C3, C30 are the constants in the definition of a P(B, D)-manifold X
and Lemma 8.1. In particular Xv c yjj,.
Proof. Let/? B^U) and y(t) the solution of Ty =/supplied by (A6),
so that yn@) e X and C.5) holds. For large t, Vt(y(t), y*(t)) = Vfyjf),
y»*@) = ^oOoc(O), 2/oo*@)) = 0. Thus Green's formula gives
(f,P'y*)dr
{Py,f*)dr
W0(y@),
Dichotomies for Solutions of Linear Equations 491
where the right side is at most \Py\D \f*\v + ^@) ||y@)|| • ||y*@)|| ^
\f\s(C3 \r\j, + |8@)C30 ||2/*@)||) by C.5). Hence the assertion follows
from Lemma 9.2.
Let y* e Y*. Then (CJ implies the existence of a unique x* e Y* such
that the linear functional V0(y, y*) on y is representable in the usual
pairing of Y, Y* as V0(y,y*) = (y,x*) for ally ? Yand \\x*\\ ^ ^@) ||y*||;
i.e., there is a unique linear map x* = Sy* of Y* into itself satisfying
A2.2)
Vo(y, y*) = (y, Sy*) for y e Y, y* e Y*.
In this section, the following will be assumed:
(C2) The (unique) bounded linear map S: Y* -*¦ Y* defined by A2.2) is
onto (and hence has a unique inverse S^1 defined on all of Y*).
It is clear that for a manifold X <= Y, we have Xv = S^X0, where X°
is the usual annihilator of X; i.e., X° = {y* e Y*:(y, y*) = 0 for all
yeX}.
Lemma 12.2. Assume (A,)-(A5); (A5)*; and (C0)-(C2). Let /*(/) ?
DJ(U*) and y*{t) a solution of T'y* =f* with ym*@) e YDV. Then
d(YDv,y*@)) ^ C4 H.S'-1!! • \f*\jy, where C4 is the constant in C.6) of
Lemma 3.3 and \\S~1\\ is the norm of the operator S from Y* to Y*.
Proof. Let y{t) be a PZ)-solution of 7^ = 0. Then KtO/(t), 2/*(t)) =
K(y(T)> 2/oo*(T)) f°r large t. By Green's relation, the last expression is the
constant V0(y@), ^""(O)) = 0. By Green's formula,
Wo(y(O), y*@))\ =
f
Jo
(Py,f*)dt
^ \Py\D \P\d-
From the inequality \Py\D ^ C4 ||j/@)|| in Lemma 3.3 and from A2.2), it
follows that \{y@), Sy*@))\ ^ C4 \\y@)\\ • \f*\D, for all y@) e YD. Thus
.Sj/*(O) considered as a bounded linear functional on YD (i.e., as an element
of the dual space YD*) has a norm not exceeding C4 l/*^- Since YD*
is the quotient space Y*IYD° in which the norm of an "element" y* is
d(YD<>,y*) it follows that </( V> ^*@)) ^ Q l/%- Hence YDV =
S-ijy implies that </(y/, j/*@)) ^ HS^H d(YD\ Sy*@)), and so
Lemma 12.2 follows.
Theorem 12.1. Assume (A MK); (C0)-(C2); (A5)*; andthat T'y =f*
has a solution y*(t) for every f*{t) e Dm'(U*). Then (DJ,B') is P-
admissible for T; in fact, YDV is a P'(D', B')-subspace for T (with
permissible constants analogous to C3, C30beingC3* = C3 + XCJO^ \S~X\
and C3*o = AC4 \S~X\ for any fixed X > 1).
Proof. Let f*(t) e DX'(U*). Then T'z* =f* has a solution z*(t)
which vanishes for large t. [For if z*(t) is any solution of T'z* =/*,
then, since zw*(t) exists by (As)*, the desired solution is z*(t) — z^*(t).]
492 Ordinary Differential Equations
Hence zx*(t) = 0. In particular, 0 = zx*@) e YDr, so that, by Lemma
12.2, </(*V, **@)) :g Q US!! • |/%<.
Let X > 1 be fixed and decompose the element z*@) e Y* into z*@) =
*i* + 2/o*. where xx* e YDV and \\yo*\\ ^ X d(YDr, z*@)). Consequently
112/0*11:^3*0
30 1
where C3*o = AQ
Since a;x* ? y^/ <= y*, by Lemma 12.1, Tx^it) = 0 has a solution
*!*(/) beginning at x1* for / = 0. Let y*{t) = z*(/) - x-^{t), so that
7V* =/* and yx*{t) = -*!*(/). Hence j/oo*@) a^* 6 YDV. In
addition j/*@) = z*@) - xx* = j/0*. By Lemma 12.1,
Consequently, I-P'j/*!^ = Cz*\f*\D., where C3* is the specified constant.
This proves the theorem.
Theorem 12.2. Assume (AjMAg); (C0)-(C2); and that for every
yo*eY* and f* e D\U*\ T'y*=f* has a solution y*{i) satisfying
j/*@) = y0*. Then{D',B')isP'-admissibleforT''. {Furthermore,permissible
constants analogous to C3, C30 are C3* = C3 + /9@)C30C4 ||S—a|| and
C3*0 = C4 US1|.)
Proof. Let /* e D'(U*). It must be shown that Ty* =/* has a P'B'-
solution y*(t). Let y(t) ejV(T) with j/@) e YD. Put
A2.3)
= | {Py,f*)dt.
0
Then \<p(y@))\ ^ \Py\D |/*|^ ^ C4 ||y@)|| • |/*|fl. by Lemma 3.3. In
other words cp(y) is a bounded linear functional on YD with norm ^
QI/*Id' which, therefore, has an extension to Y with the same norm.
Consequently, there is an element x0* e Y* such that ||a;0*|| ^ Ci\f*\D'
and ?)(j/) = <j/, x0*) for all j/ ? Y. By (C2), J/O* = S^* exists, so that
A2.4)
for all y ? y and
A2.5)
= V0(y, 2/0*)
\\y*\\ < Q US!! • \f*\n-
By assumption, 7^/* =/* has a solution j/*@ satisfying j/*@) = j/0*.
Let fe BX{U), y(t) a PZ»-solution of Ty =/supplied by Lemma 3.2.
By Green's formula applied to y^t) and y*(t),
), y*(t)) = V0(yoo@),
-I {Py*J*)dr
o
for large t. Since y{t) = y
for large t,
Dichotomies for Solutions of Linear Equations 493
) for large t, it follows from A2.3), A2.4) that,
Vt(y{t), y*(t)) =j\pyOB,f*) dr-+0 as t -* oo.
Thus Green's formula applied to y(t) and y*(t) gives
P</> P'y*) dr = P<Py,/*> ^ - F0B/@), j/*@)).
Jo Jo
»-"
Consequently, C.5) and A2.5) imply that
f
Jo
(f,P'y*)dr
^ \f\B(c3 + /9@)c30c4 us-1!!) I/*
D'-
By Lemma 9.2, P'j/* eB' and IP'y*^ ^ (C3 + /9@)C30C4 US!!) |/
In view of A2.5), this proves the theorem.
The usefulness of theorems like Theorems 12.1, 12.2 will be illustrated
by an application of Theorem 12.1 and of Theorem 11.3 with T, T
interchanged. Note that D' = (Z)J;, so that the second associate space
D" = (D'Y of D is the same as (DJ)'.
Theorem 12.3. Assume the conditions of Theorem 12.1; (B30d)*; (B3d)
for a fixed d>0 with D replaced by D"; (B4A); /?(*) ? M in A0.4).
Then YD~ induces an individual partial dichotomy for^(T), and \\y(t)\\ —*¦ oo
as t -*¦ oo ify(t) is a non-PD"-solution of Ty = 0. If in addition,
A2.6)
¦0 or \h0
¦ oo
as A
oo,
then YD, induces an individual exponential dichotomy for ^V(T).
If the condition "/S@ or B" or D' is small at oo" is assumed, then the
constants in condition (a) of the dichotomies do not depend on the solution
y{t) involved.
Exercise 12.1. Verify Theorem 12.3.
13. Applications to Differential Equations
The systems formally adjoint to @.1), @.2) are
A3.1) y*>+A*(t)y* = -g*(t),
A3.2) y*' + A\t)y* = 0,
where A*(t) is the complex conjugate transpose of A(t); cf. § IV 7. Let
7" be the operator associated with @.1) with Py{t) = y(t) and y[0] = y@)
and 7"" is the negative of the corresponding operator associated with A3.1)
494 Ordinary Differential Equations
[i.e., T'y* = —(y*r + A*(i)y*) = g*], then T, T' are associate operators
in the sense of § 10. The corresponding Green identity is
A3.3) f (Ty, y*) dr - f (y, T'y*) dr = [(y(t), y*{t))]J,
J a J a
as can be seen by differentiating with respect to t. Thus Vt(y, y*) =
(y, y*). Clearly, (C0)-(C.s) hold [with 0@ = 1 and Sy* = y*] and
Theorem 12.2 implies
Theorem 13.1 Let A(i) be a matrix of locally integrable functions for
t ^ 0. Suppose that (B, D) is admissible for @.1). Then (Dr, B') is admis-
admissible for A3.1).
Thus the theorems of § 6 become applicable to A3.1).
In order to consider the systems adjoint to @.3) and @.4), suppose that
Pk{i) is a kth integral (i.e., has k — 1 absolutely continuous derivatives).
The equations formally adjoint to @.3), @.4) are
A3.4)
=f*(i),
A3.5)
where
A3.6)
an asterisk on a matrix denotes complex conjugate transposition, and
indices in parentheses denote differentiation; cf. § IV 8(viii). For y =
(m@>, mA>, . . ., «(m>) e Y = t/(m+1> and y* = (m*@>, . . . , «*(m>) e Y*, put
A3.7) Vt(y, y*)=
2
jfc=0
where Pm+i(t) = /is the identity operator. This is a bilinear form in y, y*.
The following Green's formula is readily verified
A3.8) (\f, u*) dr - (\u,f*) dr = [Vt{y(t), y*{t))]J,
Jo Jo
if w(/), w*(/) are solutions of @.3), A3.4), respectively.
A rearrangement of the sums in A3.7) shows that
m / m—k m—k
A3.9) Vt(y,y*)=2Wk\
\
k=a \ i=0 i=i
Thus, if y* = (vM0\ ..., v*{m)) e Y* is arbitrary, the m + 1 equations
m—k m—k
I I(
«=0 i=i
Dichotomies for Solutions of Linear Equations 495
can be solved recursively for k = m, m — 1, .. . , 0 since Pm+1 = I.
For t = 0, this implies the analogue of condition (C2) in § 12.
Let T', P' be the operators associated with A3.4) in the same way that
T, P are associated with @.3) in § 7. The analogue of @.5) shows that
(B4A)* holds for T jf Qk* eM(U*) [i.e., QkeM(U)]; also, Lemma
7.2 shows that the same condition on Qk implies (B3S)*. Thus Theorem
11.3 gives
Theorem 13.2. Let Pk(t)eL@), k = 0,...,m be a kth integral;
Qk(t) e M{U)for k = 0, . .., m; and let there exist a /?(*) 6 M such that
A3.7) satisfies A0.4). Let (B, D) be admissible for @.3). Then Y%, induces
an individual partial dichotomy for ^V(T'); andifu*(t) is a non-B'-solution
o/A3.5), then \\y*(t)\\ -»• oo as t -+ oo. //, in addition, A1.9) holds, then
Y*y induces an individual exponential dichotomy for jV{J").
An immediate corollary of Theorem 12.2 is the following:
Theorem 13.3. Let Pk(t)eL(U), k = 0, .. ., m, be a kth integral and
let (B, D) be admissible for @.3). Then (D1, B') is admissible for A3.4).
Thus, under the appropriate conditions on the coefficients Qk*(t) of
A3.4), the theorems of § 7 become applicable. An application of Theorem
12.3 gives individual dichotomies for ^V{T) without a condition of the
type G.14), but with a condition on Qk*(t).
Theoreml3.4. Let the conditions ofTheorem 13.3 hold. LetPk(t) e M@),
Qk e M@)for k = 0, 1, ..., m and let there exist a /?(/) e M such that
A3.7) satisfies A0.4). Then YD~ induces an individual partial dichotomy
for ^V{T); and if u(t) is a non-D"-solution of @.4), then \\y(t)\\ -*¦ oo as
t —*¦ oo. If, in addition, A2.6) holds, then YD~ induces an individual ex-
exponential dichotomy for ^V{T).
It is clear from A3.7) that a function satisfying A0.4) is
for a suitable constant cm depending only on m. Thus if
A3.10) P(kj)(t)eM@) for 0^j<k-\, 1 ^ k ^ m
holds, then 0@ e M, also Qk e M@) for k = 1, . . ., m. Thus A3.10)
and the condition Pa, Qo e M(U) imply that the conditions of the second
sentence ofTheorem 13.4 hold.
Note that if Po e M(U) and A3.10) hold with j = k also permitted,
then Qo e M@). But in this case, Pk, Qk e Lm@) for k = \,...,m,
and Theorem 13.4 is contained in the theorems of § 7. (This statement
concerning L°° follows from the fact that if <p(t) is absolutely continuous
for / ^ 0 and f, <p' e M, then cp e L°°,)
496 Ordinary Differential Equations
Exercise 13.1. If p{t) is absolutely continuous, the equations formally
adjoint to the real equations G.24), G.25) are
A3.11) u*" -p(t)u*' + [p'(t) + q(t)]u* =f*(t),
A3.12) u*" - p{t)u*' + [p'(t) + q(t)]u* = 0.
The corresponding Green's relation is
A3.13) (Tfu* dr - [f*u dr = Vt{w(t), w*(t))\J,
J a J a
where w(t) = (u, «'), w*{i) = («*, «*'), and
A3.14) Vt(w, w*) = [p(t)u* - u*']u + u*u'.
Using the notation of Exercise 7.1, let jVt* denote the linear set of
functions
y*\t) = (u*(t), u*\i), u*(t + d), u*\t + 8)) e y*,
where u*(t) is a solution of A3.12). Let the "initial value" of y*\t) be
y*6[0] = (m*@), m*'@)) ? W*. Let p(t) be a real-valued, absolutely con-
continuous function and q{t) e L such that
A3.15) hsl(t)p(t) e D' and sup \hsl(t)p(t)\D, < oo,
and, for some constant C',
for 0<s<t<s+l.
A3.16) p(t) - p(s) + \ q(r) dr ^ C
J
Suppose, finally, that (B, D) is admissible for G.24). (a) Then, for suf-
sufficiently small 6 > 0, W%- induces a total dichotomy ioxjVb* with 0° =
0. (b) If, in addition, either \hOA\B, -* oo or A-Wh^]D, + \hsAp\D>) -> 0
as A -* oo uniformly for large s, then W%- induces an exponential dichot-
dichotomy ioxjVd* for sufficiently small 6 > 0.
Exercise 13.2. Let p(t) e M be a real-valued, absolutely continuous
function on / and q(t) e L such that there exist constants C, C" satisfying
G.27) and A3.16). Let E, Z») be admissible for G.24). Then, for small
d > 0, (a*)!^!' induces an individual partial dichotomy for^T/; and
if«*@^-S'isasolutionofA3.12), then ||2/*<5(/)|| -* ooas/-* oo; (a) WD-
induces an individual partial dichotomy for JT&; and if u(t) $ D" is a
solution of G.25), then ||^@ll -> oo as t -* oo; (Z)*) if A1.9) holds, then
W*B induces an individual exponential dichotomy for jV6*\ (b) if
F.3) holds, then WD, induces an individual exponential dichotomy for
Dichotomies for Solutions of Linear Equations 497
14. Existence of PZ)-Solutions
Lemma 12.1 has the following consequence:
Theorem 14.1. Let A(t) eL(&) and let (B, D) be admissible for @.1).
If {0.2) has no solution y{t) ^ 0 in D{ Y), then every solution y*(t) of A3.2)
is in B'(Y*).
For X= YD is {0}, hence YDV = Y* c y*,; i.e., Y* = Y%,.
In some situations, it is easy to deduce for @.2) the existence of solutions
y{t) $ D( Y). Suppose that there is a dichotomy [or exponential dichotomy]
for the solutions (e.g., let the theorems of § 6 be applicable). If all solutions
y(t) of @.2) are in D(Y), then all are bounded [or exponentially small] as
t -* oo. The same is then true of det U(t) if U(t) is a fundamental matrix
of @.2). Since
det 1/@ = [det 1/@)] exp (tr A(s) ds,
Jo
the integral must be bounded from above [or bounded from above by a
negative constant times t]. Thus if tr A(s) ds does not satisfy this
condition, then not all solutions y(t) of @.2) are in D(Y).
Analogues of Theorem 14.1 and the remarks following it hold if @.1)
is replaced by @.3). This will be illustrated for scalar second order
equations, first in the formally self-adjoint form:
A4.1) (p{t)u')' +q(t)u=f(t),
A4.2) (p(t)u')' + q(t)u = 0.
Let C denote the Banach space of complex numbers.
Theorem 14.2. Let p(t), llp(t), q(t) be locally integrable complex-valued
functions on t ^ 0. Suppose that (B, D) is admissible for A4.1). Then
either A4.2) has a solution u(t) ^ 0 in D(G) or every solution u(t) of A4.2)
is in B'(G).
This is a consequence of Lemma 12.1 for if u(t) is a solution of A4.1)
and v{t) a solution of A4.1) with/(f) replaced by g(t), then the correspond-
corresponding Green's relation is
J a
(fv - gu) dt = [P(t)(u'(t)v(t) - u(t)v'(t))]aT;
cf. the proof of Theorem 14.3.
A variation on the proof of Lemma 12.1 gives a result on non-self-adjoint
equations
A4.3) u" + p{t)u' + q(t)u =
A4.4) u" + p(t)u' + q(t)u = 0
498 Ordinary Differential Equations
Theorem 14.3. Let p(t), q(t) be locally integrable complex-valued
functions on t ^ 0. Suppose that (B, D) is admissible for A4.3). Then
either A4.4) has a solution 0 ^ u(t) e D(G) or every solution u(t) o/A4.4)
satisfies u(t) exp p(r) dr e B'(G).
Jo
Proof. Suppose no solution u(t) # 0 of A4.4) is in D(G). Let f(f) e
B„(€). Then, by assumption, A4.3) has a solution m = v(t) e Z»(C), which
is necessarily unique and vanishes for large t. Let A4.3) for u = v be
written as
(?(/>')' +
> = ?@/@. where ?@
= exp ,
Jo
dr.
If u(t) is any solution of A4.4), a corresponding relation holds with
E(t)f(t) replaced by 0. Thus Green's relation gives
f
Jo
E(r)f(r)u(r) dr = ?@)(M'@>@) - u@)v'@)).
The analogue of inequalities C.52) for y(t) = (v(t), v'(t)) gives an inequality
of the type
^ E(r)f(r)u(r) dr
where C depends only on C30, ?@), (m@), m'@)) and choice of norm on
C X C. In view of Lemma 9.2, the assertion follows.
•
Dichotomies for Solutions of Linear Equations 499
part II. Most of this part of the chapter is an adaptation of results and methods of
Schaffer [2, VI] dealing with first order systems (on arbitrary paired Banach spaces
Y, Y'). The treatment follows that of Hartman [25]. The idea of obtaining an individual
dichotomy for the "adjoint" equation as in Theorem 11.1 without using "test" functions
[say as supplied by assumptions of the type (B^)* or (B2e)*] is due to Hartman. (It
should be mentioned that "associate spaces" have been discussed by Luxemburg and
Zaanen; see Schaffer [1] for references and pertinent results.)
Notes
part I. The idea that "admissibility" of some pair B?, D) leads to some sort of a
"dichotomy" for solutions of the homogeneous equation occurs in a paper of Wintner
[18] on a self-adjoint equation of the second order with B = D = Z,2 (see also Putnam
[1] and Hartman [8]) and in a paper by Maizel' [1] dealing with first order systems and
B = D = Z,00. The main results (§ 6) on the system @.2) are due to Massera and
Schaffer [1, particularly, IV] and Schaffer [2, VI]. For the first order differential
operator Ty = y' — A{t)y, these authors have written a series of papers treating
systematically many of the questions considered in this chapter. An attempt to obtain
a unified treatment for @.1), @.3), and for other problems (such as those involving
difference-differential equations) led to the introduction of the more general operators
T of § 3 in Hartman [25]. The results (§ 7) on solutions of the higher order system
@.3) are due to Hartman [25]. The procedures and arguments of Part I are based on
those of Massera and Schaffer. (The papers of Massera, Schaffer, and Hartman just
mentioned deal with the case when dim Y ^ oo.) The classes F, •?"# of linear spaces of
§§ 1,9 are discussed by Schaffer [1]. The definitions of dichotomies in § 1 vary somewhat
from those of Massera and Schaffer. The results of § 2 are adapted from discussions
in Massera and Schaffer [1, IV] and Schaffer [2, VI]. The notion of a (B, D)-manifold
in connection with linear systems @.1) (with P = I) is used in Schaffer [2, VI].
Chapter XIV
Miscellany on Monotony
This chapter contains miscellaneous results related only by the fact that
one of the main features of either the assumptions, conclusions, or proofs
depends on the notion of "monotony."
Part I deals principally with linear systems of differential equations.
Most of the conclusions of the theorems are to the effect that some functions
of particular solutions are monotone. Some of these results, in conjunction
with the theorem of Hausdorff-Bernstein, imply that certain solutions can
be represented as Laplace-Stieltjes transforms of monotone functions.
Part II deals with a very special problem. It is concerned with a singular,
boundary value problem related to a particular third order, nonlinear
differential equation. This problem had its origins in boundary layer
theory in fluid mechanics.
Part III is a discussion of the stability in the large for a trivial or periodic
solution of a nonlinear autonomous system. An interesting feature of the
proof of Theorem 14.2 is that it essentially reduces a ^-dimensional
problem to 2-dimensional considerations by dealing only with 1-parameter
families of solutions at any one time.
PART I. MONOTONE SOLUTIONS
1. Small and Large Solutions
Consider a system of linear differential equations
A.1) y' = A(t)y
for a vector y = (y1,..., ]f) with real- or complex-valued components on
0 < t < co (^ oo). Let ||2/|| denote the Euclidean norm
A.2) 112/11 =(|2/T + --- + I/I2)'X(^O).
This section concerns systems A.1) with the property that, for every
solution 2/@' either
A.30) lim \\y(t)\\ < oo exists (and is finite)
F
Miscellany on Monotony 501
or
A.3.)
lim
< oo
exists.
(For example, a sufficient condition for A.30) or A.3^), respectively, is that
the Hermitian part AH(t) = \[A(i) + A*(t)] of A(t) be nonpositive definite
or nonnegative definite for 0 < t < co; so that \\y(t)\\ is nonincreasing or
nondecreasing.)
When A.30) [or A.3^)] holds for all solutions, it is natural to ask whether
there is a solution yo(t) satisfying
(lAcoo]) lim ||2/0@ll = 0 [orlim \\yo(t)\\ = oo].
t-><o
The next theorem gives an answer to this question.
Theorem 1.10[00]. Let A(t) be a d x d matrix with {complex-valued)
continuous entries for 0 ^ t < co (^ oo) such that A.30) [or A.3^)] holds
for every solution of A.1). Then a necessary and sufficient [or sufficient]
condition for A.1) to have a solution yo(t) ^ 0 satisfying A.40) [or A.4^)]
is that
A.50[oo]) Re tr A(s) ds ->- — oo [or oo] as t
¦ co.
Although A.50) is necessary and sufficient for the existence of a solution
yo(t) satisfying A.40), A.5.J is not necessary for the existence of a solution
satisfying A.4J. The second order equation u" + 3m/16/2 = 0 for t > 0
has the linearly independent solutions u = tVi and u = tVi\ cf. Exercise
XI 1.1 (c). Thus if this equation is written as a system A.1) for the binary
vector 2/ = (m, m'), then every solution satisfies \\y(t)\\ -»¦ oo as t ->- oo. But
for this system tr A(t) = 0, so that A.5.J does not hold.
Proof of Theorem l.l0. In this proof, "limit" means "finite limit."
Since it is assumed that A.30) holds for every solution, it follows that if
2/i@> 2/2@ is any pair of solutions of A.1), then the limit of the scalar
product 2/i@ • 2/2@ exists as t -* co. This follows from the relations
+ 2/2II2 =
2 Re 2/x-2/2
where i2 = — 1.
Let Y(t) be a fundamental matrix of A.1). Since the elements of the
matrix product Y*(t) Y(t) are complex conjugates of the scalar products of
pairs of solutions of A.1), it follows that C= lim y*@^@ exists as
t -» w. In particular, det \Y*Y\ = |det Y\2 tends to a limit as t -> co.
If c is an arbitrary constant vector, the general solution of A.1) is
2/ = Y(t)c and
II!= Y(t)c- Y{t)c= Y*{t)Y{t)c ¦ c.
500
502 Ordinary Differential Equations
Hence
Since C is Hermitian and non-negative definite, Cc0 ¦ c0 = 0 can hold
for c0 jL 0 if and only if Cc0 = 0. (Note that for a Hermitian matrix C,
the minimum of Cc-c, when ||c|| = 1, is the least eigenvalue of C.)
The equation Cc0 = 0 has a solution c0 ^ 0 if and only if det C = 0. Thus
A.1) has a solution 2/0@ ^ 0 satisfying A.40) if and only if
det C = lim |det 7@12 is 0.
In view of Theorem IV 1.2,
A.6) det Y(t) = [det 7@)] exp | tr A(s) ds.
Thus Theorem l.l0 follows.
Exercise 1.1. Prove Theorem Ll^.
The next theorem concerns a linear system of second order equations
A.7) y" + A(t)y = 0
for a vector y = (y1,. ..,yd).
Theorem 1.20[00]. Let A(t) be a d x d matrix of continuous, complex-
valued functions for 0 ^ t < a> (^ oo) with the properties that A(t) is
Hermitian, positive definite, and monotone (i.e.,
A.80[oo]) A(t)^A(s) [or A(t) ^ A(s)] for t^s
in the sense that A(t) — A(s) is non-positive [or non-negative] definite).
Then, ify(t) is a solution of A.7),
(l-W d{y ¦ y + A-Xy' ¦ y'} ^ 0 [or ^ 0]
and
A.100[oo]) d{Ay -y+y'-y'}^0 [or ^ 0].
If, in addition,
(l.ll0[oo]) det ,4@->-0 [or oo] as t -+ m,
then A.7)possesses a pair of(^0) solutions yo(t), y^t) satisfying
A.120[oo]) 2/o • 2/o + ^~V ' Vo ~* °° [or 0],
A.130[oo]) ^2/1 • 2/i + 2/i' • 2/i' -* 0 [or oo].
When A.7) represents Euler-Lagrange equations in mechanics, then
the expression {. . .} in A.10) is essentially "energy." The first part of the
theorem implies that if A{t) is positive definite and monotone, then the
"energy" is monotone along every solution. It will be undecided if
Miscellany on Monotony 503
2/oW> 2/i@ are linearly independent solutions. For the 1-dimensional case
of these theorems, see § 3. The proofs will depend on some facts about
Hermitian, positive definite matrices given by the following exercise.
Exercise 1.2. Lef A be an Hermitian positive definite matrix, (a)
Verify that A has an Hermitian, positive definite square root A'A, i.e.,
(A'AJ = A. (b) Show that A'A is unique, (c) Show that if A = A(t) is
a continuously differentiable function of t, then A'A(t) is continuously
differentiable.
Proof of Theorem 1.20[00]. Suppose first that A(i) is continuously
differentiable. Write A.7) as a system of first order differential equations
for a 2<5?-dimensional vector (y, z), where
A.14) z = A-*(t)y'
and A->A = (A*)-1 = (^~1)w. The resulting system is
A.15) y' = A*z, z' = -A*y - A-'A(A'A)'z.
It is clear that A.7) and A.15) are equivalent by virtue of A.14).
A 2<?dimensional vector solution (y(t), z(t)) of A.15) has the Euclidean
squared length
A.16) F= yy + z-z = y y + A~hf • y'.
Differentiation of F with respect to t gives F' = y' ¦ y + y ¦ y' + z' ¦ z +
z-z'-; or, by A.15),
F' [A-
since A*A and its derivative are Hermitian. Differentiation of (A'AJ = A
shows that A'A(A'A)' + (A'A)'(A'A) = A', or
A.17)
Hence
A-'A(A'A)' + (A'A)'A-'A = A-iAA'A-'A.
F' = -A-'AA'A-'Az ¦ z = -A'(A-'Az) ¦ (A^Az).
Since A' ^ 0 or A' ^0 according as A is nonincreasing or nondecreasing,
A.90[00]) follows from A.16) and A.80[oo]).
In order to verify A.100[oo1), write A.7) as a system of first order for
(x, 2/'), where
A.18) x = A'A(t)y.
This system is
A.19) *' = {A*)'A-"x + A1Ay', y" = -A'Ax.
The squared (Euclidean) length of (x, y') is
A.20) ?= x-x + y'-y' = Ayy + y'-y'.
504 Ordinary Differential Equations
Along a solution (x(t), y'{t)) of A.19), the derivative of E is
E' = [(AX)'A-* + (A-^)(A^)']x ¦ x,
so that, by A.17),
E' = A-*A'A~*x ¦ x = A'(A~'Ax) • (A~'Ax).
Thus A.100[oo]) follows from A.20) and A.80[oo]). This proves the first
part of the theorem.
It follows that the squared Euclidean lengths \\(y, z)\\, \\(x, y')\\ of solu-
solutions of A.15), A.19) tend to limits (^ oo) as t ¦— oo. The existence of
the appropriate solutions yo(t), y^t) of A.7) will be obtained by applying
Theorem l.l0[oo] to the systems A.15), A.19).
Let T{t) be the trace of the matrix of coefficients in A.15). Then T(t)
is the trace of -A~'A{A'Ay. Hence
A.21) 2 Re 7X0 tr
It will be shown that
' +
tr {A~
A.22)
[log det A(t)]' 2 Re 7X0
for 0 < t < co.
In order to prove A.22), it is sufficient to consider /-values near a fixed
t = t0. If A(t) is multiplied by a positive constant, neither side of A.22)
is affected. Hence it can be supposed that there are constants e, 6 such
that 0 < e < 6 < 1 and e \\y\\* ^ A(t)y -y^6 \\y\\2 for all vectors y and
t near t0. In particular, ||/ — ,4@11 ^ 1 — e < 1. Define a matrix, called
log A{t), by the convergent series
A.23) lo
[in analogy with log A — r) = — 2 rn/n]; cf. § IV 6. This series can be
differentiated term-by-term. Since [(/- A(t))n]' = -[A'(I - A)n^ +
(/ - A)A'{\ - A)n~2 + ¦¦•] and tr CD = tr DC for any pair of matrices,
This can be written as
[tr log A(t)]' = tr A~XA' = tr (A-1AA'A~'A),
1AA'A~'A
since
- AT-
Hence, by A.21),
A.24) [tr log ,4@]' = -2 Re 7X0-
Miscellany on Monotony 505
It is readily verified from A.23) that if, for a fixed t, X = X(t) is an eigen-
eigenvalue of A{t) and if y is a corresponding eigenvector of ,4@; then
[log A(t)]y
L n=i
= (log X)y.
Thus log X is an eigenvalue of log A and y is a corresponding eigenvector.
Hence if A1;.. ., Xd are the eigenvalues of the (Hermitian) matrix A(t),
then log A1;. . ., log Xd are those of the (Hermitian) matrix log ,4@-
Hence tr log ,4@ = log X1 + • • • + log Xd = log (XXX2. . . Xd) = log det
,4@ and A.22) follows from A.24).
Consequently,
A.25)
2 Re T(s) ds = —log det A{i) + const.
Thus the existence of yo(t) in Theorem 1.20[oo] follows from Theorem
1.1
oo[01-
If S(t) is the trace of the matrix of coefficients in A.19), then S(t) is
tr (A1A)'(A-'A). Thus Re S(t) = — Re T(t) and the existence of the solution
y^t) in Theorem 1.20[oo] follows from Theorem l.l0[oo].
This proves Theorem 1.20[oo] under the extra assumption that A(t) has a
continuous derivative. If this is not the case, A{t) can be suitably approxi-
approximated by a sequence of smooth matrix functions A^{t), A2(t), ... each of
which satisfies the assumptions of Theorem 1.20[oo]. The approximations
can be made so that A(t) — An{t) are so "small" that the solutions of A.7)
and x" + Anx = 0 are "close"; cf. § X 1. Theorem 1.20[oo] then follows
from a limit process.
Exercise 1.3. Let ,4@ satisfy the assumptions of Theorem 1.20[oo].
Let 5@ be a continuous matrix on 0 ^ t < co (^ oo) and consider
y" + B(t)y' + A(t)y = 0
in place of A.7). (a) The assertion A.90[oo]) remains valid if, in addition to
A.80[oo]), it is assumed that 5@^4@ + A(t)B*(t) ^0 [or ^ 0]. (When
,4@ is continuously differentiable, A.80[oo]) and this condition on BA +
AB* can be replaced by the single condition that A' + BA + AB* ^ 0
[or ^ 0].) Also, the assertion concerning the existence of yo(t) is valid if
(l.ll0[oo]) is replaced by
[det A(t)] exp |2 Crc tr B(s) ds~\ ->¦ 0 [or oo]
as t
¦co.
(b) The assertion A.100lor>]) remains valid if, in addition to A.8O[oo]), it is
506 Ordinary Differential Equations
assumed that B + B* ^ 0 [or ^ 0]. Also, the assertion concerning the
existence of y^t) is valid if A.110[oo]) is replaced by
[det .4@] exp —2 Re tr B(s) ds\->0 [or oo] as t ->¦ to.
2. Monotone Solutions
In contrast to the last section, the notation A > 0 or A > 0 for an
arbitrary (not necessarily Hermitian) matrix A = (alk) will mean that
ajk ^ 0 or % > 0 hold for j, k = 1, . . ., d. Similarly, y ^ 0 or y > 0
for a vector y = (y1,.. ., yd) will mean that y;' ^ 0 or y' > 0 for j = \,
Theorem 2.1.
A(t) ^ 0.
B.1)
?Ae system
continuous for 0 ^ t <C to (^oo) and satisfy
y' = -
a? least one solution y(t) ^ 0 satisfying
B.2)
^ 0, y'(t) ^0 for 0 ^ ? < co.
Remark. If the interval 0 ^ t < co (^ oo) is replaced by 0 < t < to
(^ oo) in both assumption and assertion, this theorem (and its corollaries)
remain valid. For, if 0 < a < to, then Theorem 2.1 implies the existence
of a solution satisfying 0 ^ y(t) ^ 0, y'(t) ^ 0 for a ^ t < to. But then
these inequalities also hold for 0 < t < a; cf. the proof of the theorem.
Proof. Since A(t) ^ 0, it follows from B.1) that a solution of B.1)
satisfies y'{t) ^0 on any interval on which y(t) ^ 0. In particular, if
0 < a < to and y(a) > 0, then y(t) ^ y(a) > 0 on 0 <| t fS a.
Let y0 > 0 be fixed; e.g., y0 = A, . . ., 1). Let ya0@ be the solution of
B.1) satisfying the initial condition ya0(a) = y0, where 0 < a < co. Thus
iUO ^ % > 0 on 0 ^ / ^ a. Let c(a) = ||ya0@)||, so that c(a) ^ ||yo|| > 0.
Let ya(t) = ya0(t)lc(a). Hence ya(t) is a solution of B.1), ya(t) ^ y0/
c(a) > 0 for 0 ^ ? < a, and ||yo@)|| = 1. Choose 0 < d < a2 < . . .
satisfying an —»¦ co as k —»¦ oo and
y° = lim yJO), where a = an, exists.
Then
B.3)
= 1. In addition,
y\t) = lim ya(t), where a = an,
exists uniformly on closed intervals of [0, to) and is the solution of B.1)
satisfying y°@) = y°; see Corollary IV 4.1. In view of ya(t) > 0 on
0 ^ t ^ a, B.3) implies that y°(t) ^ 0 on 0 ^ t < w. Also, y@) =
y° 7^ 0. This proves the theorem.
Miscellany on Monotony 507
Exercise 2.1. (a) Let y(t) be a solution supplied by Theorem 2.1.
Then y(co) = lim y@ as ? -> to exists and ||y'@ll <# < °°- (*) If ym(<w) >
0 for some w, 1 ^ w ^ rf, and A(t) = (ajk(t)) wherey, A: = 1, . . ., d, then
B.4) jaaim(.t)dt< co for j = 1, . . ., d.
(c) Show that if B.4) holds for some fixed m, it does not follow that
ym(to) > 0. (d) The condition B.4) for m = 1, . . ., d is necessary and
sufficient that y(t) in Theorem 2.1 can be chosen so that y(to) > 0 (i.e.,
ym(to) > 0 for m = 1, . .., d).
Exercise 2.2. The following is a theorem of Perron-Frobenius: Let
R be a constant d x d matrix satisfying R ^ 0. Then .R has at least one
real, non-negative eigenvalue A ^ 0 and a corresponding eigenvector
y>0,yr*0. Furthermore, if R > 0, then A > 0 and y > 0. Deduce
this from Theorem 2.1.
Corollary 2.1. Let A(t) be completely monotone on 0 ^ t < co [i.e.,
let A(t) e C°° for t ^ 0 W (-l)M(B)@ ^ 0/or k = 0, 1, . . . ; in other
words, A(t) > 0, A'(t) ^ 0, A"(t) ^ 0, . . .]. Then B.1) has a solution
y(t) ^ 0 which is completely monotone [i.e., (— l)V"'@ = Oforn = 0, 1,
. . .] ok 0 ^ 7 < oo.
It follows, therefore, from the theorem of Hausdorff-Bernstein that
there exist monotone nondecreasing functions a\t) ont ^ Ofory" = 1, . . .,
d, such that the components y'(t) of y(t) have representations of the form
Jo
fs da\s) for f ^ 0, where da1 ^ 0,
j = 1, . . ., d. (For the theorem of Hausdorff-Bernstein, see, e.g., Widder
[I]-)
Exercise 2.3. (a) Prove Corollary 2.1. (b) If p = (p1, . . . ,pd) is a
vector use the notation py = (/>V, . . . , /»V0. Show that the conclusion
of Corollary 2.1 is true if B.1) is replaced by
B.5)
p(t)y'(t)=-A(t)y,
where A(t) satisfies the conditions of Corollary 2.1 and p(t) e C°° for
0 ^ t < oo satisfies
B.6) />@>0 and (-l)y+1»@ ^ 0 for n = 0, 1, . ..
[i.e., p(t) > 0 and p(t) has a completely monotone derivative p'(t) for
^0]
508 Ordinary Differential Equations
Corollary 2.2. In the linear differential equation,
let the coefficients po(t),... ,pa(t) be continuous (real-valued) functions on
0 ^ t < a) (^ oo) satisfying
B.8) /70@ > 0 and pk(t) ^ 0 for k = 2, .. ., d and 0 ^ t < co
(while p^t) is arbitrary). Then B.7) few a? least one solution u(t) satisfying
B.9) u(t) > 0 and (-1) V>@ ^ 0
for n = 0, . . ., d — \. If in addition, p^t) ^ 0, then B.9) fej/A also for
n = d.
Exercise 2.4. Deduce Corollary 2.2 from Theorem 2.1.
For a different proof in the case d = 2, see Corollary XI 6.4.
Corollary 2.3. In B.7), let d ^2 anrf /e? ?fe? coefficient functions be of
class C00 /or ? ^ 0, />0@ > 0 and -po"(t), -p^'(t), p2(t),... ,pd(t) com-
completely monotone for t^0 [so that
B.10) p0>0, (—l)«+i/,t»+« ^0,
(_ l)»+i^»+D ^ 0 and (- \)np[n) ^ 0
for k = 2, . . . , d and n = 0, 1, . . ., 0 ^ t < oo]. Then B.7) has a
solution u(t) satisfying B.9) for n = \, 2, ... on 0 ^ t < co.
There is no condition onp0' or p1. In view of the theorem of Hausdorff-
Bernstein, Corollary 2.3 implies that ifp0 > 0 and — p0", —pi,p2, ¦ ¦ ¦ ,pd
have representations of the form
f
Jo
e~tsdo(s) for 0 ^ t < oo,
where ct(,s) is nondecreasing for s ^ 0, then B.7) has a solution m(?) > 0
representable in this form.
Exercise 2.5. Prove Corollary 2.3.
Exercise 2.6. (a) The differential equation for the associated Legendre
functions
is transformed into the differential equation for the toroidal functions
„ , /cosh A , f 2 1 , m2 "I .
u" + —— \u' - \n2 -- + —— \u =0
Vsinlw/ L 4 sinlr tl
by the substitution v = n — -|, fi = m, x = cosh t. If «2 ^ |, show that
this last equation has a solution u(t) > 0 completely monotone for t > 0
Miscellany on Monotony 509
(and that this solution is unique up to constant factors if and only if
n2 > i). (b) In the hypergeometric equation
x(\ - x)
[c - (a + b + 1)] — - abu = 0,
make the change of independent variables 2x — 1 = cosh t, where 1 <
x < oo, or 2x — 1 = —cosh t, where — oo < x < 0, so that 0 < t < oo.
The resulting equation is of the form
u" + Pl(t)u' - p2(t)u = 0.
Show that if ab ^ 0 and a + b ^ max Bc — 1, 0), then there exists a
completely monotone solution u > 0 on t > 0 (and that this solution is
unique up to constant factors if and only if either ab < 0 or a = b = 0).
(c) Kummer's form of the confluent hypergeometric equation
tu" + (c - t)u - au = 0,
has a completely monotone solution u(t) > Ofor? > Oifa > 0, c arbitrary;
this solution is unique up to constant factors. Also, if t is replaced by
— t, the new equation has a completely monotone solution for t > 0 if
a ^ 0 and c ^ 0 (and this solution is unique up to constant factors if and
only if a < 0). (d) Whittaker's normal form of the confluent hypergeo-
hypergeometric equation
u = 0
has a completely monotone solution m = Wkm(t) > 0 for ? > 0 if k ^ 0
and m^y^. This solution is unique up to constant factors.
Corollary 2.2 has the following generalization in which
where i,j = 1, . . ., k, denotes the Wronskian determinant of the functions
«i, ..., uk.
Corollary 2.4. Let m be fixed, 0 < m ^ d. In B.7), let the coefficients be
continuous for 0 ^ 7 < to (^ oo) a«rf Aaue the properties that
B.11) />*@^0 for fc = i»i+ 1,...,?/,
e mth order differential equation
B.12)
= o
few a set of solutions ux(t),.. ., um(t) such that
B.13) Wk(i; Ml, . . . , uk) > 0 for A: = 1,.... m on 0 ? t < to.
510 Ordinary Differential Equations
Then B.7) has a solution satisfying B.9) for n = 0, 1, . .. ,d — m on
O^t < co.
Exercise 2.7. Prove Corollary 2.4.
Exercise 2.8. Let/= (f1,. . . ,/<*) and y = (y1,. . . , ya). Assume that
f(t, y) is continuous for t ^ 0, y ^ 0; thatf(t, 0) = 0; and that/(f,«/) ^
0. Let c be any nonnegative number. Show that y' = —f(t, y) has at
least one solution y(t) for t ^ 0 satisfying ||«/@)|| = c and «/(?) ^ 0,
y\t) ^ 0 for ? > 0.
Exercise 2.9. Let «/, / be real-valued, (a) Assume that f(t, y, y') is
continuous for t > 0, y ^ 0,«/' ^ 0; that/(f, 0,0)h0; that/(',«/,«/') ^
0; and that solutions of y" = f(t, y, y') are uniquely determined by initial
conditions. Show that there exists a c0, 0 < c0 ^ oo, such that if 0 < c <
c0, then «/" = f(t, y, y') has at least one solution y(t) for t ^ 0 satisfying
y@) = c, «/@ ^ 0 and y'(t) ^ 0 for t ^ 0. This is not contained in
Exercise 2.8, where the corresponding initial condition is
(b) Show that it is not always possible to take c0 = oo in (a), (c)
Let f(t, y, y') be continuous for t ^ 0, y > 0, y' ^ 0; /(?, 0, 0) = 0;
/('> «/, 0) ^ 0 for t > 0, «/^0; for every R > 0, let there exist a
positive continuous function cp(z) = cpR(z) for z ^ 0 such that h rf«/
9?(—m) = oo and \f(t, y, z)\ ^ <p(z) for 0 ^ t ^ ^?, 0 ^ y ^ /?, « ^ 0.
Let c > 0. Show that «/" =/(?, 2/,«/') has a solution on ? ^ 0 satisfying
2/@) = c and «/(?) ^ 0, y'(t) ^ 0. (This is a special case of Theorem XII
5.2 and Exercise XII 5.3.)
3. Second Order Linear Equations
This and the next section will be concerned principally with solutions of
oscillatory equations (cf. § XI 6) of the form
C.1) u" + q(t)u = 0,
where q(t) is a monotone function of t.
Theorem 3.10[oo]. Let q(t) > 0 be continuous for 0 ^ t < oo (^ oo)
and monotone; i.e.,
C.2oM) dq ^ 0 [or ^ 0].
Then, for any solution u(t), the functions u2 + u'2\q and qu2 + u'2 are
monotone, in fact,
C.30[00]) d\u* + - «'2} = Z^lAl ^o [or ^ 0],
C.4o[.]) d{qu* + «'2} = u2dq^0 [or > 0].
Miscellany on Monotony 511
If u(i) ^ 0 has a {finite or infinite) set of zeros @ ^) t0 < ^ < . . ., then
C.50[oo]) ?„ — ?„_! is nondecreasing [or nonincreasing] with n.
Furthermore, if
C.60[oo]) g(f)->-0 [or q(t) -»¦ oo]
ai t -*¦ co, then C.1) possesses linearly independent solutions mo(O> Mi@
satisfying, as t —*¦ oo,
C.70[00]) «o2 + - «o2 -> oo [or 0],
a
C.80[oo]) qu2 + u[2 -+ 0 [or oo].
, u
dq>0
Figure 1
If C.1) is oscillatory at t = co, then ^ > 0 implies that the graph of
u = \u(t)\ in the (t, «)-plane consists of a sequence of convex arches. The
assertion C.3) implies that the successive "amplitudes" (i.e., maxima of
\u\), which occur at the points where u = 0, are monotone. Correspond-
Correspondingly, the successive maxima of \u'\, which occur at the points where
u" = 0 or, equivalently, where u = 0, are monotone by C.4). See Figure 1.
The Sturm comparison theorems imply C.5) and even more:
Exercise 3.1. Let ^@^0 be continuous and nonincreasing for
Ti = ' = T6 and let C.1) have a solution u(t) with exactly three zeros
' = Ti> T3» T6> where t1 < t3 < t6. (a) Show that u'(t) has exactly two
zeros t = t2, t4 satisfying Tj < t2 < t3 < t4 < t5 and that t;+1 — t;- ^
Ti+2 — T;(i f°r 7 = 1, 2, 3. (b) After a reflection across a vertical line
' = Tm for 7 = '> 2 or 3, the graph of m = |m@| for a quarter-wave
Ti.i ^ ' ^ Ty.a lies over the graph of u = |w(r)| for the preceding quarter-
wave ry ^ f < ryiI; i.e., |k(t,h- 01^ |M(ryil + 01 for 0^ t^rjn- rt.
512 Ordinary Differential Equations
The assertions C.3), C.4), C.7), and C.8) are consequences of Theorem
1.2, but slightly different arguments for their proofs will be indicated.
Except for the assertion C.5), either of the two cases of the theorem,
corresponding to C.2) and C.2M), is a consequence of the other. This can
be seen from the following lemma which can be interpreted as a "duality
principle" between equations C.1) and C.11) in which u,u',q,dt are
replaced by «', — (sgn^M, \jq, \q(t)\ dt, respectively:
Lemma 3.1. Let q{t) j? 0 be continuous for 0 ^ t < to. Introduce the
new dependent variable v and independent variable s defined by
v = u, ds = \q(t)\ dt, and s@) = 0.
C.9)
Then
C.10)
C.1) and the equation
C.11) ^E
dv u'
— = — = -(sgnq)u;
ds \q\
= 0, where Q(s) = -y-
q(t)
and 0 ^ s < dtj\q{t)\ ^ oo, are equivalent by virtue of C.9); finally,
Jo
C.12)
Proof. This lemma is trivial for, by C.10), dvjds + (sgn q)u = 0.
Differentiating this relation with respect to t and dividing by \q(t)\ gives
C.11). The relations C.12) follow from C.9) and C.10).
Exercise 3.2. Find the analogue of Lemma 3.1 if C.1) is replaced by
(p(t)u')' + q(t)u = 0, where p(t) > 0, q(t) * 0.
Proof of Theorem 3.1. Note that if q is monotone, the functions
m2 + u'2/q, qu2 + u'2 are clearly of bounded variation on any interval
[0, a] c [0, to). The relations C.3), C.4) follow from C.1).
In the case C.20[ooj), the existence of a solution mx [or w0] in C.8O)
[or C.7M)] implies the existence of a solution w0 [or mJ in C.7O) [or C.8M)].
This can be seen as follows: Let u0, «j be linearly independent solutions of
C.1). Then their Wronskian is a nonzero constant
— "o"i' = c 5* 0.
Since this Wronskian is
- «o«i' = c,
Miscellany on Monotony 513
it follows from Schwarz's inequality that
C-13) 0 < c2 ^ L2 + - u'o2) {qu2 + u'2).
\ q I
Hence C.7M) implies C.8M) [and interchanging u0 and Ml9 it is seen that
C.8O) implies C.7O)].
Thus in view of Lemma 3.1, it only remains to verify C.50[oo]) and the
existence of a solution u0 satisfying C.7M) in the case C.2M):
Exercise 3.3. Assuming C.2M), verify the existence of a solution
u0 ^ 0 satisfying C.7M). Apply either Theorem 1.2M or apply arguments
similar to those used in the proof of Theorem 1.1 directly to the quantity
u2 + u'2lq (instead of \\y\\2).
Exercise 3.4. State and prove the analogue of Theorem 3.10[oo] when
C.1) is replaced by (p(t)u')' + q(t)u = 0.
Corollary 3.1. Let q(t) > 0 be continuous and nondecreasing for 0 ^
t < (o (^ oo) and C.1) oscillatory at t = to. Let uo(t) be a solution o/C.1)
satisfying uo(t) —*¦ 0 as t -*¦ oo. Then
C.14) uo(s) ds = lim uo(s) ds
JO t->a> Jo
converges
{possibly conditionally).
Exercise 3.5. Verify Corollary 3.1.
Exercise 3.6. Let JJj) be the Bessel function of order ju. There exists
a constant c such that
0 ^ J^s) ds^c for all t ^ 0 and fi>Vi.
Jo
See Hartman and Wilcox [1, p. 239].
Note that if the conditions C.20[oo]) of Theorem 3.1 hold for a contin-
continuous q(t) > 0 and C.60[oo]) does not hold, then q(to) = lim q(t) as t -*¦ to
satisfies 0 < q(to) < oo. If to = oo and 0 < ^r(oo) < oo, then F.1) has a
pair of solutions «„, u1 satisfying
ft ft
u0 = cos q (s) ds + o(l), u0' = — q (co) sin q (s) ds + o(l),
Jo Jo
C.15)
«! = sin qA{s) ds + o(l), Ml' = ^(oo) cos \ qA(s) ds + o(l),
Jo Jo
as t -> oo; cf. Exercise X 17.4(a) or XI 8.4F). In particular.
C.16) "o2 + Mi2 -* 1 as ?->oo.
This will be used in the next section.
514 Ordinary Differential Equations
When q(t) tends monotonously to oo [or 0], then there exists at least one
solution uo(t) ^ 0 which tends to 0 [or is unbounded]. When co = oo
and q(t) is of sufficiently smooth growth, then all solutions tend to 0 [or
are unbounded] as can be seen from the asymptotic formula supplied, e.g.,
by Exercise XI 8.3 or XI 8.5.
A similar statement, without involving asymptotic integration, is given
by the following exercise:
Exercise 3.7. A monotone function H(t) on a ^ t < oo satisfying
H(t) -*¦ oo as t -*¦ oo will be said to be of "irregular growth" if, for every
e > 0, there is an unbounded sequence of 7-values a = t0 < t1 < . . .
such that the open sets B = (t0, tx) U (t2, t3) U . . . and C(n) = (tlt t2) U
(t3, if) U • • • U (t2n_lt t2j have the properties
f dH(t) < oo
Jb
and lim sup
—
t2n Jc
dH(t) < e.
If H(t) is not of "irregular growth," then it will be said to be of "regular
growth." Show that if q{t) is continuous and satisfies C.20[oo]) and C.60[oo]),
and if |log q(t)\ is of "regular growth" ona^K oo for large a, then all
nontrivial solutions of C.1) satisfy C.70[oo]) and C.80[oo]). See Hartman
[23].
Exercise 3.8. LetK^O. Let q(t) possess n + 1 continuous derivatives
for t ^ 0 satisfying (-1)V3+1)(O ^ 0 fory = 0,. . ., n and 0 < ^(oo) ^
oo. Letf(t) have n + 1 continuous derivatives for t ^ 0 and (— \)'fU){t) ^
0 for j = 0,..., n + 1 and /(oo) = 0. Then v" + q(t)v =f(t) has a
unique solution v(t) such that (—lK'i;C)(O = ° for y = 0,...,w and
vU)(t) -+ 0 as t -> oo for y = 0, ...,« + 2. Prove this for n = 0, 1, 2.
(The cases n ^ 2 are more complicated; see Hartman [22].)
Exercise 3.9. Let 0 ^ t < oo and assume that C.1) is nonoscillatory
at t = oo. For a solution u ^ 0, put r = m'/w and ?" = qu2 + u'2; let u0
denote a principal solution (Theorem XI 6.4), r0 = uo'/uo and Eo =
f
f
qu02 + u0'2. (a) Let q(t) ^ 0. Show that r' ^ 0 and ^(s) as ^ r@ ^
l/(' — Q f°r large '• (&) Under the additional assumptions of Theorem
3.10 with co = 00, ?->-0 as ? -»¦ 00 for every solution m@ of C.1) if and
= 00. (c) Let ^@ < 0, dq ^ 0 [or <fy ^ 0] and ^@0) = 0
/•O
only if
[or <?(oo) = —00]. Then, for the principal solution, r0 < 0, /¦„' > 0,
Eo < 0, dE0 ^ 0 [or r0 < 0, /¦„' < 0, ?0 > 0, rf?0 ^ 0] for 0 ^ / < 00
and r0(co) = 0, E0(cc) = 0 [or ro(cc) = — 00, ?0(oo) = 0]; and, for a
nonprincipal solution, /¦ > 0, r' < 0, ? > 0, dE ^ 0 [or r > 0, / > 0,
? < 0, dE ^ 0] for large t and /-(oo) = 0 [or r(cc) = 00, ?@0) = — 00].
In the case ^r(oo) = 0, ?@0) = 00 for all solutions if and only if
Miscellany on Monotony 515
-J tq(t)dt = oo. (a") Let q(t) < 0, dq ^ 0 [or dq ^ 0] and q(ao) =
— A2, where A > 0. Then, for the principal solutions r0 < —A, /¦„' > 0,
?0 < 0, dE0 ^ 0 [or r0 > -A, /¦„' < 0, Eo > 0, o"?0 ^ 0] for 0 ^ ? < oo
and /-O(oo)=-A, ?0(oo) = 0 in both cases. On the other hand,
± {exp 2 [—q(s)]1A ds} dq(t) = oo is necessary and sufficient in order
that the nonprincipal solutions satisfy E(cc) = oo [or E(cc) = — oo]; in
this case, r > A, r' < 0, E > 0, dE ^ 0 [or r < A, /•' > 0, ? < 0, dE ^ 0]
for large ?.
4. Second Order Linear Equations (Continuation)
This section concerns the function r = [uo2(t) + h12@]1/* ^ 0, where
uo(t) and Mj(?) are certain solutions of
D.1) u" + q(t)u = 0
when q(t) is monotone. The desired result will be deduced from results
analogous to those concerning C.15), C.16) and the following simple
lemma.
Lemma 4.1. Suppose that q(t) > 0 is continuous on 0 < t < co (^ oo),
satisfies
D.2) q(t) ^-1 as t->¦ co,
and is monotone,
D.3-) dq^O or D.3+) <fy ^ 0.
Then there exist real-valued solutions uo(t), ux{t) of D.1) such that the
complex-valued solution z(t) = uo(t) + fa^f) satisfies
D.4)
as t —»¦ co;
D.5-)
exp
' IZ
or
D-5+)
\z\2q ^ 1 ^ \z\-
and
and
a'|* ^ 1 ^ i
\'\\
^ 1 ^ i k
Remark 1. It will be convenient to note that in D.5+), the inequality
|«|2 ^ 1 holds for 0 < t < co if the conditions ? > 0, dq ^ 0 are relaxed to
q ^ 0 for 0 < t ^ t0 and ? > 0, dq ^ 0 for ?0 < t < co for some fixed f0,
0 < t0 < w; cf. the end of the proof of the lemma.
516 Ordinary Differential Equations
Proof. If a) — oo, the existence of solutions u0, wx satisfying D.4)
follows from Exercise X 17.4(a) or XI 8.4F); cf. C.15)-C.16). If « < oo,
then q(t) can be defined at t = u> so as to be continuous there and the
existence of u0, u^ is then trivial.
Lemma 3.1 shows that the assertions D.5±) follow when D.5+) is
proved in the case D.3+). For a fixed cp, consider the solution of D.1)
given by
D.6) u = uo(t) cos <p + «j@ sin cp = Re {e~i<pz{t)}.
Note that u = Re {e~ilfz(t)} = -Im {e~i<pz(t)} + o(l) as t -* to, by D.4).
Hence u2 + m'2 = \z\2 + o(l) = 1 + o(l) as f -* co.
By Theorem 3.1M, assumption D.3+) implies that C.3M), C.4J hold.
Since qu2 + u'2 -* 1 and u2 + m'2/^ -> 1 as t -> co by D.2), it is seen that
for 0 < t < to.
D.7) qu2 + u'2 ^ 1 < u2 + - u'2
q
In particular qu2 ^ 1. For fixed ?, choose cp so that t/(f) = \z(t)\. This
gives the first of the four inequalities in D.5 + ). Also, by D.7), u\t) > 1
if u'(t) = 0. For t fixed, choose cp so that u'(t) = 0. By Schwarz's in-
inequality, \u(t)\ ^ \z(t)\ and so \z{t)\ ^ 1. This is the second of the in-
inequalities in D.5+). The last two are obtained similarly. This completes
the proof of Lemma 4.1.
As to Remark 1 following Lemma 4.1, note that the Wronskian of
D.8) mo"i' - "o'"i = 1,
by D.4). By the lemma, D.5 + ) holds for t ^ t0, so that |z(fo)| ^ 1,
lz'('o)l = 1- Choose cp so that cos cp = Hi'(fo)/Iz'('o)l and sin cp = —uo'(to)l
\z'(to)\. Then u satisfies the initial conditions
u(t0) = —!— ^ i, «'(/„) = o.
If ^ ^ 0 for 0 < t ^ ?0, then an argument involving convexity shows that
u(t) > 0 and u"(t) ^ 0 for 0 < t ^ t0. Hence u'(t) ^ m'(?o) = ° and so
u(t) ^ m(?0) ^ 1 for 0 < t ^ ?0. As before, |z(f)| ^ «@ ^ 1. This proves
Remark 1.
Exercise 4.1. In Lemma 4.1, show that \z\2 q + \z'\2 ^ 1 + q in both
cases D.3±).
Lemma 4.2. Le? q(t) > 0 be of class C2 /or t > f° w«7A ?Ae properties
D.9)
Miscellany on Monotony 517
satisfies
D.10) G@-*l as ?->oo
a«rf ?Aa? either
D.11—) ^2^0 for ?>?"
or, for some t0 ^ ?°,
D.11+) 2<0 for t°<t<t0 and
g > 0, dQ ^ 0 for ? > ?0.
Then D.1) possesses a pair of real-valued solutions uo(t), u^t) such that
z(t) = uo(t) + iux{t) satisfies, as t -> 00,
D.12) qAz ~ exp iJtQlA(s)qA(s) ds, (qAz)' ~ i9W(gWz)
and, for t > f°, «7Aer
D.13-) q\z\*^\ or D.13 + ) ?|«|4^1,
according as D.11 —) or D.11 +)
Remark 2. Note that if q is of class C3, then Q has a continuous
derivative given by
lSggV' - 15<?-3 - 4q2q'"
164
Hence dQ ^ 0 is implied by
D.14) q>0, q'^0, q" ^ 0,
> 0.
Remark 3. If ^(?) satisfies the conditions of Lemma 4.2 and if </@ is
monotone for large t with 0 < ^r(oo) < 00, then D.10) is redundant. For
the monotony of Q implies that 1 — Q(t) does not change signs for large
t, so that q'\qVl is monotone by D.9). Hence
tends to a limit as t -*¦ 00. Since 0 < ^@0) ^ 00 and Q is monotone for
large t, the relation D.10) holds.
Proof. By the Liouville change of variables
D.15) U=uqV\ ds =
D.1) is transformed into
D.16) ^ + Q(t)U = 0, where / = t(s)
as
518 Ordinary Differential Equations
and Q is defined by D.9); cf. § XI 2(xiii). The ^-interval t° < t < oo is
transformed into some ^-interval (— oo ^) a < s < a> (^ oo).
By Lemma 4.1 and the Remark following it, the differential equation
D.16) has a pair of real-valued solutions U0(s), U^s) such that Z(s) =
Uo + iU1 satisfies, as s -»¦ a>,
Z ~ exp i Q^(s) ds, — ¦—¦ iZ
J ds
and the analogues of D.5±) if z, z', q are replaced by Z, dZ/ds, Q. In
particular |Z|2 < 1 or |Z|2 ^ 1 according as D.11-) or D.11+) holds.
By D.15), the equation D.1) has the solutions u0 = UJq1-4, mx = Ujq*4,
and z = u0 + iu^, thus qViz = Z at t = f(s). This gives Lemma 4.2.
Theorem 4.1. Let q(t) satisfy the conditions and u0, u1 the assertions of
Lemma 4.1 with co = oo.
D.17)
|z| > 0, |z|" ^0 for ? > ?°.
//", in addition, q(t) is continuous for t > 0, q(t) ^0 for 0 ^t < t°,
q(t) ^ const. > Ofor large t, then
D.18) \z\ > 0, |z|' ^ 0, |z|"^0 for 0 < t < oo.
(ii)Le?D.11+)/joW.
D.19) |z| >0,
< 0 for f > ?°.
Proof. Let r =\z\ = (u02 + u^fA ^ 0. Then two differentiations of r
show that, by virtue of D.1),
D.20)
r" = -qr + r-3 = r~3(\ - qr*).
Since D.13±) mean that 1 — qr* ^ 1 or < 1, the first and last inequalities
in D.17), D.19) follow. Also, if q ^ 0, then r" ^ 0, so that the third
inequality in D.18) holds.
From D.13—), it is seen that r(t) is bounded as t -»¦ oo ifq(t) ^ const. >
0 for large t. Thus r ^ 0 follows from r > 0, r" ^ 0 in D.18). Also
r ^ 0 follows from /¦ > 0, r" ^ 0 in D.19). This proves the theorem.
Corollary 4.1. Let q(t) be continuous for t > 0 and of class C3 for
t > t° ^ 0. Let q<0for 0 < t <:t° and D.14) hold for t > t°. Then
DAS) holds.
Exercise 4.2. Consider Bessel's equation
D.21)
Miscellany on Monotony 519
The variation of constants u = tl/iv transforms it into
D.22)
1 —-I u = 0, where a = /n2 — J,
so that a | 0 according as @ ^) ^1 \. The real-valued solutions
v = ^@. YJj) of D.21) are such that, for some real number 6,
D.23) z = OttO'-V^,, - iT^)
satisfies «~ei(, z' ~ ieu as ?^- oo. Use these facts in the following:
(a) Show that
D.24) t(j; + Y/)(l - -\ < - < t(j; + Y2) if p>\,
\ t I TT
D.25) t(j; + y,,2)(l - ^) > I > ?(^2 + Y2) if 0 ^ ^ < f
(i) Furthermore,
D.26) {t2-f*2 +
- i) > - > ?(/,2 +
t I TT
2- if
77
D.27)
? if
77
and/M<f
(c) The function r = ?^(/u2 + F,,2I^ > 0 satisfies D.18) or D.19) for
t° = 0 according as /u, > ^ or 0 ^ ^ < y%. (d) Show that
D.28)
(t2 -
77
if ^^^0.
Exercise 4.3. Let k ^ 0. Let q(t) be continuous for ? ^ 0 and possess
n + 2 continuous derivatives satisfying (— \)'qu+1) ^0 for j = 0, . . .,
n + 1 and 0 < ^(oo) ^ oo. (a) Show that D.1) has a pair of solutions
uo(t), mx@ such that mo'«i — «0"i' =1 and w= u02 + u^ satisfies
(- 1)V> ^ 0 for j = 0,. .., n + 1 and w<>> -^ 0 as t -+ oo for _/ = 1,
. . ., n + 3, while w-> I or w -> 0 as / -»• oo according as ^(oo) < oo or
^r(oo) = oo. See Hartman [22]. (b) Let u(t) ^ 0 be a real-valued solution
of D.1) and let its zeros be @ ^) t0 < tx < . . . . Let A^1^ = A(A>^);
thus A1?, = Atk = tM - tk, A\ = tk+2 - 2tk+1 + tk,.... Show that
PART II. A PROBLEM IN BOUNDARY LAYER THEORY
5. The Problem
This part deals with a generalization of a problem in boundary layer
theory in fluid mechanics. The problem concerns existence and uniqueness
520 Ordinary Differential Equations
questions for a singular boundary value problem involving the autonomous,
third order, nonlinear differential equation
E.1) u1" + uu" + A(l - u'2) = 0,
and solutions on 0 ^ (< oo satisfying the boundary conditions
E.2) w@) = oc, m'@) = /?, and u'(cc) = 1,
where A, a, /S are constants. The problem will further be restricted to the
consideration of solutions of E.1), E.2) satisfying
E.3)
0 < u'(t) < 1 for 0 < t < oo;
in particular, it will be assumed that 0 ^ /S < 1. [Questions of existence
and uniqueness without the restriction E.3) are not yet completely
settled.]
The cases A = 0 and A = Yz of E.1) are often called the Blasius and
Homann differential equations, respectively. As far as questions of
uniqueness are concerned, the cases A = 0, A > 0, A < 0 are quite
different. Although, all cases can be treated in a similar manner, a simple
different existence proof for the case A > 0 will be given. The existence
and uniqueness problems for the case A > 0 will be given in § 7, for A < 0
in § 8, and for A = 0 in § 9. The asymptotic properties of the solutions for
all cases will be given in § 10.
6. The Case X > 0
The existence theorem in the case A > 0 will be based on the following
simple topological argument:
Lemma 6.1. Let y,fbe d-dimensional vectors andf(t, y) continuous on an
open (t, y)-set Q such that solutions of initial value problems associated with
F.1) y'=f(t,y)
are unique. Let Q° be an open subset ofQ. with the properties that all egress
points from Q° are strict egress points and that the set Cle of egress points is
not connected. Let ?l( denote the set of ingress points ofQP and S a con-
connected subset of Q° U Qe U Q.t such that S n (Q° U QJ contains two
points (tu yj, (t^y^for which the solutions yt(t) o/F.1) through (t},y})
for j =1,2 leave Qo with increasing t at points of different (connected) com-
components ofQ.e. Then there exists at least one point (t0, y0) e S n (Q° Uflf)
such that the solution yo(t) of FA) determined by yo(to) = y0 remains in
Q° on its (open) right maximal interval of existence.
For the definition of egress, ingress, and strict egress point, see § III 8.
Proof. If the lemma is false, there exists a continuous map 77: 5—*¦ Qe
Miscellany on Monotony 521
where, for (t0, y0) e S, Tr(t0, y0) is the first point (t, y), t ^ t0, where the
solution through (t0, y0) meets Qe. The map tt is continuous since every
egress point of Q° is a strict egress point and solutions of F.1) depend
continuously on initial conditions (Theorem V 2.1). Consequently, the
connectedness of 5 implies that the image n(S) c Qe of S is connected.
But this contradicts the assumption concerning the existence of (tu yj,
ih> 2/2)-
Theorem 6.1. Let A > 0, — oo <oc < oo, 0 ^ /S < 1. Then there exists
one and only one solution u(t) of E.1), E.2), E.3). This solution also satisfies
F.2) u"(t) > 0 for 0 ^ t < oo.
In the application of Lemma 6.1, the following fact will be needed:
the case /S = 1 of E.1)-E.2) has the trivial solution
F.3)
u = a + t, u = 1, u" = 0.
This will imply that the set Qe below is not connected.
The uniqueness proof will be given in this section for both A > 0 and
A = 0. It will be derived from Exercise III 4.1 and uses F.2); cf. (8.4).
Proof. Existence for X > 0. Rewrite the differential equation E.1)
as a system of first order for a 3-dimensional vector y = (y1, y2, y3),
where y1 = u, y2 = «', y3 = u",
F.4)
yv = y2,
y2' =
y3' y^y3 - A[l - 0,2J]
Consider this equation on Q, the entire (t, y)-space. Introduce the open
(t, «/)-set
Q° = {(t, y): t, y1 arbitrary, 0 < y2 < 1, y3 > 0},
and the boundary sets
&1 = {(*, y)- t, y1 arbitrary, 0 < y2 < 1, y3 = 0},
^2 = {(t, y)- t, y1 arbitrary, y2 = \,y3> 0},
^3 = {U, y): t, y1 arbitrary, y2 = 1, y3 = 0},
&i = {(t, y) ¦ t, y1 arbitrary, y2 = 0, y3 > 0};
see Figure 2. It is readily verified that the set of egress points for Q° is
Q1 U Q2 and that all egress points are strict egress points. The set of
ingress points is Q^. A solution y(t) through a point (to,yo), where
Vo2 = Vo3 = 0 is not in Q° for small \t — to\, since y2' = y3 = 0 and y2" =
y3' = -A < 0 at t = 0 imply that y2(t) < 0 for small \t — to\ ^ 0. Note
that the points of LI3 are neither ingress nor egress points since they are
points on the trivial solutions F.3).
522 Ordinary Differential Equations
Thus Qe = Q1 U Q2 is not connected. Let 5 = {(t, y): t = 0, y =
(a, /S, y) and y > 0 arbitrary}, where a, /S are fixed, 0 ^ /? < 1. Thus
S c fl» u flj is connected. Let yy@ be the solution of F.4) through the
point (t, y) = @, a, /?, y).
If 0 < /? < 1, the point (?, y) = @, a, /?, 0) e Q1 is a strict egress point of
Q°. This makes it clear that if y > 0 is small, the arc (t, yy{t)) leaves Q°
1
\S
0 Q1 1
Figure 2. Projections of O.', ft,- on (y2,2/3)-plane.
for some ? > 0 at a point of Q1. The same argument is valid for /S = 0
and small y > 0.
It will be shown that if y > 0 is large, the solution arc (t, yy(t)) leaves
Q° through a point of Q2, where y2 = 1. Write the third equation of F.4)
as
F.5) y3' = -G/V)' + (y2J -
- (y2J].
the component y2 is non-
Along the arc y = yy@, with (t, yy(t)) e
decreasing (for y2' = y3^ 0). Hence
F.6) y3' ^ -B/V)' + I?2 - A(l
A quadrature gives, for ? ^ 0,
y\t) > y + a/S - ^('MO + t/?2
Since 0 ^ y2@ ^ 1 so that a ^ yx@ ^ « + '»
Consequently, if y is sufficiently large, then y3^) exceeds a given positive
constant on a large ^interval [0, t] as long as (t, yy(t)) e Q°. Thusy2' = y3
implies that (t, yy(t)) leaves Q° at a point where y2 = 1.
Miscellany on Monotony 523
By Lemma 6.1, there is a y = y0 > 0 such that (t, yy(t)) efl'l/Oj on
its right maximal interval of existence, which is necessarily 0 ^ t < oo.
On this solution, y2' = y3 > 0 for t > 0, so that y2 > 0 for large t. Also,
y3' ^ —A[l — (y2J] < 0 for large t shows that lim y\t) exists as t -*¦ oo.
This limit is 0 since y\t) < 1 for all t ^ 0. Consequently, y2 -> 1 as
7 -> oo (otherwise y3' < —const. < 0). This completes the existence
proof.
Uniqueness for X ^ 0. The proof depends on the introduction of the
following new variables along a solution u = u(t) of E.1) satisfying
u'(t) > 0: Let u be the new independent variable and z = u'2 > 0 the new
dependent variable; so that d/dt = u d\du = zA d/du or d\du = z~l/i d/dt.
Thus if a dot denotes differentiation with respect to u, then
F.7) u' = a* > 0, u" = |i, «'" = \z*z.
The equation E.1) is transformed into
F.8) z*z + uz + 2AA - z) = 0,
where z = dz/du;
the boundary conditions E.2) into
F.9) z(a) = /S2,
F.10) z(oo)= 1;
and E.3) into
F.11) 0<«<l for a < u < oo.
Let «(m) be a solution of F.8), F.9) with 0 <«(«)< 1, i(«) > 0 on some
interval (a, w0]. Let u = U(z) be the function'inverse to z= «(m). Put
F.12)
Then
F.13)
V(z) = z(U(z)).
d(-U) = _ I
while dV/dz = z dU/dz = z/V, so that F.8) gives
F.14)
dz
Then F.13), F.14) constitute a system of differential equations for
(-U, V), in which the function on the right of F.13) is an increasing
function of V(> 0) and the function on the right of F.14) is an increasing
function of — U and a nondecreasing function of V for V > 0 and A ^ 0.
Hence if ((/,(«), K,(z)), ((/,(«), ^(z)) are any two solutions of F.13)-F.14)
524 Ordinary Differential Equations
such that USJ = U2(P2) = a, V^) > K2(/S2) > 0. Then U2(z) >
tfi(z), V^z) > K2(z), and t/2(z) - U^z), V^z) - V2(z) are increasing
functions of z on any interval /S2 < z < «0 (^ 1) on which the solutions
exist; cf. Exercise III 4.l(a)-(c).
Now suppose that E.1)—E.3) has a pair of distinct solutions u^t), u2(t)
on t ^ 0 and suppose that hJ(O) > h?@). By F.2) or (8.4), mJ > 0 for
0 ^ ? < oo and wj ->0 as t -*¦ oo. Let «i(m), «2(w) be the corresponding
solutions of F.8) denned by F.7); U^z), U2(z) the functions inverse to
«!(«), z2(u) and V^z), V2(z) denned by F.12). Thus U}(z), Vt(z) > 0
are denned for /S2 < z < 1. Also G^2) = t/2(/S2) = a and ^(/S2) >
K2(/S2). Then mJ^-0 as t -* oo implies that Kj - K2^-0 as «-*l,
but V1 — V2 > 0 is increasing with z. This contradiction establishes
uniqueness.
Exercise 6.1. Let u(t, A) be the solution supplied by Theorem 6.1.
Modify the uniqueness proof, to show that if 0 < A < /u, then w"@, A) <
m"@, /<) and u(t, A) < «(?, /<), u'(t, A) < h'('> /") for ° < ' < °°-
Exercise 6.2. If a ^ 0, the uniqueness assertion of Theorem 6.1
follows by a variant of Exercise XII 4.6(a) applied to F.8) where u > a ^
0 for t > 0 [and the interval 0 ^ ? ^ p of Exercise XII 4.6(a) is replaced by
0 < t < oo].
Exercise 6.3. Let A > 0. Show that if — oo < a < oo and /? > 1,
then E.1), E.2) has one and only one solution u(t) satisfying u'(t) > 1 for
0 < t < oo. This solution also satisfies u" < 0 for 0 ^ t < oo and
m" ->¦ 0 as t -> oo.
Exercise 6.4. Give another proof of existence in Theorem 6.1 based on
the following: Put z = 1 — u . Then E.1) becomes
F.15) z" + uz' - A(l + u')z = 0, u = a + t - I «(s) rfs.
Jo
Define a sequence of successive approximations by letting z°(t) = 1,
u\t) = 0, and if z°,.. ., zn~\ u°, . . ., m" have been defined, let zn(t) be a
solution of
F.16)
z" + un-\t)z' - A[l + m"-1 \t)]z = 0
satisfying «@) = 1 - jS, z > 0, z' ^ 0 for t ^ 0; cf. Corollary XI 6.4.
The function zn(t) is unique and satisfies zn(f) -*¦ 0 as t -> oo; see Exercise
XI 6.7. Let
un(t) = x+ t -\ zn(s) ds.
Jo
Show that 1 = z° ^ z1 ^ ... for > 0 t and z1' ^ «2' ^ ... at t = 0;
Miscellany on Monotony 525
cf. Corollary XI 6.5. In a similar way, define a sequence of successive
approximations z0, zlt . . ., with
un = a + t - zjs)
Jo
starting with zo(t) = 0. Show that 0 = z0 < z1 ^ z2 ^ . . . for t ^ 0 and
Zl'@) ^ z2'@) ^ Also «X0 ^ «*@ for t ^ 0 and y, k = 0, 1, . . .
and «/@) ^ «*'@) for j, k = 1, 2, . . . . Show that the limits z(t) =
lim «"(?) and z{t) = lim «„(?) and the last part of F.15) define solutions of
E.1)—E.3). (These are the same solution by uniqueness in Theorem 6.1.)
7. The Case X < 0
In case A < 0, the analogue of Theorem 6.1 becomes
Theorem 7.1. Let A, /? be fixed, A < 0 andO ^ /? < 1. 77zert ^ere cmfs
a number A = ^4(A, /S) a«rf a continuous increasing function y(a) defined for
a.^ A with the properties that y(A) = 0 and that if u(t) is a solution of
G.1) u" + uu" + A(l - m'2) = 0,
G.2) «@) = a, «'@) = 0,
E-3) AoW ifand only ifz^AandO^ m"@) < y(a);
G.3)
m"@ > 0 for 0 < t < oo.
Thus for given A < 0 and /? on 0 ^ /? < 1, the problem E.1)-E.3) has
one and only one solution if a = A. When a < A, there is no solution and
when a > A, there is a family of solutions. In the case A < 0, the unique-
uniqueness proof of the last section breaks down, for the function on the right of
F.14) is not a nondecreasing function of V.
At one point, the proof will use the simple estimates for solutions of the
Weber equation supplied by Exercise X 17.6; cf. Exercise XI 9.7. In the
proof, the singular boundary value problem F.8)—F.11) will be considered.
If z = z(u) is a solution of this problem, a solution of problem E.1)—E.3)
is obtained by inverting the quadrature
¦ = f V
Jx
^) ds.
Note that the usual existence theorems apply to the differential equation
F.8) only if z > 0. Nevertheless, solutions of
G.4) z*z + uz + 2AA - z) = 0, where i = dz/du,
"determined" by initial conditions
G.5) s(a) = /?*, i(a) = y,
526 Ordinary Differential Equations
0 ^ ft < 1, y ^ 0, will be considered (at least for small u — a. > 0) even if
P = 0. This is to be understood in the sense that u = u(t) is the solution
of E.1) satisfying m@) = a, m'@) = p ^ 0, m"@) = \y ^ 0 and z = z(u)
is determined by F.7). In the critical case z(a) = 0, z(a) = 0 (i.e., P =
y = 0), E.1) implies that u'" = — X > 0, hence u" > 0 for small t > 0, and
so u' > 0 for small t > 0 and 2 > 0 for small u — a. > 0.
The proof of Theorem 7.1 will be divided into steps (a)-(k).
Proof, (a) A solution z(m) of G.4), G.5) with 0 ^ ?2 < 1, y ^ 0
satisfies z(m) > 0 for m > a as long as 0 < z(u) < 1.
For if there is a point ux > a, where ifa) = 0, z(u) > Ofor a. < u < ut
and 0 < z(mx) < 1, then zfai) ^ 0. But this is impossible, for by G.4),
z(u0 > 0.
(b) Let ?2 > 0 and y > 0. Then there exists an a0 = a°(A, P, y) such
that if a ^ a0, then the solution of G.4), G.5) exists on a M-interval
[a*, a], z(u) > 0 on [a*, a], and z(u) = 0 if u = a*.
Choose a0 so large that a ^ a0 implies that
ii(a) = -
+ 2AA -
is negative.
Thus, by convexity, the statement concerning z(m) is correct unless, in
decreasing u from a, we encounter a first point mx where z(mi) = 0 before
z(m) vanishes. It will be shown that such a point ux cannot exist if a0,
hence a, is sufficiently large. If ux does exist, then z{ux) > z(u) > z(a) = y
for m, < M < a. Hence
thus
J Ui
z(u) du > y(a. —
From G.4) and zfa) = 0, H^t/i) + 2A[1 - z(Ml)] = 0, so that Mj. > 0 and
«iy < Ki*("i) = 2 |A| [1 - 2(«x)] < 2 |A|.
Consequently, the last two formula lines give
i?2 + 2 |A| - ay,
which is negative if a0 (hence a ^ a0) is sufficiently large. This contra-
contradiction proves the statement (b).
(c) Let z(u) be a solution of G.4) for large u satisfying z(u) > 0 and
0 < z(u) < 1. Then z(u) -> 1 as m -> 00.
For the proof of this, use E.1) rather than G.4). A differentiation of
E.1) gives
um + mm'" + A - 2X)u'u" = 0.
Miscellany on Monotony 527
By assumption, 0 < u' < 1 andw" > 0 for large?. At points where m'" = 0,
it follows that u"" < 0. Hence u'"(f) can have at most one zero. Also,
u'" < 0 for large t (for if«'" > 0, then u" > 0 implies that u' is unbounded).
-Z2(U)
a Ux(z) U2(z) u
Figure 3
Suppose, if possible, that lim u'(t) < 1. Then E.1) shows that mm" >
c > 0 for large t and some constant c. Hence m'm" ^ cm'/m, and so it
follows that \u'2 > c log m + const. -> 00 as ? -> 00. This contradiction
proves (c).
«1
W «2
Figure 4
(i/) Let zx(u), 22(m) be two solutions of G.4) satisfying either
G.6) z2(ol) = Zi(a) = p\ 0^p <l, and 0 ^ i2(a) < ^(a),
with a = ax = a2; or
G.7) 2l(ai) = «2(a2) = i?2, 0 ^ /J < 1 and a2 > a1(
G.8) 0 < i2(a2) ^ 2l(ai);
cf. Figures 3 and 4. Let m = ?//z) be the function inverse to z = z^u) and
Vj = ijiUjiz)) for y = 1, 2. Then, as long as both solutions 2 = z/m)
satisfy ?2 < 2 < 1,
G.9) ^(s) - K,(z) > 0,
G.10) t^aCO — ^1B) > 0 is increasing in z.
528 Ordinary Differential Equations
In particular, the arcs z = zx{u) and z = z2(u) in the (u, z)-plane do not
intersect for u > a2 as long as /?2 < zf < 1.
For a given j, (JJP V}) is a solution of F.13)—F.14). Since the right sides
of these equations are increasing functions in V, — U, respectively, the
assertions G.9), G.10) follow from assertion (a) and from Exercise III
4.1(a)-(c) if V^olJ > 0; i.e., ifa) > 0 in G.8). The case z^olJ = z2(a2)
= 0 follows from continuity considerations (in which we first obtain U2
— U-l > 0 is nondecreasing in place of G.10)).
(e) Let zx(u), z2(u) be solutions of G.4) satisfying G.7) and ^(aj ^ 0.
Then there exists a positive e = e(a1; a2, /?) such that if
G.11)
0 ^ i2(«2) ^ *i(«i) + e,
then the arcs z = z^u), z = z2(u) cannot intersect for u > a2 as long as
iS2 < «i, z, < 1; cf. Figure 4.
Let z3(w) be the solution of G.4) with z3(a2) = /?2, i3(a2) = i^aj.
Let m = U3(z) be the inverse of z = z3(m) and F3 = i3(l/3(z)). Then L^ <
a2 < C/3, 0 < F3 < Ki on some small z-interval ^ < z ^ ^2 + E, in
particular, at z = /?2 + E. By continuity, if e > 0 is sufficiently small in
G.11), then Ux < a2 < C/2 and 0 < F2 < V1 at z = ft2 + <3. Assertion (e)
follows from (d) if a1; a2 are replaced by L^/?2 + <5), ?/20S2 + E), respec-
respectively.
(/) Suppose that G.4) has a solution satisfying F.9)—F.11). Then there
exists a number y* = y*(a) with the property that the solution of G.4)
determined by G.5) satisfies F.9)-F.11) if and only if 0 ^ y < y*.
It will first be shown that if y > 0 is sufficiently large, then the solution
of G.4), G.5) does not satisfy F.9)-F.11). Consider only a < u < a + 1
and suppose that 0 < z < 1 on this interval. Then G.4) shows that
zVrz > — uz ^ — (a + l)i. Divide by zX/i and integrate over [a, u] to
obtain z ^ y — 2(a + l)z'-^ ^ y — 2A + |a|). Integrating over a < u <
a + 1 gives the contradiction z > 1 at u = a + 1 if y > 2A + |a|) + 1.
By assumption, there is a y ^ 0 such that the solution of G.4), G.5)
satisfies F.11). Let y* = sup y taken over all such y. It follows from (d)
that solutions of G.4), G.5) with 0 ^ y < y* satisfy F.11) and solutions of
G.4), G.5) with y > y* do not satisfy F.11). The case y = y* follows
from the hypothesis if y* = 0 or from continuity considerations if y* > 0.
(g) There exist a,,, /30 (with 0 < ft, < 1), y0 > 0 such that the solution of
G.4) determined by
G.12) *(«„) = iV,
exists for u ^ a0 and
G.13) ,3O2 < z(m) < 1 for a0 < u < oo.
1
Miscellany on Monotony 529
Consider a Riccati equation associated with G.4) as follows: introduce
successively
G.14) w = 1 — z, r = h'/w,
so that G.4) becomes
2X- ur
G.15)
r=-r<
A -
Consider the Weber differential equation
G.16) v + uv — 2Xv = 0, where v = dvjdu.
If s denotes the logarithmic derivative of a nontrivial solution
G.17) s = v/v,
the corresponding Riccati equation is
G.18) i = -s2 + BA - us).
By Exercise X 17.6(a), G.16) has a solution v = v(u) such that s = v/v ~
-aasM^-oo. Let jj(m) > 0 for large u and let a0 be so large that
G.19) 0 < v(u) < 1 and 2X — us > 0 for m ^ a0 > 0.
Define ?„ > 0, y0 > 0 by
G.20) i302 = 1 - t<a0), y0 < -z)(a0),
and let z(m) be the solution of G.4) and G.12). Thus
G.21) u<a0) = 1 - z(a,,) = v(x0) > 0, 0 > u<a0) = -z(a0) > z)(a0);
in particular r(a0) > s(oo).
It will be verified that
G.22) r(u) > s(uy
for all u ^ a0 for which r(w) exists. On any interval a0 ^ m ^ a1; where
r(w) ^ s(u) holds, a quadrature shows that w(u) ^ y(w) > 0 by virtue of
r = w/h', s = z)/t>, and G.21). In this case, 0 < z(u) < 1 for a0 < m < ax
since z(u) = 1 — w{u) < 1.
By G.21), r > s holds at u = a0. Suppose, if possible, that there exists
a first u = <X! > <x0, where G.22) fails to hold, then 5(ax) 5: r(ax). But, by
G.18) and the last part of G.19),
[1 - h^)]74
This contradiction proves that G.22) and 0 < z(«) < 1 hold for all u ^ <x0.
(A) There exists a number Ao = v40(A, ^) with the property that if a > Ao,
then G.4) has a solution satisfying F.9)—F.11).
530 Ordinary Differential Equations
Let 0 < y < y0 and ax > max (a0, a0), where (a0, ft0, y0) is given by (g)
and a0 = a°(A, ft0, y) is given by F). Consider the solution z^u) of G.4)
determined by
G.23) z(Kl) = ft,2, z(Kl) = y.
Then, by (d), the solution z^m) and the solution zo(u), determined by G.12),
cannot intersect. Since z^u) increases as long as 0 < z^u) < 1 by assertion
(a), it follows that zx{u) exists for u ^ ax and ft02 < z^m) < zo(m) < 1 for
u > ax.
Applying F), the solution z^u) can be extended over an interval [a*, 04]
such that 2i(M) -^0as«^«*. For a given ft, 0 ^ /9 < 1, there exists a
unique M-value ^0 = ^0(A, ft), a* ^ ^0 < 00, satisfying z^o) = P2- Put
ri = ^(^o), so that y1 > 0 by (a) or F) according as ft > 0 or /9 = 0.
As before, (rf) implies that the solution z(m) of G.4) satisfying z(<x) = /?2
and z(<x) = y, where a ^ Ao and 0 ^ y ^ y1; exists and 0 < z(u) < z^m) <
1 for m > a. By (c), this proves (h).
(i) There exists a number ^(A, /?) such that the solutions of G.4) deter-
determined by z(oc) = ft2, z(a) = 0 satisfy F.9)-F.11) if a ^ ^; but if a < A,
then no solution of G.4) satisfies F.9)—F.11).
Let <x# = — \23X\~1att. It will first be shown that if z(u) is a solution of
G.4) and G.5), where a < a*, then z(u) assumes the value 1 for some
u < a. — a* < 0. In view of (d), it is sufficient to consider the case y = 0.
Suppose, if possible, that 0 < z(u) < 1 for a < u ^ a — a*. Then
w = 1 — z satisfies 0 < w < 1. Let r = w/w as in G.14), so that G.15)
holds. Note that w = 1 — ft2 and w = — y = 0 at u = a, so that r = 0
and, by G.15), r = 2A/? < 0. As long as a ^ m ^ 0, r ^ 0 and 0 <
w < 1, it is seen that r ^ — r2 + 2A. Under these circumstances, r(«) ^
.R(m), where R(u) = -\2V\A tan \2V\\u - a) is the solution of A =
-R2 + 21, R(ix) = 0 [for A = — R2 + 2A is the Riccati equation belonging
to ti - 2Xv = 0; cf. § XI 2(xiv)]. Hence
w/w ^ -|2A|H tan |2A|^(m - a),
provided that a<M^a— a!|s<0 and r = w/w ^ 0. Clearly, the
proviso r < 0 is not needed, for 0 < |2A|'^(m — a) < tt/2 on this M-range.
Since tan t is not integrable over 0 ^ t < |tt, it follows that w-»-0 as
u^>u0 for some m0 g a - ar This contradiction proves the assertion.
If a = ax has the property that the solution z(u) of G.4) satisfying
z(a) = /?2, z(a) = 0 assumes the value 1 for some u > ax. Then, by (d),
the same is true for all a < ax. Let ^4 = sup a^ Then ^4 < 00; in fact,
u.% < A < Ao, where ^40 is given by (h).
If a > A, then solutions of G.4) and z(a) = ?2, z(a) = 0 satisfy F.9)-
F.11). By continuity, the same holds for a = A. It is clear from (d) that
if a < ^, then no solution of G.4), G.5) satisfies F.9)-F.11).
1
Miscellany on Monotony 531
(j) The number A(X, fi) in (i) is also given by A = inf Ao(%, (I) taken
over the set of numbers A0{X, fi) with the property specified in (ft).
In fact, A ^ inf A0(A, fi) is obvious from the inequality A ^ Ao(%, ft) in
the last proof. Also if ax < inf A0{X, /?), then ax has the property specified
in the last proof, so that the inequality ax < inf Ao(\, /?) implies that
ol1<A; hence A ^ inf A0(X, /S).
(k) Proof of Theorem 7.1. By the characterization of A{X, /?) in (/) and
(j), equation G.4) has a solution satisfying F.9)—F.11) if and only if
a ^ ^(A, /9). By (/), for a > A{X, /9), there is a y* = y*(a) such that the
solutions of G.4), G.5) satisfy F.9)-F.11) if and only if 0 ^ y ^ y*.
It is clear that (e) implies that y*(a) is an increasing function of a. In
particular, y*(<x) > 0 for a > A [since y*(A) ^ 0].
It will now be verified that y*(A) = 0. For suppose, if possible, that
y*(A) > 0. Consider solutions z^u) and z2(u) of G.4) determined by
Zl(A) = p2, i^A) = y*(A), and z2(A) = p2, z2(A) = %y*(A), respectively.
Let u = Uj(z) be the functions inverse to z = z;(m) and Vt{z) = z([//z)).
Then U2(z) > U^z) and V^z) > V2(z) for $2 < z < 1. Let a < ^ and
z(m) be the solution of G.4) such that z(oc) = ;32, z(a) = \y*(A) and let
C/(z) be the inverse of z(m) and V(z) = z(E/(z)). Let <5 > 0 be fixed. Then,
by continuity, t/(/52 + <5) > C/^i?2 + <3), F^^2 + <5) > F(;32 + $) for small
^ — a > 0. Then, by (e), z(m) exists and z(u) < z^m) < 1 for large u.
Consequently, z(m) satisfies F.9)—F.11). Since a < A, this gives a contra-
contradiction and proves y*(A) = 0.
The argument just completed can be used to show that y*(a — 0) =
y*(a) for a > A. By considering solutions of G.4) satisfying G.5) and
z(a) = ft2, z(a) = y*(a) and applying a continuity argument, it is seen that
the solution of G.4) determined by z(<x) = ft2, z(a) = y*(a + 0) satisfies
F.9)-F.11). Hence y*(a + 0) ^ y*(a). This proves the continuity of
y*(a) for x> A. Thus Theorem 7.1 follows from the choice y(a) =
2y*(a) since u" = \z.
8. The Case X = 0
When X = 0, the differential equation E.1) reduces to
(8.1) m"' + mm"=0;
the boundary and side conditions are the same:
(8.2) m@) = a, m'@) = ft, and m'(oo) = 1,
(8.3) 0 < u\t) < 1 for 0 < t < 00.
Theorem 8.1. // 0 < ft < 1, then (8.1)-(8.3) has one and only one
solution for every a, — 00 < a < 00. If ft = 0, there exists a number
532 Ordinary Differential Equations
A ^ 0 such that (8.1)-(8.3) has a solution if and only if a. ^ A; in this case,
the solution is unique. In either case 0 < /9 < 1 or fi = 0, the solution
satisfies
(8.4) m"@ > 0 for 0 ^ t < oo.
Notice that the uniqueness has been proved in § 6 in the course of the
proof of Theorem 6.1.
Proof. The proof will be given in steps («)-(/), but the proofs of some
steps will only be sketched because of their similarity to some of the
arguments in the proofs of Theorem 6.1 and 7.1.
(a) If u{t) is a solution of (8.1), then either u"{t) = 0 or u"(t) > 0 or
m"@ < 0 for all t for which u(t) exists. For (8.1) is a first order linear
equation for u"; thus either u" = 0 or u" ^ 0.
(b) Let My(?) denote the solution of (8.1) satisfying the initial conditions
(8.5) m@) = a, m'@) = p ^ 0, and «"@) = y > 0.
Then uy(j) exists for t ^ 0, and
(8.6) m/(oo) = lim uy'(t) exists and w/(oo) > 0.
(->oo
Since u" > 0 for all t where uy(t) exists and m/@) ^ 0, it is clear that
there exists a t0 such that uy(t) exists on 0 ^ t ^ t0 and «y(f0) = 1. In
addition, uy(t) ^ 1 for all t ^ ?0 f°r which uY{t) exists. Thus, for ? ^ ^ ^ f0,
(8.7) 0 < m';@ ^ «;'(?o) exp (?0 - 0,
(8.8) 0 < m/@ - «/(?!) ^ «;'(?0)[exp (f0 - ?i) - exp (t0 - t)].
This makes the assertion clear.
(c) The limit My'(°°) is a continuous function of y > 0. It is clear that
t0 = to(y), uY"(t0) are continuous functions of y. Thus, by (8.8), uY'(t) -*
u '(°°) uniformly as t-*- oo on closed bounded intervals of 0 < y < oo.
Hence My'(oo) is a continuous function.
(rf) The limit My'(oo) is an increasing function of y > 0. This follows by
the arguments used in the proof of uniqueness in Theorem 6.1.
(e) The problem (8.1)-(8.3) has a (unique) solution for a = 0, /? = 0.
For if m@ is a solution of (8.1) and c > 0, then cu(ct) is also a solution of
(8.1). (This follows from a direct verification.) Hence, for a = /? = 0,
cm^c?) = My(?) for y = c3, thus My'(oo) = /-V(oo). Since m/(oo) ^ 0,
there is a unique y > 0 such that My'(oo) = 1 and My(?) is the desired so-
solution of (8.1)-(8.3) when a = 0 = 0.
(/) The limit Ky'(oo) tends to oo as y ->¦ oo. For a moment, denote
«y@ by «aj3y@ to show the dependence on a and ^, as well as y. It is clear
from the arguments in step (d) in the proof of Theorem 7.1 that m^/oo)
is a nonincreasing function of a and a nondecreasing function of /? and of
Miscellany on Monotony 533
y. As in the proof of (e), uaOaa(t) = u.uioi(a.t) for a > 0. Hence u'a0afi(co) =
a2Mi01(oo) —*¦ oo as a —*¦ oo. This implies (/).
(g) If, for a fixed /9 on 0 ^ /9 < 1, the problem (8.1)—(8.3) has a solution
u°(t) when a = a0, then it has a solution for a ^ 0^.
The argument in the step (d) of the proof of Theorem 7.1 shows that
the existence of a solution a = a0 implies the existence of a solution
«y@ of (8.1) with «y'(oo) < 1, where y = m°"@). Thus steps (c), (d), (/) of
this proof imply (g).
•
1
z
—7
{_
Zy(Uj
a 0 a0 u0 ux u
Figure 5
(h) If /9 = 0 and a ^ —2, then (8.1)—(8.3) has no solution. Consider
the reduction of the problem (8.1)-(8.3) to F.8)-F.11) with X = 0. The
differential equation F.8) with X = 0 is
(8.9)
'i + uz = 0, where z = dz/du,
and u is on the range a < u < oo. Let a ^ —2 and a ^ u ^ — 1, so
that z = —uz/z'A ^ i/21^. Hence i ^ y + 2zH since /? = 0. In par-
particular, i ^ 22^ and so zA ^ m — a ^ m + 2. Consequently, 2 attains
the value 1 on a < m ^ — 1 for all choices of y ^ 0.
(i) If 0 < ft < 1, then (8.1)-(8.3) has a solution.
Consider the differential equation (8.9) and let zy{t) be the solution of the
equation corresponding to the solution uY{t) of (8.1). Also, let z\t) be the
solution of (8.9) corresponding to the solution u°(t) of the problem
(8.1)-(8.3) with a = j5 = 0; cf. (e)«
Let t = tp be the unique lvalue where m°'@ = /? for 0 < /? < 1. Put
«0 = M°(^) and let ol <0,u1> a0; see Figure 5. Note that if y = 0, then
m"@ = 0, «y'@ = /S > 0, and so zY(u) = ^, zY(u) = 0. In particular, for
small y > 0, zY{u^) < 2°(m1) and 2y(«) < 2°(m) for a0 ^ u < ut. For such a
small y > 0, there is a m0, a0 < m0 < m1; where2y(«0) = 2°(m0), 2y(«0) < 2°(m0).
Thus, by the arguments of (d) in the proof of Theorem 7.1, 2y(«) < z°(u) for
u > m0. Consequently, 2^@0) ^ 1.
The existence assertion (/) for the fixed a < 0 follows from (c), (d),
534 Ordinary Differential Equations
and (/). The assertion (/) for all a (for fixed /?, 0 < /? < 1) is a consequence
of (g). This proves (;') and completes the proof of Theorem 8.1.
9. Asymptotic Behavior
In this section, the asymptotic behavior, as t -> oo, of solutions of
E.1)—E.3) will be discussed. The results will be based on the asymptotic
integrations of second order, linear differential equations.
If u(f) is a solution of E.1), put
(9.1) h(t) = 1 - u'(t).
Then h{t) satisfies the differential equation
(9.2) h" + u(t)h' - X[\ + u'(t)]h = 0.
Differentiating (9.2) gives
(9.3) h'" + u(t)h" + A - 2X)u\t)h' = 0
since h' = —u".
In order to eliminate the middle term in (9.2), put
(9.4) h = x exp — | u(s) ds,
Jo
so that x satisfies
(9.5) x" - q(t)x = 0,
where
cf. (XI 1.9)-(XI 1.10). Thus
q' = \uu + (X + i)«"
and, by E.1),
q" = —Xuu" + «'2(i + \X + X2) — X(X + 4)-
Since 0 < u' < 1, u" > 0 and u <—> 1, m ~ f as t —*¦ oo, there is a constant
C such that for large t
f 00
In addition, m" <fr is (absolutely) convergent (since u\i) -»¦ 1 as f —»¦ oo),
an tViot «
^a'2dt f00 lo"l i/f
^—55- < oo and \A < oo
so that
(9.7)
Miscellany on Monotony 535
provided that
(9.8)
f " m
J
dt
< oo.
It is easy to check (9.8), for an integration by parts (integrating u" and
differentiating u"/t6) gives
J ^-*?+/?[¦»'
by E.1). The last integral is absolutely convergent and lim inf u"(t) = 0
asf^oo. Thus (9.8) holds.
Consequently, (9.7) holds, and thus (9.5) has a principal solution
x(t) satisfying, as t -> oo,
(9.9) a; ~ cq~/l{t) exp I —
where c # 0 is a constant, while linearly independent solutions satisfy
(9.10) x~Cq-YX
cf. Exercise XI 9.6.
From the last part of (9.6) and m ~ t,
qA(t) =hu+.
hence
'A(s) ds =
2X)(?
^ + O (i),
X) log u 4
where c° is a constant. Thus (9.9), (9.10) become
X
^ + c°
x ~ ct * x exp I —
ct'exp (j'Uu +-) dt\.
In view of (9.4), the equation (9.2) has a principal solution satisfying
(9.11) h~crwexp (- | (m + -] ds), c ^ 0,
while linearly independent solutions satisfy
(9.12) h~cfAexp f —, c^O.
J u
536 Ordinary Differential Equations
By treating (9.3) as a second order equation for h' in the same way that
(9.2) was handled, it is seen that (9.3) has principal solutions satisfying
(9.11')
h' = c't exp
(-/'- 4
and that the linearly independent solutions satisfy
(9.12') h'=crr1+2\ cV 0,
as? ^oo. «„
If (9.1) satisfies (9.11), then, since u ~ t, it follows that thdt < oo;
thus J
u = t + cx + o(l),
as t -»¦ oo. Substituting this into (9.11), (9.11') gives
(9.13) 1 — u' <~ CqC1'2* exp (—\t2 — CjO, u" ~ f(l — «')
as ? —»¦ oo, where c0 > 0, cx are constants.
If (9.1) satisfies (9.12), then u ~ f implies that h = 1 - m' ~ C?2A+OA) as
f -> oo. Hence u(t) = t + 0(fM+1+f) as ? ^ oo for all e > 0. If this is
substituted into (9.12), (9.12') and if it is supposed that A < 0 (and 2A +
e < 0), then
(9.14) 1 - u' ~ c0t2\ u" ~ -2Ac0r1+2A
as t —>• oo, where c0 > 0 is a constant.
Theorem 9.1. Lef A ^ 0 and let u(t) be a solution o/E.1)-E.3). Then
there exist constants c0 > 0, cx such that (9.13) holds as t -*¦ oo.
Proof. For a given «(?), it has to be decided whether h = 1 — m'
satisfies (9.11), (9.11') or (9.12), (9.12'). If A ^ 0, (9.12) cannot hold, for
otherwise h = 1 — m'^0, t -> oo fails to hold. Thus (9.11), (9.11') are
valid and, as was seen, this gives (9.13).
Theorem 9.2. Let X < 0, 0 ^ /9 < 1, a ^ /4(A, /9), w/iere /4(A, ?), y(a)
are g«;ew &y Theorem 7.1. Le? m(?) 6e a solution o/E.1)-E.3). 77ie« 7/iere
exist constants c0 > 0, ct such that (9.13) /ioWs if and only ifu"@) = y(a);
for other solutions u(t) of E.1)-E.3), with a > A(X, 0) ani/ 0 < m"@) <
y(a), ?/ie asymptotic relations (9.14) /10W (w/7/i a suitable constant c0 > 0).
Proof, (a) If m*@ is the solution of G.1), G.2) and m*"@) = y(a), then
(9.13) holds.
Using the notation of the proof of (g) in § 7, let z*(u) be related to
u*(t) by F.7) and let v(u) be a solution of Weber's equation G.16) satisfying
v/v -~ — u as u -*¦ 00 and u(m) > 0 for large u. Let r(w) = —i*/(l — 2*)
and s(u) = v/v; cf. G.14), G.17).
Then, for large u, r*(w) ^ s(m). For suppose that r*(«) > 5(m) for some
large u = u0. In this case, r(«) > s(u) for m = m0 if z(w) = — «/(l — 2)
Miscellany on Monotony 537
belong to a solution of G.1), G.2) with m"@) = y(a) + e for small |e|.
But then, as in the proof of (g) in § 7, it follows that r(«) > s(u) for all
u ^ m0 and that u(t) satisfies E.1)—E.3). This contradicts the main property
of y(a).
Hence r*(«) ^ s(u) for large m, and so 1 — z*(u) ^ c*v(u) for large m
and some constant c* > 0. Since log v(u) <~ — |m2 as m —>- 00, it follows
that h = 1 — «*' cannot satisfy (9.14) and therefore satisfies (9.11).
This gives (9.13).
F) The problem E.1)—E.3) cannot have two distinct solutions satisfying
(9.13).
Suppose, if possible, that there exist two solutions u^t), u2(t) of E.1)—E.3)
satisfying (9.13) and, say ^"(O) > m2"@). Let z/m) be the solution of G.4)
corresponding to ut(t) by virtue of F.7) for; = 1, 2. Let U;(z) be the
function inverse to 2 = 2,(m) and F,(«) = z^U^z)).
Then z^u), z2(u) satisfy G.6) and, by (d) in § 7, the assertions G.9),
G.10) hold. By (9.13), u/'(t) -+0 and u/'(t) ~ t(\ - «/) as t -+ 00. By
virtue of F.7) and u?t) -~ ? as t -*¦ 00, the latter relation implies that
zy ~2mA — z/A) as m^ 00. Or, since 1 — 2/^ = A — z,)/(l + z/A) ~
|A — z,) as m^- 00, we have z; ~ u(l — z,) as u-> 00. Thus F,-~
?/,-(! - 2)> also F,, -^ 0 as 2 -> 1 [since u"(j) -* 0 as t -* 00].
The functions [/ = C/,, F= F,- satisfy the differential equation F.14).
Hence
AT/. TT. / 1 \
as z->-l.
dz
Consequently, as 2 -*¦ 1,
dz
It/!/ \C/27
By G.10), [4B) — C/^2) > 0 is increasing, and so there exists a constant
c > 0 such that C/2B) — [^B) ^ c > 0 for 2 near 1. Also Ufa) -> 00 as
z -^ 1. Therefore, d(Vx — V2)/dz > \c > 0 for 2 near 1, so that F^z) —
F2B) is increasing for 2 near 1. Since Vx(z) — V2(z) > 0 by G.10), this
contradicts the fact that F/2) -*¦ 0 as 2 -»¦ 1 and proves F) and Theorem
9.2.
PART HI. GLOBAL ASYMPTOTIC STABILITY
10. Global Asymptotic Stability
Consider a real autonomous system of differential equations
A0.1) y' =f{y)
in which solutions arc uniquely determined by initial conditions. Let
538 Ordinary Differential Equations
yo(t) be a solution for t ^ 0. This solution is said to be globally asymp-
asymptotically stable when the system A0.1) has the property that if y(t) is a
solution for small t > 0, then y(t) exists for all t > 0 and y{t) — yo(t) —>¦ 0
as t —> co.
This contrasts with the notion of asymptotic stability of § III 8 in that
it is not assumed here that the initial point y@) is near the initial point
yo(O) of yo(t). It will often be assumed that
A0.2)
/@) = 0
and that yo(t) is the solution yo(t) = 0, as in § III 8.
Let the function f(y) have continuous first order partial derivatives and
let J(y) denote the Jacobian matrix (df/dy) = (df'/dy10), where/, k = 1,
..., d. The criteria for global asymptotic stability to be obtained below
reduce in simple cases to conditions involving one of the two inequalities
A0.3)
or
A0.4)
w
J(y)x ¦ x ^
/(If)
0
1 • f(y
if
x-f(y)
= 0,
where a dot denotes scalar multiplication. It is very curious that both
conditions A0.3) and A0.4), which in a certain sense are complementary,
lead to stability.
The condition A0.3) which states that J{y)x ¦ x ^ 0 whenever the vector
x is in the direction of ±f(y) can be replaced by the condition that
J(y)f(y) ¦ x ^ 0 whenever x = Gf(y), and G is a constant d X d, positive
definite Hermitian matrix; i.e., by
A0.5) GJ(y)f(y) ¦ f(y) ^ 0.
Correspondingly, A0.4) can be replaced by
A0.6)
GJ(y)x ¦ x ^ 0 if Gx -f(y) = 0,
where G is the same as in A0.5). Actually, the conditions A0.5), A0.6)
are not more general than A0.3), A0.4) in the following sense:
Exercise 10.1. In A0.1), let/(«/) be a function of class C1 satisfying
A0.5) [or A0.6)], where G = G* is positive definite. Let G'A be the
self-adjoint, square root of G; cf. Exercise 1.2. Introduce the new
dependent variable z = G'^y in A0.1) and show that the resulting system
for z satisfies the analogue of A0.3) [or A0.4)].
The general criteria to be obtained will actually be generalizations of
A0.3) or A0.4) involving nonconstant, positive definite Hermitian matrices
G(y).
Miscellany on Monotony 539
11. Lyapunov Functions
Recall that if V(y) is a real-valued function having continuous partial
derivatives, then its trajectory derivative V(y) with respect to the system
(ii-i) y'=f(y)
is given by the scalar product
(H-2) V(y) =f(y) ¦ grad V(y).
Lemma 11.1. Let f(y) be continuous on an open set E and such that
solutions of A1.1) are uniquely determined by initial conditions. Let V(y) be
a real-valued function on E with the following properties: (i) V e C1 on E;
(ii) V(y) and its trajectory derivative V(y) satisfy
A1.3)
on E. Let y(t) be a solution of A1.1) for t ^ 0. Then the co-limit points of
y(t), t ^ 0, in E, if any, are contained in the set Eo = {y: V(y) = 0}.
Proof. Let tn < tn+1 ->¦ oo, y(tn) -+y0 as n -»• oo and y0 e E. Then
V(y(tn)) -* V(y0) as « ^ oo and V(y(t)) ^ V(y0) for / ^ 0. Suppose, if
possible, that y0 $ Eo, so that V(y0) < 0. Let yo(t) be the solution of A1.1)
satisfying yo(G) = y0. Consider yo(t) for 0 < t ^ e, where e > 0 is small.
Then V(yo(t)) < V(y0) for 0 < / ^ e.
The continuous dependence of solutions on initial values (Theorem
V 2.1) implies that \\y(t + tn) — yo(t)\\ is small for 0 ^ t ^ e and large n.
Hence | V{y(t + tj) - V(yo(t))\ is small for 0 ^ t ^ e and large n. In
particular V(y(tn + e)) < V(y0) for large n. But this contradicts V(y(t)) ^
V(y0) for t ^ 0 and shows that y0 e Eo.
Corollary 11.1. Let f V be as in Lemma 11.1, where E is the y-space,
and let V(y) -»• oo as \\y\\ ->¦ oo. Then all solutions y = y(t) of A1.1)
starting at t = 0 exist for t ^ 0 and are bounded [in fact, y = y(t) is in the
set {y:V(y) < V(y@))} for t ^ 0]. If, in addition, there exists a unique
point y0, where V(y0) = 0 (i.e., ifE0 reduces to the point y0), then \\f(y)\\ ^ 0
according as \\y — yj ^ 0, and the solution yo(t) = y0 of A1.1) is globally
asymptotically stable.
Exercise 11.1. Verify Corollary 11.1.
Corollary 11.2. Let f(y) e C1 for all y and let f(y0) = 0. Let G = G*
be a real, constant, positive definite, Hermitian matrix and let the Jacobian
matrix J(y) = (df/dy) satisfy GJ(y)x ¦ x < 0 for all y ^ y0 and all vectors
x jt. 0. Then the solution yo(t) = y0 of A1.1) is globally asymptotically
stable [and, in particular, f(y) ^ Ofor y ^ y0].
Proof. Put V(y) = G(y - y0) • (y - y0), so that V(y) ^ 0 according as
\\V - Veil ^ 0 and V(y) -* oo as ||y|| -* oo. Also %) = 2G(y - y0) -f(y).
540 Ordinary Differential Equations
It will be verified that V(y) ^ 0 according as \\y —yo\\ ^ 0. To this
end, we have
= J(Vo + s(y - yo))(y - y0) ds
Jo
as can be seen by noting that f(y0) = 0 and that the derivative of
f(y0 + s(y - %))
with respect to s is J(y0 + s(y — yo))(y — «/0)- Hence
Jv
= GJ{y - y0) ¦ (y - y0) ds,
J
where the argument of / is the same as before. This shows that V(y) =
2G(y — y0) •/(«/) ^ 0 according as \\y — yo\\ ^ 0 and proves the corollary.
Exercise 11.2. Let/(«/) e C1 for all y and let GJ(y)x ¦ x < 0 for all y
and all vectors x, where G = G* is real, positive definite. Let y1(t), y2(t) be
two distinct solutions of A1.1) starting at t = 0. Then y^t), y2(t) exist and
G(y2(t) - 2/iCO) • (%@ - 2/i@) decreases for ? > 0.
Corollary 11.3. Let f(y) e C1 for all y and \\f(y)\\ -> oo as \\y\\ -> oo.
Let G = G* be a real, positive definite matrix and let J(y) = (df/dy) satisfy
GJ(y)f(y) •/(«/) ^ 0 according as \\y - yo\\ ^ 0. Then f(y0) = 0 anrf ?/ie
solution yo(t) = y0 of (II.I) is globally asymptotically stable.
Exercise 11.3. Prove Corollary 11.3 by choosing V(y) = Gf(y) •/(«/).
The condition \\f(y)\\ -> oo as ||y||-> oo in Corollary 11.3 can be
considerably weakened. Also, the constant matrix G can be replaced by
suitable matrix functions G(y) in Corollaries 11.2 and 11.3. This type of
result will be considered in the §§ 12-13; cf. Corollary 12.1 and Theorem
13.1.
12. Nonconstant G
Let ? be a connected open y-set. Let G(y) = G*(y) be a (real) positive
definite matrix and let G(y) be continuous on E. We can associate with
the matrix G(y), the Riemann element of arclength
A2.1)
d d
ds2 = G(y) dy ¦ dy =2 2 Sikiv) dtf dT
3 = 1 1=1
if G = (gik(y)). By this is meant that if C:y = y(t), a ^ t < b, is an arc
of class C1 in E, its Riemann arclength L(C) with respect to A2.1) is
defined to be
A2.2) L(C) = C[G(y(t))y'(t) • */'@]^ ^'-
Ja
This is readily seen to be independent of any C1 parametrization of C.
Miscellany on Monotony 541
We can also introduce a new metric r(yu y2) on E by putting
A2.3)
= inf L(Q
taken over all arcs C:y = y(t),a ^ t ^ b, in ?, of class C1 joinings =
and «/2 = «/(*)• The function r(yx, y2) satisfies the usual conditions for a
metric: r(yx, y2) = r(y2, yx); r(ylt y2) ^ 0 according as H^ - y2\\ ^0;
and the triangular inequality
A2.4)
^ r(yx,
, y2).
Remark. Since G(y) is continuous and positive definite, it follows that
if Ex is a compact subset of E, then there exist positive constants cx, c2 such
that d dydy^ ds2 ^ c2dy ¦ dy. Thus if C is an arc of class C1 in Elt
then CiLXQ ^ L(C) ^ c2Le(C), where L(C) is the Riemann and Le(C)
the Euclidean arclength of C. In particular, if y° is an arbitrary point of
E and e > 0, there exists a 6 = d(y°, e) > 0 with the property that if
ll«/i - «/oll ^ <5> II2/s - yo\\ ^ <5. then, in determining r(yx, y2) in A2.3), it
suffices to consider arcs C in \\y0 — yo\\ < e. Hence if y° is an arbitrary
point of E, then there exists a small E = d(y°) > 0 and a pair of positive
constants c10 and c20, depending on y°, such that cw\\yx — y2\\ ^ r(yuy2) <
c20 Il2/i - ftll if 112/,- - 2/oN ^ <5 (or if r(y,, y0) ^ d) for; = 1, 2.
The Riemann element of arclength ds will be called complete on the
«/-set ? is it has the property that the convergence of the integral in A2.2)
for a half-open arc C:y = y(t) of class C1 in E defined on a half-open
interval a ^ ? < b (^ 00), implies that «/(&) = lim «/(?) as t ->- 6 exists and
is in ?; i.e., dy is complete if half-open arcs C of finite length A2.2) have
an endpoint in E.
This concept of "complete" is equivalent to the usual notion that the
set ? considered as a metric space with the metric A2.3) be complete. But
the fact will only be used in § 13. The following simple lemma will be used
subsequently.
Lemma 12.1 Let E be the y-space or the part of y-space \\y\\ ^ a > 0
exterior to a ball. Let G(y) be of class C1 on E and G(y) = G*(y) positive
definite. Then ds in A2.1) is complete on E if and only if every unbounded
arc C:y(t), a ^ t < b (^00) of class C1 in E has an infinite Riemann
arclength L(C).
Exercise 12.1. (a) Verify Lemma 12.1. (b) If, in Lemma 12.1, G(y) =
p2(y)I, where p(y) > 0 is a function of class C1, then a sufficient condition
for A2.1) to be complete on ?is thatp(y) ^ c > 0, or that \\y\\ p(y) ^ c >
0, or, more generally, that
A2.5)
min p(y)
J Lvi-.< J
du
00.
542 Ordinary Differential Equations
Let f(y) be of class C1 on E and let y(t) be a solution of
A2.6) y' = /(*/)
on some ^-interval. Let x(j) be a solution of the equations of variation of
A2.7) along A2.3), i.e., a solution of the linear system
A2.7) x = J(y(t))x,
where J(y) is the Jacobian matrix J(y) = (df/dy). Let G(y) e C1 on E and
consider the function
A2.8) tit) = G(y(t))x(t) ¦ x(t).
Its derivative with respect to t is easily seen to be given by
A2.9) v\t) = 2B(y(t))x(t) ¦ x(t),
where B(y) = (bjk(y)) is the d X rf matrix with elements
A2.10) bik-i8jp+
m=l O«T 2 m
In particular,
A2.11) V(y) = G(y)f(y)-f(y)
implies that V(y) = 2B(y)f(y) ¦ f{y)
since «/'@ = /(M0) is a solution of A2.7).
The matrix B has occurred in (V 7.11) and Lemma V 9.1 for a similar
purpose, where G = A*A. (For readers familiar with Riemann geometry,
it can be mentioned that if f(y) is considered as a contra variant vector
field; fkj the components of its covariant derivative; and B°(y) = (b?k(y))
is defined by bfk = 2 g}mf™, then 5 — B° is a skew-symmetric matrix.
Thus A2.9) is not affected if B is replaced by B°.)
Note that if G(y) = G is a constant matrix, then ?(«/) = GJ(y).
Theorem 12.1. Let f{y) be of class C1 on an open connected set E
containing y = 0. Let G(y) = G*{y) be of class C1 on E, positive definite
for each y, and such that ds in A2.1) is complete on E. Let <p{f) > 0 be
nonincreasing for r ^ 0 and satisfy
A2.12)
Jo
<p(r)dr=co.
Finally, let r(y) = r(y, 0) [cf A2.3)] and
A2.13) B(y)f(y) ¦ f(y) ^ -f(.r(y))G(i,)f(y) ¦ f{y).
Then (i) every solution y{t) of y' =/(«/) starting at t = 0 exists for t ^ 0;
(ii) «/(oo) = lim«/(?) ex/stt as ? -»• oo W is a stationary point, /(«/) = 0
aty = y{co);
A2.14)
v(t) = G(.y(t))f(y(t)) -f(y(t))
Miscellany on Monotony 543
is a decreasing function for t ^ 0 and tends to 0 as t —>¦ oo; (iv) the set of
stationary points [i.e., zeros off(y)] is connected; hence (v) if the stationary
points of j{y) are isolated (e.g., if det B(y) ^ 0 whenever f(y) = 0), then
f(y) has a unique stationary point y0 and the solution yo(t) = y0 is globally
asymptotically stable.
The proof will give a priori bounds for solutions y(t). Let
A2.15)
p, ¦
Jo
)da
and let T(r) be the function inverse to <D(r), then it will be seen that
A2.16) r(y(t)) ^ c, where c = T(d>[rB/@)]) + v*
In addition, r(y(t)) ^ c implies that
A2.17) 0 ^ v{t) ^ v@)e-2<pU)t for t ^ 0;
and since
we have
A2.18)
r(y(t), ,@0)) ^
for ? > 0.
If A2.12) does not hold but the initial point y@) of a particular solution
y{t) is such that the definition of c in A2.16) is meaningful, then assertions
(i)-(iii) are valid for this y{t).
Exercise 12.2. Using the example of the binary system where f(y) =
(—y1, 0), G = I, and E is the «/-space, show that the additional assumption
in (v) concerning isolated stationary points cannot be omitted.
Proof, (i)-(iii). Let y(t) be a solution of y'= /starting at t = 0. Then
the Riemannian length of the solution arc y = y(t) over [0, t] is the integral
of »H@, where v(t) is given by A2.14). Put r(t) = r(y(t)) = r(y(t), 0) and
A2.19)
u{t) = r@)+ vA(o)do.
By the triangular inequality A2.4) and by A2.3), it is clear that r(t) ^ u(t).
Since y(t) is a solution of y' =/ its derivative x = y'(t) =f(y(t)) is a
solution of the equations of variation A2.7). Thus A2.8)-A2.9) hold and
so, by A2.13), v\t) < -2<p(r(t))v(t). Consequently,
A2.20) WA(t)]' ^ — <p(r(t))[v1A{t)].
By A2.19), u' = v'<* and u" = [v*]' ^ -f(r(t))u. Since <p(r) is non-
increasing, r(t) ^ u(t) implies that
544 Ordinary Differential Equations
Integrating over [0, t] gives
u'(t) ^ "'@) - <f>M dw.
J»(o>
Since m@) = r@) and m' = t^, this can be written as
vA(t) <| d^@) + O(r@)) - O(m@),
by A2.15). This inequality, ^ ^ 0, and r(t) <? w@ show that
Consequently, A2.16) holds on any interval [0, ?] on which y(t) exists.
Thus the monotony of <p and A2.20) give
[vH(t)]f ^ -<p{c)WA(t)],
and so the inequalities in A2.17) hold on any interval [0, t] on which
y(t) exists. Consequently,
A2.21)
9?(c) cp{c)
and if 0 is 7 < co (^?oo) is the right maximal interval of existence of
2/@, then the last integral converges as t -»• m. Since this integral is the
Riemann arclength of the arc y = y(t), 0 <; f < co, the completeness of
ds implies that y(m) = lim y(t) exists as t -*¦ m and is in E. But then
co = oo by Lemma II 3.1. This proves (i), the existence of «/(co) in (ii),
and (iii). The fact that f(y) = 0 at y = y(co) follows from (iii) since the
integral in A2.21) is convergent as t -*¦ oo.
Proof, (iv)-(v). Let Eo be the set of zeros of /(*/). In order to show
that Eo is connected, define a mapP : E->E0 of E onto Eo as follows:
if y(t) is an arbitrary solution of y' =/for t 2: 0, put P«/@) = «/(co). It
is clear that the range of P is Eo. Since continuous maps send connected
sets into connected sets, it will follow that Eo is connected if it is verified
that P is continuous.
Let y° be an arbitrary point. It will be shown that P is continuous at
y = y°. If \\y — j/°|| is small, there exist positive constants c10, c20 such
that c10 \y - 2/°|| ^ r(y, y°) <; c20 \\y - 2/1; cf. the Remark following
A2.4). Thus, in proving the continuity of P : E -»• Eo at y = y°, it can be
supposed that ? carries the metric defined by r(yly y2) in A2.3).
Let y\t) be the solution of y(t) satisfying y°@) = y° and Ma the Riemann
sphere r(y°, y) ^ 6. Since c in A2.16) depends only on the initial point
«/@) of the solution y(t), it follows that the inequalities A2.17)—A2.18)
hold with a constant c > 0 which can be chosen independent of y@) e M6.
Hence if e > 0 is fixed, there exists a number t€ independent of y@) e M6
such that r(y(t), «/(oo)) < e if t > t?. Let <5 = <5(e) > 0 be so small that
Miscellany on Monotony 545
r(y(t), y°(t)) <e for 0 < ? < te if r(«/°, «/@)) < <5(e). Consequently,
r(y°(<x>), y(co)) < 3e if r(y, «/@)) < <5(e). This proves the continuity of P
at y = y° and completes the proof of (iv).
The main part of (v) follows from (iv). As to the parenthetical part of
(v), note that if f(y0) = 0, then B(y0) = G(yo)J(yo) by A2.10), where
J(y) = (df/dy)- This completes the proof of Theorem 12.1.
Corollary 12.1. Consider a map T of the y-space into itself given by
T'-Vi =f(y\ where f(y) is of class C1 for all y. Let the Jacobian matrix
Ay) = (Sf/dy) satisfy detJ(y) # 0 and J{y)x ¦ x <; -(p(\\y\\)\\x\\2 for all x
andy, where <p{r) > 0 is nonincr easing for r ^ 0 and satisfies A2.12). Then
T is one-to-one and onto [i.e., T has a unique inverse T-1: y = f^y^) defined
for all y^]. In particular, there is a unique point y0 where f(y) = 0; further-
furthermore, the solution yo(t) = y0 of y' =/(«/) is globally asymptotically stable.
Proof. Let E be the «/-space and G = / in Theorem 12.1, and replace
f(y) ty/(*/) — y° for a fixed y°.
If x0 is fixed and the condition on J(y) is relaxed to /(«/)(/(«/) — *0)"
(/(«/) - *o) ^ ~<P i\\y\\) Wfiy) - *oP» then it follows from Theorem 12.1
that the equation f(y) = x0 has at least one solution y.
Exercise 12.3. In Corollary 12.1, show that Tis one-to-one and onto
if "the assumption that/G/) is of class C1 and the condition on J(y)" is
relaxed to the following: "/(«/) is continuous and satisfies
lAVi) -fCtJ] ¦ B/i - 2/2) ^ -<Kr) II2/i - 2/2II2
for all yx, y2 in the sphere \\y\\ ^ r." (This generalizes the first part of
Corollary 12.1; cf. the proof of Corollary 11.2.)
Exercise 12A. (a) Let/(«/) e C1 for all y; (p(r) as in Theorem 12.1,
p(y) > 0 of class C1 for all y and satisfying A2.5). If/*(«/) = /(«/) • grad/?(«/)
and J(y) = (df/dy), assume that
A2.22) (Jf-f)p +p ||/||2^ -M " \minp(x)]du)p ||/||2.
\JO Llla:ll = » J /
Show that assertions (i)-(v) of Theorem 12.1 are valid with G(y) = p\y)I.
(b) Verify that if \\f(y)\\ ^ 0 according as \\y\\ ^ 0 and p(y) satisfies all
conditions of (a) except that p@) = 0, that p(y) is merely continuous at
y = 0, and that A2.22) holds only for y # 0, then the conclusions of (a)
still hold, (c) What are the conditions on f(y) in order that part (b) be
applicable with p(y) = ||/(y)||?
13. On Corollary 11.2
In order to obtain an analogue of Corollary 11.2 in which G is replaced
by a matrix function G(y), a property of complete Riemann elements of
arclength will be needed.
546 Ordinary Differential Equations
A set of points y on E will be said to be bounded with respect to the
metric r{y^ y^ if for some (and/or every) fixed point y° sE, there is a
constant such that r{y, y°) ^ c for all y in the set.
Lemma 13.1. Let G(y) = G*(y) be continuous on a connected open y-set
E and positive definite for each y e E. If ds in A2.1) is complete on E,
then every subset ofE which is bounded with respect to the metric r(yu y2) has
at least one cluster point in E, hence a compact closure in E. In particular,
every such subset ofE is bounded {with respect to the Euclidean metric on E).
The converse of this assertion is clear. Lemma 13.1 will be used only
for the proof of Theorem 13.1. Its use can be avoided, of course, by
making the redundant assumption in Theorem 13.1 that ds has the
property specified in Lemma 13.1, as well as being complete. In most
applications, this fact will be clear. For a proof of Lemma 13.1, see
Hopf and Rinow [1].
Theorem 13.1 Letf{y) e C1 on an open connected y-set E. Let G(y) =
G*(y) be of class C1 on E, positive definite for fixed y, and such that ds in
A2.1) is complete on E. Let the matrix B(y) defined by A2.10) satisfy
A3.1) B{y)x ¦ x < 0 for all vectors x j? 0.
Then every solution y(t) of y' = f{y) starting at t = 0 exists for t ^ 0;
furthermore, if y-^f), y2(t) are two distinct solutions for t ^ 0, then
A3.2) K</i@> y^f)) is decreasing.
In particular, if there exists a stationary point y0, f(y0) = 0, then every
solution y(t) ^ y0 satisfies
A3.3) r(y(t),y0) decreases and r(y(t), y0) -> 0 as t-+cc
(andf(y) ^Ofory* y0).
The following proof could be simplified by using the known fact that if
ylt y2 are two points of E, then there exists a geodesic arc C of class C2
joining them such that r(yu y2) is the Riemann arclength L{C) of C.
Proof. Let y^t) be a solution of y =/for 0 ^ t ^ T. Let y1 = y^O),
y2 5* ylt and r0 = r(yly y2). The set of points ET: {y:r(y, y^i)) < r0 + 1
for some t, 0 ^ t ^ T) has a compact closure in E, by Lemma 13.1. Thus,
by the Remark following A2.4) and a similar remark applied to the form
B(y)x • x in A3.1), it follows that there is a constant c > 0 such that
A3.4) B{y)x ¦ x ^ —cG(y)x • x for y e ET and all x.
Let C0:y = z(u), 0 ^ u ^ 1, be an arc of class C1 satisfying z@) = ylt
KO = ^2 and HCo)is so near to ro = %i> y%) that
A3.5) L(C0) <ro+l and e~cTL(C0) < r0.
Thus it follows that Co c ET.
Miscellany on Monotony 547
Let y(t, u) be the solution of y' = / satisfying y@, u) = z(u), so that
y(t, 0) = y^t), and y2(t) = y(t, 1) is the solution starting at y2 for t = 0.
Then y(t, u) is of class C1 on its domain of existence; Theorem V 3.1.
By Peano's existence theorem, there is an S > 0 independent of u such
that the solution y = y(t, u) exists for 0 < t ^ S for every fixed u,
0 ^ u ^ 1. It will be shown that y(t, u) exists for 0 < t ^ T. This is clear
for small u ^ 0 by Theorem V 2.1. Suppose, if possible, there is a least
w-value c, 0 < c ^ 1, such that if the right maximal interval of y(t, e) is
0 ^ t < co, then to ^ T.
For fixed t, 0 ^ t < co, let L(t) be the length L(C(t)) of the arc C(t):
y = y(t,u),0 <; u 5! e; i.e.,
A3.6)
= \[G(y(t, u))yu(t, u) • yu(t, u)
Jo
du,
where yu = dy/du. Note that x = yu(t, u) is a solution of the equations of
variation A2.7) with y(t) = y(t, u). Hence the integrand in A3.6) is
vA(t, u) where, for fixed u, v(t, u) is given by A2.8) with y(t) = y(t, u),
<t) = yu(t, u). By A2.9),
>(
= v-*(t,u)B(y)yu-yudu.
Jo
For small fixed t > 0, the arc y(t, u) is in ET since y@, u) = z(«) is. In
this case, A3.4) implies that
Hi) ^ -c (CvA(t, u) du cL(t).
Jo
Consequently, by A3.5),
A3.7)
L{t) <: e~ctL@)
r0
as long as y(t, u) e ET. The inequality A3.7) shows that as t increases
from 0 to co (^ T), y(t,u) cannot leave ET. Thus A3.7) is valid for
0 ^ / < co.
It follows that the integral A3.6) with t = co is convergent. Thus, by
the completeness of ds in A2.1), lim y(t, u) exists as u -*¦ e and is in E for
t = co (as well as for 0 ^ t < co). This limit is y(t, e), so that this solution
exists for 0 ^ t ^ co. This contradicts the fact that 0 ^ t < co is the right
maximal interval of existence of y(t, e). Consequently, the solution
y(t, u) exists for 0 ^ t ^ Tfor every u, 0 ^ u ^ 1.
In particular, y2(?) = y(f, 1) exists for 0 ^ f ^ T. Also, if c = 1 and
/ = rinA3.6),andA3.7),thenitfollowsfromA2.3)thatr(y1(r),y2(r)) ^
e-^L(C0). Hence, by A3.5), r{yi{T\ yt(T)) < r0 = r(^@), ^@)). Since
r can be replaced in this argument by any /-value 0 < / ^ T, it follows
that A3.2) holds on any interval on which yx{t) exists.
548 Ordinary Differential Equations
It will now be shown that y^t) exists for t > 0. If yx{T) = ^@), then
y^t) is periodic and exists for all T. liy^T) j? ^(O), apply the argument
just completed with y2 = y^T). Then y2(t) = yx{t + T) exists for 0 ^ t <
T; i.e., yi(t) exists for 0 ^ t ^ 2T. Repetitions of this argument show
that yx(t) exists for t ^ 0.
If y0 is a stationary point and y{t) ^ 2/0 is a solution of y = f then
Ky@» ^o) is decreasing. In particular,/^) ^ 0 for y ^ yo and r(y(t), y0)
is bounded for t ^ 0. Hence C+: y = y(t), t ^ 0, has a compact closure in
isby Lemma 13.1.
Note that V(y) = G(y)f(y) ¦ f(y) ^ 0 satisfies V(y) = 2B(y)f(y) ¦ f(y) <
0 by A2.11) and A3.1). Consequently, by Lemma 11.1, the co-limit points
of C+ are zeros of V{y). But A3.1) shows that V{y) = 0 if and only if
f(y) = 0. Hence y(t) -*¦ yo as t -*¦ oo. This proves Theorem 13.1.
Exercise 13.1. In Theorem 13.1, let assumption A3.1) be relaxed to
B(y)x • x ^ 0 for i/e? and for all x. Show that the following analogue of
the first part of Theorem 13.1 is valid: (a) r(y\{t),yi{t)) is nonincreasing
and (b) if y(t) is a solution of y' = f(y) starting at t = 0, then y(t) exists
for t ^ 0.
14. On
^ 0 if x /0>) = 0"
The last three sections have been concerned with the condition
J{y)f{y) ' f(y) = 0 an(l its generalizations. In this section, the condition
"J(y)x • x ^ 0 if a; • f(y) = 0" and generalizations will be considered.
Iff(y) in
A4.1) y =f{y)
is of class C1 on a set E, and ds2 = G(y) dy ¦ dy a positive definite Riemann
element of arclength with G(y) = G*(y) of class C1, the generalized
condition is
A4.2) B(y)x-x^0 if G(y)f(y) ¦ x = 0,
where B(y) is defined in A2.10).
In the first theorem, it will be supposed that
A4.3) /@) = 0 and f(y) 5* 0 for y^O
and that
A4.4) y = 0 is a locally asymptotically stable solution of A4.1);
cf. § III 8. By the domain of attraction ofy = 0 is meant the set of points
y0 of E such that solutions y(t) of A4.1) starting at y0 for / = 0 exist for
t ^ 0 and y(t) -»¦ 0 as ? -»¦ oo. If ? is open, the domain of attraction is
open.
Miscellany on Monotony 549
Theorem 14.1. (i) Let f{y) be of class C1 on a connected open y-set E
containing y = 0 such that A4.3) and A4.4) hold, (ii) Let e > 0 be so
small that \\y\\ ^ e is in the domain of attraction ofy = 0 and let Ec be the
set of points y e E satisfying \\y\\ ^ e. (iii) On Ee, let G(y) = G*(y) be of
class C1, positive definite for each y and such that ds2 = G(y) dy • dy is
complete on Ec. (iv) Finally, let A4.2) hold for y e Ee and all x. Then y = 0
is globally asymptotically stable.
Before proceeding to the proof, it will be of interest to formulate some
corollaries.
Corollary 14.1. Let (i), (ii) of Theorem 14.1 hold with E the y-space.
Let there exist a function p{y) > 0 of class C1 on Ee: \\y\\ > e satisfying
A4.5)
min p(y) du =
J Liwi=u J
oo
and, ifp{y) =f(y) • grad p{y) andJ(y) = (dfjdy), then
A4.6) p(y) \\x\\*+ p(y)J(y)x ¦ x < 0 when x ¦ f(y) = 0.
Then y = 0 is globally asymptotically stable.
Note that ifp = 1, A4.6) reduces to J{y)x -x^Q when x -f(y) = 0.
Exercise 14.1. Verify Corollary 14.1 by choosing G(y) = p\y)I; cf.
Exercise 12.4.
Exercise 14.2. Verify that Corollary 14.1 is applicable with p{y) =
II/GOII if
A4.7) PTmin
J Lii»ii=u
A4.8) [J(y)f(y) ¦ f(y)] ||z||2 + \\f\\*J(y)x -x^0 when x ¦ f(y) = 0.
Corollary 14.2. Let (i) in Theorem HA hold with E the y-space and let
A4.7) hold. Let the eigenvalues of the Hermitian part JH(y) = \{J + J*)
ofj(y) be Uy) ^ • ^ ld{y) and let
A4.9) Uy) + X2{y) ^ 0.
du = oo
Then y = 0 is globally asymptotically stable. If d = 2, then A4.9) is
equivalent to
A4.10)
tr.%) ^ 0.
Exercise 14.3. Verify this corollary by showing that A4.9) implies
A4.8).
Theorem 14.1 will be deduced from the following result dealing with a
solution yo(t), not necessarily yo(t) = 0.
Theorem 14.2. Let /(?/) ^ 0 be of class C1 on an open y-set E. Let
G{y) = (?*(?/) be of class C on E and positive definite for y e E and let
550 Ordinary Differential Equations
A4.2) hold. Let yo(t) be a solution o/A4.1) on the right maximal interval
0 ^ ? < co (^ oo) with the property that there exists a number a > 0 such
that r(yo(t), dE) > a > 0. Then there are positive constants d, Ksuch that,
for any solution y(t) of D.1) with r(yo(O), y@)) < 6, there exists an increasing
function s = s(t), 0 ^ t < to, such that s@) = 0, 0 ^ t < s(ca) (^ oo) is
the right maximal interval of existence of y(t), and r(y(s(t)), ya(t)) ^
Kr(y@),y0@))for0<t <to.
In this theorem r(yu y^) is the metric associated with ds2 = G{y) dy ¦ dy.
It is not assumed that ds is complete on ?and the assumption Hyo{t), dE) >
a means that if C:y = x(u), 0 ^ u < 1, is a half-open arc of class C1
starting at the point yo(t), i.e., x@) = yo(t), and the Riemann length L(C)
is finite and L(C) ^ a, then x(l) = lim x(m) exists as u -> 1 and x(m) e ?.
Roughly speaking, r(yo(t), dE) > a. means that yo(t) is at least a distance
a (in the r-metric) from the boundary dEof E [i.e., the set {y:r(y, y(t)) ^ a
for some t, 0 ^ ? < to} is in ?]. Theorem 14.2 will be proved in the next
section and Theorem 14.1 in § 16.
Exercise 14.4 Let f(y)^ 0 be of class C1 on a bounded, connected
open set E. Lety(t),t> 0, be a solution of A4.1) such that dist(y(t), dE) >
a > 0, where "dist" refers to the Euclidean metric, (a) Let
y(y) = max J(y)x • x for ||z|| = 1, x-f(y) = 0
satisfy y(y) ^ — c < 0 for some constant c > 0. Then the set of co-limit
points of y(t), t ^ 0, is a periodic solution yo(t) of A4.1) which has d — 1
characteristic exponents with negative real parts (and so is asymptotically
stable, in fact, Theorem IX 11.1 is applicable). See Borg [3]. (b) Show
that the condition y(y) ^ — c < Ofory eE can be relaxed to the condition
T(t) — T(s) ^ C — c(t — s) for 0 ^ s < t < oo and a pair of constants
C, c > 0, where V(t) = y{y(u)) du. See Hartman and Olech [1].
Jo
15. Proof of Theorem 14.2
In this proof, notions of the length of a vector a; at a point y or orthog-
orthogonality of vectors xu x2 at y refer to the Riemann geometry; i.e.,
(G(y)x • xfA or G(y)x1 • x2 = 0. Similarly, arclength of an arc C refers to
its Riemann arclength L(C); cf. A2.2).
Let y0 = yo(O) and v the piece of the hyperplane
ir-G(yo)f(yo)-(y-yo) = O
through y0 orthogonal to f(y0) with a parametrization ir.y = z(p, u)
for 0 ^ p ^ px, where u is any unit vector orthogonal to f(y0) at y0,
z(p, u) = y0 + pu. It is clear that all solutions of A4.1) with initial points
near y0 cross v.
Miscellany on Monotony 551
For a fixed u, let y = y(t,p) be the solution of A4.1) determined by the
initial condition y@,p) = z(p, u). Let 0 ^ t < co(p) ^ oo be the right
maximal interval of existence of y(t, p). Thus y(t, 0) = yo(t) and co@) = to.
For fixed u, consider the 2-dimensional surface S: y = y(t,p) defined
on a (t,p)-set containing 0 ^ t < «(/?), 0 ^/> ^ pv On S, consider the
differential equation for the orthogonal trajectories to the parameter arcs
p = const, [i.e., to the solution paths of A4.1) on S] determined by the
y(T(p,0),p)=x@,p)
~y(T(p,q),p)~x(q,p)
y - ylt, p)
Figure 6
relations G(y)f(y) ¦ dy/dp = 0, where y = y(t,p) and t = t(p). Let t =
T{p, q) be the solution of this differential equation,
A51) ii=_
dp~ G(y)f(y)-f(y)
with initial condition
A5.2)
where y = y(t, p),
W,q)=q
(so that the corresponding orthogonal trajectory starts at the point
y = y»{<i)Y> see Figure 6. In A5.1) and later, subscripts/?, q denote partial
differentiation.
Since the right side of A5.1) has a continuous partial derivative with
respect to the dependent variable t, the solution t = T(p,q) of A5.1),
A5.2) is of class C1 and has a continuous second mixed derivative Tm =
Tm; see Corollary V 3.1. Furthermore, as a function of/?, Tq(p,q)
satisfies a homogeneous linear differential equation by Theorem V 3.1,
hence
A5.3)
Tq(p, q) > 0
552 Ordinary Differential Equations
since A5.2) implies Tq@,q) = 1 > 0. The reparametrization of S given
by
A5.4) S: y = x(q, p) = y{T(p, q), p)
will be used.
Let D be the open subset of E which is the union of the "spheres"
r{y, 2/0@) < «/2 for 0 <; t < w. Thus r(y, BE) ^ a/2 if y e D. There is
a constant /8 > 0, independent of w, such that T(p, 0) exists for 0 ^ /? ^ (8
for every u and
A5.5) (\g(x@, p))xv{Q, p) ¦ xv@, p)]'A dp < a/2.
Jo
Thus the orthogonal trajectory starting at y0 reaches, for every fixed u,
the solution path of A4.1) through x@, (S) and has an arclength satisfying
A5.5). Since r(yo,x(O,p)) is less than or equal to the integral A5.5) for
0 ^p ^ (8, it is seen that x@,p) eDforO^p^fi.
The set of q-values for which t = T(p, q) exists for 0 ^ p ^ ft is open by
Theorem V 2.1. Let q0, 0 < q0 < oo be the least upper bound for this set.
Put
A5.6) L(q, a, r) = P [G(x(q, p))xv(q, p) ¦ xv{q, p)} *A dp
Jo
for 0 ^ a ^ t < ,8 and 0 ^ <? < q0. It will be shown that L(q, a, t) is
nonincreasing with respect to q for fixed a, r. Let v(q, p) denote the square
of the integrand. It will suffice to show that dvjdq ^ 0. To this end, note
thatA5.4) implies thatz4= ra/(a;) and, hence that x4J) = TQJ{x)xv+ TQvf(x).
In addition, G{x)f{x) ¦ xv = 0 by the definition of T(p, q). Using these
facts, A2.10), and
dq k=\
give dvjdq = 2TQB(x)xp • xp. Consequently, dvjdq ^ 0 by A4.2), A5.3).
Thus L(q, 0, /?) <; ?@, 0, ,8) if 0 <| q < ?0- Since the integral in A5.5)
is ?@, 0, (8), it follows that L(q, 0, (8) < a/2 and so, x(q, p) e D for 0 ^ ^ <
It will now be shown that
A5.7) T(q,p)-+co(p) as ?->?<,
for 0 ^ /? ^ (8, where 0 ^ t < co(/?) is the right maximal interval of
existence of y(t,p). Suppose, if possible, that A5.7) fails to hold for some
p = p<>} 0 ^ p° ^ (8. Since the arguments to follow do not depend on the
position ofp° on [0, 0], assume that/?0 = 0. Thus A5.7) fails to hold for
p = (8. In particular, y° = lim y(^5 (8) exists as q -> ^0> and y° is in the
Miscellany on Monotony 553
closure D of D. There exists an orthogonal trajectory y = x(p) on S such
that x(j8) = y°, and z(/?) is defined on some interval @ ^) a < /? ^ (8. In
particular, the solution y = y(t,p) of A4.1) for o<p^f) crosses
2/ = ic(/?) with increasing ? near r(^0, (8).
From the continuous dependence of solutions on initial conditions, it
follows that x{p) [and hence xv(p)] is the uniform limit of x(q,p) [and
xp(q,p), respectively] asq ->q0 on every closed interval (o <) t ^ p ^ /3.
Thus L(q, t, (S) is continuous at q = q0 if ?(^0, t, ,8) is defined by
dp
L{qa, r,p)=\ [G(x(p))xv(p) . Xl
and a < t < 0. By the monotone property of ?, ?(^0, t, ,8) ^ ?@, 0, ,8) <
a/2. Thus the arc y = x(p), a < /? ^ ,8, has a finite arclength < a/2 and
z(/?) e 5, so that Ka^/?), dE) > a/2. Consequently, x(cr) = lim x(p) exists
as p -*¦ a + 0 and x(a) e 5.
The limit relation x(q, p) -»¦ x(^) as ^ -> ^0 holds uniformly on the closed
interval a ^ p ^ (8 if x(q,p) is equicontinuous with respect to /? on
cr ^/? ^ /8, for 0 ^ ^ ^ ^0; see Theorem I 2.2. In order to verify the
equicontinuity, note that
A5.8) r(x(q,Pl), x{q,pj) ^ L(q,Pl,p2) ^ L@,pup2)
and ?@, pu p2) -> 0 as p2 - p1 -+ 0.
It is easy to see that x = x(p) can be continued over the interval 0 ^ p ^
,8. For if a > 0, the arguments above can be applied top = a, instead of
p = (8, to obtain an extension to an interval ^ ^ p ^ (8, where 0 ^ ^ <
cr. Furthermore, the set of p = a1 which can be so reached is open and
closed relative to 0 ^ p < (8, so that p = 0 can be reached.
This means that y = x(q,p) can be defined for 0 ^ q ^ q0, 0 ^ p ^ (}
and hence for 0 ^ q ^ q0 + e, 0 ^ p ^ f) for some e > 0. But this
contradicts the definition of q0. Thus the assumption that A5.7) fails to
hold for some/? = p° is untenable. In particular, q0 = co.
By A5.8) with p1 = 0, p2 = p, and the definition of x(q,p) in A5.4),
A5.9) r(y(T(p,q),p),yo(q))^L(O,O,p) for 0 ^ q < co.
By the continuity of C(y) at y = y0, there exist constants (S > 0, ct > 0,
c2 such that ?@, 0, /?) ^ c2/> if 0 ^ p ^ ,8 and r(z(/?, u), y0) ^ c^; cf. the
Remark following A2.4). Furthermore, (8, clt c2 can be chosen independent
of u.
Thus, if K = cjci, then
A5.10) r(y(T(p, t),p), yo(t)) ^ Kr(y@,p), yo)
for 0 < / < (-> and 0 < p < fi. Thus if y(t) is a solution of A4.1) with
554 Ordinary Differential Equations
initial point y(O)=y@,p) for some/? (and w), 0 ^ p ^ fi, then the
assertions of Theorem 14.2, except for 5@) = 0, follows with s{t) = T{p, t).
On the other hand, if 2/@) is near y0, then there exists a small |^| such that
y{t) crosses it at t = t1 near y0; i.e., y{t^ = y@,p) for some smallp ^ 0
and some u. Also, it is clear that r(y@,p), yo) is majorized by a constant
times r(y@), yo). Thus, if K is suitably altered,
riyih + T{p, 0), 2/0@) ^ *W), J/o)
forO ^ t < co. Thus the assertions of Theorem 14.2, except for 5@) = 0,
hold with s(t) = ?i + T(p, t). In either of the two cases just considered,
the modification of s(t) so as to satisfy 5@) = 0 is trivial. This proves
Theorem 14.2.
16. Proof of Theorem 14.1
Since the domain of attraction of y = 0 is an open set containing the
sphere 2(e): \\y\\ ^ e, there exists an a > 0 such that 2 (e + a): \\y\\ ^
e + a is also in the domain of attraction. Let E* denote the open set
E — 2(e) obtained by deleting 2(e) from E. Then/(j/) ^ 0 on ?* and the
metric ds* = G(y) dy • dy satisfies A4.2) on E*.
Suppose, if possible, that Theorem 14.1 is false, then there exists a point
y0 e E* on the boundary of the domain of attraction of y = 0. Let
y = yo(t) be the solution of A4.1) satisfying j/0@) = y0, hence
r(j/0@, dE*) > ol
on the right maximal interval 0 ^ t < co.
Thus Theorem 14.2 is applicable with E replaced by E*. Hence all
solutions y{t) starting at points y@) near y0 remain close to y = yo(t) in
the sense of Theorem 14.2. In particular y{t) e E* on its right maximal
interval of existence. But this contradicts the fact that y0 is on the bound-
boundary of the domain of attraction of y = 0 and proves the theorem.
Notes
section 1. Theorem 1.1 is due to Hartman [2], [23] and generalizes a result of
Milloux [1] on an equation of the second order; see last part of Theorem 3.1. Results
of the type A.9), A.10) in the first part of Theorem 1.2 were given for the Bessel equations
by Watson [1] (cf. [3, pp. 488-489]) and for general scalar second order equations by
Milne [1] but, as the proof shows, these are consequences of older theorems on first
order systems. (Szego [1] attributes the result to Sonine.) The last part of Theorem 1.2
concerning A.11)—A.13) and Exercise 1.3 are in Hartman [23].
section 2. Most of the results of this section are due to Hartman and Wintner. For
Theorem 2.1, Corollaries 2.1 and 2.2, see [13]; for Corollaries 2.3 and 2.4, see [14];
for Exercise 2.6 [except part (d)], see [14], [7]; and for Exercise 2.9, see [8]. For
Exercise 2.6(d), see Wintner [19]. For a generalization of Theorem 2.1 to Banach
Miscellany on Monotony 555
spaces, see Coffman [1]. Exercise 2.8 is a modification of a result of Hartman and
Wintner [16] and was suggested by Coffman [3]. For related results on a third order,
linear equation, see Gregus [1].
section 3. The part of Theorem 3.1 concerning existence in C.70c) is due to Milloux
[1]. The reduction of the proof of the theorem via Lemma 3.1 is in Hartman [23]; cf.
Wintner [23]. For Corollary 3.1 and Exercise 3.5, see Hartman and Wintner [4]. The
result of Exercise 3.7 was stated by Armellini; it was proved independently by Sansone
and Tonelli, see Sansone [1, pp. 61-67]; a simple proof is given by Hartman [23]. For
Exercise 3.8, see Hartman [22]. For Exercise 3.9, see Hartman and Wintner [15].
section 4. The main results of this section are in Hartman [22]. Parts of Exercise
4.2 are given by Schafheitlin [1]. Exercise 4.3(a) is a particular case of a result on
nonlinear differential equations of arbitrary order; see Hartman [22]. For Exercise
4.3F), see Lorch and Szego [1].
section 5. Cf. Weyl [3] who proved an existence theorem for E.1)—E.3) for X g: 0
and a = /? = 0.
section 6. Theorem 6.1 is due to Iglisch [1], [2]. The existence proof in the text
and Exercises 6.1-6.3 are due to Coppel [1]; the uniqueness proof is that of Iglisch [2].
section 7. Theorem 7.1 is a result of Iglisch and Kemnitz [1].
section 8. Cf. Grohne and Iglisch [1]. The arguments in the text are adapted from
Iglisch and Kemnitz [1].
section 9. For Theorem 9.1, see Coppel [1]; for Theorem 9.2, see Hartman [29].
section 11. For Lemma 11.1, see LaSalle [3]. Corollary 11.2 is in Hartman [24]
and is a generalization of a result of Krasovskii [1], [2], [3] (cf. [4] and Hahn [1, pp.
31-32]). For related results on nonautonomous systems, see Wintner [8, pp. 557-559],
Zubov [1], and Hartman [24, pp. 486-492].
sections 12-13. On the notion of completeness of ds, see Hopf and Rinow [1].
Theorems 12.1 and 13.1 are due to Hartman [24]; see Markus and Yamabe [1] for
weaker results. Corollary 12.1 is contained in Hadamard [4] (the proof in the text is in
Hartman [24]); the generalization in Exercise 12.3 is due to F. E. Browder [1] with a
proof valid for Hilbert space. Theorem 13.1 is related to inequalities of Lewis [2], [3];
cf. Opial [8].
section 14. A condition of the type "J(y)x ¦ x < 0 if x -f(y) = 0" was introduced by
Borg [3]; cf. Exercise 14.4(a). The main results of this section are in Hartman and
Olech [1]. They were suggested in part by the 2-dimensional case of Corollary 14.2
in Olech [3].
Hints for Exercises
Chapter II
2.3. Consider the initial value problem y' =f~Hy)i, 2/@) = 0. Since
\fo\y)%\ ^Af|||, Theorem 2.1 implies that this initial value problem has a
solution y = Y(t, I) for \t\ ^ b/M\$\. By the implicit function theorem, this
solution is unique and satisfies/(Y(jt, I)) = t$. (Why?) In particular, if ||| = 1,
then y = Y(t, I) exists for \t\ ^ b/M. Replace I by cS, where c > 0, so that
y = Y(t, cl) satisfies y' =fv~\y)cS, 2/@) = 0 for \t\ ^ b/Mc |l|. By uniqueness,
Y(t, cl) = Y(ct, I) for |f| ^ bjMc |||. Let |l| = 1 and t = 1 and rename c to
t obtaining Y{\, t$) = Y(f, I) for |!| = 1, |f| ^ b/M. Put 2/ =^), where
g(x) = Y(\,x) for |a;| ^ 6/M. Thus f(g(z)) = x and, by the implicit function
theorem, g(x) is of class C1 for |»| ^ 6/M. The Jacobian matrix (dg/dx) is
/,T1B/) at 2/ = ^(») and hence is nonsingular. Let Z»o denote the open 2/-set which
is the image of \x\ < bjM under the map y = g(x), so that g(f(y)) = y for y
on the closure Z>0 of Do. Then a; =f(y) gives a one-to-one map of Do onto
|a| ^ b/M(for if there is an a;0, |a;0| ^ 6/M, such that/B/) = x0 has two solutions
2/ = 2/i,2/2e ^o. then^(a;) is not single-valued; i.e., j<a;0) = yx and^(a;0) = 2/2)-
In order to show that Do contains Dx :\y\ ^ b\MMx, apply the result just proved
for x =f(y) on D:\y\ ^ b to the map y = g(x) on |»| < b/M; this shows that
there exists a domain Z5j in \x\ < 6/M such that y = g(x) is a continuous one-
to-one map of the closure of D1 onto \y\ ^ b\MMx.
(c,y2l
y=yx(t)
4.1. Consider the continuation to the left of a solution y' =f(t,y), y(c) =
y/°; see diagram.
4.2. Let <p(r) be a continuous function which is 0 for 0 ^ r ^ l/?> 1 for
% ^ *¦ ^ Y\, 0 for r ^ %; in particular <p(l) = 1 and <p(r) is bounded. Let
557
558 Ordinary Differential Equations
U(B) be periodic of period In, odd, and UF) = 28'A^ - 8fA > 0 for 0 ^
8 ^ it. Consider the differential equation for y = (y\ y2) given by y' =f(y1,y2),
where / = (—y2<p{r)U(8),y1(p{r)U(d)), y1 = r cos 8, y2 = r sin 8, and consider
the initial point (y1, y2) = A, 0). Find the differential equations for r, 8. Note
that 8' = U(8), 8@) = 0 has the maximal solution B^t) = ir sin21 for
0 ^ t ^ \-a, 6$) = it for t ^ \-rt and the minimal solution 8 = —B-^t). Thus
Sc, for c ^ Jw, is the circle (y1J + (y2J = 1.
4.3. (a) Show that the set of solutions y(t) of D.1) is uniformly bounded for
'o = ' = c m which case the proof of Theorem 4.1 is applicable.
Chapter III
4.1. (a) Consider z*(a) ^ «/0* and Z^W ?fk(t, z(t)), k = \,...,d. Let
«/e@ be a solution of the system yk' =f\t, y) + e, 2/(a) = 2/0 for small
e > 0. Suppose that c is such that a < c ^ b and that y (t) exists for a ^ t :g c
for all small e. Suppose, if possible, that z\i) ^ 2/e @ does not hold for
a ^ t ¦& c,k = \,... ,d. Then there is a largest ?0, a < t0 < c such that
z*@ ^ 2/e*@ for« = ' = >o> k = 1,..., dbxxt, for some/, z'(t) > yjit) holds for
some ^-values, t > t0 and t arbitrarily near to t0. In particular, z\t0) = yj(to).
But then ZV('o) ^/3(?0, z('o)) ^/*('<» 2/e('o)) < 2/^Co). so that z>(f) < ye3"@ for
small t - t0 > 0. Contradiction. Thus z»(?) ^ 2/?3(') for a ^ t ^ c. Let e tend
toO.
4.1. (Jb) Let y^t) be the solution of y' =f(t,y), yi(a) = z(a). It suffices to
show that y^it) < yok(t) for 0 <t^b,k = \,...,d, since zk ^ y/ ^ y0*.
It is clear that y^ < yok or that (yok - yf)' > 0 at t = a for every k ^ j; hence
2/i* < 2/o* f°r small ? — a > 0 for ? ^ j and for ? = j. If there is a first f-value
t0 > a at which 2/x^q) = ^/q^Co) for some k, then B/0* - y^)' < 0 at t = t0,
but 2/!(?0) 5^ 2/o('o) by uniqueness, and the monotony of /* implies that
B/o* - 2/i*)' > ° at t = to-
6.1. Let <p@ be defined for 0 < t ^ 1 and satisfy ?2 < <p(t) < 4t2/3, <p'(t) is
continuous, (f'it) ^ 2? and lim <p'(t) does not exist as t -> 0. Put co(?, «) =
<p'(t)ul<p(t), so that co(?, «) ^ 3«/2?. Let 2/,/be scalars and put/(?, y) = 0, 42//3?,
4/5^/3 according as y < 0, 0 ^ 2/ ^ ^ or y > t^ and t ^ 0.
6.2. Let /"(?, 2/) be 0, A + e)y/t or A + e)?e according as 2/ ^ 0, 0 < y < tl+c,
or y ^ ?1+^.
6.5. Put cyo(?, m) = sup |/(r, 2/1) -/(', 2/2I for I2/1 - 2/2I ^ «• Then co0C. «)
is continuous for t0 ^ ? ^ t0 + a and 0 ^ u ^ 2b (why?); nondecreasing in
u (for fixed t); o^t, 0) = 0; \f(t, yj - f(t, 2/2)| ^ cyo(?, 1^ - 2/2|); a>0(t, u) ^
w(t, u) for t0 < t ^ t0 + a, and 0 ^ m ^ 26. Suppose that u' = a>0(t, u),
u(t0) = 0 has a solution u°(t) ? 0 on some interval [t0, t0 + e]. Then
u0' = <ao(t, m°) ^ w(t, m°). But the proof of Theorem 6.1 shows that this is
impossible.
6.6. (a) Suppose that there are two solutions on 0 ^ ( g e and let u(t) denote
their difference. Then « = «'=•••= u{a) = 0 at t = 0 and |«D)@l ^ ¦*('),
where
A@ = 2 **(<* - A:)! rd+* |
Successive integrations of uw(t) give
ft
(d — k - l)!«(*'(rt = \ (t - sy-^u1
Jo
for k = 0,... ,d - 1;
Hints for Exercises 559
so that
ds.
(d - k)\rd+k u{k\t)\ ^ (d - k)rl \ A -
Jo
Multiplying this relation by ck and adding gives
X(t) ^f'kid ~ k)rl f A - sltY-^Ks) ds.
*=o Jo
Note that X(t) ^ 0 is continuous for 0 < t ^ e and A@) = 0. Suppose that
u(t) ? 0, then A(?) ^ 0. Choose t in the last relation so that k(t) > 0 is the
maximum value of I on [0, e]. Then replacing A(s) by Mj) on the right gives the
inequality
< KOJ,
- k)r* f A -
Jo
f-x-1 ds.
The factor of A(?) on the right is S e*. = 1. Contradiction.
6.7. (a) Suppose that there are two solutions y = 2/1W, 2/2W on 0 ? < ^ e.
Put m(r) = I2/1W - y2it)\. Then l^wWI ^ (oj(t,m(t)). Suppose that there
exists a ( = j, 0 < j g t, such that m(s) > a.(s). Then the minimal (unique, by
Corollary 6.3) solution of u = w^t, u), u(s) = m(s) satisfies u(t) ^ m(t) on its
left maximal interval. Then u(t) can be extended over @, s] and m(+0) = 0.
This contradicts u(s) = m(s) > a(s). Consequently, m(j) ^ a(?) for 0 ^ t ^ e.
Similarly, if w(e) > 0, then the minimal (= unique) solution of v' = a>2(t, v),
v(c) = w(e) exists on 0 < t ^ e and satisfies 0 ^ u(?) < w(?). Thus 0 ^
lim K0//3W ^ lim m(f)IP(f) ^ lim a@//3@ =0as(^0. Thus v(t) = 0, but
(
6.8. Let d(t) = \\y2(t) - 2/1WII and follow proof of Theorem 6.2.
7.1. Consider the scalar initial value problem u' = 3u%, u(tj) = ux.
7.2. Consider the differential equation u' = g(u), whereg(u) > 0 is continuous
for all u. The solutions are the level curves U(t, u) = const, of U(t, u) =
"I"
Jo
dslg(s).
8.1. It can be supposed that V@) = 0 and that V has a strict minimum at
y = 0 for otherwise V(y) can be replaced by ±[V(y) — V@)]. Although it is
not assumed that V is of class C\ V has the trajectory derivative V = 0. Hence
the proof of Theorem 8.1 is applicable.
Chapter IV
2.2. (a) By the standard orthogonalization process, there exists a nonsingular,
triangular Q(t) such that Z(t) = Y(t)Q(t) is unitary. This can be verified by
first showing that if ylt. .., ya are d linearly independent (Euclidean) vectors,
then there exists a nonsingular, triangular d x d matrix R = (rik) with /•„ ^ 0,
rJk = 0 if k > j, such that the (column) vectors dlt..., 6a are orthonormal,
where 8} = rny} + ¦ ¦ ¦ + rJYy}. In fact, let 61 = y1/||j'1||, so that ru = 1/llyJ.
In order to obtain d2, subtract the component of y2 along dx from y2 and normalize
the result to be a vector of length 1; i.e., <52 = [y2 - (y2 ¦ djdj]/1| y2 - (y2 ¦ SJS^,
where a dot denotes scalar multiplication. Thus 62 is a linear combination
r2iYi + r22y2 of ylt y2 with rM ^ 0. This process continues. If F is the matrix
with the columns y , yd and A the matrix with columns <5 , <5rf, then
560 Ordinary Differential Equations
A = RT is unitary. Hence A* = V*R* is unitary. This can be applied to
r = Y*(t) to give ?>@ = R* and Z(t) = A*. The construction shows that Q(t)
is continuously differentiable. The desired result follows from Exercise 2.1
for if C(t) = (cjk) and Z(t) has the vectors z^t),.. . , zd(jt) as columns, then since
CH = Z*AHZ and cik = 0 if j > k, it follows that cj} = \zt ¦ AHzi and zik =
zi' AHzk ify" < k. [The construction of Q(t) shows incidentally that the diagonal
elements of Q(t) are real-valued; also that Q(t) has only real entries if Y(jt) has
only real entries.]
2.2. (b) If y^t), . . . , yd(t) are the columns of Y(t), let
Q(t) = diag
in Exercise 2.1.
8.2. (c) Induction on d — g. If d — g = 0, then h(j) = \, the roots
A(l),. . . , A(W) are distinct, and Y@) is the Vandermonde matrix (M.j)d'lc~1) for
y, k = 1, ...,</, so that det Y@) = JJ [A(/) - X(j)] and (8.11) holds. Assume
(8.11) for values of d — g less than a given d — g > 1. Assume that /*A) > 1.
Let X * X(\), ..., %)and replace the first column of Y(p) by (Xd~\ Xd~2,. . . , 1).
The resulting matrix, say K(X), corresponds to the case where the distinct
roots of (8.9) are A, X(l),. . ., X(g) with corresponding multiplicities 1, h{\) — 1,
h{2),. . . , h(g). Thus by the induction hypothesis, det K(X) is obtained from the
right side of (8.11) by replacing h{\) by h(\) — 1 and multiplying by
[X - ^l)]71'1'-1!! [A - X(j)]h{i). If the first column of K(X) is differentiated
h{\) - 1 timeswithrespecttoAaU = A(l), the resulting matrix is [h{\) -l]!y@).
Hence [A(l) - 1]! det Y@) = aftA) det AXtySA*1*-1 at X = X{\).
8.3. (a) Determine u/J) as a solution of (8.1) satisfying u} =«/ = ••• =
„(.<«—j-i> = o, «J<*—'> = 1 at t = a. Then Wx(t\ ux) = u^t) ^ 0 on [a, b). Assume
that 1 ^ k < d and that fP3@ = wAr> «i, ¦¦-,«,) ^ 0 on [a, 6) for j =
1, .. ., k. Consider the kin order linear differential equation with solutions
« = «!,...,«* given by
(*)
W(t;u,Ul,...,uk)IWk(t)) =0.
This is an equation of the form uw + • • ¦ = 0. Let a < t0 < b; it will be
shown that Wk+1(t0) ^ 0. Since Wk(t0) ^ 0, it is possible to find constants
clt. .., ck such that u0 = c^it) + • • ¦ + c^jlt) + uk+1(t) has a zero of
multiplicity of at least k at t = t0 and, of course, has a zero of multiplicity
d — k — \ at t = a. Hence, by assumption, u{ok)(to) ¦? 0. It is clear that
me, uk+1, «!,..., uk)\Wk{t) = W(t; «0, «! uk)/Wk(t) = «<*>(?) ^ 0 at t =
t0. Hence Wk+l(t0) ^ 0 for a < t0 <b.
8.3. (b) Define Mf) by A@ = L[v]. Then (8.19) holds if v is written in place
of u. Since aou has d + 1 zeros on (a, b), its derivative (aou)'. hence a^v)', has
</ zeros on (a, b). Thus [«i(aou)']'. hence a2[ai(oou')]', has rf — 1 zeros on (a, b).
Continuing this argument, we find a zero t = 8 for A.
8.3. (c) Let m(?) ^ 0 be a solution of (8.1) having d zeros on (a, b). Then
m@ = c^it) + • ¦ ¦ + cdud(jt) for some constants c1; . . . , q. Suppose that
ckx_ j ^ 0 but Cfc+2 = • • ¦ = cd = 0 for some k,0 ^ k ^ d — 1. Let L&[«]
denote the function on the left of (*) in the Hint to part (a). Then, by part (b),
Lk[u] = 0 at some point t = 6, a < 6 < b. This contradicts Wk+1{B) ^ 0.
8.3. (d) Constants c1;..., ca such that u = c^^t) + • • ¦ + cdu?t) has the
desired property satisfy a set of d linear, inhomogeneous equations. These
Hints for Exercises 561
equations have a unique solution since the only solution for the corresponding
homogeneous system is c1 = . . . = cd = 0 by part (c).
8.3. (e) Let uo(t) be any solution of L[u] =1. By (d), there is a solution
ifiit) of (8.1) such that u(t) = uo(t) + u°(t) has the desired properties.
8.3. (/) Let a < t0 < b and suppose that t0 ¦? tlt... , tk. Then uo(to) ^ 0
byF)sinceL[M0] =1^0. Choose the number a such that v(t0) = u(t0) + a«0(?0).
Then w(t) = vit) - u(t) - a«0(?) has at least d + 1 zeros on [y, 6] if [y, 6]
contains t0, tlt. . ., tk. Hence L[w]F) = 0 for some 6 on (y, d) by (b). But
L[w]F) = L[v]F) - a.
9.1. (a) Let F(X, t) = det [XI - R - G(t)], F(X) = det (XI - R) be the
characteristic polynomials of R + G(t\ R, respectively; so that F(X, t) ->¦ F(X)
as t -* co uniformly on any bounded A-set. In the complex A-plane, let the disc
|X — X,-\ ^ e contain only one zero, X = Xit of F(X). It follows from the theorem
of Rouche (in the theory of functions of a complex variable) that, for large t,
F(X, t) has exactly one zero A>(?) in \X — Xj\ ^ e and A,(f) ->• Xt as t ->• oo. By
residue calculus,
X}(t) = BW)"
t) dX\F(X, t),
where Fx = dF/dX and the line integral is taken Over the circle \X — X}\ = e.
9.1. (b) Let Ro = Q^RQo, G0(t) = Q^G(t)Q0, and consider the equation
for an eigenvector of Ro + G0(t) belonging, say, to X^t) and having its first
component z\t) = 1. Let Rv be the (d — 1) x (d — 1) matrix diag
[Aj,. .., ld, ii0] and similarly G^t) the matrix obtained from G0(t) by deleting
the first row and column. Then det [Ro + G0(t) - X^t)!] = 0, but det
[R + G^t) - VO/d-i] ^ ° for large '• Hence, if z1 is a (d - l)-dimensional
vector, then z=(l,z1) satisfies [/Jo + G0(t) - ^(t)]z = 0 if and only if
[i?! + Gjit) - hWa-^ = -gl(t\ where gl(t) =(g210(t), . . . ,gdw(t)) and
(^¦110.^210. • • • ,gdio) is the first column of G0(t). Thus yx(t) = B0z@. where
«(/) = A, «!(/)) and Zl@ = -[i?! + CxW - MO/^J-ViW-
9.1. (c) Let Q^t) be the nonsingular matrix with columns y^t), . .., j/j.(
• ¦ . 2^; so that Q^[R + G(t)]Q1 has the form
\
0
0
0
0
0
\
*22«/
where R22(t) is a (d - k) x (d - k) matrix. Let A(i) = Ql\R + G)Q1 -
Aj(O/ and let v.s = A(t)ej. Thus v1 =0, u2 = [A2@ - X^ty\e2, . . . , vk =
[4@ - Aj@K. Note that /12(?J/ = 0 implies A(t)y = 0, hence y = (const.) ev
for A = 0 is a simple eigenvalue of A(f). This implies that ex,. . . , efc> vk+1,. . . , vd
are linearly independent, for a linear combination of these vectors is of the form
cxex + A(c2e2 + ¦ ¦ ¦ + cded). Let wr = A(t)vr(t) for r = 2, . . . , d and write
wr(t) as the linear combination wr(t) = hnex + ¦ ¦ • + hrkek + hrk+1vk^1 + • • • +
hrilva. Then hn = 0 (otherwise, there is a vector y such that Ay = e,). Thus,
with respect to the basis e{, et ek, vk,,,..., vd, the linear transformation
562 Ordinary Differential Equations
A(t) corresponds to the matrix
B(t)
'0 0
0 hit) - h
0 0
0
0
\
0
0
hit) - hit)
0
0
0
0
0
hit) - hit)
0 ... 0\
H12it)
H22(t),
where H22(jt) is a (d — k) x (d — k) matrix. This means that if Q2(t) is the
matrix with the columns (elt e2 ek, vk+1(t) vd(t)) then Q2lA(t)Q2 =
Bit), i.e., (QiQJ^lR + G@](QiQa) = Bi0 + hW- The desired conclusion
follows from this in the case k = 1. If k > 1, repeat these operations on the
(d — 1) x (d — 1) matrix in the lower right corner of B(t) + hW-
11.1. What is the relation between Q(t) and Q0(t) in the proof of Theorem
11.2?
11.3. The equation t2u" + C? — \)u' +u = 0 has the formal solution
u = 1 + 1! t + 2! t2 + ¦ ¦ ¦ + k\ tk +
12.3. (a) If fi, is not an integer, then there are two solutions J± ^(t) of the form
A2.12) with Jp = j[ {-\f{^tJ1c+>'lk\ T(k + p + 1). If /* ^ 0 is an integer,
there is the solution JJj) and a linearly independent solution obtained from
s-1 J-2 (s) ds which can be taken of the form
u = J (t) log t +t-e% a2kt2k,
o
where the last power series can be determined by a substitution into the Bessel
differential equation.
13.2. (a) Apply Lemma 11.2 to X{t). Make the variation of constants
rj = Z(t)y and multiply the resulting system by P~\t) to obtain a system
tDrj' = ... to which Theorem 13.1 applies.
Chapter V
5.1. Use Lemma 5.1 and its proof.
5.2. Use Stokes' formula E.3) when Sis a rectangle ax ^y1 ^bltaj ^ y> ^b}
and y* = const, for i ^ \,j. Differentiate the result with respect to bj, then bt.
6.1. Number the components of y so that (ij id) becomes @,. . ., 0,
i;+1 id), where 0 ^j ^ d and ii+l,.. ., id are distinct integers on the range
1 ^ ij. ^ d. Write y" =/as a system of first order, say j/' = z,z' =f(t, y, z)
or dy — z dt = 0, «fe -/fifr = 0. Choose the 2d x Id matrix A = A(t, y, z)
so that
A =
Q
0\
Id+} is the unit (d +j) x (d + j) matrix, P = diag [2S+1,..., zu], and the last
d—j components of A(dy — z dt, dz —fdt) are zhdzk — fk dyh for k =
y + 1 d.
Hints for Exercises 563
6.2. (a) Write the differential equations F.11) as a system of first order
i f 2/diil ( ) A d 1*) i h
q y
equations for a 2</-dimensional vector (x, y) = (x1, . . ., xd, y1
form dxi - u* dt = 0, dy* + ZZT)kyiyk dt = 0, where 1 = 1,
the Id x Id matrix A = A(x, y) to be
y*) in the
d. Choose
-G ")¦
where / is the unit d x d matrix and B = (bjk) is the d x d matrix with the
elements bjk = ^V{mym. Apply Theorem 6.1.
6.2. (c) Write y1 = x, y2 = y, and ds2 = h(y)(dx2 + dy2), where h = 1 + 9y%.
If x is used as the independent variable, the differential equations for geodesies
become d2y/dx2 = ?[1 + (,dy/dxJ]H(y), where H(y) = dXoghjdy. The initial
value problem y@) = 0, dy@)/dx = 0 has the solutions y = 0 and y = Xs.
6.3. (a) Let Wj = Pndy1 + pj2dy2 and fifa^ —qjdy1 i\dy2. Determine
^12 =Pidy1 +p2dy2 by the equations ^^ -/>2/>2i = ft, -/>i/>i2 +/>2/>ii = ?a
6.3. F) It can be supposed that pupi2 r* 0 at 2/ = 0, otherwise renumber
wlt <a2. Let j/1 = rj\y2, u1) be the solution of the initial value problem dy1jdy2 =
-P\iiP\\ and y1 = u1 at y2 = 0. Thus y1 = r)\y2, u1) is of class C1 and
drj\O, 0)/3m1 5* 0. Hence y1 = ^(y2, u1) can be solved for u1 = UHy1, y2) for
small \\y\\. In terms of(y2, M^-coordinates, w1 is of the form co1 = T1du1, where
71! ?s 0 is a continuous function of (j/2, m1), hence of (y1, y2). Similarly, let
2/2 = nKv1, u2) be the solution of dy2\dyx = —p2jp22 and y2 = u2 when 2/1 = 0
and let u2 = U2^1, y2) be the inverse function. In terms of (y1, u2), w2 has the
form w2 = T2 du2 and T2 ^ 0. The transformation u> = UKy1, y2) is of class
C1 with a nonvanishing Jacobian at y = 0 and has an inverse y' = Y'iu1, u2).
In (m1, M2)-coordinates, co3- = 7} ^m3 where T, is a continuous function of (m1, m2).
Thus a = ITdu1 du2 and T = TXT2 ^ 0. It can now be supposed that
T(u\ 0) = T@, u2) = 1 for otherwise (u1, u2) are replaced by
v1 = ± | T@,
Jo
T(s, 0) fife,
Jo
r@,
where ±T@, 0) > 0. Under the (^-transformation 2/-<¦ m, the property of
having a continuous exterior derivative is not lost. Hence dTjdu2, dTJdu1 exist
and are continuous; Exercise 5.2. Let h(ux, u2) = (TJTJX > 0. Thus w1 =
hTjX du\ w2 = (T'A/h) du2, and hence,
«ii = -[log(rH//,)L,</«i + [logG^//OLs du2.
Since </co12 = Ka>1 A to2, Stokes' relation is of the form (pcoi2 = KTdu1 du2.
Apply this to the rectangle with vertices @, 0), (m1, 0), (m1, u2), @, m2) to obtain
-log 7V, «2) = KTdu^du2.
Jo Jo
8.1. Cf. the proof of Theorem 8.1 and Exercise III 7.3.
Chapter VI
1.2. Let xoeE. Choose the enumeration of the components of x and of
Ylt. . ., Yt such that Bu = (btj(x0)), where /, y = 1,..., e, is nonsingular and
let B21 = (btJ(i:)), where i' •¦ e + 1,. .. , e + d and y = 1 d. Write
564 Ordinary Differential Equations
x = (z1, .. . , z«, y\ . . ., yd) and H(y, z) = B21(x)Bf1\x). Then the system (*)
can be written as {dujdz)B11 + (du/dy)B21 = 0 or equivalently as A.12). Hence
(*) is complete on a neighborhood of x0 = (z0, y0) if and only if <a = dy —
H(y, z) dz is completely integrable at (y0, z0). Define the (d + e) x (d + e)
matrix Bo by
Let Bo = (&*(*)) and B = (Sik(x)), where j, k = \,...,d + e. Thus Yk[u] =
Z.jPJk(x) Bit/ dx'fork = l,...,d + e, let a, = Zk8jk(x) dxk for j = 1 ,...,</ + e.
Thus yt = • • • = y, = 0 is complete at x0 if and only if ae+1 a^e is
completely integrable at x0. For any function u(x) of class C1, the total differ-
differential du = Y.{duldxi) dxi can be written as
d + e
A)
du =
since (Ert) = (ft^T1. If «(») is of class C2, a simple calculation of the exterior
derivative of du gives
B) i,Yk[u-\ d*k + ^ 2Yt[Y?u)]«t a a, = 0.
There exist (unique) continuous functions e^^x) = —e.jik(x) such that
d + ed+e
C) «*<**= j ^ eiik(x)ai A ai
is the expression for da.k in terms of the base a1(. . . , a^. Consequently, A)
and B) give
d + e
D) Yt[Y{u)] - YflYtu)] + ^ em Yk[u] = 0
*=i
for any function u of class C2. By Theorems 3.1 and 3.2, the system
ae+1 a.e+d is completely integrable if and only if o.e+1 = • • • = a.e+i = 0
implies that doce+1 = • • • = </ae+(i = 0; this is the case if and only if em = 0
for i,j = 1 e and k = e + 1 e + d. In view of D), the desired result
follows with cijk = —em for i,j, k = 1 d.
3.2. Use Lemma V 5.1.
3.4. See the proof of Lemma 3.1.
3.5. If dw exists and w = (w1 (od), then d<ak is of the form da>k =
-I.I.(dhkj/8yi)dyi Adz1 + I,I,pkijdzi Adz1, where pkii = -pkH. What are
the conditions onpkij for C.1)? Use Exercise V 5.1.
6.1. (b) See Exercise 3.5.
6.2. Show that if such an A exists, the solution of F.2) for fixed z0 and r\ = y0
is of the form y = T(z)t] + y^z), where Y(z) is a nonsingular matrix of class C1
and yx(z) is a vector of class C1 for z near z0. Also, the map (z,»?) -> (z, ?/) gives
dy - H dz = Y(z) drj. Since the form Y(z) dr] in fify, </z has a continuous ex-
exterior derivative, the same is true of the form dy — H dz in dy, dz. Apply Lemma
3.1.
8.1. (a) Let u0 = u(y0), p0 = uv(y^). By Lemma V 3.1, there are continuous
i bk k f ( ) f H I l I l I l
functions a, bk, ck of («
for / = 1,2, such that
vy^ y
«2» 2/2.^2) for smaH IM» ~ "ol, I?/.- - 2/ol» I/7* ~ /'ol
Hints for Exercises 565
A) F(u2,y2,p2) -
= (u2 - Ul)a
+ I B/2* -
and that a = Fu, bk = <>F\dyk,ck = dF/dpk if (u^y^pj = (u2,y2,p2). Let
h > 0 be small. Apply A) to (u1,y1,p1) = (u(y),y, «„(«)) and (u2,y2,p2) =
(u(y + Ay), y + Ay, uy(y + Ay)), where Ay is the vector with itsyth component
equal to 0 or h according asy ^ m ory = m. It follows from G.1) that
B)
a Au/h + b7
(&pk/h)ck = 0.
Let yh(t) be a solution of y' = c, y@) = y0, where c = (c1 cd) and the
argument of c is (u(y), y, uy(u), u(y + Ay), y + Ay, uy(y + Ay)) for fixed Ay.
It is clear that if e > 0 is small and fixed and h1 > h2 > . . . is a suitably chosen
sequence, then y(t) = limyh(t), as h = hn -> 0, exists uniformly for \t\ ^ e and
is a solution of y' = FP(u(y), y, uy(y)), y@) = y0. Note that if phm(t) =
[«B/a@ + A2/) - uiVhVWh and y is replaced by yh(t) in B), then B) can be
written as phm\t) = -aAu/h - bm since (yh(t) + Ay)' = yh'(t) = c. Thus, as
h = hn --0, /7Am@ -> du(y(t))/dym and /V'@ -> -FJu/dy™ - dF/dym uni-
uniformly for \t\ ^ e.
8.1. (c) The condition Fv(u0,y0,p0) 7* 0 implies that G.1) can be written in
the form G.17), where y is a real variable if d = 2. Application of part (a) gives
the desired result; cf. Exercise 7.2 (a) for the corresponding equations of the
characteristic strips.
8.2. (b) Let u(f), y(t),p(jt) be a characteristic strip for F = 0, i.e., a solution of
G.9). Then the relation G(u(t), y(t),p(t)) = 0 can be differentiated with respect
to t to give Guu' + Gy-y' + Gvp' = 0. Hence H(u(t), y(t),p(t)) = 0 follows
from G.9).
9.1. Introduce the new variables (x, x + y) instead of (x, y).
Chapter VII
3.1. Use C.1) and a circle/.
3.2. Consider the deformation A - s)fo(y) + sf(y) =fo(y) + sf^y), 0 ^
s ^ 1 and small \\y\\.
7.2. Make a real linear change of variables to bring A into a suitable normal
form.
10.2. Construct C as follows: Let a > 0 be large. C consists of the part of
the arc Ca: %v2 + G(u) = a. for u ^ a, the line segments v = ±y for \u\ ^ a
with y > 0 and \y2 + G(a) = a. and the part of the arc Q: Ju2 + G(u) =0
for u ^ -a with C = Jy2 + G(-a).
11.1. Put z(t, y) =
gaT\y)), where y = (y\ y2) and z = (z1, z2). Thus
z(?, 2/) is of class C1 and, for small |f| and \h\, z(t + h, y) = z(t, z(h, y)). Thus
dz(t, y)\dt = [dz(t + h, y)jdK\h^ = S (dz(t, y)/dy')[dz\h, y)/dh]h=0.
11.2. (c) By (b), dN is empty or dN = N. Correspondingly, N° = N or N°
is empty. In the first case, N is open and closed, hence N = M since M is
connected. In the second case, N" is empty implies that N is nowhere dense
since N is closed.
13.1. Let 1 > 0 be arbitrarily fixed. Let i,j be such that 0 < y, — y{ < f.
566 Ordinary Differential Equations
Then translates of the interval yl ^y < y} by m{j — i)a + h for h, m =
0, ±1, ... have endpoints of the form not + k and cover -co < y < co.
14.1. If the torus is cut along the circle r, it becomes a piece of cylindrical
surface. If a half-orbit remains in this piece of cylindrical surface for / ^ 0, its
set of co-limit points is a closed orbit; cf. the proof of Theorem 4.1.
14.2. There is an m0 such that ro = C+(m0) is a Jordan curve by Theorem
14.1. ro cannot be contracted to a point; cf. the proof of Lemma 14.1. If the
torus is cut along ro, it becomes a piece of cylindrical surface on which the
arguments in the proof of Theorem 4.1 become applicable.
Chapter VIII
2.2. Replace B.1) by dy\dx = Y/X; cf. the proof of Theorem III 6.1.
3.1. Introduce polar coordinates; cf. C.14) where a = 0, F1 = xh(r),
F2 = yh(r).
3.2. Let F(z) = (-yh(r), xh(r)), r = (x2 + y2fA ^ o and h(r) a suitably
chosen continuous function for 0 ^ r ^ 1 with h@) = 0.
3.3. Let F(z) = (-h(r)k(B)y, h(r)k{B)x), where r = (x2 + y2)^ ^ 0, h(r) =
|logr|-1 or 0 according as 0 < r |= \ or r = 0; and for the respective cases
(a), (b), or (c), let kF) = 1, |cos 6\lA, or 0.
3.4. The only characteristic values (mod 2w) are 6 = 0, -n. Apply Theorem
2.1.
3.6. Cf. Exercise 2.2.
4.2. Consider x' = — yx* — ys — q>(x, y)y, y' = xy* — x3y2 + (p{x, y)x, where
<KX, y) = 0 according as x2 + y2 ^ 0.
4.3. (b) Note that if y>(r) = a(log l/r)-*/(s-1), then D.36) has a positive
solution 6 = <r(log \jr)~1l(k~Vl provided that there exists an e>0 satisfying
e/(k — 1) — cek ^ a. This is the case if a > 0 does not exceed the maximum
of el(k - 1) - cek.
4.4. (b) Let yir) = a(log l/r)-*/'*-1' and let 6 = M(log l/r)-1'**-1' in D.38).
Then r(log \\r)du\dr = a - \u\{k - 1) - cuk] ^ const. > 0 if a exceeds the
maximum of uj(k — 1) — cuk.
4.6. (d) Put 0
O =
9?(j) ds + y/f-1 for a suitable constant y to be determined
4.6. (e) Put 60 =
"-" P
ds.
Chapter IX
5.3 (b) Suppose that n = Ei. + F(|) has no solution I. Let 0 < r1 < r2 be
such that F(|) = Oif Hill > rx, ||| + ?^AI1 < r2if ||||| ^ r1>and \E~\\\ < r2.
Then I -+?-1(?'| + F(|)) = I + E^FiS) maps the sphere ||||| < r2 into itself,
reduces to the identity on the boundary, and omits the point Erxr\. This is
impossible (Brouwer).
5.4. (d) If not, \\zn* — znl| > const, (c — 2c5)n for large n [by part (c) applied
to Sn* instead of SJ.
5.4. (e) Let (yn,zj = Tn(y0,z0), where z = g(y0); so that 2/nB/0), zB0/o) are
functions of %. Let ynh = [2/nB/0 + he,) - yn(y0)]lh, znh = [zn(y0 + he,) -
*»(»o)]/A. Then (»„»,*„») = (/<»»_! + Y^_x + Y2\z*n_x, GJL1 + Z*^ +
•^2n2n-i). where the matrices Yfn, Y2hn, Z?n, Z2\ tend to the Jacobian matrices
Hints for Exercises 567
3yiZ Yn, Zn evaluated at (yn-\, 2n-i)» as h -> 0. By (d), the functions j/B*, z,,* of
y0 are bounded and equicontinuous (for fixed n) and satisfy ||zm*|| ^ Il2/n'l|| for
n = 0, 1,.... Hence, they have limits (un, vn), as h tends to 0 through suitably
selected subsequences; (wn, vn) = T**(u0, v0), u0 = e,-, ||fn|| ^ ||«B||. By unique-
uniqueness, selection is unnecessary.
11.2. If the assertion is false, apply Stokes' formula to <j> / 2(l) dS1 -f\S) dS2.
Chapter X
1.4. (d) In order to obtain the sufficiency of the first criterion, make the change
of variables ? = [I — G0(t)]t] for large / and apply part (c). The sufficiency of
the second criterion follows from that of the first and from part (b) or can be
obtained from (c) by the change of variables S = [I + G0(j)\'1rj.
rt ct ft
4.1. (a) For the inequality involving a(t), write = + in D.18) and apply
Jo Jo Jt
integration by parts to the second integral.
4.1. (b) Note that a(j) is a solution of the differential equation a' + (c — e)a =
V and integrate this equation over the interval [s, t].
4.2. (a) Multiply the differential equation a' + (c — e)a = y for a by tr",
integrate over [0, /], and apply Holder's inequality to the integral on the right.
4.6. (b) Without loss of generality, it can be supposed that G(t) is continuously
differentiable, otherwise G(t) can be approximated arbitrarily closely by such
matrices. By Exercise IV 9.1, for large /, there is a continuously differentiable,
•/
nonsingular Q(j) such that \\Q'(j)\\dt < oo, lim 2@ as / ->- <x> is nonsingular;
Q-X[E + G(t))Q = diag [X^t),. . ., Xk(t), E22(t)l where E22(t) is a (d - k) x
(d — k) matrix and Eo = lim E22(j) as / ->- oo, is of the form Eo = diag (Ex, E2);
Ex is an m x m matrix with eigenvalues having positive real parts; E2 is an
n x n matrix with eigenvalues having negative real parts. The change of
variables I = Q(t)rj gives a system
V' = Q-H.01E + G(.t)]Q(t)r, + Q-\t)QV)v.
If r\ = (y, x, z), where y is a ^-dimensional, x an m-dimensional, and z an n-
dimensional vector, then the system for r\ has the form
Exx + Gu(.t)x + G12(t)y + G13(t)z,
diag [A^O, • • •, Xk(t)]y + G21(t)x +
E2y + G31(t)x + G32(t)y + G33(t)z,
G23(t)z,
where Gjk is a rectangular matrix. There exist continuous functions y(t), yo(/)
such that y(/) -»¦ 0 as / -+ oo and y>0(/) dt < oo and ||Gw@ll = v(') + Vo(')
for i,j = 1, 2, 3 and, in addition, ||G2//)|| ^ vo(') for j = l> 2» 3- II follows
that solutions rt(t) = (?/(/), x(t), z(t)) for which x(t), z(t) = o(,\\y(,t)\\) as / -»¦ oo
are bounded.
4.6. (c) If Tj(t) * 0 is a solution of the system in (b) such that x(t), z(t) =
o(ll.'/@II), then ? - Q(t)>)(t) has the desired form.
568 Ordinary Differential Equations
4.6. (d) If y = (y1,. . . , yk) in (b), introduce the new variables u = (u1, . . . , uk)
ft
in place of y, where y' = u' exp A,(j) ds fory = 1, . . . , k. Apply Lemma 4.3.
4.7. Use Exercise IX 5.3 (a).
4.8. (a) Let Kn = {{x,y,z): \\z\\ g, \e(\\x\\ + \\y\\)}, Kn0 = {(x,y,z): \\x\\ =
6 ||2/||}, and S = {(x,y,z): x = x0, y = y0, \\z\\ ^ ?e(||zo|| + \\yo\\)}. Apply
Exercise 4.7. To verify that Kn n Tn(S) is not empty, use Exercise IX 5.3 (b)
and show that there exists a z(n) such that the z-coordinate of TJx0, y0, z(n)) is 0.
4.9. (a) Note that \\yn+1 - yn\\ =: yo(//)(l + 26) \\yj, hence \\yn+1\\ ^
hull [1 + Vo(«)(l +26)]. Thus \\yj ^ c0 \\yj , where c0 = U [1 + V0("H +
26)] is a convergent (infinite) product. Hence 2 |l2/n+i — 2/n|| is majorized by
cbll»ollO + 26J yo(//).
4.9. (b) From the arguments in part (a) and 1 — y>0(n)(\ + 26) ^ 1 —
3yo(n) > 0, note that there exists a constant cx such that if the inequality in
D.44) holds for n =0,1,.. . ,y, then \\yk\\ ^ cx ||2/,|| and \\yk+1 - y,\\ ^c1 \\y,\\sk
for k = 0, 1,...,/ - 1, where sk = yo(A;) + yo(A; + 1) + y>0(k + 2) + ¦ ¦ ¦ .
Show that there exists a (yU), z(J)) such that if TJx0, y{i), zU)) = (xn, yn, zn), then
yj = yx and z> = 0. As in Exercise 4.8, ||zn|| =: %6(\\xn\\ + \\yn\\) for n =
0, 1,... ,/. For large j, we have \\x'\\ ^ 6 \\y>\\ = 6 U2/JI, hence ||x»|| = 6 ||j/n||
for n = 0, 1 y. [For suppose, if possible, that i|a;Ji| > 6 \\y>\\, then 6 <
' for A; = 0, 1 y. But then H^+^l =
+ d)/2 < 1, and so \\x'\\ ^ 60» ||x°||. For
and ||z01| = \\x°\\ = 6 11^11.] Thus
= 0,... ,y. It follows that \\yk - yx\\ ^
B/o» 2o) be a limit point of the sequence
A - a) 6/3 implies" that
60 liar*" '
large
ll-rfcll
where 60 = a + 6B + 6)A
y, this contradicts ||xJ|| > 6
|| for A;
j. Let
..fcll, llz*ll^ eii2/
ci II2/00H *s-i for * = 1,. .
B/A). 2A)). B/B)> 2B))>
11.2. Use Exercise 4.2.
17.1. Since R is not in a Jordan normal form if h < d, let y be a constant
matrix such that Y~*RY = E = diag [J(\),. . . ,J(g)] is a Jordan normal form.
It can be supposed that the diagonal elements of /(I) are A(l) = 0, so that
= h. Since ad_h+1 = ¦ ¦ ¦ = ad = 0, Y can be chosen of the form
<::¦)¦
where Ih is the unit h x h matrix and 0 is the rectangular (d — h) x h zero
matrix; cf. Exercise IV 8.2 where Y = Y@). The change of variables I = Yr\
replaces A7.4) by
V' = Er, + Y
Note that the only nonzero elements of G(t) Y are in the first row. The first h
elements of the first row of G Y are —pa-h+n ¦ ¦ ¦ > ~Pd> and the other elements
of this row are linear combinations of plt.. . ,pd with constant coefficients.
It is then necessary to consider the factor G(j)YQ(t) of Qr\t)Y^Gfj)YQ(t),
where Q(t) is described by A4.1)-A4.2). Actually, Q(t) is of the form
Q(t) =
o2 \
2W
Q2W
where Q-Jj) is an A x h matrix corresponding to the matrix Q(t) in A4.1) with
d replaced by h; Q2(t) = tx~PD2 where /3 = h - j + a and D2 is a constant
diagonal matrix; and 0u 02 are rectangular zero matrices.
Hints for Exercises 569
17.3. (a) Consider the differential equation for the function u'ju and integrate
this equation over the interval [/, s].
17.4. (a) and (b) This can be obtained from Exercise 4.6. It is more easily
obtained directly by assuming that q(t) has a continuous derivative and replacing
the second order equation by a first order system for y = (y1, y2) where y1 =
u' +f^u, y9' = u' -f*u, and /(/) = 1 + q(t) in case (a) and /(/) = 1 - q(t)
in case (b). Introduce a new independent variable s, ds = f ^ dt.
17.5. Write A7.23) as a linear system y' = H(t)y for the binary vector
y = («' +f'-^u, u' — f'^u). For parts (a) and (b), introduce the new independent
ft
variable j = |Re/^(r)| dr. There are solutions satisfying y% = oO?/1!) and
yl = o(\y2\) as t ->¦ co. For part (c), assume that / has a continuous second
derivative and introduce the new dependent variable 2 = (z1, z2), where y = Q(t)z
and QrxHQ is a diagonal matrix, say
Q
\-bl(a
-bj(a + c)'
bj(a + c) 1
')¦
where a =/*•*, b =f'j^f, c = (a2 + b2I^. Introduce the new independent
ft
variables = \Ref'-^(r)\ dr. Note that the assumptions imply that bj(a + c) ->¦ 0
as / -* 00 and
+ c)]'\ dt < 00.
17.6. Introduce the new dependent variable z defined by u = ze~t2/i; cf.
(XI 1.9). Apply Exercise 17.5 (b) to the resulting equation for z.
Chapter XI
4.1. If «(/) is a solution of D.U), D.2), then u(t) is a solution of D.U), D.2).
Apply Green's relation B.10) to/ = -Xu(,t),g = -A«(/).
4.3. (a) Denote by D.19n) the equation which results if u is replaced by un
in D.19). Multiply D.19n+1) by u0, D.190) by Un+1, subtract, and obtain vn' =
—(K+i — ^o)roun+iuo- Divide this relation by ro«o2 and differentiate to get
D.20) with v = vn = '¦oMo2(Mn+i/«o)'- The last part follows from vn'jv0' =
(K+i - \>)(h - K)~\uu+ilu<i) and L'Hopital's rule.
4.3. Q>) Note that vn = /vvK+iK)'- Write uot = «i@ for« = 0 k - 1
and uu = Vi(t) for / = 1,. ., k — 2. If u^if) for i = 0, . . . , k — j — 1, and
Pi > 0, rs > 0 have been defined, put pi+1 = l/r,H20. Consider u =
couo H + (??_!«?_! or rather u = comOo + ' ' ' + cfc-i«o,s-i- Then
Hence/JiM^I/JoMopCw/woo)'/"^]' = c2«2o + • • • + ct_1«%,fc_s. Continuing this proc-
process gives the desired result with a0 = 1/m00, «! =/'o«oo/«io. a2 = />i«2o/M2o>
4.4. (c) There is a constant c such that ?/„(/) = cZ>(/, /0, . . ., t^^, and
D(t, t0,.. ., /n_j) is positive or negative according as the number of inversions
in /, /„,.. ., /„_! is even or odd.
5.1. Choose a = 0, b — 1, 0 < * < 1. It suffices to construct a non-negative
continuous q(j) — q(j\ m, s) such that E.1) has a solution u(t) ^ 0 satisfying
570 Ordinary Differential Equations
Jo
= mA) =0 and m(t)q(j)dt ^ [m(s) +
J
- s). To this end, let
0 < d < min (s, 1 - s), u(t) a function having a continuous, nonpositive second
derivative on 0 ^ / ^ 1, such that m(/) = /andM(/) = A - t)(s — 8IA — s — S)
onO ^ t ^s - 6 and s + d ^ / ^ 1, respectively. Let q(t) = 0, -u"ju, or 0
according asO ^t ^s - d, s-S^t^s + 8, or s + 8 ^ t ^ I. Then
m@?@ dt = -
o Jg
dtlu(t).
Use
m(/) ^ ot(j) + <r if \t - s\ < 8 and d > 0 is small, u(t) ^ * - 8 for
|f - j| < <5, and -u"(t) ^ 0.
5.3. (a) It can be supposed that / =/+ ^ 0. (Why?) Put w = u(t) and
h = -gu' -fu,a=0 in B.18) and differentiate B.18) to obtain a formula
for u'(t). Let a = max \u'(t)\ and note that \u(t)\ ^ a min (/, b - t). The
P f6
function G(t) = \ s \g\ ds + F - j) |^| efc is nonincreasing or nondecreasing
Jo Jt
according as 0 ^ / ^ bj2 or b\2 ^ t ^ b.
5.3. (b) Multiply the differential equation by m; use uu" = (uu'Y — u'2 and
integrate.
5.3. (d) It can be supposed that u = Oat/ = 0, b. LetO = ^ < • • • < ad = b
be d zeros of u. If a = max |«(d)@l, then \u(t)\ ^ an |/ - ak\jd\ [This is a
consequence of Newton's interpolation formula but can be proved directly:
there exists a 0 = 0(/o) such that u(t0) = M(d)@)n(/o - ak)jd\ as can be seen by
considering v(t) = u(t) - CU(t - ak)jd\, where C = d\ u(t^jll(t0 - ak). Hence
v = 0 at / = /„, Oj,..., ad, so that its c?th derivative vanishes at some / = 0].
Since flx = 0, ad = b and 0 ^ t ? b, it follows that n |/ - a}\ ^ /*F - t)d~k
when ak ^ t ^ ak+i. For 0 ^ / ^ 6 and A; = 1, . . . , d — 1, one has
tk(b - k)k ^ max [t(b - 0d~\ t*-\b - t)] ^ bd(d - l)"-1^" by differential
calculus. Hence \u(t)\ ^ a.(bllld\)[(d - l)d/c?d]. In addition, u' has at least
c? - 1 zeros, say a{, . . . , a'd_i- Thus \u'\ ^ an |/ - a/\j(d - 1)! ^ af/
(d - 1)! Similarly, \u"\ ^ txbd-2/(d - 2)\,. . . , l«(rf—1)| ^ afc/1!. Choose /* in
the differential equation so that |mw)(/*)| = a.
5.5. (a) Use B.43) where p(t) = 1.
5.5. F) Use § 2(xiii) and part (a).
5.8. Use y@ = arc tan k'/11' = arc tanpu'jpv' and note that ^ ^ 0 implies
that the zeros of m and u' separate each other since u" ^ 0 [or u" ^ 0] if m ^ 0
[or m ^ 0].
6.1. Assume first that J: t1 ^ t ^ t2 and let m1; m2 > 0 in F.2). Next, if J is
not a closed bounded interval, let u^t), u^t) be linearly indpendent solutions of
F.1), so that by the first step, for any [tlt t2]eJ, there are constants c1; c2 such
that
y p y
+ c22 = 1 and the solution u =
+ c2u2(t) > 0 on [tv t2]. Use a
limit process involving [tlt t2]
6.2. Use the device A.7) to reduce F.1) to an equation of the form d2ujdsz +
qo(s)u = 0 on an interval 0 ^ j < co. From qo(s) ^ 0, it follows that
d2ujds2 ^ 0 when u ^ 0.
6.3. Suppose that the assertion is false. Then F.1) has a solution u(t) ^ 0
with two zeros a, b in J. Suppose that b is not an endpoint of J. Then m(/)
changes sign at t = b. Let F.1?) be the differential equation obtained by
replacing q(t) by q(t) — e, e > 0, and /?(»?, a, /3) the corresponding functional.
Hints for Exercises 571
Then I((tj; a, /S) is positive definite for e > 0 on ^2(a, C) for all [a, C] c7. Let
m?(/) be the solution of F.1?) satisfying uc(a) = 0, ue'(a) = u'(a). Then ue(t) ^ 0
for / # a. But ue(t) —¦ u(t) as <r -<¦ 0 uniformly on compact intervals of J, and so,
for small e, ue(t) = 0 at a / near b.
6.4. Show that if m2(/) # 0 on t1 S / ^ t2 [i.e., if C.12) is disconjugate on
h = ' ^ '2].tnen C.1i) is disconjugate on /j ^ / ^ /2. Compare the functionals
belonging to C.1!) and C.12).
6.5. (a) Let u(t) * 0 for T ^ t < co. It can be supposed that u(t) is a non-
principal solution; cf. first part of Corollary 6.3. Then
f
Jt
«or(O =
cf. F.14).
6.5. (b) Consider J:0 ^ / < -n and the differential equation u" + u = 0. A
principal solution is u = sin /. There is no principal solution satisfying m@) = 1.
6.8. This is a consequence of Exercise 6.5(a), but can be obtained directly as
follows: Note that uT(t) > 0, uT'(t) ^ 0 for a ^ t < T. (Why?) Also, if
a < S < T < eo, then us(t) ^ uT(t) for a ^ t ^S since us(t) * uT(t) for
/ > a and mt(/) / 0 for a ^ r < T. Hence mt'@) ^ 0 is a nondecreasing
function of T, so that lim mt'@) exists as T -* co. This implies the existence of
uo(t) satisfying F.19). Also uo(t) is a principal solution by (iii) of Theorem 6.4.
6.10. Make the change of variable u = vu^t) in F.302) and apply Corollary
6.4.
6.11. For 0 ^ / ^ \j-n, put M30 = 1 +y'2sin tjj; p} = 1 and q, = —uj0"ju}0 =
(sin tjj)j(\ +f sin tjj). For \j-n < / < co, extend the definitions of q}, uj0 so
that M;0 becomes the principal solution of F.333), with/j, = 1. Thus q-(t) —¦ 0,
/ —¦ co, uniformly on bounded intervals of 0 ^ / < co, but ui0(t) -* co as
j -*¦ co for every / > 0.
7.2. (b) Introduce the new dependent variable v = u exp — c ^(j) & and note
that, in the resulting differential equation, v" + G(t)v' + F(t)v = 0, we have
Fit) ^ 0.
7.3. Let q(t) = fijt2 and Q(t) = njt.
1A. Let u(t) be the solution of G.1) satisfying m@) = 1, m'@) = 0. Then
r = u'ju satisfies the Riccati equation G.4) and r@) = 0. Suppose u(t) > 0
for 0 ^ / ^ b. A quadrature of G.4) gives r = R - Q, where R(t) =
- r2(s) ds and R' = -r2. Since Q ^ 0 for a ^ t ^ b, R' = -r2 =
Jo
-fR - CJ ^ -(^2 + g2) and J? ^ 0 for a ^ / ^ 6. If 2 ?* 0, its logarthmic
derivative r1 = z'jz satisfies r{ = -(rj2 + Q2) and r^a) = 0. Also, rx(t) —¦ - 00
as / -<¦ /0 if / = /„ is the first zero of z greater than a. By Theorem III 4.1,
R ^ ^1 on the interval [a, /„).
7.9. Make the change of variables m = w}(t)z in G.35,), where w//) =
exp Qj(s) ds, use Theorem 7.4 with y = 2 and Exercise 7.2(a).
Jo
8.1. Introduce the new dependent variable u = t'-^v in Bessel's equation.
8.2. Use B.22), B.28), and Exercise X l.4(d).
8.3. (a) Use § 2(xiii).
8.4. (a) Write (8.8,) as a first order system for (u,pju'), and apply a linear
572 Ordinary Differential Equations
change of variables. It can be supposed that po(.uo'vo - uovo') exp (rolpo) ds =
1; so that neither 2 \p<po'vo\ nor 2 \p^ao'\ exceed | \po(uo + vo)(uo + vo)'\ +
ft
J \po(uo - vo)(uo - vo)'\ + |exp - J (rolpo) ds\.
8.4. (b) Assume that q(t) has a continuous derivative. For a fixed choice of
±, note that v = exp ±i q^(s) ds satisfies the differential equation u" +
Jo
[q T \iq-l/iq'\u = 0. Apply (a), identifying this equation with (8.8!) and (8.1)
with (8.8O). (In order to show that the conditions on q, including the paren-
parenthetical conditions, imply that a solution u of (8.1) and its derivative u' are
bounded, let q = g + ih, whereg(t), h(t) are real-valued. If E = g \u\2 + \u'\2,
then E' = g' |w|2 + ih(uu' - u'u), and so \E'\ ^ const. (|^'| + \h\)E for large /.
This implies that E, hence u and u', are bounded.) For another proof when q
is real-valued, see Exercise X 17.4(a).
8.5. Let z = uf'A, so that (8.12) becomes (f-^z')' + (,f^(t)g(t))z = 0, where
g(t) = 1 +/74/3/* - 5//2/16/5^; cf. B.34M2.35). If w denotes either of the
functions in (8.11), then w' = ±if'^hl^w, where h(t) = 1 +/'2/16/3 Thus
(f-\4w')' = —fXhw ±i(hM)'w. Using ±iw = w'lf^h1-^, it follows that w
satisfies (f~Mw')' — (fWf~Xfrl/?w' +fXfnv = 0. Identify the equations for w
and z with (8.8O) and (8.8!). Apply Exercise 8.4(a).
8.6. (a) and (b) Consider the case u(J)u'(J) ^ 0, say u'(J) > 0, u(J) ^ 0.
Integrate (8.13) over [T, t] to get
u'(t) ^ W(T)
-\\f\dr-\ ^
JT JT
Note that
[l
+
du(r).
Hence if T ^ / ^ U ^ T + 6jC and u' ^ 0 on [r, /], / ^ S, then
(*) u'(t) ^ M'(r) - | I/I c?r - Cm(/).
JT
Suppose that there is a (first) t = T*,T <T* ^U, where w' = 0, so that
(**)
^ f I/I
JT
+ Cu(T*)
at / = T. This is valid for T ^ t ^ T*. Integrate over [T, T*] to get an
estimate for u(T*), and hence by (**) with / = T, for u'(J) > 0. If no T* exists,
then (*) is valid for T ^ / ^ t/, and an integration over [T, t/] gives the desired
result.
8.6. (d) The proof of part (a) shows that if b - a = A < Bjc and a ^ / ^ b,
then |w'@l is majorized by either
f
Ja
\f\dr +C\lKT)\l(l ~
Hints for Exercises 573
or by one of the two quantities
r
Ja
Ja
|/| dr + r(a) or I |/| dr + r{b).
Ja
Thus, in any case,
A - 6) |k'@I ^ I/I dr + C \u(T*)\ + r(a) + r(b),
Ja
where \u(J*)\ = max \u(r)\ for a ^ r ^ b. Integrating this inequality over
[a, T*] gives an estimate for \u(T*)\ which, together with the last inequality,
gives the desired result.
8.7. Case 1: g(b) - g(a) ^0. Let a ^ c ^ b. PutA(/) = P(t) - N(t) + h(c)
where P(c) = N(c) = 0 and P(t), N(t) are the positive and negative variation of
h [so that P(J), N(t) are nondecreasing and var h = P(b) + N(b) — P(a) — N(a)]
Write
I
\h(t) - h(c)] dg(t) = \" P(t)d[g(t) -
Ja
Ja
N(t)d[g(t) -
Integrating by parts and using P(a) ^ 0 ^ N(b) gives (8.21) with inf h replaced
by h(c). Case 2: g(b) - g(a) = 0. The arguments just used show that, in this
case, (8.21) can be improved to
f
Ja
h(t) dg (t) ^ (var h) sup dg (t) for a ^ a < 0 ^ b.
Case 3: g(b) — g(a) < 0. It is sufficient to consider the case that g(t) is a
polynomial, for otherwise g can be approximated by polynomials and a limit
process applied to (8.21). For a polynomial^/) with^(a) >g(b), the interval
[a, 6] can be divided in a finite number of subintervals, a = t0 < tx <•••</„ =
b, so that g(j) is either decreasing on \th t1+1] or g(tj+1) = g(tj) for each j. If
[y, 6] = [tjt tl+1], then, by Case 2,
i;
h dg ^ 0 or
\
hdg -^ (var A) sup
[«]
for y ^ a < p ^ E.
8.8. Reduce (8.20) to the form (8.13) by introducing the new independent
C I C \
variable j = s(t) = exp I — d(t) dr I dr.
JT \ JT I
9.6. Apply § 2 (xiii) and Corollary 9.2 with A = 1.
9?9. (a) Use Lemma 9.5 and Corollary 9.3.
9.9. (b) Use Lemma 9.5 and Theorem X 17.5 [cf. (X 17.22)].
10.1. Suppose that /(??; a, b) ^ 0 for all [a, b] <= J and r) e A3/i(a, b) but that
A0.2) is not disconjugate on J. Let (*(/), y(t)) ^0 be a solution of A0.2)
and x(t) = 0 at points / = a, b of J. Suppose b is not an endpoint of J, say
t = peJ and p > b. Put %(/) = x(t) for a ^ / ^ b, ??0(/) = 0 for b ^ / ^ /3;
thus rj0 e AzJ^a, p). For an arbitrary r\ e A2(a, b), put ^(z) = rH(t) + ???(/) and
7@ = Kiel a, P). Thus /(c) ^ 0, /@) = /(»?„; a, b) = 0. Hence rfy(c)/rfc = 0 at
e = 0. Using A0.22) and integrations by parts, it is easy to see that (dj/de)e=0 =
2P(b)x'(b) ¦ rib). If nib) = x'Qj), it follows that x'(b) = 0, hence x(t) = 0.
Contradiction.
574 Ordinary Differential Equations
10.2. If (PyK')' + Qjx = 0 has a solution x(t) ^ 0 vanishing at two points
t = a,p, where a ^ a < p ^ b then /(»?, a, 0) = (P2V ¦ rf - Q2V ¦ rj) dt is
not positive definite on A2[tx, p]. Let ??(/) = x(t).
10.4. Note that B = P. Let B'^ denote the self-adjoint, positive definite
square root of B; cf. Exercise XIV 1.2. Since M is nonsingular and M =
f
L dt, it is seen that
(U^BU*-^ ¦ c) dt = \ \\Bl<*U*-lc
J J
From the relation H2 = (U^B^B-^Uc ¦ c = B~^Uc ¦ B^V*~xc and
Schwarz's inequality ||c||2 ^ \\B-^Uc\\ ¦ ||flW?/*-ic|], it follows that
' \\B-KUc\\-2 ^ WB^U'-icflWcW*.
Finally, Hfi-^t/cf = B~xUc ¦ Uc = Px ¦ x.
10.5. Cf. The proof of Theorem 10.4.
10.7. Without loss of generality, it can be supposed that ?/0(a) = U(a) = /.
Since V(a)U-\a) = U*~\a)V*(a), it follows that A = Ko in A0.12) Hence
A0.21) holds with j = a and Kx = /. If *:„ = A ^ 0, it follows that {. . .} in
A0.21) is positive definite, hence nonsingular, and so det U(t) jt 0 for / ^ a.
Conversely, suppose that Koc ¦ c > 0 for some vector c ^ 0 and suppose, if
possible, that det U(t) *0 for t ^ a. Then Sa(t), defined by A0.34) with
j=fland T(r) = I, satisfies A0.48) with s = a. But M = -*:„ ^ 0. This is a
contradiction.
Chapter XII
3.2. Write C.12) as a first order system for the 2c?-dimensional vector
(-«' - F*z,z).
3.3. (a) Use the Sturm comparison theorems, e.g., Corollary XI 3.1.
3.3. (b) Repeat the proof of Theorem 3.3 with r = ||a;(/)||2 + x(s)-K(s)x(s) ds.
Jo
4.1. (b) Introduce x — txjp as a new dependent variable; see Theorem 4.1
and Remark 1 following it.
4.2. Introduce x - txjp as a new dependent variable in D.1).
4.3. Let i?! > 0 be arbitrary. Then ||/|| ^ M, where M = aR^ + b if
llx'H ^ i?!. Show that if i?x = r*, then the two inequalities in D.20) hold.
4.4. If there exist two solutions, let x(t) denote their difference and r(t) =
\\x(t)\\2. Show that r'\t) ^ -B0O + WWO; cf-the proof of Theorem 3.3.
4.6. (a) If xx(t), x2(t) are two solutions, the difference x1 — x2 cannot have a
positive maximum at a point /„, 0 < t0 < p. Otherwise x{ = x2', (xx - x2)" > 0
at t = t0.
4.6. (b) The boundary value problem x" = -7>(x'YA, «@) = a<2) = 0 has the
solutions x = 0 and x = [1 - A - /L]/4.
4.6. (c) Suppose that there are two solutions x^t), x2(t) and that x(t) =
x^t) — x2(t) has a positive maximum at a point /„, 0 < /„ </j. It can be supposed
that »(/) > 0 for 0 < t < p, otherwise @,p) is replaced by the largest interval
Hints for Exercises 575
containing / = t0 on which x(j) > 0. Put
or a(/) = 0 according as xx'(t) - x2'(t) * 0 or = 0 and /?(/) = /(/, xx(t), x2'(t)) -
/(/, a;^/), x2'(t)). Then «(/) satisfies a differential equation of the form x" =
oi(t)x' + /?(/), where oc(/) is bounded and measurable and /?(/) > 0. Hence, if
y(t) = exp - a(j) ds, then (y(t)x')' = y(/)/3(/) ^ 0. Introduce the new
Jo
independent variable s, where ds = y(/) c?/, so that dxjds = y»' and d2x/ds2 ^ 0.
4.7. (a) If there are two, say xx(t) and x2(t), let a; = x1 — x2. Then x"(j) >
0 [ <0] at any / where x has a positive maximum [negative minimum].
4.7. (b) At a point where x(j) has a maximum, x'(t) = 0 and x"(t) ^0. At a
point where x'(t) has a maximum, »"(/) = 0.
4.7. (c) If A = 0, there is the solution x(t) = 0. Use Theorem 2.3 to show that
the set of A is open on 0 ^ A ^ 1 and use part (b) to show that the set of A is
closed.
4.8. Let h(t) be periodic of period p and of class C1. Then
t, h(t), x')
t, h(t), x')
has a unique solution x(t) = T0[h] of period/j by Exerise 4.7. Show the applica-
applicability of Schauder's fixed point theorem.
5.2. On 0 ^ / ^ 1, consider the family of real-valued functions x =
1 + m\t - I///L or x = 1 according as 0 ^ / ^ \\n or \\n ^ / ^ 1.
5.3. If E.21) is assumed instead of E.29), then r(t) has no maximum on
0 <t <p. Hence if 0 ^ /„ ^ / ^p, then r(j) < max(r(/0), r(p)) = r(t0); i.e.,
r(/) is nonincreasing, so that E.30) holds. When E.29) holds, the proof of
Theorem 5.1 shows that r = |]a;?@ll2 satisfies E.30). But the inequalities E.30)
are not lost in the limit process e = e(n) ->¦ 0.
5.4. Introduce the new dependent variable x = u — «o(/), then the boundary
value problem becomes x" =f(t,x,x') and x@) = «o - «o(O), x(p) = uv - uo(p),
where /(/, x, x') = h(t, x + «„(/), x' + uo'(t)) - h(t, u^t), uo'(t)). Note that
±/(/, ±R, 0) ^0 for R > 0 since h is nondecreasing in u. Also, |/| ^
9?(|a;'| + c) for a constant c ^ |«b'@l- Apply Corollary 5.2.
5.5. (a) Let w = 1, 2, By Theorem 5.1, E.26) has a solution on
0 5; / ^ w satisfying »@) = »„, ^w) = 0. For any p > 0, the sequences of
functions «„(/), xm'(t) for m ^ /j are uniformly bounded and equicontinuous on
0 ^t ^p.
5.6. Note that r = 11»(/)]|2 satisfies r" ^ 0 for large /. Thus if the assertion is
falsfc, there is an m > 0 such that 0 < m2 ^ r(t) ^ i?2 for large /. Define
q(t) = 2[x -fit, x, x') + \\x'\\2y\\x\\2, where x = x(t). Hence r" - q(t)r = 0 and
q(t) ^ 2h{t)\m2. Apply the last part of Corollary XI 9.1.
5.9. (b) Define a function xx(x0) of x0 for ||a;0|| ^ i? as follows: xx = x(p, x0).
This gives a map of the ball \\xo\\ ^ R into itself. Apply Brouwer's fixed point
theorem.
8.2. Let?-(/) be a bounded, measurable function for / ^ 0. Then
2/@ = f G(t, s)g(s) ds + U(t)y0
Jo
is a bounded solution of F.1) satisfying /^(O) - y0. For x(t) a measurable
576 Ordinary Differential Equations
function on t ^ 0 such that ||x(/)|| ? p, letgQ) = /(/, *(/)) and let 2/@ = To[x(t)]
be defined by the last formula line. Let 93 = 2) = LX(Y) and consider To as a
map of 2p into ?.
9.1. It can be supposed that 2/@ = 0, otherwise introduce y — y^t) as a new
dependent variable in (9.5). The problem is reduced to showing existence of
solutions [for the transformed equation (9.5)] which exist for large t and tend
to 0 as / ->- co.
Chapter XIII
2.1. Suppose first that B.26), hence B.28), holds. This is equivalent to
max (||z(r)||, \\y(t)\\) ^ K\\z(s) - y(s)\\ for 0 ^ r ^ s ^ t
if 2/@) e Wo, z@) e fVv If z@) ? Wo, let z@) = z°@) + 2/@), where 2/@) 6 PF?.
z°@)e WX. Follow the proof of Lemma 2.2 to obtain analogue of B.7). This
gives the "if" part under the assumption B.26). The proof under the assumption
B.27) follows by a modification of the proof of Lemma 2.5.
6.3. Let 0 ^ y(t) e B and
y@
rt+i
=
Jt
yds,
ft
= \ e
Jo
=
Jt
ds.
It suffices to show that v>i, y2 e ^! cf ¦ the proof of Theorem 6.4. Let the integer
n ^ 0 be such that n ^ / < n + 1. Then
Vi@
= Y e-M
i=\ Jt-i
n rt-j+i ' rt-n
^ y e-Ui-i) <p ds + e~Xn <f
j=l J«-i JO
ds
where y>'(t) = 0 or y>'(t) = y>(t — j) according as / < / or / g; j. Thus y>te D;
in fact, Iv^U ^ S e~3A |v>Id by condition (d) for Z) e&~. Similarly, a use of the
first part of F.9) shows that f2 e D.
7.1. Use Exercise XI 8.8 and Theorems 5.3, 5.4.
13.1. Use Theorem 12.1 and Exercise 7.1.
13.2. Use Theorem 12.3.
Chapter XIV
1.1. If the assertion is false, then A.30) holds for all solutions. Hence
|det Y(t)\2 has a finite limit as t -+ a>. This contradicts A.5oo) and A.6).
1.2. (a) Let U be a unitary matrix such that U*AU = D is diagonal,
say D = diag [Ax2,..., A/], where A, > 0 for / = 1, . .., d. Put D^ =
diag (Aj, . . . , Ad). Then D ^ D'^D'^ and A = UDU* = (UD^U*)(UD^U*).
Let A^ = UD^U*.
1.2 F) It is sufficient to suppose that A = D is diagonal, say Z> =
diag [Aj2, . . ., A/]. (Why?) Let D^ be any Hermitian, positive definite square
root of D. Let the vector y and eigenvalue A of ZI-^ satisfy D'-^y = ty. Then
Hints for Exercises 577
Dy = A22/, so that y is an eigenvector of D belonging to the eigenvalue A2. It
follows that DM = diag [Alt . . ., Ad]. (Why?)
1.2. (c) Consider only / near a fixed t0. Then there exists an e > 0 such that
A(t)y ¦ y ^ e||2/||2 for all vectors y. If ^(/) is multiplied by a positive constant
c > 0, its square root is multiplied by c'A. Hence it can be supposed that
e\\VII2 ^ AQ}y ¦ y ^ 6\\y\\2, where 0 < c < 6 < 1. Then
( 2 / - ^(OlT » ? A - 0)\\y\\2 and ]|/ - A(t)\\ ^ 1 - c < 1.
= B(/), where
f if A - r)H = 1 - J V-
Show that
Note that (S(/)J = A(t) follows from [A - rI^]2 = 1 - r since the powers of
/ — A(f) commute. Also B(t) is Hermitian and positive definite since an > 0,
Oj + a2 + ¦ ¦ ¦ = 1. (This gives another proof of the existence of A1^.) Finally,
the series for B(j) = A'-^(t) can be differentiated term-by-term.
2.1. (c) Consider the binary system y1' = —y2,y2' = —yjt2 for / ~g. 1 and
Exercise XI 1.1 (c).
2.1. (d) For necessity, cf. (b); for sufficiency, cf. the proof of Theorem X 1.1.
2.2. Cf. §IV 5.
2.3. (a) Let y(t) be solution given by Theorem 2.1. Verify (- l)nyin)(t) ^ 0
by induction on n.
2.4. Write the sum of the first two terms po(t)u{d) +/J1(/)«(d-1) of B.7)
as po(t)or\t)(ai(t)uid'1)y, where ot(/) = exp p^s) ds/po(s) > 0. Divide the re-
Jo
suiting equation by po(t)ja.(t) and consider the result as a first order system for
2/ = (m, -«', «',..., (-l)d-Vd-2),(-l)d-1a(/)M<d-1»),i.e.,for2/ = (y\ ..., yd),
where y' = (-l)'-V'-1' for j = 1, . . . , d - 1 and yd = (-l)d~1ix(t)u{d-1K
Theorem 2.1 implies the existence of a solution satisfying B.9) for n = 0,
1, .. ., d-\. (Why u > 0?) From B.7), it follows that (-l)d(/>ow(d> +
2.5. Prove B.9) by induction on n. To this end, first show that if d > 2 and
B.9) holds for n = 1,. .., d - 2, then it holds for n = d - 1. Note that
(-l)d[pou{d) +/>1H(d-1)] ^0, hence [(-l)""^^-1' cxp\ (pjpo) ds]' ^0.
Jo
that if (-l)d-Vd-1)(a) < 0 for some a on 0 ^ a < oo, then
(-l)d-1K<d-1>@ <0
an*-(-l)d-V<i-2)(/) is increasing for t ^ a. If c? ^ 3, this is incompatible
with (-i)<*-2M<<*-2) ;> o and the fact that lim u{d-3)(t) exists as / -+ oo.
2.6. (a) Use analogue of Corollary 2.3 for 0 < / < oo and the Remark
following Theorem 2.1; for uniqueness, use Exercise XI 6.7.
2.7. There exist positive continuous functions ao(/),. .. , am(/) such that the
expression on the left of B.12) can be written as
/'0am(am-l{am-2[- • ¦(«(>«)'¦ • -I'}')'!
§IV8(ix). Thus B.7) is
Show
= 0.
k~m\ 1
578 Ordinary Differential Equations
Write this as a first order system for y = (y1,. . ., yd), where
j/* = (-l)*-1^*-1) for k = 1, . . . , d - m
and yo-™+* = (-l)i-m+k-1oLk_1{oL^2[. . ,(a0M<d-m))'. . .]'}' for k = 1, . . . , m.
Apply Theorem 2.1.
2.8. Suppose first that (*) if T > 0, then there exists a solution yT(t) of
y' = -/ on 0 ^ / ^ T satisfying \\yT((i)\\ = c and y^t) ^ 0, yT\t) ^ 0 for
0 5; / ^ T. Choose a sequence of T-values T1 < T2 < . . . such that Tn -* 00
and 2/@) = lim 2/y@) exists as n -* 00, where T = r(n). Then ?/(/) = lim yT(t),
T = T(n) and n -* 00, exists uniformly on bounded /-intervals (why?) and is a
solution with the desired properties. Thus it only remains to prove (*) for every
fixed T. To this end, suppose first that solutions of y' = —/ are uniquely
determined by initial conditions and let y(j, y0) be the solution satisfying y = y0
at / = T. Since y = 0 is a solution, it follows that y(t, y0) exists on 0 ^ / ^ T
if \\yo\\ is sufficiently small; Theorem V 2.1. Since f ^ 0 and y0 ^ 0, it follows
that y(t, y0) ^ y0 on 0 ^ / ^ T and so ||2/@,2/0)|'| ^ H2/0II. Thus if c > 0 is
sufficiently small, \\ij@, yo)\\ = c holds for some small \\yo\\. Let S be the set of
numbers c0 such that, for every c on 0 < c < c0, there is a 2/0 with the property
that \\y@, yo)\\ = c. It is clear that S is closed relative to the half-line 0 < c0 < 00
(why?) and also open relative to 0 < c0 < 00 (why?). Hence S contains all
c0 > 0. This proves (*) in the case that the solutions of y' = -/are uniquely
determined by initial conditions. In other cases, approximate / by smooth
functions.
2.9. (a) Let yo(t, a) be the solution of y" = / satisfying y0 = 0, y0' = a ^ 0
at/ = 1. This solution exists for 0 ^ / ^ 1 if -a ;> 0 is sufficiently small. Also
Vo > 0. Vo' ^ 0.2/0" ^ 0 on 0 ^ / ^ 1, since y" = f ^ 0. Let c = 2/0@, a) > 0
for some small — a > 0 and y(t, P) be the solution of y" = /satisfying 2/@) = c,
2/'@) = p 5; 0. For c fixed, let S be the set of p < 0 for which there exists a
'0 = 'o(^) > 0 such that ?/(/, C) exists on 0 ^ / ^ /0, y{t, P) > 0 for 0 ^ C ^ /<,,
2/(>o, ?) = 0. Then ?/'(>, C) ^ 0, y"(t, P) ^ 0 for 0 ^ / ^ /„. Show that S is not
empty and is open. Let p0 = sup p for p e S and show that 2/(/, ^0) is the
desired solution.
2.9. (b) Consider y" = 3A + y'%f+xy2l2K where 0 < A < 1. This has no
solution satisfying 2/@) > 1 and y(j) ^ 0, y (t) ^ 0 for all / ^ 0; cf. Hartman
and Wintner [8, pp. 396-397].
3.1. Use Theorem XI 3.2; cf. diagram for Exercise 3.5.
3.2. Introduce the new independent variable j where ds = dtjp(t) and s@) = 0.
3.5. Use Sturm's second comparison Theorem XI 3.2 to show that if the arch
of the graph of u = \uo(t)\ on /„ ^ / f= tn+1 is reflected across the line / = /n+1,
then it lies "over" the arch of u = |wo(/)| on tn+1 ^ / ^ tn+2\ see diagram. Use
"alternating series" argument to prove C.14).
3.8. Let 0 ^ $ < 00 and let G(s, /) be the solution of u" + q(j)u = 0 satisfying
the initial conditions u = 0, u' = 1 at / = j [e.g., if q(j) = 1, then G(s, t) =
sin (/ —s)]. Show that v(s) = G(s, /)/(/) dt is uniformly convergent on bounded
Hints for Exercises 579
j-intervals and is a solution of v" + qv = f. Note that when q(j) > 0, then
\G(s, /)|2^ \Gt(s, sTIqit) < llq(t) and \Gs(s, t)\2 ^ \Gs(s, s)\2 = 1 for / ^ s by
Theorem 3.1,30. The proof of v ^ 0 in the case n = 0 depends on arguments
similar to those in the proof of Corollary 3.1. When n = 1, write dv/ds =
("gAs, /)/(/) dt and Gs(*, /)/(/) asq(t)Gs(s, /)[/(/)/?(/)] = -Gstt(s,
apply integration by parts twice. For n = 2, use the fact that w = v"jq is a
solution of w" + qw = (flq)", for w + v = fjq.
3.9. (a) Cf. Exercises XI 7.5 and XI 7.6. Use r' + r2 + q = 0, hence
r' + q ^ 0, r' + r2 ^ 0.
3.9. (b) Cf. Corollary XI 9.1 for "only if." For the "if" part, use dE =
(•CO
u2 dq ^0 and that tq(t) dt = 00 implies that u'( 00) = 0 for all solutions.
Suppose that ?@0) > 0 for some nonprincipal u, so that r' + Eju2 = 0 gives
—r' ^ c/w2, c > 0. Integrate over [/, 00), multiply by u to obtain u'(t) ^
cuo(t) ^ const. > 0.
3.9. (c) First prove the assertions concerning r0(oo), r(oo) by using Corollaries
XI 6.4 and 6.5.
3.9. (d) For the first part, cf. the hint for part (c). For the second part, use
Exercise X 17.4 (b) and dE = m2 dq.
4.1. Let u be as in D.6), then u2q + u'2 is a quadratic form in (cos <p, sin <p) for
fixed /. In the case D.3 +), D.7) holds and shows that the eigenvalues A of this
form satisfy A ^ 1. Using D.8), it is seen that the equation for the eigenvalues is
A2 — M\z\2 q + \z'\2) + q = 0. The desired inequality merely expresses the fact'
that the largest root A of this equation satisfies A ^ 1. The case D.3—) follows
from D.3+); cf. Lemma 3.1.
4.2. (d) Make the change of independent variables / = es in Bessel's equation
D.21); then the change of dependent and independent variables V = (t2 — i^f^-v,
da = (t2 - n*yA ds for / > /u.
4.3. (b) Take a number a such that, up to a constant factor, u = mo(/) cos a +
w^Osina. It can be supposed that u = u0, for otherwise replace u0 and ux
by u0 cos a + t/j sin a and — u0 sin a + ux cos a, respectively. Define a contin-
continuous 0(/) by 0(/) = arc tan mo(')/«i(O and 0(/o) = 0, so that 0(/n) = w. Then
B\t) = l/(w02 + Uj2) > 0. Let / = /@) be the inverse function of 0 = 0(/).
Then (-l)'+V3/@)/rf0» ^ 0 for j = 1, . . . , n + 2 and tn = /(//tt) for n = 0,
1, .... Note that a mean value theorem of Holder implies that if a function
/@) has / ^ 1 continuous derivatives, then A3/(//tt) = 7r3(c?3'//c?03)e=), where
0 = y is some number satisfying w ^ y ^ (n + j)tt. This fact is easily verified
from A>t(mr) = [A3'-1 c?/@ + m
J c?0! c?02 . . . rf0y.
6.3. The existence proof is similar to that of Theorem 6.1 where fl° is chosen
to be {(/, 2/): /, y1 arbitrary, y2 > 1, ys < 0}. For uniqueness, use a variant of
Exercise III 4.\(d).
12.1 (a) Cf. the Remark following A2.4).
12.1 (b) Introducing polar coordinates with r = \\y\\, it is seen that ds ^
p(y) dr ^.po(r)dr if po(r) = min p(y) for \\y\\ = r. Use Lemma I 4.1.
12.4. (a) The ds is complete by Exercise 12.1F). If y(t), 0 ^ / ^ 1, is any C1
dd
s (" -t- pir. inis i
= f" • • \\dh(m
Jo Jo
+ e1 +
+
580 Ordinary Differential Equations
r\\v}\
arc joining y = 0 and y, then its Riemann length is not less than I Po(u) du
Jo
if po(u) = min p(y) for \\y\\ = u; cf. the proof of Exercise 12.1F). Hence
K0. y) is n°t less than this integral.
14.3. If a(y) = Aj + A2, verify that, for fixed y and arbitrary vectors x, z,
{Jx ¦ x)\\z\\* + \\x\\2(Jz ¦ z) - (x ¦ z)[Jx -z +x-Jz]^ <x(\\x\\* jj2|j2 - (x ¦ zf).
To this end, note that the inequality is not affected if J is replaced by JH and x,
z are subject to an orthogonal transformation; thus it can be supposed that
JH = (jiag [^i; A2, . A] Th lft ide f the desired inequality is
d d
2 2 ti&x
or, equivalently,
i 21 ^ + Wxiz1 - xhif = ia 2 2 (icizJ' -
This gives the stated inequality. Let z =f(y) and x ¦ f(y) = 0.
g p
]. The left side of the desired inequality is
d d
+ ^Vz'V - Ix^z^) =22
;=ij=i
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598 Ordinary Differential Equations
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Index
Adjoint, dih. order equation, 66
linear system, 62
system of total differential equations,
119
see also Associate operator
Admissible, 438, 462, 481
Angular distance, 451
Arzela theorems, 4
Ascoli theorems, 4
Associate operator, 486
Asymptotic integration, boundary layer
theory, 534
difference equation, 241(Ex. 5.4),
289(Ex. 4.8), 290(Ex. 4.9)
rfth order equation, 314, 447(Ex.
9.2(c»
perturbed linear system, 212, 259,
273, 445, 447(Ex. 9.3)
second order equation, 319(Ex. 17.4,
17.5), 320(Ex. 17.6), 369, 375,
446,447(Ex. 9.2)
Asymptotic lines, 107(Ex. 6.3(c))
Asymptotic phase, 254
Asymptotic stability, see Stability
Attractor, 160,213
domain of attraction, 548
Autonomous system, 38, 144, 202
Banach spa^s, admissible, 438, 462,
481
associate, 484
class 3, 3#, 452, 484
lean at w, 439
Lp,L°°,L0«>,M, 436,453
open mapping theorem, 405, 437,
439, 464
quasi-full, 453, 467, 471, 474, 475
Schauder fixed point theorem, 405,
414, 425
stronger than L, All, 453
Bessel equation, 87(Ex. 12.3(a))
asymptotic behavior of solutions,
371 (Ex. 8.1)
integral of 7^,513 (Ex. 3.6)
Jf + K*, 518(Ex. 4.2), 519(Ex.
4.3)
zeros of solutions, 336(Ex. 3.2(</)),
519(Ex. 4.3)
Blasius differential equation, 520
Bohl theorem, 199
Boundary layer theory, 519
Boundary value problems, 337, 407,
418,519
adjoint, 410
linear, first order, 407
linear, second order, 418
nonlinear, second order, 422, 433,
434(Ex. 5.4)
periodic, nonlinear, 412, 435(Ex.
5.9F))
singular, third order, 519
Sturm-Liouville, 337
Bounds for solutions, 30
autonomous system, 543
derivatives, 428
equation of variation, 110
second order equation, 373(Ex. 8.6),
374(Ex. 8.8)
see also Differential inequalities
Brouwer fixed point theorem, 278
Cantor selection theorem, 3
Cauchy, characteristic strips, 133, 135
initial value problem, 131, 137, 140
Center, 159, 215, 216(Ex. 3.1)
Characteristic direction, 209, 220
Characteristic equation (polynomial),
65
Characteristic roots and exponents, 61,
252, 253
607
608
Index
Characteristic strips, 133, 135
Comparison theorem, principal solu-
solutions, 358
second order equations, 333
second order systems, 391(Ex. 10.2)
Sturm, 333
Complete integrability, 118, 123, 128
Complete Riemann metric, 541, 546
Complete system of linear partial dif-
differential equations, 119, 120(Ex.
1.2), 124
Conjugate, points (Jacobi's theorem),
391(Ex. 10.1)
solutions, 385, 386, 399
Constant coefficients, rfth order equa-
equation, 65
linear system, 57
second order equation, 324 (Ex. 1.1)
Continuum, 16
Curvature, quadratic form, 106(Ex. 6.2,
6.3F))
lines of, 107(Ex. 6.3(rf))
Riemann tensor, 106(Ex. 6.2)
Cycles, see Limit cycles
Derivate, 26
Dichotomies, adjoint systems, 484
definitions, 453
first order systems, 474
Green's functions, 461, 476, 477
higher order systems, 478
second order equations, 483(Ex. 7.1),
496(Ex. 13.1, 13.2)
Difference equations, 241 (Ex. 5.4), 289
(Ex. 4.8),290(Ex. 4.9)
Differential forms, 101, 120
Differential inequalities, 24
equations of variation, 110
linear systems, 54
partial, 140, 141<Ex. 10.1), 142(Ex.
10.2)
Disconjugate equation of second order,
351,362
positive solutions, 351, 352(Ex. 6.2)
variational principles, 352
see also Nonoscillatory
Disconjugate system of second order,
384
Disconjugate system of second order,
criterion, 388, 390, 391 (Ex. 10.1,
10.2), 420, 421(Ex. 3.3)
Domain of attraction, 548
Egress point, 37, 175, 202, 278, 281,
520
Eigenfunction, 338, 342
interpolation, 344(Ex. 4.4)
IAapproximation (completeness),
338
Equations of variation, 96, 110
bounds for solutions, 110
Equicontinuity, 3
Equivalence, differential equations, 258,
271(Ex. 14.1)
maps, 258
Ergodic, 193, 194, 198, 199
Euler differential equation, 85
Existence in large, 29
for linear systems, 31, 45
Existence theorems, boundary value
problems, 337, 407, 418, 519
Cauchy problem, 137
invariant manifolds, 234, 242, 296
linear system of partial differential
equations, 124
maximal solution, 25
monotone solution, 357, 506, 514
(Ex. 3.8)
PD-solution, 497
Peano,10
Picard-Lindelof, 8
see also Periodic solution, Solutions
tending to 0
Extension theorem, 12
Exterior, derivatives, 102
forms, 101, 120
Fixed point theorems, see Brouwer,
Schauder, Tychonov
Floquet theory, 60, 66, 71, 302(Ex.
11.3)
Focus, 160, 215, 216(Ex. 3.1)
Formal power series, 78, 79, 261
Frechet space, 405, 436
Frobenius, factorization, 67
Perron-Frobenius theorem, 507(Ex.
2.2)
total differential systems, 117, 120
Index
609
Fuchs, theorem, 85
type of differential equation, 86(Ex.
12.2)
Fundamental matrix or solution, 47
adjoint system, 62
analytic system, 70
characteristic exponents and roots, 61
Floquet theory, 60, 71, 302(Ex.
11.3)
system with constant coefficients, 57
unitary, 62(Ex. 7.1)
Geodesies, 106(Ex. 6.2)
Global asymptotic stability, 537
Green's formula, 62, 67, 327, 385
Green's functions, 328, 328(Ex. 2.1,
2.2), 338, 409(Ex. 1.2), 439, 441,
461, 476, 477
Gronwall inequality, 24
generalized, 29
systems, 29 (Ex. 4.6)
Haar's lemma, 139
Half-trajectory, 202
Hermitian part of a matrix, 55, 420
Homann differential equation, 520
Hypergeometric equation, 509 (Ex.
Rummer's confluent form, 509 (Ex.
2.6(c))
Whittaker's confluent form, 509 (Ex.
Implicit function theorem, 5, 11 (Ex.
2.3)
Index, see Jordan curve, Stationary
point
Indicial equation, 85
Inhomogeneous equations, see Varia-
Variation of constants
Integrability conditions, 118, 119, 123,
128
Integral, first, 114, 124
{m +l)st, 478
Invariant manifold, 228, 234, 242, 296
Invariant set, 184, 184(Ex. 11.2, 11.3)
Jacobi, system of partial differential
equations, 120
Jacobi, theorem on conjugate points,
391 (Ex. 10.1)
Jordan curve, flow on, 190
index and vector field, 149, 173
theorem, 146
Jordan normal form, 58, 68
Kamke uniqueness theorem, 31
van Kampen uniqueness theorem, 35
application, 113
Kneser, H., theorem, 15
Kronecker theorem, 194
Lagrange identity, 62, 67, 327
Legendreequation, 87(Ex. 12.3F))
associated equation, 87(Ex. 12.3(c)),
508(Ex. 2.6(a))
Lettenmeyer theorem, 87, 91 (Ex. 13.2)
Lienard equation, 179
Limit cycle, 145, 151, 152, 156, 178,
181 (Ex. 10.4, 10.5), 190,253
Limit points, w- and a-, 145, 184
set of w-limit points, 145, 154, 155,
158, 184(Ex. 11.3, 11.4), 190, 193
Linearizations, differential equations,
244, 257(Ex. 12.1), 258
maps, 194, 245, 257
Linearly independent solutions, 46, 64,
326
Liouville, formula, 46
Sturm-Liouville problems, 337
substitution, 331
volume preserving map, 96(Ex. 3.1)
Lipschitz, continuity, 3
S- and L-Lipschitz continuity, 107
Lyapunov, asymptotic stability, 38, 40,
539
function, 38, 40, 539
order number, 56, 294, 301, 303
stability, 38, 40
theorem on second order equation,
346
uniform stability, 40
Manifold, definition of 2-manifold, 182
flow on, 183
invariant, 228, 234, 242, 296
P(B,D)-manifold, 484
stable and unstable, 238, 244, 255
610
Index
Maps, associated with general solution,
94,96,231
associated with periodic solution, 251
equivalence, 258
invariant manifolds, 234
linearization, 194, 245, 257
volume preserving, 96(Ex. 3.1)
Matrix, exponential of, 57
factorization of analytic, 75
Jordan normal form, 58, 68
logarithm of, 61
norm, 54
trace (tr), 46
see also Fundamental matrix
Maximal interval of existence, 12
Maximal solution, 25
Minimal set, 184, 184(Ex. 11.2, 11.4),
185, 190, 193
Monotony, 500
functions of solutions, 500, 510, 518,
519(Ex. 4.3)
solutions, 357, 506
Nagumo theorems, 32, 428
Node, 160, 216, 216(Ex. 3.2, 3.3), 219
Nonoscillatory equations of second or-
order, 350, 362
criteria, 362
necessary conditions, 367, 368
see also Disconjugate
Norm, matrix, 54
on R<*, 3
Open mapping theorem, 405, 437, 439,
464
Operator, associate, 486
closed, 405
graph, 405
null space, 462
Order number, 56, 294, 301, 303
Oscillatory, 333, 351, 369, 510
see also Nonoscillatory
Osgood uniqueness theorem, 33
Peano, differentiation theorem, 95
existence theorem, 10
Periodic solution, associated map on
transversal, 251
characteristic roots, 61, 252
Periodic solution, existence, 151, 178,
179(Ex. 10.1 (b)), 179, 181 (Ex.
10.4), 198, 407, 412, 415, 416,
435(Ex. 5.9F))
Floquent theory, 60, 66, 71, 302(Ex.
11.3)
see also Limit cycle, Stability
Perron, singular differential equation,
91(Ex. 13.1)
theorem of Perron-Frobenius, 507
(Ex. 2.2)
Perturbed linear systems, 212, 259, 273
invariant manifolds, 242, 296
linearization, 244, 257(Ex. 12.1)
see also Asymptotic integration
Picard-Lindelof theorem, 8
Poincare-Bendixson theorem, 151
generalized, 185
van der Pol equation, 181 (Ex. 10.3)
Polya's mean value theorem, 67 (Ex.
8.3)
Principal solution of second order
equation, 350, 355
comparison, 358
continuity, 360
definition and existence, 355, 357
(Ex. 6.5)
monotone, 357
Principal solution of second order sys-
system, 392, 398 (Ex. 11.4)
Priifer transformation, 332
Quasi-full Banach space, 453, 467, 471,
474, 475
Reduction of order, 49
second order equation, 64, 327
Regular growth, 514(Ex. 3.7)
Retract, 278
quasi-isotopic deformation, 280(Ex.
2.1)
Riccati, equations, 331, 364
differential inequality, 362, 364, 368
generalized equation, 226(Ex. 4.6)
Riesz, F., theorem, 387
Rotation number, 191, 198
Rotation point, 158, 173(Ex. 9.1)
see also Center
Saddle-point, 161, 216, 218
Sauvage theorem, 73
partial converse, 74
Schauder fixed point theorem, 405, 414
425
Schwartz, A. J., theorem, 185
Second order, linear equation, 322
linear system, 384,418
nonlinear, 174, 422
see also Asymptotic integration,
Boundary value problems, Bounds,
Comparison theorem, Dichotomies,
Disconjugate, Liouville, Monot-
Monotony, Nonoscillatory, Principal so-
solution, Priifer, Riccati, Solutions
tending to 0, Sturm, Variation of
constants, Zeros
Sectors (elliptic, parabolic, hyperbolic),
161
index of stationary point, 166
level curves, 173 (Ex. 9.1)
Self-adjoint, operator, 342
dth order equation, 398
Singular point, simple, 73, 78(Ex.
11.2), 84, 86(Ex. 12.2)
regular, 73, 78(Ex. 11.2), 85, 86
(Ex. 12.2)
Small at oo, 486
Solution, definition, 1, 46(Ex. 1.2)
continuity with respect to initial con-
conditions or parameters, 94
differentiability with respect to initial
conditions or parameters, 95, 100,
104, 115
D-solution, 437
PD-solution, 462
Solutions tending to 0, binary systems,
161, 208, 209, 211, 220
linear systems, 500
perturbed linear systems, 259, 294,
300, 304, 445
second order equations, 510, 514,
(Ex. 3.7)
see also Stability
Spirals, 151, 159, 190, 211, 216, 220
see also Focus
Square root, non-negative Hermitian
matrix, 503(Ex. 1.2)
Index 611
Square root, differential operators, 354,
392
Stability, asymptotic, 38, 40, 537
global, 537
Lyapunov, 38, 40
orbital, 157, 254
periodic solutions, 158, 178, 179,
253, 302(Ex. 11.3)
uniform, 40
Stationary point, 144, 183, 209, 212, 220
index, 149, 166, l73(Ex. 9.1)
see also Center, Focus, Node, Saddle-
point
Sturm, comparison theorems, 333, 362
majorant, 334
separation theorem, 335
Sturm-Liouville problems, 337
Successive approximations, 8, 40, 45
(Ex. 1.1), 57, 236, 247, 296(Ex.
8.1)
bracketing, 42(Ex. 9.2), 43(Ex. 9.3,
9.4)
general theorem, 404
Superposition principle, 46, 63, 326
Topological arguments, 203, 278, 520
Toroidal function, 508(Ex. 2.6(a))
Torus, 185
flow on, 195
Total differential equations, 117, 120
adjoint, 119
complete integrability, 118, 123, 128
Trace (tr),46
Transversal, 152, 184, 196
Tychonov fixed point theorem, 405, 414,
425, 444
Umlaufsatz, 147
Uniqueness theorems, 31, 109
dth order equation, 33(Ex. 6.6)
Kamke's general theorem, 31, 33(Ex.
6.5)
van Kampen, 35
L-Lipschitz continuity, 109
Nagumo, 32
null solutions, 211, 212(Ex. 2.3, 2.4)
one-sided, 34, 110
612
Index
Uniqueness theorems, Osgood, 33
second order boundary value prob-
problems, 420, 421 (Ex. 3.3), 423, 425,
427(Ex. 4.6)
(«,v)-subset, 281, 291, 293
Variation of constants, 48, 64
second order equations, 328, 329
Variational principles, 352, 390, 399
Wazewski theorem, 280
Weber equation, 320(Ex. 17.6), 382
(Ex. 9.7), 529
Wintner theorem on existence in large,
29
Wirtinger inequality, 346(Ex. 5.3F))
Wronskian determinant, 63, 326
Zeros of solutions of second order equa-
equation, monotony, 519 (Ex. 4.3)
number of, 344
see Disconjugate, Nonoscillatory,
Oscillatory, Sturm