[Philip Hartman] Ordinary Differential Equations ((BookZa.org)

Ordinary Differential Equations

SIAM's Classics in Applied Mathematics series consists of books that were previousl allowed to go out of print. These books are republished by SIAM as a professional service because they continue to be important resources for mathematical scientists. Editor-in-Chief Robert E. O'Malley, Jr., University of Washington Editorial Board Richard A. Brualdi, University of Wisconsin-Madison Herbert B. Keller, California Institute of Technology Andrzej Z. Manitius, George Mason University Ingram Olkin, Stanford University Stanley Richardson, University of Edinburgh Ferdinand Verhulst, Mathematisch Instituut, University of Utrecht Classics in Applied Mathematics C. C. Lin and L. A. Segel, Mathematics Applied to Deterministic Problems in the Natural Sciences Johan G. F. Belinfante and Bernard Kolman, A Survey of Lie Groups and Lie Algebras with Applications and Computational Methods James M. Ortega, Numerical Analysis: A Second Course Anthony V. Fiacco and Garth P. McCormick, Nonlinear Programming: Sequential Unconstrained Minimization Techniques F. H. Clarke, Optimization and Nonsmooth Analysis George F. Carrier and Carl E. Pearson, Ordinary Differential Equations Leo Breiman, Probability R. Bellman and G. M. Wing, An Introduction to Invariant Imbedding Abraham Berman and Robert J. Plemmons, Nonnegative Matrices in the Mathematical Sciences Olvi L. Mangasarian, Nonlinear Programming *Carl Friedrich Gauss, Theory of the Combination of Observations Least Subject to Errors: Part One, Part Two, Supplement. Translated by G. W. Stewart Richard Bellman, Introduction to Matrix Analysis U. M. Ascher, R. M. M. Mattheij, and R. D. Russell, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations K. E. Brenan, S. L. Campbell, and L. R. Petzold, Numerical Solution of InitialValue Problems in Differential-Algebraic Equations Charles L. Lawson and Richard J. Hanson, Solving Least Squares Problems J. E. Dennis, Jr. and Robert B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations Richard E. Barlow and Frank Proschan, Mathematical Theory of Reliability *First time in print.

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Richard Bellman, Introduction to Matrix Analysis, Second Edition Beresford N. Parlett, The Symmetric Eigenvalue Problem Richard Haberman, Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic Flow Peter W. M. John, Statistical Design and Analysis of Experiments Tamer Ba§ar and Geert Jan Olsder, Dynamic Noncooperative Game Theory, Second Edition Emanuel Parzen, Stochastic Processes Petar Kokotovic, Hassan K. Khalil, and John O'Reilly, Singular Perturbation Methods in Control: Analysis and Design Jean Dickinson Gibbons, Ingram Olkin, and Milton Sobel, Selecting and Ordering Populations: A New Statistical Methodology James A. Murdock, Perturbations: Theory and Methods Ivar Ekeland and Roger Temam, Convex Analysis and Variational Problems Ivar Stakgold, Boundary Value Problems of Mathematical Physics, Volumes I and II J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables David Kinderlehrer arid Guido Stampacchia, An Introduction to Variational Inequalities and Their Applications F. Natterer, The Mathematics of Computerized Tomography Avinash C. Kak and Malcolm Slaney, Principles of Computerized Tomographic Imaging R. Wong, Asymptotic Approximations of Integrals O. Axelsson and V. A. Barker, Finite Element Solution of Boundary Value Problems: Theory and Computation David R. Brillinger, Time Series: Data Analysis and Theory Joel N. Franklin, Methods of Mathematical Economics: Linear and Nonlinear Programming, Fixed-Point Theorems Philip Hartman, Ordinary Differential

Equations, Second Edition

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Ordinary Differential Equations Second Edition Philip Hartman The Johns Hopkins University Baltimore, Maryland

Society for Industrial and Applied Mathematics Philadelphia

Copyright © 2002 by the Society for Industrial and Applied Math This SIAM edition is an unabridged, corrected republication of the edition published by Birkhauser, Boston, Basel, Stuttgart, 1982. The original edition was published by John Wiley & Sons, New York, 1964.

10987654321 All rights reserved. Printed in the United States of America. No part of this book may be reproduced, stored, or transmitted in any manner without the written permission of the publisher. For information, write to the Society for Industrial and Applied Mathematics, 3600 University City Science Center, Philadelphia, PA 19104-2688.

Library of Congress Cataloging-in-Publication Data Hartman, Philip, 1915Ordinary differential equations / Philip Hartman. p. cm. - (Classics in applied mathematics ; 38) Previously published: 2nd ed. Boston : Birkhauser, 1982. Originally published: Baltimore, Md., 1973. Includes bibliographical references and index. ISBN 0-89871-510-5 (pbk.) 1. Differential equations. I. Title. II. Series. QA372 .H33 2002 515'.352-dc21

siam is a registered t

2002017641

To the memory of my parents To the patience of Sylvia, Judith, and Marilyn

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Contents Foreword to the Classics Editio Preface to the First Edition Preface to the Second Edition Errata

xv xsvii xviii xix

I. Preliminaries 1. Preliminaries, 1 2. Basic theorems, 2 3. Smooth approximations, 6 4. Change of integration variables, 7 Notes, 7

1

II. 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

8

III.

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

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

24

45

x

Contents 7. Adjoint systems, 62 8. Higher order linear equations, 63 9. Remarks on changes of variables, 68 APPENDIX. ANALYTIC LINEAR EQUATIONS, 70 10. 11. 12. 13.

V.

Dependence on initial conditions and parameters

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. VI.

Fundamental matrices, 70 Simple singularities, 73 Higher order equations, 84 A nonsimple singularity, 87 Notes, 91 93

Preliminaries, 93 Continuity, 94 Differentiability, 95 Higher order differentiability, 100 Exterior derivatives, 101 Another differentiability theorem, 104 S- and L-Lipschitz continuity, 107 Uniqueness theorem, 109 A lemma, 110 Proof of Theorem 8.1, 111 Proof of Theorem 6.1, 113 First integrals, 114 Notes, 116

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 (1.1), 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

Contents VII. The Poincare-Bendixson theory 1. Autonomous systems, 144 2. Umlaufsatz, 146 3. Index of a stationary point, 149 4. The Poincaré-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

xi 144

APPENDIX. POINCARÉ-BENDIXSON THEORY ON 2-MANiFOLDS, 182 11. 12. 13. 14.

Preliminaries, 182 Analogue of the Poincaré-Bendixson theorem, 185 Flow on a closed curve, 190 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 Tt 231 3. Modification of F(ξ), 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

228

APPENDIX. SMOOTH EQUIVALENCE MAPS, 256 12. Smooth linearizations, 256 13. Proof of Lemma 12.1, 259 14. Proof of Theorem 12.2, 261 APPENDIX. SMOOTHNESS OF STABLE MANIFOLDS, 271 Notes, 271 L

xii Contents X.

Perturbed linear systems

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

XL

273

The case £ = 0, 273 A topological principle, 278 A theorem of Wazewski, 280 Preliminary lemmas, 283 Proof of Lemma 4.1, 290 Proof of Lemma 4.2, 291 Proof of Lemma 4.3, 292 Asymptotic integrations. Logarithmic scale, 294 Proof of Theorem 8.2, 297 Proof of Theorem 8.3, 299 Logarithmic scale (continued), 300 Proof of Theorem 11.2, 303 Asymptotic integration, 304 Proof of Theorem 13.1, 307 Proof of Theorem 13.2, 310 Corollaries and refinements, 311 Linear higher order equations, 314 Notes, 320

Linear second order equations 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

322

APPENDIX. DISCONJUGATE SYSTEMS, 384 10. Disconjugate systems, 384 11. Generalizations, 396 Notes, 401 XII.

Use of implicit function andfixedpoint 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. 7. 8. 9.

Basic facts, 435 Green's functions, 439 Nonlinear equations, 441 Asymptotic integration, 445 Notes, 447

XIII. Dichotomies for solutions of linear equations

.

.

.

.

.

450

PART I. GENERAL THEORY, 451 1. 2. 3. 4. 5. 6. 7. 8.

Notations and definitions, 451 Preliminary lemmas, 455 The operator T, 461 Slices of ||Py(t)|| 465 Estimates for ||y(t)||, 470 Applications to first order systems, 474 Applications to higher order systems, 478 P(B, D)-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. 2. 3. 4.

Small and large solutions, 500 Monotone solutions, 506 Second order linear equations, 510 Second order linear equations (continuation), 515

500

xiv

Contents

PART II. A PROBLEM IN BOUNDARY LAYER THEORY, 51 5. 6. 7. 8. 9.

The problem, 519 The case A > 0, 520 The case λ < 0, 525 The case A = 0, 531 Asymptotic behavior, 534

PART III. GLOBAL ASYMPTOTIC STABILITY, 537 10. 11. 12. 13. 14. 15. 16.

Global asymptotic stability, 537 Lyapunov functions, 539 Nonconstant G, 540 On Corollary 11.2, 545 On "J(y)x • x ^ 0 if x -f(y) = 0", 54 Proof of Theorem 14.2, 550 Proof of Theorem 14.1, 554 Notes, 554

HINTS FOR EXERCISES, 557 REFERENCES, 581 INDEX, 607

Foreword to the Classics Edition Those of us who grew up having this tome as our text or supplementary guide often referred to it with a certain reverence as being the ultimate authority. Some of my colleagues simply called it "The Bible." This is not a book that trots out the usual topics and examples in a formulaic way. Instead it is written by one of the world's leading practitioners, whose purpose appears to have been to give the serious student all the necessary tools for work at the forefront and to introduce research topics which, at the time of the book's original publication in 1964, were hot off the press. Thus, the book is comprehensive in fundamental theory but goes far beyond that in its presentation of several important and interesting topics as well as of some proofs which had never before appeared in print. I am delighted that SIAM has undertaken to reprint Hartman's wonderful text, not only because of its historical significance but because it contains almost everything one would want in a modern advanced course in ordinary differential equations. The topological ideas and techniques of functional analysis that are so important today are clearly explained and applied. The basic theory of ordinary differential equations is given in sufficient generality that the book serves as a valuable reference, whereas many other texts supply only the standard simplified theorems. What makes the text outstanding, in my mind, are the chapters on invariant manifolds, perturbations, and dichotomies. The idea of first establishing the existence of invariant manifolds for maps, and then applying the theory to solution operators for ordinary differential equations suitably modified outside a neighborhood of a critical point, has become a standard approach, now also followed for equations in infinite-dimensional spaces. One finds a complete proof of the HartmanGrobman theorem on transforming a nonlinear to a linear flow in the neighborhood of a hyperbolic equilibrium. Theorems on smooth equivalence and on the smoothness of invariant manifolds are presented—these being important for perturbation and normal form theory. Poincare sections are used to show persistence of hyperbolic periodic orbits under

xvi

Foreword

perturbations in the system—again, a technique that is now widely used. All of this has found considerable use in the geometrical theory of differential equations and dynamical systems, such as in helping to resolve questions of structural stability. But it is also true that invariant manifold theory has become ubiquitous in both science and engineering. Typically the theory is used to deduce qualitative information about solutions to large, possibly infinite-dimensional, systems by reducing the essential dynamics to a small-dimensional system. Readers will find many parts of this outstanding book as essential today as they were when the book was first published. So it is that in Philip Hartman's book we find the ever-useful classical theory and what could be considered the modern theory of ordinary differential equations and dynamical systems. Peter Bates, Brigham Young University

Preface to the First Edition This book is based on lecture notes of courses on ordinary differential equations which I have given from time to time for advanced 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; §§ 1-3 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 XIII. 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 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

xviii

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

Preface to the Second Editio This edition is essentially that of 1964 (John Wiley and Sons, Inc.) with some corrections and additions. The only major changes are in Chapter IX. A few pertinent additions have been made to the bibliography as a Supplement, but no effort has been made to bring it up to date. I should like to thank Professors T. Butler, K.T. Chen, W.A. Coppel, L Lorch, M.E. Muldoon, L. Nicolson and C. Olech for corrections and/or suggestions, and Mrs. Margaret A. Einstein for able secretarial assistance. P.H. (10/73)

Errata This book has been photographically reproduced from the edition published by Birkhauser, Boston, Basel, Stuttgart, 1982. The following are corrections for typographical errors in that edition. Page 262: In lines 6 and 2 from the bottom, replace ^ K+ — K° by G K~. Page 264: At the end of the line following (14.8), replace ^ by 6. Page 264: Replace the last five lines of page by the following: (d) There is a constant C > 1 such that

This is clear for j — I from the normalizations following (4.1), and the cases j > I follow by induction. Page 265: Replace the part beginning at Proof by the following: Proof. If 0 < r2 < ri,f G K+(r2] - M+ and r, = T~k(^£, then, by the property of ri,Tlr) G S'(ro) for i = 0 , . . . , fc(0- Put

We show by induction that r2 can be chosen so that

for 0 < i < fc(0- Assume (14.14) for 0 < i < j - 1 < fc(£). Then CO') = (TiTT^-n - T^-1^} + (TiT^r? - TT^- 1 ^), so that

XX

Errata

Thus Cax < 1 implies that

If we define r? = ^(A) by r% — minfri, r\/(C\ + 2), 1], then N > X gives both ||C(j)|| < CAr2A < Cxr2 < ri and

This completes the proof of the induction for (14.14) and of assertion (14.12). The definition of £(j) and the choice j ~ k(£) in (14.15) give (14.13). Page 266: Delete the first two lines. Page 267: Line 4 should read "respectively, for suitably small r^ — ^(A). Note that" Page 267: Line 10 should read "In view of (e) and its proof, this makes the validity of (h) clear." Page 271G: Innf;B of the last line, replace £ by 77. Page 271J: In the third line following (4.6), in {Tn} replace T by r. Page 272: Add the following to the text: The translators for the Russian edition noted errors in the original arguments of steps (d) and (e) in the proof of Theorem 12.2; see P. Hartman, Obyknovennye Differencial'nye Uravneniya, IzdatePstvo "Mir," Moscow (1970), translated by I. H. Sabitov and Yu. B. Egorov, edited by V. M. Alexeev. Compare the corrections for pages 262-267 in the errata, pages xix-xx. Page 501: In the ninth line from the bottom insert a 2 before Im.

Chapter I Preliminaries

1. Preliminaries Consider a system of d first order differential equations and an initial condition where y' = dy\dt, y — (y1,...,/) and/= (/*,... ,/d) are J-dimensional vectors, and/(f, y) is defined on a (d + l)-dimensional (/, y)-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 = 0tis called a solution of the initial value problem (1.1) if y(t 0) = y0, (t, y(t)) e £, y(i) is differentiable, and y'(i) — /(/, y(/)) for t e /. It is clear that y(t) then has a continuous derivative. These requirements on y are equivalent to the following: y('o) = 2/o. (t, 3/(0) 6 E> J/(0 ^ continuous and

for/eJ. An initial value problem involving a system of equations of /wth order, where z u) = d'z\dt\ z and F are e-dimensional vectors, and F is defined on an (me + 1 )-dimensional set E, can be considered as a special case of (1.1), where y is a d = me-dimensional vector, symbolically, y = (z, z(1), , z(m~l}) (or more exactly, y = (z 1 ,..., z% z 1 ',..., z " , . . . , z'*"*-1')); correspondingly,/(f, y) = (z^,..., a*1"-", F(r, y)) and y0 = (z0, z,..., z*^1'). For example, if e = 1 so that z is a scalar, (1.3) becomes

1

2

Ordinary Differential Equations

The first set of questions to be considered will be (1) local existenc (does (1.1) have a solution y(t) defined for t near ?0?); (2) existence in the large (on what /-ranges does a solution of (1.1) exist ?); and (3) uniqueness of solutions. The significance of question (2) is clear from the following situation: Let y,f be scalars, /(/, y) defined for 0 ^ t ^ 1, \y\ ^ 1. A solution y = y(t) of (1.1), with (/ 0, y0) = (0,.D), 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 / > J. Or consider the following scalar case where/(/, y) is defined for all (t, y): It is easy to see that y = c/(l — ci) is a solution of (1.4), but this solution exists only on the range — oo < / < 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 This has more than one solution, in fact, it has, e.g., the solution y(i) s 0

Figure 1.

and the 1-parameter family of solutions defined by y(t) = 0 for t ^ c, y(t) = (t — c)2/4 for / ^ c, where c ^ 0; see Figure 1. This situation is typical in that if (1.1) has more than one solution, then it has a "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 3

The symbols O, o will be used from time to time where, e.g., /(/) = O(g(tJ) as / -»• oo means that there exists a constant C such that |/(f)l = C |g(f)l for large /, while/(O = o(g(0) as / -> oo means that C> 0 can be chosen arbitrarily small (so that if g(t) 7* 0, f(t)lg(t) -»• 0 as t -* oo). "Function" below generally means a map from some specified set of a vector space R* into a space Rd, not always of the same dimension. Rd denotes a normed, real ^-dimensional vector space of elements y = (y 1 ,..., y*1) with norm |y|. Unless otherwise specified, \y\ will be the norm

and ||y|| the Euclidean norm. If y0 is a point and £ a subset of Rd, then dist (yc, £), the distance from y0 to £, is defined to be inf |y0 — y\ for y e E. If £lt £2 arc two subsets of Rd, then dist (£1} £2) is defined to be inf \yl — y2| for yl E Elt y2 e £2, and is called the distance between El and £2. If £1 (or £2) is compact and Elt £2 are closed and disjoint, then dist (£lf £2) > 0. If £ is an open set or a closed parallelepiped in Rd,/e Cn(£), 0 ^ n < oo, means that/(y) is continuous on £ and that the components of/have continuous partial derivatives of all orders k is n with respect toy 1 ,..., y*. A function /(y, 2) =/(y 1 ,..., y4*, z 1 ,..., 2') defined on a (y, «)-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

withy =1,2. Any constant K satisfying (2.1) is called a Lipschitz constant (for/on £). (The admissible values of K depend, of course, on the norm's in the/- and y-spaces.) A family F of functions/(y) defined on some y-set £ cr Rd is said to be equicontinuous if, for every e > 0, there exists a 6 = 0 such that \f(Vi) -/(y2)l ^ « whenever ylt y2 e £, ^ - y2| ^ (5 and /e F. The point of this definition is that (5 does not depend on / but is admissible for all/e F. The most frequently encountered equicontinuous families F below will occur when all/e F are uniformly Lipschitz continuous on £ and there exists a K> 0 which is a Lipschitz constant for all/6F; in which case, d can be chosen to be d — t\K. Lemma 2.1. If a sequence of continuous functions on a compact set E is uniformly convergent on E, then it is uniformly bounded and equicontinuous. Cantor Selection Theorem 2.1. Let /i(y),/2(y),... be a uniformly bounded sequence of functions on a y-set E. Then for any countable set .. convergent on D. D c: £, there exists a subsequence fn(l)(y),fn(2)(y),.

4

Ordinary Differential Equations

In order to prove Cantor's theorem, let D consist of the points ylt yz, • • • • Also assume that/n(y) is real-valued; the proof for the case that/B(y) = (fnl(y)> • • • >/«%)) is a ^-dimensional vector is similar. The sequence of numbers /i(y),/2(y),... is bounded, thus, by the theorem of BolzanoWeierstrass, there is a sequence of integers /^(l) < «1(2) < ... such that lim/^yj) exists as k -> oo, where n = n^(k). Similarly there is a subsequence n2(l) < «2(2) < ... of/ijO), «j(2),... such that \imfn(y^) 2 exists as k -*• oo for n = nz(k). Continuing in this fashion one obtains successive subsequences of positive integers, such that if /*,(!) < nf(2) < ... is the yth one, then lim/^) exists on k -*• oo, where n — nf(k) and / = 1,... ,y. The desired subsequence is the "diagonal sequence" «j(l) < «2(2) < «3(3) < ... . 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, /ef/iG/),/2(y),.. . be a sequence of functions which is equicontinuous and convergent on a dense subset of E. Thenfi(y),fz(y),... converges uniformly on E. Selection Theorem 2.3. On a compact y-set E c: Rd, Ietf v(y\f4y\... be a sequence of functions which is uniformly bounded and equicontinuous. Then there exists a subsequence ff n(1)(y), fn(2)(y), y) . . . which is uniformlyu 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, y0eE and/0 is a cluster point of the sequence/^o),/2(y0),..., then the subsequence/n(1,(y),/n(2,(y),... in the assertion can be chosen so that the limit function/(y) satisfies/(y0) =/0. Remark 2. If, in Theorem 2.3, it is known that all (uniformly) convergent subsequences of/i(y),/2(y),... have the same limit, say/(y), then a selection is unnecessary and/(y) is the uniform limit of/i(y),/2(y),.... This follows from Remark 1. Theorem 2.3 and the following consequences of it will be used repeatedly. Theorem 2.4. Let y,fe Rd andf0(t, y^ftt, y),/a(f, y\...bea sequence of continuous functions on the parallelepiped R : t0 ^ / 5s /0 + a, \y — y0\ ^ b such that uniformly on R.

Let yn(t) be a solution of

Preliminaries

5

on [t0, /o + a]> where n = 1 , 2 , . . . , and

(2.5)

* n -^> b. It is clear that/*(f, y) is continuous for t0 ^ / ^ tlt y arbitrary, and that |/*(/, 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, y) =/*(/, y)9>°(|y|), where 0 depending only on q>(s) and the dimension d, such that for every e > 0,

wher« ||y||' = (S |y*|2)!/* is the Euclidean length of y. Put

where »? = (^ x ,..., rf), so that

Since/ (/, y) is an "average" of the values of/0 in a sphere \\TJ — y|| ^ e for a fixed t, it is clear that/* ->/° as c -+ 0 uniformly on /0 £! / ^ /lt y arbitrary. Note that |/«| ^ A/ for all e > 0 and that /*(f, y) « 0 for |y| ^ A + 1 + «. Furthermore,/*(?, y) has continuous partial derivatives of all orders with respect to y 1 ,..., y*. The last formula can be used to show that if f°(t, y) has continuous partial derivatives of order k with respect to y 1 ,..., y*1, then the corresponding partial derivatives of /*(f, y) tend uniformly to those off°(t, y) as € -> 0.

Preliminaries

7

4. Change of Integration Variables

In order to avoid an interruption to some arguments later, it is 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 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 (4.1) holds for a ^ t ^ b if u(t) has a continuous derivative. For in this case, both integrals in (4.1) vanish at t = a and have the same derivative U(u(t))u'(i). If u(i) does not have a continuous derivative, let ii^r), uz(t),... be a sequence of continuously differentiable functions on a ^ t ^ b satisfying A ±± un(i) ^ B, un(t) -*• w(/) as n -»• oo uniformly on [a, b], and such that the sequence of total variations of Mx(0, «z(0» • • • over ta> ^1 ^s bounded. (The existence, of functions u^(i), uz(t),... follows from the last section.) Then (4.1) holds if u(t) is replaced by un(i). Term-by-term integration theorems applied to both sides of the resulting equation lead to (4.1). Notes SECTION 2. Theorems 2.2 and 2.3 go back to Ascoli [1] and Arzela [1], [2].

Chapter II Existence

1. Hie Picard-Lindelof Theorem

Various types of existence proofs will be given. One of the most simple and useful is the following. Theorem 1.1. Let y,/eR d ; f(t,y) continuous on a parallelepiped R:tQ ^ / ^ /„ + a, \y — y0\ ^ b and uniformly Lipschitz continuous with respect to y. Let Mbea bound for \f(t, y}\ on R; a = min (a, b/Af). Then 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 t0 — a ^ t ^ f0, \y — y0\ ^ b. It is also clear from these "right" and "left" existence theorems that if R is replaced by \t — 'ol = a-> \y — y\ = b, then (1.1) has a unique solution on \t — /0| ^ 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 requirement a ^ bjM is dictated by the fact that if y = y(t) is a solution of (1.1) on [/0, t0 + a], then \y'(t)\ ^ M implies \y(t) - y0\ < M(t - tQ), which does not exceed b if t — t0 51 bjM. Remark 1. Note that in Theorem 1.1, \y\ can be any norm on Rd, not necessarily the norm (2.1) or the Euclidean norm. For another proof of "uniqueness," see Exercise III 1.1. Proof by Successive Approximations. Let y0(/) = y0. Suppose that yk(t) has been defined on [t0, t0 + a], is continuous, and satisfies \yk(i) — y0| ^ b for k - 0 , . . . , n. Put Then, since/(/, yn(t)) is defined and continuous on [t0, t0 + a], the same holds for yn+1(0- It is also clear that

8

Existence

9

Hence y0(t), y^(t),... are defined and continuous on [r0, f 0 -j- a], and

\yn(t)-y»\^b.

It will now be verified by induction that

n = 0, 1 , . . . , where £is a Lipschitz constant for/. Clearly, (1.30) holds. Assume (1.3 0 ),..., (l.S^). By (1.2),

for « ^ 1. Thus, the definition of A" implies that

and so, by (U^),

This proves (1.3W). In view of (1.3J, it follows that

is uniformly convergent on [f0, /„ + a]; that is,

Since /(f, y) is uniformly continuous on R, f(tt yn(t)) -*•/(?, y(0) as n -> oo uniformly on [r0, /„ 4- a]. Thus term-by-term integration is applicable to the integrals in (1.2) and gives

Hence (1.4) is a solution of (1.1). In order to prove uniqueness, let y = z(/) be any solution of (1.1) on ['o, >0 + «]. Then

An obvious induction using (1.2) gives

10

Ordinary Differential Equations

and n = 0, 1, If n -* oo in (1.7), it follows from (1.4) that \y(t) — z(/)| ^ 0; i.e., y(t) = z(t). This proves the theorem. Remark 2. Since z == y, (1.7) gives an estimate of the error of approximation

Exercise 1.1. Show that if, in addition to the conditions of the theorem, /(/, y) is analytic on R (i.e., in a neighborhood of every point (t°, y°) E R, f(t, y) is representable as a convergent power series in t — t°, y1 — y 0 1 ,..., y&>_ and t-> o>+. The extension of y(t) need not be unique and, correspondingly, a>± depends on the extension. To say, e.g., that y(t) tends to dE as / -> a>+ is interpreted to mean that if £° is any compact subset of £, then (t, y(t)) $ £° when / is near a>+. Proof. Let Elt £ 2 ,... be open subsets of £ such that £ = U£n; the closures £lt £z,... are compact, and £„ !/«}). Corollary 2.1 implies that there exists an en > 0 such that if (?0, y0) is any point of En, then all solutions of (3.1) through (/0, j/0) exist on \t — r0| ^ en. Consider a given solution y = y(f) of (3.1) on an interval J. 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 «(1) be so large that (60, y(60)) e ^nu)- Then y(t) can be extended over an interval [b0, b0 + «n(i)]. If (b0 + en(1), y(b0 +  n(1))) e EnW, then y(0 can be extended over another interval [bQ +  n(1), bQ 4- 2en(1)] of length  n(1). Continuing this argument, it is seen that there is an integer /(I) ^ 1 such that y(t) can be extended over a ^ t ^ bl} where b± — bo +y(l)^n(i) and (blt y(M ^ £B(1). Let n(2) be so large that (blt yfa)) e ^ n(2 )- Then there exists an integer j(2) > 1 such that y(t) can be extended over a ^ t ^ bz, where bz = *i +./(2K(2> and (ba, y(b£) $ £n(2). Repetitions of this argument lead to sequences of integers «(1) < «(2) < . . . and numbers b0 < b± < ..-. such that y(t) has an extension over [a, to+), where !, yC^)), (fc2, y(bz)),... is either unbounded or has a cluster point on the boundary dE of £. To see that y(t) tends to dE as / -> o>+ on a right maximal interval [a, a>+), it must be shown that no limit point of a sequence (tlt y(tj), (t2, y(t%)),..., where tn -*• co+, can be an interior point of £. This is a consequence of the following: Lemma 3.1. Letf(t, y) be continuous on a (t, y)-set £. Let y = y(t) be a solution o/(3.1) on an interval [a, 6), 6 < oo, for which there exists a sequence tlt t2,.. . such that a ^ tn-+8 as n-*-oo and y0 = lim y(tn) exists. If f(t, y) is bounded on the intersection of E and a vicinity of the point (d, y0), then

14

Ordinary Differential Equations

If, in addition, f (8, y0) is or can be defined so that f(t, y) is continuous at ( then y(t) £ Cl[a, 6] and is a solution of (3.1) on [a, d]. Proof. Let e >0 be so small and Af > 1 so large that \f(t, y)\ ^ M€ for (/, y) on the intersection of E and the parallelepiped 0 ^ d — t ^ e, \y - yQ\ _, cu+) and a sequence of positive integers «(1) < »(2) < . . . with the property that if _ < w+ ^ lim inf o>n+

as H = «(&) -»> oo.

Proof. Let £1? £ 2 ,... be open subsets of E such that £ = U£M, the closures Elt £2,... are compact and En c: En+1. Suppose that (/„, y0) e £t and hence that (tn, yn0) e ^ for large n. In tfce proof, y(t) will be constructed only on a right maximal interval of existence [/0, o>+). The construction for a left maximal interval is similar. By Corollary 2.1, there exists an clt independent of n for large n, such that any solution of (3.5) [or (3.7)] for any point (tn, yn0) e £x [or (?0, y0) e £J exists on an interval of length ^ centered at t — tn [or t = f0]. By Arzela's theorem, it follows that if «(1) < n(2) < ... are suitably chosen, then the limit (3.8) exists uniformly for t0 ^ t 5J t0 +et and is a solution of (3.7). If the point (/0 + el5 y(t0 + e^) e Et the sequence n(l) < n(2) < . . . can be replaced by a subsequence, again called n(l) < n(2) < . . . , such that the limit (3.8) exists uniformly for /0 +  i = ' = 'o + 2«i and is a solution of (3.1). This process can be repeated/ times, where (f0 -f /wel5 y(t0 -f we!)) 6 Ev for m = 0 , . . . ,y — 1 but not for w —j. In this case, let tl = t0 + yei and choose the integer r > 1 so that (fi> y('i))e ^r- Repeat the procedure above using a suitable cr > 0 (depending on r but independent of n for large «) to obtain y(t) on an interval tl ^ f ^ fi + ;'i r» where ('i + m r» y(/! + w r)) 6 £r for m = 0, . . . _ / ! — 1 but not for m =/!. Put /2 = 'i + ji€rRepetitions of these arguments lead to a sequence of f-values f 0 < 'i < - • «and a sequence of successive subsequences of integers:

such that (3.8) holds uniformly for /„ < t ^ / m if n(k) = nm(A:). Put a>+ = lim /m (^ oo). Since (/m+1, y(/m+1)) ^ £TO, for m = 1,2,. . . , [t0, co+) is the right maximal interval of existence for y(t). The usual diagonal process supplies the desired sequence «(1) < «(2) < . . . . 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 on R:tQ ^ / < /0 + a,\y — y0\ ^ b. Let |/(/, y)| < M,'y. = min (a, b/M) and t0 < c ^ tQ + a. F/«flr//7, /e/

16

Ordinary Differential Equations

Sc be the set of points ycfor which there is a solution y = y(t) of on [f0, c] such that y(c) = yc; i.e., yc e Sc means that ye is a point reache

at t — c by some solution of (4.1). Then Se is a continuum, i.e., a close

connected set. Exercise 4.1. If y is a scalar, Theorem 4.1 has a very simple proo even without the assumption t0 ^ c ^ t0 + a. The conclusion is that S is either empty, a point, or a closed y-interval. Prove Theorem 4.1 in thi case by first showing that if ylf y2 e Sc, so that (4.1) has solutions y/r) o [/„, c] such that y,(c) = yj for/ =1,2, and if yx < y° < y2, then y° e Se Proof. Let 2 denote the set of solutions of (4.1). These exist o 0[t, c] and Sc is the set of points y(c), where y(t) e £. To see that the set eSis closed, let yne -* yc, H -+ oo, and ync e Sc. Then ync = yn(r)'for some yn(i) e S. By Theorem 12.4, y^f), y2(0> • • - has a subsequence which is uniformly convergent to some y(t) E £ on [t0, c]. Clearly, yc = y(c) e E. Suppose that the assertion is false, then Sc is not connected and is therefore the union of two nonempty, disjoint closed sets 5"°, S1. Since Scc is bounded, 6 = dist (5°, S1) > 0, where dist (S°, S1) = inf |y° - y1! for y° E 5°, y1 E S1. For any y, put e(y) = dist (y, 5°) - dist (y, S1), so that e(y) ^ 0 if y e S1 and e(y) ^ -^ < 0 if y 6 5°. The function