Differential and Integral Inequalities: Ordinary Differential Equations v. 1: Theory and Applications: Ordinary Differential Equations

405 Pages·1969·5.44 MB·English

DIFFERENTIAL
AND
INTEGRAL
INEQUALITIES
Theory and Applications
Volume I
ORDINARY DIFFERENTIAL EQUATIONS
This is Volume 55 in
R’IATHEMA’I’ICS IN SCIENCE AND ENGINEERING
A series of monographs and textbooks
Edited by RICHARD BELLMAN, University of Southern California
A complete list of the boolts in this series appears at the end of this volume.
DIFFERENTIAL
AND
INTEGRAL
INEQUALITIES
Theory and Applications
Volume I
ORDINARY DIFFERENTIAL EQUATIONS
V. LAKSHILIIKANTHAM and S. LEELA
{Jnioersity of Rho& Islmiii
Kiiqstow, Rliotlc Islaiid
A C AD E RI I C P R E S S New J’ork and London 1969
COPYRIGH0T 1 969, BY ACADEMIPCR ESS, INC.
ALL RIGHTS RESERVED.
NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM,
BY PHOTOSTAT, MICROFILM, OR ANY OTHER MEANS, WITHOUT
WRITTEN PERMISSION FROM THE PUBLISHERS.
ACADEMIC PRESS, INC.
111 Fifth Avenue, New York, New York 10003
United Kingdom Edition published by
ACADEMIC PRESS, INC. (LONDON) LTD.
Berkeley Square House, London W.l
LIBRAROYF CONGRESCSA TALOCGA RDN UMBER6:8 -8425
PRINTED IN THE UNITED STATES OF AMERICA
Preface
This volume constitutes the first part of a monograph on theory and
applications of differential and integral inequalities. 'The entire work,
as a whole, is intended to be a research monograph, a guide to the
literature, and a textbook for advanced courses. The unifying theme of
this treatment is a systematic development of the theory and applications
of differential inequalities as well as Volterra integral inequalities. The
main tools for applications are the norm and the Lyapunov functions.
Familiarity with real and complex analysis, elements of general topology
and functional analysis, and differential and integral equations is
assumed.
The theory of differential inequalities depends on integration of
differential inequalities or what may be called the general comparison
principle. The treatment of this theory is not for its own sake. 'The
essential unity is achieved by the wealth of its applications to various
qualitative problems of a variety of differential systems.
The material of the present volume is divided into two sections. The
first section consisting of four chapters deals with ordinary differential
equations while the second section is devoted to Volterra integral
equations. The remaining portion of the monograph, which will appear
as a second volume, is concerned with differential equations with time
lag, partial differential equations of first order, parabolic and hyperbolic
respectively, differential equations in abstract spaces including nonlinear
evolution equations and complex differential equations types.
The vector notation and vectorial inequalities are used freely through-
out the book. Also, because of the several allied fields covered, it becomes
convenient to use the same letter with different meanings in different
situations. This, however, should not cause confusion, since it is spelled
out wherever necessary.
The notes at the end of each chapter indicate the sources which have
V
vi PREFACE
been consulted and those whose ideas are developed. Some sources
which are closely related but not included in the book are also given for
guidance.
We wish to express our warmest thanks to our colleague Professor
C. Corduneanu for reading the manuscript and suggesting improvements.
Our thanks are also due to Professors J. Hale, N. Onuchic, and C. Olech
for their helpful suggestions.
We are immensely pleased that our monograph appears in a series
inspired and edited by Professor R. Bellman and we wish to express our
gratitude and warmest thanks for his interest in this book.
V. LAKSHMIKANTHAM
S. LEELA
Kingston, Rhode Island
December, I968
Cont ents
PREFACE V
ORDINARY DIFFERENTIAL EQUATIONS
Chapter 1. 1.0. Introduction 3
1.1. Existence and Continuation of Solutions 3
1.2. Scalar Differential Inequalities 7
1.3. Maximal and Minimal Solutions 11
1.4. Comparison Theorems 15
1.5. Finite Systems of Differential Inequalities 21
1.6. Minimax Solutions 25
1.7. Further Comparison Theorems 27
1.8. Infinite Systems of Differential Inequalities 31
1.9. Integral Inequalities Reducible to Differential Inequalities 37
1.10. Differential Inequalities in the Sense of Caratheodory 41
1.11. Notes 44
Chapter 2. 2.0. Introduction 45
2.1. Global Existence 45
2.2. Uniqueness 48
2.3. Convergence of Successive Approximations 60
2.4. Chaplygin’s Method 64
2.5. Dependence on Initial Conditions and Parameters 69
2.6. Variation of Constants 76
2.7. Upper and Lower Bounds 79
2.8. Componentwise Bounds 84
2.9. Asymptotic Equilibrium 88
2.10. Asynlptotic Equivalence 91
2.11. A Topological Principle 96
2.12. Applications of Topological Principle 100
2.13. Stability Criteria 102
2.14. Asymptotic Behavior 108
2.15 Periodic and Almost Periodic Systems 120
2.16. Notes 129
vi i
...
Vlll CONTENTS
Chapter 3. 3.0. Introduction 131
3.1. Basic Comparison Theorems 131
3.2. Definitions 135
3.3. Stability 138
3.4. Asymptotic Stability 145
3.5. Stability of Perturbed Systems 155
3.6. Convcrse Theorems 158
3.1. Stability by the First Approximation 177
3.8. Total Stability 186
3.9. Integral Stability 191
3.10. I,”-S ta hi I i ty 199
3.1 I. Partial Stability 205
3. I?. Stabilit) of Differential Inequalities 209
3.13. Boundcdness and Lagrange Stability 212
3.14. Eventual Stability 222
3.15. Asymptotic Behavior 229
3.16. Relative Stability 24 1
3.11. Stability with Respect to a Manifold 244
3.18. .-\lmost Periodic Systems 245
3.19. Uniqueness and Estimates 254
3.20. Continuous Dependence and the Method of Averaging 257
3.21. Notes 264
Chapter 4. 4.0. Introduction 267
4.1. Main Comparison Theorem 267
4.2. Asymptotic Stability 269
4.3. Instability 273
4.4. Conditional Stability and Boundedness 277
4.5. Converse Theorcms 284
4.6. Stability in Tube-like Domain 293
4.7. Stability of Asymptotically Self-Inwriant Sets 291
4.8. Stability of Conditionally Invariant Sets 305
4.9. Existence and Stability of Stationary Points 308
4.10. Notes 311
VOLTERRA INTEGRAL EQUATIONS
Chapter 5. 5.0. Introduction 31 5
5.1. Integral Inequaiitics 315
5.2. Local and Global Existence 319
5.3. Comparison Theorems 322
5.4. Approximate Solutions, Bounds, and Uniqueness 324
5.5. Asymptotic Behavior 327
5.6. Perturbed Integral Equations 333
5.7. Admissibility and Asymptotic Behavior 340
5.8. Integrodifferential Inequalities 350
5.9. Notes 354
CONTENTS ix
Bibliography 355
AUTHORI NDEX 385
SUBJECITN DEX 388