Differential and Integral Inequalities: Functional Partial, Abstract and Complex Differential Equations v. 2: Theory and Applications: Functional

335 Pages·1969·4.22 MB·English

by  Lakshmikantham

DIFFERENTIAL
AND
INTEGRAL
INEQUALITIES
Theory and Applications
Volume I1
FUNCTIONAL, PARTIAL, ABSTRACT, AND
COMPLEX DIFFERENTIAL EQUATIONS
This is Volume 55 in
MATHEMATICS IN SCIENCE AND ENGINEERING
A series of monographs and textbooks
Edited by RICHARD BELLMAN, University of Southern California
A complete list of the books in this series appears at the end of this volume.
DIFFERENTIAL
AND
INTEGRAL
INEQUALITIES
Theory and Applications
Volume I1
FUNCTIONAL, PARTIAL, ABSTRACT, AND
COMPLEX DIFFERENTIAL EQUATIONS
V. LAKSHMIKANTHAM and S. LEELA
University of Rhode Island
Kingston, Rhode Island
A C A D E MI C P RES S New York and London 1969
0
COPYRIGHT 1969, BY ACADEMIPCR ESSI, NC.
ALL RIGHTS RESERVED.
NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM,
BY PHOTOSTAT, MICROFILM, RETRIEVAL SYSTEM, 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 WlX 6BA
LIBRAROYF CONGRESCSA TALOCGA RDN UMBER68: -8425
AMS 1968 SUBJECT CLASSIFICATIO3N4S01 , 3501
PRINTED IN THE UNITED STATES OF AMERICA
Preface
The first volume of Differential and Integral Inequalities: Theory
and Applications published in 1969 deals with ordinary differential
equations and Volterra integral equations. It consists of five chapters
and includes a systematic and fairly elaborate development of the
theory and application of differential and integral inequalities.
This second volume is a continuation of the trend and is devoted to
differential equations with delay or functional differential equations,
partial differential equations of first order, parabolic and hyperbolic
types respectively, differential equations in abstract spaces including
nonlinear evolution equations and complex differential equations. To
cut down the length of the volume many parallel results are omitted
as exercises.
We extend our appreciation to Mrs. Rosalind Shumate, Mrs. June
Chandronet, and Miss Sally Taylor for their excellent typing of the
entire manuscript.
V. LAKSHMIKANTHAM
S. LEELA
Kingston, Rhode Island
August, 1969
V
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Contents
PREFACE V
FUNCTIONAL DIFFERENTIAL EQUATIONS
Chapter 6. 6.0. Introduction 3
6.1. Existence 4
6.2. Approximate Solutions and Uniqueness 9
6.3. Upper Bounds 13
6.4. Dependence on Initial Values and Parameters 18
6.5. Stability Criteria 21
6.6. Asymptotic Behavior 24
6.7. A Topological Principle 29
6.8. Systems with Repulsive Forces 32
6.9. Functional Differential Inequalities 34
6.10. Notes 42
Chapter 7. 7.0. Introduction 43
7.1. Stability Criteria 43
7.2. Converse Theoreins 49
7.3. Autonomous Systems 58
7.4. Perturbed Systems 62
7.5. Extreme Stability 66
7.6. Almost Periodic Systems 72
7.7. Notes 80
Chapter 8. 8.0. Introduction 81
8.1. Basic Comparison Theorems 81
8.2. Stability Criteria 87
8.3. Perturbed Systems 97
8.4. An Estimate of Time Lag 100
8.5. Eventual Stability 101
8.6. Asymptotic Behavior 105
8.7. Notes 110
vii
...
Vlll CONTENTS
PARTIAL DIFFERENTIAL EQUATIONS
Chapter 9. 9.0. Introduction 113
9. I. Partial Differential Inequalities of First Order 113
9.2. Comparison Theorems 118
9.3. Upper Bounds 127
9.4. Approximate Solutions and Uniqueness 134
9.5. Systems of Partial Differential Inequalities of First Order 136
9.6. Lyapunov-Like Function 144
9.7. Notes 148
Chapter 10. 10.0. lntroduction 149
10. I. Parabolic Differential Inequaliies in Bounded Domains 149
10.2. Comparison Theorems 155
10.3. Bounds, Under and Over Functions 163
10.4. Approximate Solutions and Uniqueness 170
10.5. Stability of Steady-State Solutions 174
10.6. Systems of Parabolic Diffcrential inequalities in Bounded
Domains 181
10.7. Lyapunov-Like Functions 186
10.8. Stahility and Boundedness 190
10.9. Conditional Stahility and Boundedness 200
10.10. Parabolic Differential Inequalities in Unbounded Domains 205
10.11. Uniqueness 210
10.12. Exterior Boundary-Value Problem and Uniqueness 21 3
10.13. Notes 219
Chapter 11. I 1.0. Introduction 22 1
1 1.1, Hyperbolic Diflerential Inequalities 22 1
1 1.2. Uniqueness Criteria 223
11.3. Upper Bounds and Error Estimates 229
11.4. Notes 233
DIFFERENTIAL EQUATIONS IN ABSTRACT
SPACES
Chapter 12. 12.0. Introduction 237
12. I. Existence 231
12.2. Norilocal Existence 24 1
12.3. Uniqueness 243
12.4. Continuous Dependence and the Method of Averaging 247
12.5. Existence (continued) 249
12.6. Approximate Solutions and Uniqueness 254
12.7. Chaplygin’s Method 258
12.8. Asymptotic Behavior 264
12.9. Lyapunov Function and Comparison Theorems 267
12.10. Stability and Boundedness 269
12.11. Notes 272
CONTENTS ix
COMPLEX DIFFERENTIAL EQUATIONS
Chapter 13. 13.0. Introduction 215
13.1. Existence, Approximate Solutions, and Uniqueness 215
13.2. Singularity-Free Regions and Growth Estimates 219
13.3. Componentwise Bounds 284
13.4. Lyapunov-like Functions and Comparison Theorems 286
13.5. Notes 288
Bibliography 289
Author Index 315
Subject Index 318