From the authors' preface: "This second volume is devoted to differential equations with delay, partial differential equations, differential equations in abstract spaces and complex differential equations. To cut down the length of the volume many parallel results are left as exercises.'' AMS Review by A. Halanay
Author(s): V. Lakshmikantham and S. Leela
Publisher: Academic Press
Year: 1969
Language: English
Pages: 334
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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 Theorems 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
PARTIAL DIFFERENTIAL EQUATIONS
Chapter 9. 9.0. Introduction 113
9.1. 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. Introduction 149
10.1. 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 Differential Inequalities in Bounded Domains 181
10.7. Lyapunov-Like Functions 186
10.8. Stability and Boundedness 190
10.9. Conditional Stability and Boundedness 200
10.10. Parabolic Differential Inequalities in Unbounded Domains 205
10.11. Uniqueness 210
10.12. Exterior Boundary-Value Problem and Uniqueness 213
10.13. Notes 219
Chapter 11. 11.0. Introduction 221
11.1. Hyperbolic Differential Inequalities 221
11.2. 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
12.1. Existence 237
12.2. Nonlocal Existence 241
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
COMPLEX DIFFERENTIAL EQUATIONS
Chapter 13. 13.0. Introduction 275
13.1. Existence, Approximate Solutions, and Uniqueness 275
13.2. Singularity-Free Regions and Growth Estimates 279
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
