Differential and Integral Inequalities - Theory and Applications: Functional, Partial, Abstract, and Complex Differential Equations

Front Cover......Page 1
Differential and Integral Inequalities......Page 4
Copyright Page......Page 5
Preface......Page 6
Contents......Page 8
PART 1: FUNCTIONAL DIFFERENTIAL EQUATIONS......Page 14
6.0 Introduction......Page 16
6.1 Existence......Page 17
6.2 Approximate Solutions and Uniqueness......Page 22
6.3 Upper Bounds......Page 26
6.4 Dependence on Initial Values and Parameters......Page 31
6.5 Stability Criteria......Page 34
6.6 Asymptotic Behavior......Page 37
6.7 A Topological Principle......Page 42
6.8 Systems with Repulsive Forces......Page 45
6.9 Functional Differential Inequalities......Page 47
6.10 Notes......Page 55
7.1 Stability Criteria......Page 56
7.2 Converse Theorems......Page 62
7.3 Autonomous Systems......Page 71
7.4 Perturbed Systems......Page 75
7.5 Extreme Stability......Page 79
7.6 Almost Periodic Systems......Page 85
7.7 Notes......Page 93
8.1 Basic Comparison Theorems......Page 94
8.2 Stability Criteria......Page 100
8.3 Perturbed Systems......Page 110
8.4 An Estimate of Time Lag......Page 113
8.5 Eventual Stability......Page 114
8.6 Asymptotic Behavior......Page 118
8.7 Notes......Page 123
PART 2: PARTIAL DIFFERENTIAL EQUATIONS......Page 124
9.1 Partial Differential Inequalities of First Order......Page 126
9.2 Comparison Theorems......Page 131
9.3 Upper Bounds......Page 140
9.4 Approximate Solutions and Uniqueness......Page 147
9.5 Systems of Partial Differential Inequalities of First Order......Page 149
9.6 Lyapunov-Like Function......Page 157
9.7 Notes......Page 161
10.1 Parabolic Differential Inequaliies in Bounded Domains......Page 162
10.2 Comparison Theorems......Page 168
10.3 Bounds, Under and Over Functions......Page 176
10.4 Approximate Solutions and Uniqueness......Page 183
10.5 Stability of Steady-State Solutions......Page 187
10.6 Systems of Parabolic Differential inequalities in Bounded Domains......Page 194
10.7 Lyapunov-Like Functions......Page 199
10.8 Stahility and Boundedness......Page 203
10.9 Conditional Stability and Boundedness......Page 213
10.10 Parabolic Differential Inequalities in Unbounded Domains......Page 218
10.11 Uniqueness......Page 223
10.12 Exterior Boundary-Value Problem and Uniqueness......Page 226
10.13 Notes......Page 232
11.1 Hyperbolic Diflerential Inequalities......Page 234
11.2 Uniqueness Criteria......Page 236
11.3 Upper Bounds and Error Estimates......Page 242
11.4 Notes......Page 246
PART 3: DIFFERENTIAL EQUATIONS IN ABSTRACT SPACES......Page 248
12.1 Existence......Page 250
12.2 Nonlocal Existence......Page 254
12.3 Uniqueness......Page 256
12.4 Continuous Dependence and the Method of Averaging......Page 260
12.5 Existence (continued)......Page 262
12.6 Approximate Solutions and Uniqueness......Page 267
12.7 Chaplygin’s Method......Page 271
12.8 Asymptotic Behavior......Page 277
12.9 Lyapunov Function and Comparison Theorems......Page 280
12.10 Stability and Boundedness......Page 282
12.11 Notes......Page 285
PART 4: COMPLEX DIFFERENTIAL EQUATIONS......Page 286
13.1 Existence, Approximate Solutions, and Uniqueness......Page 288
13.2 Singularity-Free Regions and Growth Estimates......Page 292
13.3 Componentwise Bounds......Page 297
13.4 Lyapunov-like Functions and Comparison Theorems......Page 299
13.5 Notes......Page 301
Bibliography......Page 302
Author Index......Page 328
Subject Index......Page 331