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

Front Cover......Page 1
Differential and Integral Inequalities: Theory and Applications......Page 4
Copyright Page......Page 5
Contents......Page 8
Preface......Page 6
PART 1: ORDINARY DIFFERENTIAL EQUATIONS......Page 14
1.1. Existence and Continuation of Solutions......Page 16
1.2. Scalar Differential Inequalities......Page 20
1.3. Maximal and Minimal Solutions......Page 24
1.4. Comparison Theorems......Page 28
1.5. Finite Systems of Differential Inequalities......Page 34
1.6. Minimax Solutions......Page 38
1.7. Further Comparison Theorems......Page 40
1.8. Infinite Systems of Differential Inequalities......Page 44
1.9. Integral Inequalities Reducible to Differential Inequalities......Page 50
1.10. Differential Inequalities in the Sense of Caratheodory......Page 54
1.11. Notes......Page 57
2.1. Global Existence......Page 58
2.2. Uniqueness......Page 61
2.3. Convergence of Successive Approximations......Page 73
2.4. Chaplygin's Method......Page 77
2.5. Dependence on Initial Conditions and Parameters......Page 82
2.6. Variation of Constants......Page 89
2.7. Upper and Lower Bounds......Page 92
2.8. Componentwise Bounds......Page 97
2.9. Asymptotic Equilibrium......Page 101
2.10. Asymptotic Equivalence......Page 104
2.11. A Topological Principle......Page 109
2.12. Applications of Topological Principle......Page 113
2.13. Stability Criteria......Page 115
2.14. Asymptotic Behavior......Page 121
2.15 Periodic and Almost Periodic Systems......Page 133
2.16. Notes......Page 142
3.1. Basic Comparison Theorems......Page 144
3.2. Definitions......Page 148
3.3. Stability......Page 151
3.4. Asymptotic Stability......Page 158
3.5. Stability of Perturbed Systems......Page 168
3.6. Converse Theorems......Page 171
3.7. Stability by the First Approximation......Page 190
3.8. Total Stability......Page 199
3.9. Integral Stability......Page 204
3.10. L"-Stability......Page 212
3.11. Partial Stability......Page 218
3.12. Stability of Differential Inequalities......Page 222
3.13. Boundcdness and Lagrange Stability......Page 225
3.14. Eventual Stability......Page 235
3.15. Asymptotic Behavior......Page 242
3.16. Relative Stability......Page 254
3.17. Stability with Respect to a Manifold......Page 257
3.18. Almost Periodic Systems......Page 258
3.19. Uniqueness and Estimates......Page 267
3.20. Continuous Dependence and the Method of Averaging......Page 270
3.21. Notes......Page 277
4.1. Main Comparison Theorem......Page 280
4.2. Asymptotic Stability......Page 282
4.3. Instability......Page 286
4.4. Conditional Stability and Boundedness......Page 290
4.5. Converse Theorems......Page 297
4.6. Stability in Tube-like Domain......Page 306
4.7. Stability of Asymptotically Self-Invariant Sets......Page 310
4.8. Stability of Conditionally Invariant Sets......Page 318
4.9. Existence and Stability of Stationary Points......Page 321
4.10. Notes......Page 324
PART 2: VOLTERRA INTEGRAL EQUATIONS......Page 326
5.1. Integral Inequalities......Page 328
5.2. Local and Global Existence......Page 332
5.3. Comparison Theorems......Page 335
5.4. Approximate Solutions, Bounds, and Uniqueness......Page 337
5.5. Asymptotic Behavior......Page 340
5.6. Perturbed Integral Equations......Page 346
5.7. Admissibility and Asymptotic Behavior......Page 353
5.8. Integrodifferential Inequalities......Page 363
5.9. Notes......Page 367
Bibliography......Page 368
Author Index......Page 398
Subject Index......Page 401