Nonoscillation and Oscillation: Theory for Functional Differential Equations

Nonoscillation and Oscillation: Theory for Functional Differential Equations......Page 11
Preface......Page 13
Contents......Page 14
Bibliography......Page 370
1.2. Initial Value Problems......Page 17
1.3. Definition of Oscillation......Page 20
1.4. Some Fixed Point Theorems......Page 21
1.5. Notes......Page 25
2.2. Equations with a Single Delay: General Case......Page 26
2.3. Equations with Variable Delay: Critical Case......Page 37
2.4. Equations with Constant Delay......Page 43
2.5. Equations with Several Delays......Page 56
2.6. Equations with Piecewise Constant Argument......Page 58
2.7. Notes......Page 68
3.2. Comparison Theorems and Oscillation......Page 69
3.3. Oscillation of Equations with Variable Coefficients (I)......Page 77
3.4. Oscillation of Equations with Variable Coefficients (II)......Page 84
3.5. Existence of Nonoscillating Solutions......Page 91
3.6. Classification Schemes of Positive Solutions......Page 96
3.7. Positive Solutions of Neutral Perturbed Equations......Page 109
3.8. Notes......Page 121
4.2. Oscillation of Superlinear Equations......Page 122
4.3. Oscillation of Sublinear Equations......Page 128
4.4. Oscillation of Nonlinear Equations......Page 130
4.5. Forced Oscillation of Nonlinear Equations......Page 144
4.6. Positive Solutions of Nonlinear Equations......Page 154
4.7. Oscillation of Half-Linear Equations......Page 163
4.8. Notes......Page 172
5.2. Nonoscillation of Half-Linear Equations......Page 173
5.3. Classification Schemes for Iterative Equations......Page 179
5.4. Nonoscillation of Nonlinear Equations with…......Page 189
5.5. Nonoscillation of Nonlinear Equations with…......Page 196
5.6. Notes......Page 204
6.1. Introduction......Page 205
6.2. Comparison Theorems and Oscillation......Page 206
6.3. Oscillation Criteria for Neutral Equations......Page 212
6.4. Asymptotic Behavior of Nonoscillatory Solutions......Page 221
6.5. Positive Solutions of Nonlinear Equations......Page 226
6.6. Classifications of Nonoscillatory Solutions......Page 234
6.7. Asymptotic Trichotomy for Positive Solutions......Page 243
6.8. Existence of Nonoscillatory Solutions......Page 249
6.9. Notes......Page 257
7.2. Oscillation of Nonlinear Systems......Page 258
7.3. Oscillation of Nonlinear Systems with Forcing......Page 262
7.4. Classification Schemes of Positive Solutions (I)......Page 265
7.4.1. The Case…......Page 266
7.4.2. The Case…......Page 271
7.5.1. The Case…......Page 279
7.5.2. The Case…......Page 280
7.5.4. The Case…......Page 282
7.6. Positive Solutions of Second Order Systems......Page 284
7.6.1. The Case…......Page 285
7.6.2. The Case…......Page 286
7.6.3. The Case…......Page 289
7.6.4. The Case…......Page 293
7.7. Nonoscillation of Emden–Fowler Systems......Page 296
7.8. Notes......Page 301
8.1. Introduction......Page 302
8.2. The Time Scales Calculus......Page 303
8.3. Oscillation of Second Order Nonlinear Dynamic Equations......Page 309
8.3.1. The Case…......Page 311
8.3.2. The Case…......Page 315
8.4. Oscillation of Perturbed Nonlinear Dynamic Equations......Page 317
8.5. Positive Solutions of Nonlinear Dynamic Equations......Page 327
8.6. Oscillation of Emden–Fowler Equations......Page 333
8.7. Oscillation of First Order Delay Dynamic Equations......Page 348
8.8. Oscillation of Symplectic Dynamic Systems......Page 357
8.9. Notes......Page 369