1 Introduction to Oscillation Theory ........................... 1
1.1 Introduction ............................................ 1
1.2 Nonoscillation of Autonomous Delay Equations with
Positive Coefficients ................................... 1
1.3 Nonlinear Equations of Mathematical Biology ............. 9
1.3.1 Linearization of Nonlinear Delay Equations ....... 9
1.3.2 Hutchinson's Equation ........................... 10
1.3.3 Lasota-Wazewska Equation ........................ 12
1.3.4 Nicholson's Blowflies Equation .................. 13
1.3.5 Mackey-Glass Equations .......................... 15
1.4 Impulsive Equations .................................... 16
1.5 Some Other Classes of Equations ........................ 18
1.6 Discussion and Open Problems ........................... 20
2 Scalar Delay Differential Equations on Semiaxes ............. 23
2.1 Introduction ........................................... 23
2.2 Preliminaries .......................................... 24
2.3 Nonoscillation Criteria ................................ 25
2.4 Comparison Theorems .................................... 28
2.5 Nonoscillation Conditions, Part 1 ...................... 32
2.6 Nonoscillation Conditions, Part 2 ...................... 37
2.7 Oscillation Conditions ................................. 43
2.8 Estimations of Solutions ............................... 46
2.9 Positivity of Solutions ................................ 49
2.10 Slowly Oscillating Solutions for Delay Differential
Equations .............................................. 51
2.11 Stability and Nonoscillation ........................... 52
2.12 Discussion and Open Problems ........................... 52
3 Scalar Delay Differential Equations on Semiaxis with
Positive and Negative Coefficients .......................... 59
3.1 Introduction ........................................... 59
3.2 Nonoscillation Criteria ................................ 59
3.1 Nonoscillation Conditions, Part 1 ...................... 66
3.4 Nonoscillation Conditions, Part 2 ...................... 71
3.5 Equations with an Oscillatory Coefficient .............. 77
3.6 Discussion and Open Problems ........................... 79
4 Oscillation of Equations with Distributed Delays ............ 83
4.1 Introduction ........................................... 83
4.2 Preliminaries .......................................... 84
4.3 Existence of a Positive Solution—General Results ....... 86
4.4 Comparison Theorems .................................... 92
4.5 Nonoscillation Criteria for Some Autonomous
Integrodifferential Equations .......................... 97
4.6 Explicit Nonoscillation and Oscillation Conditions .... 101
4.7 Slowly Oscillating Solutions .......................... 107
4.8 Equations with Positive and Negative Coefficients ..... 108
4.9 Discussion and Open Problems .......................... 118
5 Scalar Advanced and Mixed Differential Equations on
Semiaxes ................................................... 123
5.1 Introduction .......................................... 123
5.2 Advanced Equations .................................... 123
5.3 Mixed Equations with Positive Coefficients ............ 132
5.4 Mixed Equation with Negative Coefficients ............. 134
5.5 Positive Delay Term, Negative Advanced Term ........... 135
5.6 Negative Delay Term, Positive Advanced Term ........... 141
5.7 Discussion and Open Problems .......................... 144
6 Neutral Differential Equations ............................. 149
6.1 Introduction and Preliminaries ........................ 149
6.2 Nonoscillation Criteria ............................... 151
6.3 Efficient Nonoscillation Conditions ................... 156
6.4 Explicit Oscillation Conditions ....................... 160
6.5 Positivity of Solutions ............................... 164
6.6 Slowly Oscillating Solutions .......................... 165
6.7 Neutral Equations with Positive and Negative
Coefficients .......................................... 166
6.8 Discussion and Open Problems .......................... 168
7 Second-Order Delay Differential Equations .................. 171
7.1 Introduction .......................................... 171
7.2 Preliminaries ......................................... 171
7.3 Nonoscillation Criteria ............................... 172
7.4 Comparison Theorems ................................... 176
7.5 Explicit Nonoscillation and Oscillation Conditions .... 182
7.6 Slowly Oscillating Solutions .......................... 187
7.7 Existence of a Positive Solution ...................... 188
7.8 Discussion and Open Problems .......................... 190
8 Second-Order Delay Differential Equations with Damping
Terms ...................................................... 193
8.1 Introduction .......................................... 193
8.2 Preliminaries ......................................... 193
8.3 Nonoscillation Criteria ............................... 195
8.4 Comparison Theorems ................................... 200
8.5 Explicit Nonoscillation Conditions .................... 203
8.6 Discussion and Open Problems .......................... 204
9 Vector Delay Differential Equations ........................ 207
9.1 Introduction .......................................... 207
9.2 Preliminaries ......................................... 208
9.3 Main Results .......................................... 210
9.4 Comparison Results .................................... 214
9.5 Higher-Order Scalar Delay Differential Equations ...... 217
9.6 Positivity and Solution Estimates ..................... 220
9.7 Positive Solutions and Stability ...................... 223
9.8 Systems of Differential Equations with a Distributed
Delay ................................................. 229
9.8.1 Nonnegativity of Fundamental Matrices .......... 229
9.8.2 Comparison Results and Positivity of
Solutions ...................................... 232
9.8.3 Solution Estimates ............................. 233
9.8.4 Nonoscillation and Stability ................... 236
9.9 Discussion and Open Problems .......................... 238
10 Linearization Methods for Nonlinear Equations with
a Distributed Delay ........................................ 241
10.1 Introduction .......................................... 241
10.2 Preliminaries ......................................... 241
10.3 Linearized Oscillation ................................ 244
10.4 Applications .......................................... 248
10.4.1 Logistic Equation .............................. 249
10.4.2 Lasota-Wazewska Equation ....................... 250
10.4.3 Nicholson's Blowflies Equation ................. 253
10.5 "Mean Value Theorem" for Equations with
a Distributed Delay ................................... 258
10.6 Discussion and Open Problems .......................... 261
11 Nonlinear Models—Modifications of Delay Logistic
Equations .................................................. 263
11.1 Introduction .......................................... 263
11.2 Generalized Logistic Equation with Several Delays ..... 265
11.3 Multiplicative Delay Logistic Equation ................ 277
11.3.1 Preliminaries .................................. 277
11.3.2 Nonoscillation Criteria ........................ 278
11.3.3 Multiplicative Logistic Equation—Main Results .. 281
11.4 Discussion and Open Problems ......................... 282
12 First-Order Linear Delay Impulsive Differential Equations .. 285
12.1 Introduction .......................................... 285
12.2 Preliminaries ......................................... 286
12.3 Nonoscillation Criteria for Impulsive Equations ....... 287
12.4 Explicit Nonoscillation Tests and Comparison
Theorems .............................................. 292
12.5 Reduction to Equations Without Impulses ............... 295
12.6 Impulsive Equations with a Distributed Delay .......... 296
12.7 Discussion and Open Problems .......................... 299
13 Second-Order Linear Delay Impulsive Differential
Equations .................................................. 301
13.1 Introduction .......................................... 301
13.2 Preliminaries ......................................... 302
13.3 Nonoscillation Criteria ............................... 303
13.4 Comparison Theorems ................................... 308
13.5 Explicit Nonoscillation and Oscillation Conditions .... 309
13.6 Impulsive Equations with Damping Terms ................ 315
13.7 Discussion and Open Problems .......................... 318
14 Linearized Oscillation Theory for Nonlinear Delay
Impulsive Equations ........................................ 319
14.1 Introduction .......................................... 319
14.2 Preliminaries ......................................... 319
14.3 Oscillation and Nonoscillation ........................ 321
14.4 Applications to Equations of Mathematical Biology ..... 329
14.4.1 Logistic Equation: Theoretical Results ......... 329
14.4.2 Logistic Equation: Numerical Simulations ....... 333
14.4.3 Generalized Lasota-Wazewska Equation ........... 334
14.5 Discussion and Open Problems .......................... 336
15 Maximum Principles and Nonoscillation Intervals ............ 339
15.1 Introduction .......................................... 339
15.2 Preliminaries ......................................... 340
15.3 Maximum Principles in the Case of Positive Volterra
Operator (-B) ......................................... 346
15.4 Nonoscillation and Positivity of Green's Functions
for Positive Volterra Operator В ...................... 350
15.5 Nonoscillation on the Semiaxis ........................ 356
15.6 Positivity Tests for Green's Functions Through
Choice of v(t) ........................................ 357
15.7 The Generalized Periodic Problem for Positive
Volterra Operator В ................................... 360
15.8 Regular Behavior of the Green's Function to a
One-Point Boundary Value Problem ...................... 362
15.9 Positivity of Green's Functions for Equations
Including Difference of Positive Operators ............ 363
15.10 Positivity of the Cauchy and Green's Functions ....... 367
15.11 Equations with an Oscillating Coefficient ............ 376
15.12 Positivity of the Cauchy Function and Exponential
Stability ............................................ 384
15.13 General Boundary Value Problems ...................... 388
15.14 Discussion and Open Problems ......................... 391
16 Systems of Functional Differential Equations on Finite
Intervals .................................................. 399
16.1 Introduction .......................................... 399
16.2 Nonnegativity and Nonpositivity of Green's Matrices ... 401
16.3 Positivity of the n-th Row of the Cauchy Matrix ....... 408
16.4 Positivity of the Fixed и-th Row of Green's Matrices .. 417
16.5 Nonpositivity Conditions for the n-th Row of Green's
Matrices .............................................. 420
16.6 Discussion and Open Problems .......................... 426
17 Nonoscillation Intervals for n-th-Order Equations .......... 429
17.1 Introduction .......................................... 429
17.2 Homogeneous Functional Differential Equations of the
n-th Order ............................................ 430
17.3 Wronskian of the Fundamental System ................... 432
17.4 Nonoscillation of Functional Differential Equations ... 434
17.5 Nonoscillation and Regular Behavior of Green's
Functions ............................................. 437
17.6 Tests for Differential Equations with Deviating
Arguments ............................................. 444
17.7 Discussion and Open Problems .......................... 451
Appendix A Useful Theorems from Analysis ..................... 455
A.1 Vector Spaces ......................................... 455
A.2 Functional Spaces ..................................... 456
A.3 Sets in Functional Spaces ............................. 458
A.4 Linear Operators in Functional Spaces ................. 458
A.5 Nonlinear Operators ................................... 462
A.6 Gronwall-Bellman and Coppel Inequalities .............. 463
Appendix В Existence and Uniqueness Theorems, Solution
Representations ............................................ 465
B.l Linear Functional Differential Equations .............. 465
В.1.1 Differential Equations with Several
Concentrated Delays ............................. 465
B.l.2 Mixed Equations with an Infinite Number of
Delays .......................................... 468
B.1.3 Equations with a Distributed Delay .............. 472
B.l.4 Equations of Neutral Type ....................... 475
B.l.5 Higher-Order Scalar Delay Differential
Equations ....................................... 477
B.2 Estimations of the Fundamental Matrix ................. 480
B.3 Nonlinear Delay Differential Equations ................ 482
B.4 Linear Delay Impulsive Differential Equations ......... 490
B.4.1 First-Order Impulsive Equations ................ 490
B.4.2 Second-Order Impulsive Equations ................ 494
B.5 Bohl-Perron Theorems .................................. 499
References ................................................. 503
Index ......................................................... 517
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