Oscillation theory of delay differential equations

Cover ......Page 1
Oxford mathematical monographs ......Page 2
Title page ......Page 4
Preface ......Page 6
Acknowledgements ......Page 8
CONTENTS ......Page 10
1.1 Some basic existenc and uniqueness theorems ......Page 14
1.2 Exponential boundedness of solutions ......Page 23
1.3 The Laplace transform ......Page 26
1.4 The z-transform ......Page 28
1.5 Some basic lemmas ......Page 30
1.6 Some basic lemmas on differential inequalities ......Page 36
1.7 Some useful theorems from analysis ......Page 40
1.8 Notes ......Page 44
OSCILLATIONS OF LINEAR SCALAR DELAY EQUATIONS ......Page 45
2.1 Necessary and sufficient conditions for oscillations. The autonomous case ......Page 46
2.2 Explicit conditions for oscillations ......Page 50
2.3 Sufficient conditions for oscillations and for non-oscillations. The non-autonomous case ......Page 55
2.4 Oscillations of delay differential equations with  asymptotically constant coefficients ......Page 59
2.5 Nicholson\'s blowflies ......Page 64
2.6 Oscillations in non-autonomous equations with positive and negative coefficients ......Page 67
2.7 Notes ......Page 68
2.8 Open problems ......Page 69
3.1 A generalized characteristic equation ......Page 71
3.2 Differential inequalities and comparison results for positive solutions ......Page 77
3.3 Existence of positive solutions ......Page 83
3.4 Sufficient conditions for the oscillation of non-autonomous equations ......Page 87
3.5 Notes ......Page 92
3.6 Open problems ......Page 93
LINEARIZED OSCILLATION THEORY AND APPLICATIONS TO MATHEMATICAL BIOLOGY ......Page 94
4.1 Linearized oscillation theory ......Page 95
4.2 Time lags in a food-limited population model ......Page 97
4.3 Oscillations in the delay logistic equation with several delays ......Page 100
4.4 The Lasota-Wazewska model for the survival of red blood cells ......Page 102
4.6 Open problems ......Page 103
OSCILLATIONS OF SYSTEMS OF DELAY EQUATIONS ......Page 104
5.1 Necessary and sufficient conditions for the oscillation of linear autonomous systems ......Page 105
5.2 Explicit conditions for oscillations and non-oscillations of linear autonomous systems ......Page 108
5.3 Sufficient conditions for the oscillation of linear non-autonomous systems ......Page 113
5.4 Oscillations of a system of delay logistic equations ......Page 115
5.6 Open problems ......Page 123
OSCILLATIONS OF NEUTRAL DIFFERENTIAL EQUATIONS ......Page 124
6.1 Oscillations and asymptotic behaviour of scalar neutral delay differential equations ......Page 125
6.2 Oscillations of scalar neutral equations with mixed arguments ......Page 128
6.3 Necessary and sufficient conditions for the oscillation of systems of neutral equations ......Page 136
6.4 Oscillations of scalar neutral equations with variable coefficients ......Page 138
6.5 Differential inequalities and comparison theorems ......Page 143
6.6 Linearized oscillations for neutral equations ......Page 148
6.7 Existence of positive solutions ......Page 153
6.8 Oscillations in neutral delay logistic differential equations ......Page 161
6.9 Oscillations in non-autonomous equations with several delays ......Page 164
6.10 Oscillations in a system of neutral equations ......Page 169
6.11 Notes ......Page 172
6.12 Open problems ......Page 173
OSCILLATIONS OF DELAY DIFFERENCE EQUATIONS ......Page 175
7.1 Necessary and sufficient conditions for the oscillation of linear systems of difference equations ......Page 176
7.2 Explicit necessary and sufficient conditions for oscillations ......Page 179
7.3 Sufficient conditions for oscillations ......Page 182
7.4 Linearized oscillations for delay difference equations ......Page 185
7.5 A sharp condition for the oscillation of difference equations ......Page 191
7.6 Difference inequalities ......Page 197
7.7 Equations with positive and negative coefficients ......Page 200
7.8 Existence of positive solutions ......Page 203
7.9 Oscillations in a discrete delay logistic model ......Page 207
7.11 Open problems ......Page 209
OSCILLATIONS OF EQUATIONS WITH PIECEWISE CONSTANT ARGUMENTS ......Page 211
8.1 Necessary and sufficient conditions for the oscillation of linear systems with piecewise constant arguments ......Page 212
8.2 Linearized oscillations and the logistic equation with piece-wise constant arguments ......Page 218
8.3 Oscillations in non-autonomous equations with piecewise constant arguments ......Page 222
8.4 Oscillations in equations of alternately retarded and advanced type ......Page 227
8.5 Oscillations in equations with continuous and piecewise constant arguments ......Page 230
8.6 Notes ......Page 235
8.7 Open problems ......Page 236
9 OSCILLATIONS OF INTEGRODIFFERENTIAL EQUATIONS ......Page 238
9.1 Oscillations of linear integrodifferential equations ......Page 239
9.2 Comparison results for integrodifferential equations of Volterra type ......Page 244
9.3 Positive solutions of integrodifferential equations with unbounded delay ......Page 248
9.4 Oscillation in Volterra\'s integrodifferential equation ......Page 253
9.6 Open problems ......Page 260
10 OSCILLATIONS OF SECOND- AND HIGHER-ORDER EQUATIONS ......Page 261
10.1 Linearized oscillations for second-order differential equations—oscillations of the sunflower equation ......Page 262
10.2 Linearized oscillations for second-order differential equations—continued ......Page 268
10.3 Differential inequalities of higher order ......Page 273
10.4 Sufficient conditions for the oscillation of higher-order neutral equations ......Page 276
10.5 A comparison theorem for neutral differential equations of odd order ......Page 283
10.6 A linearized oscillation theorem for neutral equations of odd order ......Page 287
10.7 A comparison theorem for neutral equations of even order ......Page 293
10.8 A linearized oscillation theorem for neutral equations of even order ......Page 295
10.10 Open problems ......Page 299
11.1 Global attractivity in population dynamics ......Page 301
11.2 Global attractivity in models of haematopoiesis ......Page 310
11.3 Global attractivity in a discrete delay logistic model ......Page 317
11.4 Global stability of solutions of linear non-autonomous difference equations ......Page 320
11.5 Necessary and sufficient conditions for global attractivity in a perturbed linear difference equation ......Page 327
11.6 Notes ......Page 334
11.7 Open problems ......Page 335
12.1 Non-existence of slowly oscillating periodic solutions ......Page 336
12.2 Wright\'s equation has no solutions of period four ......Page 338
12.3 Oscillations in neutral equations with periodic coefficients ......Page 342
12.4 Asymptotic behaviour of rapidly oscillating solutions ......Page 354
12.6 Open problems ......Page 356
BIBLIOGRAPHY ......Page 357
INDEX ......Page 378