 | Abbas S. Topics in fractional differential equations / S.Abbas, M.Benchohra, G.M.N'Guérékata. - New York: Springer, 2012. - xiii, 396 p.. - (Developments in mathematics; vol.27). - ISBN 978-1-4614-4035-2; ISSN 1389-2177
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1 Introduction ................................................. 1
2 Preliminary Background .................................... 11
2.1 Notations and Definitions ................................. 11
2.2 Properties of Partial Fractional Calculus ................. 12
2.3 Properties of Set-Valued Maps ............................. 18
2.4 Fixed Point Theorems ...................................... 21
2.5 Gronwall Lemmas ........................................... 23
3 Partial Hyperbolic Functional Differential Equations ...... 25
3.1 Introduction .............................................. 25
3.2 Partial Hyperbolic Differential Equations ................. 25
3.2.1 Introduction ....................................... 25
3.2.2 Existence of Solutions ............................. 26
3.2.3 An Example ......................................... 32
3.3 Perturbed Partial Differential Equations .................. 33
3.3.1 Introduction ....................................... 33
3.3.2 Existence of Solutions ............................. 33
3.3.3 An Example ......................................... 39
3.4 Neutral Partial Differential Equations .................... 40
3.4.1 Introduction ....................................... 40
3.4.2 Existence of Solutions ............................. 40
3.4.3 An Example ......................................... 46
3.5 Discontinuous Partial Differential Equations in Banach
Algebras .................................................. 47
3.5.1 Introduction ....................................... 47
3.5.2 Existence of Solutions ............................. 47
3.5.3 Existence of Extremal Solutions .................... 51
3.5.4 An Example ......................................... 55
3.6 Upper and Lower Solutions Method for Partial Hyperbolic
Differential Equations .................................... 56
3.6.1 Introduction ....................................... 56
3.6.2 Main Result ........................................ 56
3.7 Partial Functional Differential Equations with Infinite
Delay ..................................................... 60
3.7.1 Introduction ....................................... 60
3.7.2 The Phase Space В .................................. 60
3.7.3 Main Results ....................................... 62
3.7.4 An Example ......................................... 71
3.8 Partial Hyperbolic Differential Equations with State-
Dependent Delay ........................................... 73
3.8.1 Introduction ....................................... 73
3.8.2 Existence of Solutions for Finite Delay ............ 74
3.8.3 Existence of Solutions for Infinite Delay .......... 83
3.8.4 Examples ........................................... 94
3.9 Global Uniqueness Results for Partial Hyperbolic
Differential Equations .................................... 98
3.9.1 Introduction ....................................... 98
3.9.2 Global Result for Finite Delay .................... 100
3.9.3 Global Result for Infinite Delay .................. 105
3.9.4 Examples .......................................... 112
3.10 Notes and Remarks ........................................ 114
4 Partial Hyperbolic Functional Differential Inclusions .... 115
4.1 Introduction ............................................. 115
4.2 Partial Hyperbolic Differential Inclusions ............... 115
4.2.1 Introduction ...................................... 115
4.2.2 The Convex Case ................................... 116
4.2.3 The Nonconvex Case ................................ 121
4.2.4 An Example ........................................ 124
4.3 Existence Results for Partial Hyperbolic Differential
Inclusions ............................................... 125
4.3.1 Introduction ...................................... 125
4.3.2 Existence of Solutions ............................ 125
4.3.3 Qualitative Properties and Topological Structure
of the Solution Set ............................... 130
4.4 Upper and Lower Solutions Method for Partial
Differential Inclusions .................................. 143
4.4.1 Introduction ...................................... 143
4.4.2 Main Result ....................................... 144
4.5 Partial Functional Differential Inclusions with
Infinite Delay ........................................... 150
4.5.1 Introduction ...................................... 150
4.5.2 Main Results ...................................... 150
4.5.3 An Example ........................................ 159
4.6 Fractional Order Riemann-Liouville Integral Inclusions
with two Independent Variables and Multiple Time Delay ... 159
4.6.1 Introduction ...................................... 159
4.6.2 Existence of Solutions ............................ 160
4.6.3 An Example ........................................ 168
4.7 Notes and Remarks ........................................ 169
5 Impulsive Partial Hyperbolic Functional Differential
Equations ................................................ 171
5.1 Introduction ............................................. 171
5.2 Impulsive Partial Hyperbolic Functional Differential
Equations ................................................ 171
5.2.1 Introduction ...................................... 171
5.2.2 Existence of Solutions ............................ 172
5.2.3 An Example ........................................ 178
5.3 Impulsive Partial Hyperbolic Differential Equations at
Variable Times ........................................... 179
5.3.1 Introduction ...................................... 179
5.3.2 Existence of Solutions ............................ 179
5.3.3 Nonlocal Impulsive Partial Differential
Equations ......................................... 186
5.3.4 An Example ........................................ 187
5.4 Impulsive Discontinuous Partial Hyperbolic Differential
Equations on Banach Algebras ............................. 188
5.4.1 Introduction ...................................... 188
5.4.2 Existence of Solutions ............................ 189
5.4.3 Existence of Extremal Solutions ................... 193
5.4.4 An Example ........................................ 198
5.5 Impulsive Partial Hyperbolic Differential Equations
with Variable Times and Infinite Delay ................... 199
5.5.1 Introduction ...................................... 199
5.5.2 Main Result ....................................... 200
5.5.3 An Example ........................................ 209
5.6 Impulsive Partial Hyperbolic Functional Differential
Equations of Fractional Order with State-Dependent
Delay .................................................... 211
5.6.1 Introduction ...................................... 211
5.6.2 Impulsive Partial Differential Equations with
Finite Delay ...................................... 212
5.6.3 Impulsive Partial Differential Equations with
Infinite Delay .................................... 215
5.6.4 Examples .......................................... 220
5.7 Impulsive Partial Hyperbolic Functional Differential
Equations with Variable Times and State-Dependent Delay .. 222
5.7.1 Introduction ...................................... 222
5.7.2 Impulsive Partial Differential Equations with
Finite Delay ...................................... 223
5.7.3 Impulsive Partial Differential Equations with
Infinite Delay .................................... 231
5.7.4 Examples .......................................... 241
5.8 Upper and Lower Solutions Method for Impulsive Partial
Hyperbolic Differential Equations ........................ 243
5.8.1 Introduction ...................................... 243
5.8.2 Main Result ....................................... 244
5.9 Notes and Remarks ........................................ 249
6 Impulsive Partial Hyperbolic Functional Differential
Inclusions ............................................... 251
6.1 Introduction ............................................. 251
6.2 Impulsive Partial Hyperbolic Differential Inclusions ..... 251
6.2.1 Introduction ...................................... 251
6.2.2 The Convex Case ................................... 252
6.2.3 The Nonconvex Case ................................ 259
6.2.4 An Example ........................................ 263
6.3 Impulsive Partial Hyperbolic Differential Inclusions
with Variable Times ...................................... 265
6.3.1 Introduction ...................................... 265
6.3.2 Existence of Solutions ............................ 265
6.3.3 An Example ........................................ 275
6.4 The Method of Upper and Lower Solutions for Partial
Hyperbolic Fractional Order Differential Inclusions
with Impulses ............................................ 276
6.4.1 Introduction ...................................... 276
6.4.2 Main Result ....................................... 277
6.5 Notes and Remarks ........................................ 285
7 Implicit Partial Hyperbolic Functional Differential
Equations ................................................ 287
7.1 Introduction ............................................. 287
7.2 Darboux Problem for Implicit Differential Equations ...... 287
7.2.1 Introduction ...................................... 287
7.2.2 Riemann-Liouville and Caputo Partial Fractional
Derivatives ....................................... 288
7.2.3 Existence of Solutions ............................ 289
7.2.4 An Example ........................................ 294
7.3 A Global Uniqueness Result for Implicit Differential
Equations ................................................ 295
7.3.1 Introduction ...................................... 295
7.3.2 Existence of Solutions ............................ 295
7.3.3 An Example ........................................ 298
7.4 Functional Implicit Hyperbolic Differential Equations
with Delay ............................................... 299
7.4.1 Introduction ...................................... 299
7.4.2 Existence Results with Finite Delay ............... 300
7.4.3 Existence Results for Infinite Delay .............. 305
7.4.4 Existence Results with State-Dependent Delay ...... 308
7.4.5 Examples .......................................... 312
7.5 Darboux Problem for Implicit Impulsive Partial
Hyperbolic Differential Equations ........................ 316
7.5.1 Introduction ...................................... 316
7.5.2 Existence of Solutions ............................ 316
7.5.3 An Example ........................................ 325
7.6 Implicit Impulsive Partial Hyperbolic Differential
Equations with State-Dependent Delay ..................... 326
7.6.1 Introduction ...................................... 326
7.6.2 Existence Results with Finite Delay ............... 327
7.6.3 Existence Results with Infinite Delay ............. 332
7.6.4 Examples .......................................... 337
7.7 Notes and Remarks ........................................ 339
8 Fractional Order Riemann-Liouville Integral Equations .... 341
8.1 Introduction ............................................. 341
8.2 Uniqueness Results for Fredholm-Type Fractional Order
Riemann-Liouville Integral Equations ..................... 341
8.2.1 Introduction ...................................... 341
8.2.2 Main Results ...................................... 342
8.3 Fractional Order Riemann-Liouville Integral Equations
with Multiple Time Delay ................................. 353
8.3.1 Introduction ...................................... 353
8.3.2 Existence of Solutions ............................ 353
8.3.3 Examples .......................................... 358
8.4 Nonlinear Quadratic Volterra Riemann-Liouville Integral
Equations of Fractional Order ............................ 359
8.4.1 Introduction ...................................... 359
8.4.2 Existence of Solutions ............................ 359
8.4.3 An Example ........................................ 364
8.5 Asymptotic Stability of Solutions of Nonlinear
Quadratic Volterra Integral Equations of Fractional
Order .................................................... 365
8.5.1 Introduction ...................................... 365
8.5.2 Main Results ...................................... 366
8.5.3 An Example ........................................ 373
8.6 Attractivity Results for Nonlinear Fractional Order
Riemann-Liouville Integral Equations in Banach Algebras .. 374
8.6.1 Introduction ...................................... 374
8.6.2 Main Results ...................................... 375
8.6.3 An Example ........................................ 381
8.7 Notes and Remarks ........................................ 382
References .................................................... 383
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