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  Место хранения:   013 | Институт математики СО РАН | Новосибирск | Библиотека

Оглавление / Contents

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