Basic theory of fractional differential equations

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Başlık:

Basic theory of fractional differential equations

Yazar:

Zhou, Yong, 1964-

ISBN:

9789813148161

Ek Yazar:

Zhou, Yong, 1964-

Edition:

Second edition

Yayım Bilgisi:

New Jersey : World Scientific, 2017

Fiziksel Tanım:

xii, 367 s. ; 25 cm.

Contents:

Preliminaries -- Fractional functional differential equations -- Fractional ordinary differential equations in Banach spaces -- Fractional abstract evolution equations -- Fractional impulsive differential equations -- Fractional boundary value problems -- Fractional partial differential equations.

Konu Terimleri:

Fractional differential equations.

Kesirli diferansiyel denklemler

Differential equations.

Diferansiyel denklemler

Fractional calculus.

Fraksiyonel hesap

Added Author:

Wang, JinRong (Mathematics professor)

Zhang, Lu (Mathematics professor)

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Searching... Pamukkale Merkez Kütüphanesi	Kitap	0093300	QA372 Z47 2017	Searching... Unknown

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This invaluable monograph is devoted to a rapidly developing area on the research of qualitative theory of fractional ordinary and partial differential equations. It provides the readers the necessary background material required to go further into the subject and explore the rich research literature. The tools used include many classical and modern nonlinear analysis methods such as fixed point theory, measure of noncompactness method, topological degree method, the technique of Picard operators, critical point theory and semigroup theory. Based on the research work carried out by the authors and other experts during the past seven years, the contents are very recent and comprehensive.In this edition, two new topics have been added, that is, fractional impulsive differential equations, and fractional partial differential equations including fractional Navier-Stokes equations and fractional diffusion equations.

Preface to the Second Edition	p. v
Preface to the First Edition	p. vii
1 Preliminaries	p. 1
1.1 Introduction	p. 1
1.2 Some Notations, Concepts and Lemmas	p. 1
1.3 Fractional Calculus	p. 3
1.3.1 Definitions	p. 4
1.3.2 Properties	p. 9
1.3.3 Mittag-Leffler functions	p. 12
1.1 Some Results from Nonlinear Analysis	p. 13
1.4.1 Sobolev Spaces	p. 13
1.4.2 Measure of Noncompactness	p. 14
1.4.3 Topological Degree	p. 15
1.4.4 Picard Operator	p. 17
1.4.5 Fixed Point Theorems	p. 18
1.4.6 Critical Point Theorems	p. 20
1.5 Semigroups	p. 22
1.5.1 C o -semigroup	p. 22
1.5.2 Almost Sectorial Operators	p. 23
2 Fractional Functional Differential Equations	p. 27
2.1 Introduction	p. 27
2.2 Neutral Equations with Bounded Delay	p. 28
2.2.1 Introduction	p. 28
2.2.2 Existence and Uniqueness	p. 28
2.2.3 Extremal Solutions	p. 33
2.3 p-Type Neutral Equations	p. 42
2.3.1 Introduction	p. 42
2.3.2 Existence and Uniqueness	p. 44
2.3.3 Continuous Dependence	p. 55
2.4 Neutral Equations with Infinite Delay	p. 58
2.4.1 Introduction	p. 58
2.4.2 Existence and Uniqueness	p. 60
2.4.3 Continuation of Solutions	p. 67
2.5 Iterative Functional Differential Equations	p. 71
2.5.1 Introduction	p. 71
2.5.2 Existence	p. 72
2.5.3 Data Dependence	p. 78
2.5.4 Examples and General Cases	p. 79
2.6 Notes and Remarks	p. 86
3 Fractional Ordinary Differential Equations in Banach Spaces	p. 87
3.1 Introduction	p. 87
3.2 Cauchy Problems via Measure of Noncompactness Method	p. 89
3.2.1 Introduction	p. 89
3.2.2 Existence	p. 89
3.3 Cauchy Problems via Topological Degree Method	p. 98
3.3.1 Introduction	p. 98
3.3.2 Qualitative Analysis	p. 98
3.4 Cauchy Problems via Picard Operators Technique	p. 102
3.4.1 Introduction	p. 102
3.4.2 Results via Picard Operators	p. 102
3.4.3 Results via Weakly Picard Operators	p. 109
3.5 Notes and Remarks	p. 113
4 Fractional Abstract Evolution Equations	p. 115
4.1 Introduction	p. 115
4.2 Evolution Equations with Riemann-Liouville Derivative	p. 116
4.2.1 Introduction	p. 116
4.2.2 Definition of Mild Solutions	p. 117
4.2.3 Preliminary Lemmas	p. 120
4.2.4 Compact Semigroup Case	p. 126
4.2.5 Noncompact Semigroup Case	p. 131
4.3 Evolution Equations with Caputo Derivative	p. 134
4.3.1 Introduction	p. 134
4.3.2 Definition of Mild Solutions	p. 134
4.3.3 Preliminary Lemmas	p. 136
4.3.4 Compact Semigroup Case	p. 140
4.3.5 Noncompact Semigroup Case	p. 143
4.4 Nonlocal Problems for Evolution Equations	p. 145
4.4.1 Introduction	p. 145
4.4.2 Definition of mild solutions	p. 145
4.4.3 Existence	p. 147
4.5 Abstract Cauchy Problems with Almost Sectorial Operators	p. 153
4.5.1 Introduction	p. 153
4.5.2 Properties of Operators	p. 158
4.5.3 Linear Problems	p. 164
4.5.4 Nonlinear Problems	p. 169
4.5.5 Applications	p. 177
4.6 Notes and Remarks	p. 179
5 Fractional Impulsive Differential Equations	p. 181
5.1 Introduction	p. 181
5.2 Impulsive Initial Value Problems	p. 182
5.2.1 Introduction	p. 182
5.2.2 Formula of Solutions	p. 182
5.2.3 Existence	p. 185
5.3 Impulsive Boundary Value Problems	p. 190
5.3.1 Introduction	p. 190
5.3.2 Formula of Solutions	p. 190
5.3.3 Existence	p. 193
5.4 Impulsive Langevin Equations	p. 197
5.4.1 Introduction	p. 197
5.4.2 Formula of Solutions	p. 198
5.4.3 Existence	p. 206
5.5 Impulsive Evolution Equations	p. 213
5.5.1 Introduction	p. 213
5.5.2 Cauchy Problems	p. 214
5.5.3 Nonlocal Problems	p. 216
5.6 Notes and Remarks	p. 222
6 Fractional Boundary Value Problems	p. 223
6.1 Introduction	p. 223
6.2 Solution for BVP with Left and Right Fractional Integrals	p. 223
6.2.1 Introduction	p. 223
6.2.2 Fractional Derivative Space	p. 226
6.2.3 Variational Structure	p. 231
6.2.4 Existence Under Ambrosetti-Rabinowitz Condition	p. 238
6.2.5 Superquadratic Case	p. 243
6.2.6 Asymptotically Quadratic Case	p. 247
6.3 Multiple Solutions for BVP with Parameters	p. 250
6.3.1 Introduction	p. 250
6.3.2 Existence	p. 251
6.4 Infinite Solutions for BVP with Left and Right Fractional Integrals	p. 261
6.4.1 Introduction	p. 261
6.4.2 Existence	p. 262
6.5 Solutions for BVP with Left and Right Fractional Derivatives	p. 271
6.5.1 Introduction	p. 271
6.5.2 Variational Structure	p. 272
6.5.3 Existence of Weak Solutions	p. 275
6.5.4 Existence of Solutions	p. 279
6.6 Notes and Remarks	p. 283
7 Fractional Partial Differential Equations	p. 285
7.1 Introduction	p. 285
7.2 Fractional Navier-Stokes Equations	p. 285
7.2.1 Introduction	p. 285
7.2.2 Preliminaries	p. 287
7.2.3 Global Existence	p. 290
7.2.4 Local Existence	p. 297
7.2.5 Regularity	p. 301
7.3 Fractional Euler-Lagrange Equations	p. 309
7.3.1 Introduction	p. 309
7.3.2 Functional Spaces	p. 311
7.3.3 Variational Structure	p. 314
7.3.4 Existence of Weak Solution	p. 317
7.4 Fractional Diffusion Equations	p. 321
7.4.1 Introduction	p. 321
7.4.2 Preliminaries	p. 324
7.4.3 Existence and Regularity	p. 327
7.5 Fractional Schrödinger Equations	p. 336
7.5.1 Introduction	p. 336
7.5.2 Preliminaries	p. 337
7.5.3 Existence and Uniqueness	p. 340
7.6 Notes and Remarks	p. 342
Bibliography	p. 343
Index	p. 363

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