This accessible monograph is devoted to a rapidly developing area on the research of qualitative theory of fractional ordinary differential equations and evolution equations. It is self-contained and unified in presentation, and 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, Picard operators technique, critical point theory and semigroups theory. This book is based on the research work done so far by the author and other experts, and contains comprehensive up-to-date materials on the topic.
In this third edition, four new topics have been added: Hilfer fractional evolution equations and infinite interval problems, oscillations and nonoscillations, fractional Hamiltonian systems, fractional Rayleigh–Stokes equations, and wave equations. The bibliography has also been updated and expanded.
Author(s): Yong Zhou
Edition: 3
Publisher: World Scientific
Year: 2023
Language: English
Pages: 516
Tags: Fractional Calculus, Fractional Differential Equation, Banach Space, Fractional Evolution Equation, Fractional Impulsive Differential Equations, Fractional Boundary Value Problems, Fractional Hamiltonian Systems, Fractional Partial Differential Equations
Contents
Preface to the Third Edition
About the Author
1. Preliminaries
1.1 Introduction
1.2 Some Notations, Concepts and Lemmas
1.3 Fractional Calculus
1.3.1 Definitions
1.3.2 Properties
1.3.3 Mittag-Leffler Functions
1.4 Some Results from Nonlinear Analysis
1.4.1 Laplace and Fourier Transforms
1.4.2 Sobolev Spaces
1.4.3 Measure of Noncompactness
1.4.4 Topological Degree
1.4.5 Picard Operator
1.4.6 Fixed Point Theorems
1.4.7 Critical Point Theorems
1.5 Semigroups
1.5.1 C0-Semigroup
1.5.2 Almost Sectorial Operators
2. Fractional Functional Differential Equations
2.1 Introduction
2.2 Neutral Equations with Bounded Delay
2.2.1 Introduction
2.2.2 Existence and Uniqueness
2.2.3 Extremal Solutions
2.3 p-Type Neutral Equations
2.3.1 Introduction
2.3.2 Existence and Uniqueness
2.3.3 Continuous Dependence
2.4 Neutral Equations with Infinite Delay
2.4.1 Introduction
2.4.2 Existence and Uniqueness
2.4.3 Continuation of Solutions
2.5 Iterative Functional Differential Equations
2.5.1 Introduction
2.5.2 Existence
2.5.3 Data Dependence
2.5.4 Examples and General Cases
2.6 Oscillations and Nonoscillations
2.6.1 Introduction
2.6.2 Preliminaries
2.6.3 Oscillation of Neutral Differential Systems
2.6.4 Existence of Nonoscillatory Solutions
2.6.5 Fractional Partial Functional Differential Equations
2.6.5.1 Oscillation of Fractional ODEs
2.6.5.2 Boundary Value Problem (2.104) and (B1)
2.6.5.3 Boundary Value Problem (2.104) and (B2)
2.6.5.4 Boundary Value Problem (2.104) and (B3)
2.6.5.5 Example
2.7 Notes and Remarks
3. Fractional Ordinary Differential Equations in Banach Spaces
3.1 Introduction
3.2 Cauchy Problems via Measure of Noncompactness Method
3.2.1 Introduction
3.2.2 Existence
3.3 Cauchy Problems via Topological Degree Method
3.3.1 Introduction
3.3.2 Qualitative Analysis
3.4 Cauchy Problems via Picard Operators Technique
3.4.1 Introduction
3.4.2 Results via Picard Operators
3.4.3 Results via Weakly Picard Operators
3.5 Notes and Remarks
4. Fractional Abstract Evolution Equations
4.1 Introduction
4.2 Evolution Equations with Riemann-Liouville Derivative
4.2.1 Introduction
4.2.2 Definition of Mild Solutions
4.2.3 Preliminary Lemmas
4.2.4 Compact Semigroup Case
4.2.5 Noncompact Semigroup Case
4.3 Evolution Equations with Caputo Derivative
4.3.1 Introduction
4.3.2 Definition of Mild Solutions
4.3.3 Preliminary Lemmas
4.3.4 Compact Semigroup Case
4.3.5 Noncompact Semigroup Case
4.4 Nonlocal Problems for Evolution Equations
4.4.1 IntroductionThe nonlocal condition has a
4.4.2 Definition of Mild Solutions
4.4.3 Existence
4.5 Optimal Controls of Fractional Evolution Equations
4.5.1 Introduction
4.5.2 Preliminaries
4.5.3 Existence of α-Mild Solutions
4.5.4 Existence of Fractional Optimal Controls
4.6 Abstract Cauchy Problems with Almost Sectorial Operators
4.6.1 Introduction
4.6.2 Properties of Operators
4.6.3 Linear Problems
4.6.4 Nonlinear Problems
4.6.5 Applications
4.7 Evolution Equations with Hilfer Derivative
4.7.1 Introduction
4.7.2 Preliminaries
4.7.3 Some Lemmas
4.7.4 Existence Results
4.8 Infinite Interval Problems with Hilfer Derivative
4.8.1 Introduction
4.8.2 Preliminaries
4.8.3 Lemmas
4.8.4 Existence on Infinite Interval
4.9 Notes and Remarks
5. Fractional Impulsive Differential Equations
5.1 Introduction
5.2 Impulsive Initial Value Problems
5.2.1 Introduction
5.2.2 Formula of Solutions
5.2.3 Existence
5.3 Impulsive Boundary Value Problems
5.3.1 Introduction
5.3.2 Formula of Solutions
5.3.3 Existence
5.4 Impulsive Langevin Eq
5.4.1 IntroductionIn 1908, Langevin introduced
5.4.2 Formula of Solutions
5.4.3 Existence
5.5 Impulsive Evolution Equations
5.5.1 Introduction
5.5.2 Cauchy Problems
5.5.3 Nonlocal Problems
5.6 Notes and Remarks
6. Fractional Boundary Value Problems
6.1 Introduction
6.2 Solutions for BVP with Left and Right Fractional Integrals
6.2.1 Introduction
6.2.2 Fractional Derivative Space
6.2.3 Variational Structure
6.2.4 Existence under Ambrosetti-Rabinowitz Con
6.2.5 Superquadratic Case
6.2.6 Asymptotically Quadratic Case
6.3 Multiple Solutions for BVP with Parameters
6.3.1 Introduction
6.3.2 Existence
6.4 Infinite Solutions for BVP with Left and Right Fractional Integrals
6.4.1 Introduction
6.4.2 Existence
6.5 Solutions for BVP with Left and Right Fractional Derivatives
6.5.1 Introduction
6.5.2 Variational Structure
6.5.3 Existence of Weak Solutions
6.5.4 Existence of Solutions
6.6 Notes and Remarks
7. Fractional Hamiltonian Systems
7.1 Introduction
7.2 Existence and Multiplicity of Homoclinic Solutions (I)
7.2.1 Fractional Derivative Space
7.2.2 Some Lemmas
7.2.3 Existence of Homoclinic Solutions
7.3 Existence and Multiplicity of Homoclinic Solutions (II)
7.3.1 Introduction
7.3.2 Some Lemmas
7.3.3 Existence and Multiplicity
7.4 Notes and Remarks
8. Fractional Partial Differential Equations
8.1 Introduction
8.2 Fractional Navier-Stokes Equations
8.2.1 Introduction
8.2.2 Preliminaries
8.2.3 Global Existence
8.2.4 Local Existence
8.2.5 Regularity
8.3 Fractional Rayleigh-Stokes Equations
8.3.1 Introduction
8.3.2 Preliminaries
8.3.2.1 Space Settings
8.3.2.2 Solution Representation
8.3.3 Globally Lispchitz Source Term
8.3.4 Locally Lispchitz Source Term
8.3.4.1 Existence of the Mild Solution
8.3.4.2 Continuation and Blow-Up Alternative
8.4 Fractional Euler-Lagrange Equations
8.4.1 Introduction
8.4.2 Functional Spaces
8.4.3 Variational Structure
8.4.4 Existence of Weak Solution
8.5 Fractional Diffusion Equations
8.5.1 Introduction
8.5.2 Regularity and Unique Existence
8.6 Fractional Wave Equations
8.6.1 Introduction
8.6.2 Preliminaries
8.6.3 Approximation Solution
8.6.4 Energy Estimates
8.6.5 Well-Posedness and Regularity
8.7 Notes and Remarks
Bibliography
Index
