Fractional-Order Equations and Inclusions

Product Description


This book presents fractional difference, integral, differential, evolution equations and inclusions, and discusses existence and asymptotic behavior of their solutions. Controllability and relaxed control results are obtained. Combining rigorous deduction with abundant examples, it is of interest to nonlinear science researchers using fractional equations as a tool, and physicists, mechanics researchers and engineers studying relevant topics. ContentsFractional Difference EquationsFractional Integral EquationsFractional Differential EquationsFractional Evolution Equations: ContinuedFractional Differential Inclusions


Review


Table of Content:

Introduction

1 Fractional Difference Equations 1.1 Fractional difference Gronwall inequalities 1.1.1 Introduction 1.1.2 Caputo like fractional difference 1.1.3 Linear fractional difference equation 1.1.4 Fractional difference inequalities 1.2 S-asymptotically periodic solutions 1.2.1 Introduction 1.2.2 Preliminaries 1.2.3 Non-existence of periodic solutions 1.2.4 Existence and uniqueness results
2 Fractional Integral Equations 2.1 Abel-type nonlinear integral equations 2.1.1 Introduction 2.1.2 Preliminaries 2.1.3 Existence and uniqueness of non-trivial solution in an order interval 2.1.4 General solutions of Erdélyi-Kober type integral equations 2.1.5 Illustrative examples 2.2 Quadratic Erdélyi-Kober type integral equations of fractional order 2.2.1 Introduction 2.2.2 Preliminaries 2.2.3 Existence and limit property of solutions 2.2.4 Uniqueness and another existence results 2.2.5 Applications 2.3 Fully nonlinear Erdélyi-Kober fractional integral equations 2.3.1 Introduction 2.3.2 Main result 2.3.3 Example 2.4 Quadratic Weyl fractional integral equations 2.4.1 Introduction 2.4.2 Preliminaries 2.4.3 Some basic properties of Weyl kernel 2.4.4 Existence and uniform local attractivity of 2π-periodic solutions 2.4.5 Example

3 Fractional Differential Equations 3.1 Asymptotically periodic solutions 3.1.1 Introduction 3.1.2 Preliminaries 3.1.3 Non-existence results for periodic solutions 3.1.4 Existence results for asymptotically periodic solutions 3.1.5 Further extensions 3.2 Modified fractional iterative functional differential equations 3.2.1 Introduction 3.2.2 Notation, definitions and auxiliary facts 3.2.3 Existence 3.2.4 Data dependence 3.2.5 Examples 3.3 Ulam-Hyers-Rassias stability for semilinear equations 3.3.1 Introduction 3.3.2 Ulam-Hyers-Rassias stability for surjective linear equations on Banach spaces 3.3.3 Ulam-Hyers-Rassias stability for linear equations on Banach spaces with closed ranges 3.3.4 Ulam-Hyers-Rassias stability for surjective semilinear equations between Banach spaces 3.4 Practical Ulam-Hyers-Rassias stability for nonlinear equations 3.4.1 Introduction 3.4.2 Main results 3.4.3 Examples 3.5 Ulam-Hyers-Mittag-Leffler stability of fractional delay differential equations 3.5.1 Introduction 3.5.2 Preliminaries 3.5.3 Main results 3.5.4 Examples 3.6 Nonlinear impulsive fractional differential equations 3.6.1 Introduction 3.6.2 Preliminaries 3.6.3 Existence results for impulsive Cauchy problems 3.6.4 Ulam stability results for impulsive fractional differential equations 3.6.5 Existence results for impulsive boundary value problems 3.6.6 Applications 3.7 Fractional differential switched systems with coupled nonlocal initial and impulsive conditions 3.7.1 Introduction 3.7.2 Preliminaries 3.7.3 Existence and uniqueness result via Banach fixed point theorem 3.7.4 Existence result via Krasnoselskii fixed point theorem 3.7.5 Existence result via Leray-Schauder fixed point theorem 3.7.6 Existence result for the resonant case: Landesman-Lazer conditions 3.7.7 Ulam type stability results 3.8 Not instantaneous impulsive fractional differential equations 3.8.1 Introduction 3.8.2 Framework of linear impulsive fractional Cauchy problem 3.8.3 Generalized Ulam-Hyers-Rassias stability concept 3.8.4 Main results via fixed point methods 3.9 Center stable manifold