Asymptotic Integration And Stability: For Ordinary, Functional And Discrete Differential Equations Of Fractional Order

by Dumitru Baleanu, Octavian G Mustafa

Overview

This volume presents several important and recent contributions to the emerging field of fractional differential equations in a self-contained manner. It deals with new results on existence, uniqueness and multiplicity, smoothness, asymptotic development, and stability of solutions. The new topics in the field of fractional calculus include also the Mittag-Leffler and Razumikhin stability, stability of a class of discrete fractional non-autonomous systems, asymptotic integration with a priori given coefficients, intervals of disconjugacy (non-oscillation), existence of Lp solutions for various linear, and nonlinear fractional differential equations.

Product Details

ISBN-13:	9789814641098
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	03/13/2015
Series:	Series On Complexity, Nonlinearity And Chaos , #4
Pages:	208
Product dimensions:	6.10(w) x 9.10(h) x 0.80(d)

Table of Contents

Preface vii

1 The Differential Operators of Order 1 + α and Then- Integral Counterparts 1

1.1 The Gamma Function 2

1.2 The Riemann-Liouville Derivative 3

1.3 The Abel Computation 19

1.4 The Operators. The Caputo Differential 21

1.5 The Integral Representation of the Operators. The Half-line Case 23

2 Existence and Uniqueness of Solution for the Differential Equations of Order α 27

2.1 A Lovelady-Martin Uniqueness Result for the Equation (2.2) 34

2.2 A Nagumo-like Uniqueness Criterion for the Fractional Differential Equations with a Riemann-Liouville Derivative 39

2.3 A Wintner-type Existence Interval for the Equation (2.2) 45

3 Position of the Zeros, the Bihari Inequality, and the Asymptotic Behavior of Solutions for the Differential Equations of Order α 51

3.1 A Fite-type Length Criterion for Fractional Disconjugacy 51

3.2 The Bihari Inequality 55

3.3 Asymptotic Integration of the Differential Equations of Orders 1 and α 61

3.4 The Bihari Asymptotic Integration Theory of the Differential Equations of Second Order 76

4 Asymptotic Integration for the Differential Equations of Order 1 + α 79

4.1 An Asymptotic Integration Theory of Trench Type 79

4.2 Asymptotically Linear Solutions 90

4.3 A Bihari-Like Result 95

4.4 Convergent Solutions 104

4.5 L^p-Solutions of the Equation (4.3) 109

5 Existence and Uniqueness of Solution for Some Delay Differential Equations with Caputo Derivatives 121

6 Existence of Positive Solutions for Some Delay Fractional Differential Equations with a Generalized N-Term 129

6.1 The Existence Theorem 130

6.2 Existence and Uniqueness for the Solution 137

7 Stability of a Class of Discrete Fractional Nonautonomous Systems 139

7.0.1 Preliminaries 139

7.0.2 The Lyapunov Method for Discrete Fractional Nonautonornous Systems 140

8 Mittag-Leffler Stability of Fractional Nonlinear Systems with Delay 145

9 Razumikhin Stability for Fractional Systems in the Presence of Delay 151

10 Controllability of Some Fractional Evolution Nonlocal Impulsive Quasilinear Delay Integra-Differential Systems 157

10.1 Prehimnaries 157

10.2 The Problem 161

10.3 A Controllability Result 162

11 Approximate Controllability of Sobolev Type Nonlocal Fractional Stochastic Dynamic Systems in Hilbert Spaces 169

11.0.1 The Problem 170

11.0.2 Approximate Controllabihty 175

Bibliography 181

Index 195