Weak Solutions for Nonlinear Fractional Differential Equations in Banach Spaces

Wen-Xue Zhou,Hai-Zhong Liu,Ying-Xiang Chang

doi:10.1155/2012/527969

Abstract

We discuss the existence of weak solutions for a nonlinear boundary value problem of fractional differential equations in Banach space. Our analysis relies on the Mönch′s fixed point theorem combined with the technique of measures of weak noncompactness.

Highlights

This paper is mainly concerned with the existence results for the following fractional differential equation: cD0α u t f t, u t, t ∈ J : 0, T, 1.1 u 0 λ1u T μ1, u 0 λ2u T μ2, λ1 / 1, λ2 / 1, where 1 < α ≤ 2 is a real number, cD0α is the Caputo’s fractional derivative, λ1, λ2, μ1, μ2 ∈ R. f : J × E → E is a given function satisfying some assumptions that will be specified later, and E is a Banach space with norm u
It should be noted that most of the books and papers on fractional calculus are devoted to the solvability of initial value problems for differential equations of fractional order
A function x ∈ C J, Eω is said to be a solution of the problem 1.1 if x satisfies the equation cD0α u t f t, u t on J and satisfies the conditions u 0 λ1u T μ1, u 0 λ2u T μ2

Summary

Introduction

This paper is mainly concerned with the existence results for the following fractional differential equation: cD0α u t f t, u t , t ∈ J : 0, T , 1.1 u 0 λ1u T μ1, u 0 λ2u T μ2, λ1 / 1, λ2 / 1, where 1 < α ≤ 2 is a real number, cD0α is the Caputo’s fractional derivative, λ1, λ2, μ1, μ2 ∈ R. f : J × E → E is a given function satisfying some assumptions that will be specified later, and E is a Banach space with norm u. To investigate the existence of solutions of the problem above, we use Monch’s fixed point theorem combined with the technique of measures of weak noncompactness, which is an important method for seeking solutions of differential equations. This technique was mainly initiated in the monograph of Banasand Goebel and subsequently developed and used in many papers; see, for example, Banasand Sadarangani , Guo et al , Krzyska and Kubiaczyk , Lakshmikantham and Leela , Monch 25 , O’Regan 26, 27 , Szufla 28, 29 , and the references therein. We give an example for the illustration of the theories established in this paper

Preliminaries and Lemmas

Main Results

Γα t 0 t

An Example