Existence of Weak Solutions for Nonlinear Fractional Differential Inclusion with Nonseparated Boundary Conditions

Wen-Xue Zhou,Hai-Zhong Liu

doi:10.1155/2012/530624

Abstract

We discuss the existence of solutions, under the Pettis integrability assumption, for a class of boundary value problems for fractional differential inclusions involving nonlinear nonseparated boundary conditions. Our analysis relies on the Mönch 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 inclusion with non-separated boundary conditions: cDαu t ∈ F t, u t, t ∈ J : 0, T, T > 0, 1.1 u 0 λ1u T μ1, u 0 λ2u T μ2, λ1 / 1, λ2 / 1, where 1 < α ≤ 2 is a real number, cDα is the Caputo fractional derivative
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
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

Summary

Introduction

This paper is mainly concerned with the existence results for the following fractional differential inclusion with non-separated boundary conditions: cDαu t ∈ F t, u t , t ∈ J : 0, T , T > 0, 1.1 u 0 λ1u T μ1, u 0 λ2u T μ2, λ1 / 1, λ2 / 1, where 1 < α ≤ 2 is a real number, cDα is the Caputo fractional derivative. 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’s , O’Regan 25, 26 , Szufla 27, 28 , and the references therein. An example is given to illustrate our main result

Preliminaries and Lemmas

Main Results

An Example