Existence of solutions of boundary value problems for nonlinear fractional differential equations with integral conditions

Ahcene Boukehila

doi:10.22199/issn.0717-6279-4271

Abstract

In this work we investigate the existence and uniqueness of solutions of boundary value problems for fractional differential equations involving the Caputo fractional derivative with integral conditions and the nonlinear term depends on the fractional derivative of an unknown function. Our existence results are based on Banach contraction principle and Schauder fixed point theorem. Two examples are provided to illustrate our results.

Highlights

Fractional differential equations are generalizations of the ordinary differential equations to an arbitrary non-integer order
In [3], Benchohra et al established sufficient conditions for the existence of solutions to a class of boundary value problem for fractional differential equations using the techniques of some fixed point theorems
It is desirable to extend this condition which leading to a more difficult and complicated case, in which the aim of the present paper is to overcome these difficulties. Inspired by these works in [3, 6] and the references therein, the present study was aimed to investigate the existence and uniqueness of solutions of boundary value problems for the fractional differential equations with integral conditions and the nonlinear term depends on the fractional derivative of an unknown function

Summary

Introduction

CD0q+u(t) = f (t, u(t),c D0σ+u(t)), 0 < t < 1, Z1 u(0) − u0(0) = η1(s)u(s) ds, Z1 u(1) − u0(1) = η2(s)u(s) ds, where cD0q+ is the Caputo fractional derivative of order q, 1 < q ≤ 2 and 0 < σ < 1, f : [0, 1] × R × R → R and η1, η2 are the continuous functions which will be specified later. We firstly derive the corresponding Green’s function. Existence of solutions of boundary value problems for ... 1119 using Banach contraction principle and Schauder fixed point theorem the existence and uniqueness of solutions are obtained. The other parts of the paper are organized as follows: In Section 2, we list some definitions and lemmas to be used later. At the end of this section, two examples are provided to illustrate the main results

Preliminaries

Main results

Examples

Example