Existence results for fractional differential equations of arbitrary order with nonlocal integral boundary conditions

Bashir Ahmad,Mohammed M Matar,Ravi P Agarwal

doi:10.1186/s13661-015-0484-4

Abstract

In this paper, we investigate the existence of solutions for fractional differential equations of arbitrary order with nonlocal integral boundary conditions. The existence results are obtained by applying Krasnoselskii’s fixed point theorem and Leray-Schauder degree theory, while the uniqueness of the solutions is established by means of Banach’s contraction mapping principle. The paper concludes with illustrative examples.

Highlights

1 Introduction The study of boundary value problems of fractional differential equations has gained considerable attention and several interesting results involving a variety of boundary conditions have appeared in the recent literature on the topic
The tools of fractional calculus have been effectively employed to improve the mathematical modeling of several phenomena occurring in scientific and engineering disciplines such as viscoelasticity [ ], electrochemistry [ ], electromagnetism [ ], biology [, ], optimal control [, ], diffusion process [ – ], economics [ ], chaotic theory [ ], variational problems [ ], etc
There has been a great emphasis on studying fractional differential equations supplemented with integral boundary conditions; for instance, see [ – ]

Summary

Introduction

The study of boundary value problems of fractional differential equations has gained considerable attention and several interesting results involving a variety of boundary conditions have appeared in the recent literature on the topic. Ahmad et al Boundary Value Problems (2015) 2015:220 where J = [t , T], CDαt is the Caputo fractional derivative of order α, n = [α] + , bk ∈ R, and f , gk : J × R → R are given continuous functions. We can solve different kinds of boundary value problems involving integral (classical) and multi-point boundary conditions by applying the method of proof used for Lemma. This implies that is equicontinuous on J In consequence, it follows by the Arzela-Ascoli theorem that the operator is completely continuous. Proof Let us define a set Br = {x ∈ C(J, R) : x ≤ r}, where r is a positive constant satisfying the inequality r ≥ |T–t |α (α + ). For x ∈ Br, we have x(t) ≤ n– |t – θ |k k!

Cf x

Df x

Here α