Boundary value problem for nonlinear fractional differential equations with delay

Azmat Ullah Khan Niazi,Pang Denghao,Mujeeb Ur Rehman,Jiang Wei

doi:10.1186/s13662-017-1090-9

Abstract

We discuss the existence and uniqueness of a solution of a boundary value problem for a class of nonlinear fractional functional differential equations with delay involving Caputo fractional derivative. Our work relies on the Schauder fixed point theorem and contraction mapping principle in a cone. We also include examples to show the applicability of our results.

Highlights

1 Introduction Fractional calculus is an emerging field in applied mathematics that deals with derivatives and integrals of arbitrary orders and their applications
Much research has been done on boundary value problems of fractional ordinary differential equations [ – ] and initial value problems of fractional functional differential equations [ – ]
There are relatively exiguous results dealing with boundary value problems of fractional functional differential equations with time delays

Summary

Introduction

Fractional calculus is an emerging field in applied mathematics that deals with derivatives and integrals of arbitrary orders and their applications. The existence and uniqueness of solutions to boundary value problems for fractional differential equations have gained these days interest of many authors. We refer to [ , ], where the authors discuss existence results for coupled systems and uniqueness for nth-order differential equations, respectively. The research on fractional functional differential equations with delay is relatively sparse [ – ]. It is necessary to consider the time-delay effect in the mathematical modeling of fractional differential equations. There are relatively exiguous results dealing with boundary value problems of fractional functional differential equations with time delays. Zhao and Wang [ ] studied the existence and uniqueness of a solution to the integral boundary value problem for the nonlinear fractional differential equation.

The space X equipped with the norm x

Define the constants

Proof From the definition of T we get

Here γ