Existence of solutions for nonlinear singular fractional differential equations with fractional derivative condition

Rengui Li

doi:10.1186/1687-1847-2014-292

Abstract

In this paper, we investigate a boundary value problem for singular fractional differential equations with a fractional derivative condition. The existence and uniqueness of solutions are obtained by means of the fixed point theorem. Some examples are presented to illustrate our main results.

Highlights

1 Introduction Differential equations of fractional order have recently been addressed by many researchers of various fields of science and engineering such as physics, chemistry, biology, economics, control theory, and biophysics; see [, ]
Fractional differential equations serve as an excellent tool for the description of memory and hereditary properties of various materials and processes
Much attention has been focused on the study of the existence and uniqueness of solutions for boundary value problem of fractional differential equations with nonlocal boundary conditions by the use of techniques of nonlinear analysis; see [ – ]

Summary

Introduction

Differential equations of fractional order have recently been addressed by many researchers of various fields of science and engineering such as physics, chemistry, biology, economics, control theory, and biophysics; see [ , ]. Much attention has been focused on the study of the existence and uniqueness of solutions for boundary value problem of fractional differential equations with nonlocal boundary conditions by the use of techniques of nonlinear analysis (fixed point theorems, Leray-Schauder theory, the upper and lower solution method, etc.); see [ – ]. In [ ], Agarwal et al investigated the existence of solutions for the singular fractional boundary value problems. In [ ], Yan et al studied the existence and uniqueness of solutions for a class of fractional differential equations with integral boundary conditions. Motivated by all the works above, this paper deals with the existence and uniqueness of solutions for the singular fractional boundary value problem with a fractional derivative condition,.

Lemma and

Simple calculus gives