Existence of multiple positive solutions for singular boundary value problems of nonlinear fractional differential equations

Wen-Xue Zhou,Jie-Mei Li,Jian-Gang Zhang

doi:10.1186/1687-1847-2014-97

Abstract

In this paper, we consider the properties of the Green’s function for the nonlinear fractional differential equation boundary value problem , , , where is a real number, and is the standard Riemann-Liouville differentiation. As an application of the Green’s function, we give some multiple positive solutions for singular boundary value problems, and we also give the uniqueness of solution for a singular problem by means of the Leray-Schauder nonlinear alternative, a fixed-point theorem on cones, and a mixed monotone method.

Highlights

1 Introduction This paper is mainly concerned with the existence and multiplicity of positive solutions of the nonlinear fractional differential equation boundary value problem (BVP for short)
Some basic theory for the initial value problems of FDE involving Riemann-Liouville differential operator has been discussed by Lakshmikantham [ – ], Babakhani and Daftardar-Gejji [ – ] and Bai [ ], and so on
There are some papers which deal with the existence and multiplicity of solutions for nonlinear FDE

Summary

Introduction

This paper is mainly concerned with the existence and multiplicity of positive solutions of the nonlinear fractional differential equation boundary value problem (BVP for short). Zhang [ ] considered the existence and multiplicity of positive solutions for the nonlinear fractional boundary value problem cDq +u(t) = f t, u(t) , < t < , u( ) + u ( ) = , u( ) + u ( ) = , where < q ≤ is a real number, f : [ , ] × [ , +∞) → [ , +∞) and cDq + is the standard Caputo’s fractional derivative. The authors proved the existence of one positive solution by using the Guo-Krasnosel’skii fixed-point theorem and the nonlinear alternative of Leray-Schauder type in a cone and assuming certain hypotheses on the function f. ). Using the Leray-Schauder nonlinear alternative theorem and the Guo-Krasnosel’skii fixed-point theorem, we give some new existence criteria for the singular boundary value problem By continuity of G(·, s), using condition (H ), and the Arzela-Ascoli theorem guarantees that A : K ∩ ( \ ) → K is compact

Now we prove that

Let t x

So we have