The Existence of Positive Solution to a Nonlinear Fractional Differential Equation with Integral Boundary Conditions

Meiqiang Feng,Hanying Feng,Xiaofang Liu

doi:10.1155/2011/546038

Abstract

The expression and properties of Green's function for a class of nonlinear fractional differential equations with integral boundary conditions are studied and employed to obtain some results on the existence of positive solutions by using fixed point theorem in cones. The proofs are based on the reduction of the problem considered to the equivalent Fredholm integral equation of the second kind. The results significantly extend and improve many known results even for integer-order cases.

Highlights

Fractional calculus is an area having a long history, its infancy dates back to three hundred years, the beginnings of classical calculus
For the basic theory and recent development of the subject, we refer the reader to a text by Lakshmikantham et al 6
Being directly inspired by 11, 13, 15, we intend in this paper to study the following boundary value problems of fractional order differential equation

Summary

Introduction

Fractional calculus is an area having a long history, its infancy dates back to three hundred years, the beginnings of classical calculus. The theory of boundary value problems for nonlinear fractional differential equations is still in the initial stages, and many aspects of this theory need to be explored. In 13 , Bai and Luused the fixed point theorems to show the existence and multiplicity of positive solutions to the nonlinear fractional boundary value problem. In 15 , Zhang showed the existence and multiplicity of positive solutions of the fractional boundary value problem. Being directly inspired by 11, 13, 15 , we intend in this paper to study the following boundary value problems of fractional order differential equation. Feng et al 25 considered the existence and multiplicity of positive solutions to boundary value problem 1.5 by applying the fixed point theory in a cone for strict set contraction operators. I0α D0α u t u t C1tα−1 C2tα−2 · · · CN tα−N , 1.8 for some Ci ∈ R, i 1, 2, . . . , N, where N is the smallest integer greater than or equal to α

Expression and Properties of Green’s Function

Preliminaries

Existence of Positive Solutions

Discussion

Conclusions