Existence of Positive Solutions for a Kind of Fractional Boundary Value Problems

Hongjie Liu,Xiao Fu,Liangping Qi

doi:10.1155/2014/602604

Abstract

We are concerned with the following nonlinear three-point fractional boundary value problem:D0+αut+λatft,ut=0,0<t<1,u0=0, andu1=βuη, where1<α≤2,0<β<1,0<η<1,D0+αis the standard Riemann-Liouville fractional derivative,at>0is continuous for0≤t≤1, andf≥0is continuous on0,1×0,∞. By using Krasnoesel'skii's fixed-point theorem and the corresponding Green function, we obtain some results for the existence of positive solutions. At the end of this paper, we give an example to illustrate our main results.

Highlights

Fractional differential equations have been of great interest recently
We present the Green function of fractional differential equation boundary value problem
Let F(t, s, η) = [t(1 − s)]α−1 − (t − s)α−1(1 − βηα−1); it is obvious that function F(t, s, η) is monotonically increasing in η, when η = 0, F(t, s, η)min = F (t, s, 0)

Summary

Introduction

Fractional differential equations have been of great interest recently. With the development of nonlinear science, the researchers found that nonlinear fractional differential equations could describe something’s changing rules more accurately. Many researchers paid attention to existence and multiplicity of solution of the boundary value problem for fractional differential equations with different boundary conditions, such as [1,2,3,4,5,6,7,8,9,10]. Bai and Lu [1] investigated the existence and multiplicity of positive solutions for nonlinear fractional boundary value problem: D0α+u (t) + f (t, u (t)) = 0, 0 < t < 1, (1). Investigated the existence of positive solutions for the singular fractional boundary value problem: D0α+u (t) + f (t, u (t) , Dμu (t)) = 0, 0 < t < 1, (2). By means of a fixed point theorem on cone, the existence of positive solutions is obtained. By using Krasnoesel’skii’s fixed-point theorem, we get the existence of at least one positive solution

Background

Main Results

An Example