Abstract and Applied Analysis
Volume 2009 (2009), Article ID 314656, 18 pages
doi:10.1155/2009/314656
Research Article

The Existence of Positive Solution to Three-Point Singular Boundary Value Problem of Fractional Differential Equation

Yuansheng Tian1 and Anping Chen1,2

1Department of Mathematics, Xiangnan University, Chenzhou 423000, China
2School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, China

Received 13 May 2009; Accepted 23 June 2009

Academic Editor: Yong Zhou

Copyright © 2009 Yuansheng Tian and Anping Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We investigate the existence of positive solution to nonlinear fractional differential equation three-point singular boundary value problem: Dqu(t)+f(t,u(t))=0, 0<t<1, u(0)=0, u(1)=αD(q−1)/2u(t)|t=ξ, where 1<q≤2 is a real number, ξ∈(0,1/2], α∈(0,+∞) and αΓ(q)ξ(q−1)/2<Γ((q+1)/2),Dq is the standard Riemann-Liouville fractional derivative, and f∈C((0,1]×[0,+∞),[0,+∞)),lim⁡t→+0f(t,⋅)=+∞ (i.e., f is singular at t=0). By using the fixed-point index theory, the existence result of positive solutions is obtained.