International Journal of Differential Equations
Volume 2011 (2011), Article ID 328394, 10 pages
http://dx.doi.org/10.1155/2011/328394
Research Article

Existence of Positive Solutions for Fractional Differential Equation with Nonlocal Boundary Condition

Hongliang Gao and Xiaoling Han

Department of Mathematics, Northwest Normal University, Lanzhou 730070, China

Received 29 May 2011; Revised 24 August 2011; Accepted 28 September 2011

Academic Editor: Om Agrawal

Copyright © 2011 Hongliang Gao and Xiaoling Han. 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

By using the fixed point theorem, existence of positive solutions for fractional differential equation with nonlocal boundary condition 𝐷 𝛼 0 + 𝑢 ( 𝑡 ) + 𝑎 ( 𝑡 ) 𝑓 ( 𝑡 , 𝑢 ( 𝑡 ) ) = 0 , 0 < 𝑡 < 1 , 𝑢 ( 0 ) = 0 , ∑ 𝑢 ( 1 ) = ∞ 𝑖 = 1 𝛼 𝑖 𝑢 ( 𝜉 𝑖 ) is considered, where 1 < 𝛼 ≤ 2 is a real number, 𝐷 𝛼 0 + is the standard Riemann-Liouville differentiation, and 𝜉 𝑖 ∈ ( 0 , 1 ) , 𝛼 𝑖 ∈ [ 0 , ∞ ) with ∑ ∞ 𝑖 = 1 𝛼 𝑖 𝜉 𝑖 𝛼 − 1 < 1 , 𝑎 ( 𝑡 ) ∈ 𝐶 ( [ 0 , 1 ] , [ 0 , ∞ ) ) , 𝑓 ( 𝑡 , 𝑢 ) ∈ 𝐶 ( [ 0 , 1 ] × [ 0 , ∞ ) , [ 0 , ∞ ) ) .