Abstract and Applied Analysis
Volume 2011 (2011), Article ID 165641, 12 pages
http://dx.doi.org/10.1155/2011/165641
Research Article

Existence and Uniqueness of Positive Solution for a Boundary Value Problem of Fractional Order

J. Caballero, J. Harjani, and K. Sadarangani

Departamento de Matemáticas, Universidad de Las Palmas de Gran Canaria, Campus de Tafira Baja, 35017 Las Palmas de Gran Canaria, Spain

Received 10 February 2011; Revised 9 April 2011; Accepted 16 May 2011

Academic Editor: Elena Litsyn

Copyright © 2011 J. Caballero et al. 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 are concerned with the existence and uniqueness of positive solutions for the following nonlinear fractional boundary value problem: 𝐷 𝛼 0 + 𝑢 ( 𝑡 ) + 𝑓 ( 𝑡 , 𝑢 ( 𝑡 ) ) = 0 , 0 ≤ 𝑡 ≤ 1 , 3 < 𝛼 ≤ 4 , 𝑢 ( 0 ) = 𝑢 ( 0 ) = 𝑢 ( 0 ) = 𝑢 ( 1 ) = 0 , where 𝐷 𝛼 0 + denotes the standard Riemann-Liouville fractional derivative. Our analysis relies on a fixed point theorem in partially ordered sets. Some examples are also given to illustrate the results.