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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 803417, 18 pages
http://dx.doi.org/10.1155/2012/803417
Research Article

Existence and Uniqueness of Positive Solutions for a Singular Fractional Three-Point Boundary Value Problem

Department of Mathematics, University of Las Palmas de Gran Canaria, Campus de Tafira Baja, 35017 Las Palmas de Gran Canaria, Spain

Received 12 June 2012; Accepted 29 July 2012

Academic Editor: Juan J. Trujillo

Copyright © 2012 I. J. Cabrera 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 investigate the existence and uniqueness of positive solutions for the following singular fractional three-point boundary value problem 𝐷 𝛼 0 + 𝑢 ( 𝑡 ) + 𝑓 ( 𝑡 , 𝑢 ( 𝑡 ) ) = 0 , 0 < 𝑡 < 1 , 𝑢 ( 0 ) = 𝑢 ( 0 ) = 𝑢 ( 0 ) = 0 , 𝑢 ( 1 ) = 𝛽 𝑢 ( 𝜂 ) , where 3 < 𝛼 4 , 𝐷 𝛼 0 + is the standard Riemann-Liouville derivative and 𝑓 ( 0 , 1 ] × [ 0 , ) [ 0 , ) with l i m 𝑡 0 + 𝑓 ( 𝑡 , ) = (i.e., 𝑓 is singular at 𝑡 = 0 ). Our analysis relies on a fixed point theorem in partially ordered metric spaces.