Abstract and Applied Analysis
Volume 2012 (2012), Article ID 826580, 15 pages
http://dx.doi.org/10.1155/2012/826580
Research Article

Positive and Nondecreasing Solutions to an m-Point Boundary Value Problem for Nonlinear Fractional Differential Equation

I. J. Cabrera, J. Harjani, and K. B. Sadarangani

Departamento de MatemΓ‘ticas, Universidad de Las Palmas de Gran Canaria, Campus de Tafira Baja, 35017 Las Palmas de Gran Canaria, Spain

Received 30 September 2011; Accepted 15 November 2011

Academic Editor: Shaher M. Momani

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 are concerned with the existence and uniqueness of a positive and nondecreasing solution for the following nonlinear fractional m-point boundary value problem: 𝐷 𝛼 0 + βˆ‘ 𝑒 ( 𝑑 ) + 𝑓 ( 𝑑 , 𝑒 ( 𝑑 ) ) = 0 , 0 < 𝑑 < 1 , 2 < 𝛼 ≀ 3 , 𝑒 ( 0 ) = 𝑒 β€² ( 0 ) = 0 , 𝑒 β€² ( 1 ) = π‘š βˆ’ 2 𝑖 = 1 π‘Ž 𝑖 𝑒 β€² ( πœ‰ 𝑖 ) , where 𝐷 𝛼 0 + denotes the standard Riemann-Liouville fractional derivative, 𝑓 ∢ [ 0 , 1 ] Γ— [ 0 , ∞ ) β†’ [ 0 , ∞ ) is a continuous function, π‘Ž 𝑖 β‰₯ 0 for 𝑖 = 1 , 2 , … , π‘š βˆ’ 2 , and 0 < πœ‰ 1 < πœ‰ 2 < β‹― < πœ‰ π‘š βˆ’ 2 < 1 . Our analysis relies on a fixed point theorem in partially ordered sets. Some examples are also presented to illustrate the main results.