Abstract and Applied Analysis
Volume 2010 (2010), Article ID 501230, 15 pages
doi:10.1155/2010/501230
Research Article

Positive Solutions to Nonlinear Higher-Order Nonlocal Boundary Value Problems for Fractional Differential Equations

Mujeeb Ur Rehman1 and Rahmat Ali Khan2

1Centre for Advanced Mathematics and Physics, National University of Sciences and Technology, Sector H-12, Islamabad 46000, Pakistan
2University of Malakand, Chakdara Dir(L), Khyber Pakhutoonkhwa, Pakistan

Received 28 June 2010; Revised 21 September 2010; Accepted 28 October 2010

Academic Editor: Paul Eloe

Copyright © 2010 Mujeeb Ur Rehman and Rahmat Ali Khan. 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 study existence of positive solutions to nonlinear higher-order nonlocal boundary value problems corresponding to fractional differential equation of the type 𝑐 𝒟 𝛿 0 + 𝑢 ( 𝑡 ) + 𝑓 ( 𝑡 , 𝑢 ( 𝑡 ) ) = 0 , 𝑡 ∈ ( 0 , 1 ) , 0 < 𝑡 < 1 . 𝑢 ( 1 ) = 𝛽 𝑢 ( 𝜂 ) + 𝜆 2 , 𝑢 ( 0 ) = 𝛼 𝑢 ( 𝜂 ) − 𝜆 1 , 𝑢 ( 0 ) = 0 , 𝑢 ( 0 ) = 0 ⋯ 𝑢 ( 𝑛 − 1 ) ( 0 ) = 0 , where, 𝑛 − 1 < 𝛿 < 𝑛 , 𝑛 ( ≥ 3 ) ∈ ℕ , 0 < 𝜂 , 𝛼 , 𝛽 < 1 , the boundary parameters 𝜆 1 , 𝜆 2 ∈ ℝ + and 𝑐 𝐷 𝛿 0 + is the Caputo fractional derivative. We use the classical tools from functional analysis to obtain sufficient conditions for the existence and uniqueness of positive solutions to the boundary value problems. We also obtain conditions for the nonexistence of positive solutions to the problem. We include examples to show the applicability of our results.