Abstract and Applied Analysis
Volume 2012 (2012), Article ID 707631, 15 pages
http://dx.doi.org/10.1155/2012/707631
Review Article

The Existence of Solutions to a System of Discrete Fractional Boundary Value Problems

Yuanyuan Pan, Zhenlai Han, Shurong Sun, and Yige Zhao

School of Mathematical Sciences, University of Jinan, Jinan, Shandong 250022, China

Received 7 December 2011; Revised 26 December 2011; Accepted 1 January 2012

Academic Editor: Ferhan M. Atici

Copyright © 2012 Yuanyuan Pan 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 study the existence of solutions for the boundary value problem − Δ 𝜈 𝑦 1 ( 𝑡 ) = 𝑓 ( 𝑦 1 ( 𝑡 + 𝜈 − 1 ) , 𝑦 2 ( 𝑡 + 𝜇 − 1 ) ) , − Δ 𝜇 𝑦 2 ( 𝑡 ) = 𝑔 ( 𝑦 1 ( 𝑡 + 𝜈 − 1 ) , 𝑦 2 ( 𝑡 + 𝜇 − 1 ) ) , 𝑦 1 ( 𝜈 − 2 ) = Δ 𝑦 1 ( 𝜈 + 𝑏 ) = 0 , 𝑦 2 ( 𝜇 − 2 ) = Δ 𝑦 2 ( 𝜇 + 𝑏 ) = 0 , where 1 < 𝜇 , 𝜈 ≤ 2 , 𝑓 , 𝑔 ∶ ℝ × ℝ → ℝ are continuous functions, 𝑏 ∈ ℕ 0 . The existence of solutions to this problem is established by the Guo-Krasnosel'kii theorem and the Schauder fixed-point theorem, and some examples are given to illustrate the main results.