Journal of Applied Mathematics
Volume 2012 (2012), Article ID 569313, 15 pages
http://dx.doi.org/10.1155/2012/569313
Research Article

Positive Solutions to a Generalized Second-Order Difference Equation with Summation Boundary Value Problem

Thanin Sitthiwirattham1,2 and Jessada Tariboon1

1Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok 10800, Thailand
2Centre of Excellence in Mathematics, CHE, Sri Ayutthaya Road, Bangkok 10400, Thailand

Received 1 December 2011; Accepted 22 February 2012

Academic Editor: Yansheng Liu

Copyright © 2012 Thanin Sitthiwirattham and Jessada Tariboon. 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

By using Krasnoselskii's fixed point theorem, we study the existence of positive solutions to the three-point summation boundary value problem Ξ” 2 𝑒 ( 𝑑 βˆ’ 1 ) + π‘Ž ( 𝑑 ) 𝑓 ( 𝑒 ( 𝑑 ) ) = 0 , 𝑑 ∈ { 1 , 2 , … , 𝑇 } , βˆ‘ 𝑒 ( 0 ) = 𝛽 πœ‚ 𝑠 = 1 𝑒 ( 𝑠 ) , βˆ‘ 𝑒 ( 𝑇 + 1 ) = 𝛼 πœ‚ 𝑠 = 1 𝑒 ( 𝑠 ) , where 𝑓 is continuous, 𝑇 β‰₯ 3 is a fixed positive integer, πœ‚ ∈ { 1 , 2 , . . . , 𝑇 βˆ’ 1 } , 0 < 𝛼 < ( 2 𝑇 + 2 ) / πœ‚ ( πœ‚ + 1 ) , 0 < 𝛽 < ( 2 𝑇 + 2 βˆ’ 𝛼 πœ‚ ( πœ‚ + 1 ) ) / πœ‚ ( 2 𝑇 βˆ’ πœ‚ + 1 ) , and Ξ” 𝑒 ( 𝑑 βˆ’ 1 ) = 𝑒 ( 𝑑 ) βˆ’ 𝑒 ( 𝑑 βˆ’ 1 ) . We show the existence of at least one positive solution if 𝑓 is either superlinear or sublinear.