Abstract and Applied Analysis
Volume 2012 (2012), Article ID 172939, 30 pages
http://dx.doi.org/10.1155/2012/172939
Research Article

Positive Solutions of a Second-Order Nonlinear Neutral Delay Difference Equation

Zeqing Liu,1 Wei Sun,1 Jeong Sheok Ume,2 and Shin Min Kang3

1Department of Mathematics, Liaoning Normal University, Liaoning, Dalian 116029, China
2Department of Mathematics, Changwon National University, Changwon 641-773, Republic of Korea
3Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea

Received 15 August 2012; Accepted 6 November 2012

Academic Editor: Norio Yoshida

Copyright © 2012 Zeqing Liu 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

The purpose of this paper is to study solvability of the second-order nonlinear neutral delay difference equation . By making use of the Rothe fixed point theorem, Leray-Schauder nonlinear alternative theorem, Krasnoselskill fixed point theorem, and some new techniques, we obtain some sufficient conditions which ensure the existence of uncountably many bounded positive solutions for the above equation. Five nontrivial examples are given to illustrate that the results presented in this paper are more effective than the existing ones in the literature.