Abstract and Applied Analysis
Volume 2011 (2011), Article ID 328914, 24 pages
http://dx.doi.org/10.1155/2011/328914
Research Article

Solvability of a Second Order Nonlinear Neutral Delay Difference Equation

Zeqing Liu,1 Liangshi Zhao,1 Jeong Sheok Ume,2 and Shin Min Kang3,4

1Department of Mathematics, Liaoning Normal University, Dalian, Liaoning 116029, China
2Department of Mathematics, Changwon National University, Changwon 641-773, Republic of Korea
3Department of Mathematics, Gyeongsang National University, Jinju 660-701, Republic of Korea
4Research Institute of Natural Science, Gyeongsang National University, Jinju 660-701, Republic of Korea

Received 2 August 2011; Accepted 9 October 2011

Academic Editor: Sergey V. Zelik

Copyright © 2011 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

This paper studies the second-order nonlinear neutral delay difference equation Δ [ 𝑎 𝑛 Δ ( 𝑥 𝑛 + 𝑏 𝑛 𝑥 𝑛 − 𝜏 ) + 𝑓 ( 𝑛 , 𝑥 𝑓 1 𝑛 , … , 𝑥 𝑓 𝑘 𝑛 ) ] + 𝑔 ( 𝑛 , 𝑥 𝑔 1 𝑛 , … , 𝑥 𝑔 𝑘 𝑛 ) = 𝑐 𝑛 , 𝑛 ≥ 𝑛 0 . By means of the Krasnoselskii and Schauder fixed point theorem and some new techniques, we get the existence results of uncountably many bounded nonoscillatory, positive, and negative solutions for the equation, respectively. Ten examples are given to illustrate the results presented in this paper.