Abstract and Applied Analysis
Volume 2013 (2013), Article ID 245012, 12 pages
http://dx.doi.org/10.1155/2013/245012
Research Article

Unbounded Positive Solutions and Mann Iterative Schemes of a Second-Order Nonlinear Neutral Delay Difference Equation

Zeqing Liu,1 Xiaochuan Hou,2 Jeong Sheok Ume,3 and Shin Min Kang4

1Department of Mathematics, Liaoning Normal University, Dalian, Liaoning 116029, China
2Fushun No. 10 High School, Fushun, Liaoning 113004, China
3Department of Mathematics, Changwon National University, Changwon 641-773, Republic of Korea
4Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea

Received 12 May 2012; Accepted 12 December 2012

Academic Editor: Alexander I. Domoshnitsky

Copyright © 2013 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 is concerned with solvability of the second-order nonlinear neutral delay difference equation . Utilizing the Banach fixed point theorem and some new techniques, we show the existence of uncountably many unbounded positive solutions for the difference equation, suggest several Mann-type iterative schemes with errors, and discuss the error estimates between the unbounded positive solutions and the sequences generated by the Mann iterative schemes. Four nontrivial examples are given to illustrate the results presented in this paper.