Advances in Difference Equations
Volume 2010 (2010), Article ID 767620, 14 pages
Research Article

Solvability of a Higher-Order Nonlinear Neutral Delay Difference Equation

School of Sciences, Liaoning Shihua University, Fushun, Liaoning 113001, China

Received 19 March 2010; Revised 10 July 2010; Accepted 5 September 2010

Academic Editor: S. Grace

Copyright © 2010 Min Liu and Zhenyu Guo. 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.


The existence of bounded nonoscillatory solutions of a higher-order nonlinear neutral delay difference equation Δ(aknΔ(a2nΔ(a1nΔ(xn+bnxn-d))))+f(n,xn-r1n,xn-r2n,,xn-rsn)=0, nn0, where n00, d>0, k>0, and s>0 are integers, {ain}nn0  (i=1,2,,k) and {bn}nn0 are real sequences, j=1s{rjn}nn0, and f:{n:nn0}×s is a mapping, is studied. Some sufficient conditions for the existence of bounded nonoscillatory solutions of this equation are established by using Schauder fixed point theorem and Krasnoselskii fixed point theorem and expatiated through seven theorems according to the range of value of the sequence {bn}nn0. Moreover, these sufficient conditions guarantee that this equation has not only one bounded nonoscillatory solution but also uncountably many bounded nonoscillatory solutions.