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.

Abstract

The existence of bounded nonoscillatory solutions of a higher-order nonlinear neutral delay difference equation Δ(akn⋯Δ(a2nΔ(a1nΔ(xn+bnxn-d))))+f(n,xn-r1n,xn-r2n,…,xn-rsn)=0, n≥n0, where n0≥0, d>0, k>0, and s>0 are integers, {ain}n≥n0  (i=1,2,…,k) and {bn}n≥n0 are real sequences, ⋃j=1s{rjn}n≥n0⊆ℤ, and f:{n:n≥n0}×ℝs→ℝ 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}n≥n0. Moreover, these sufficient conditions guarantee that this equation has not only one bounded nonoscillatory solution but also uncountably many bounded nonoscillatory solutions.