Existence for nonoscillatory solutions of higher-order nonlinear neutral difference equations

In this paper, we consider the following forced higher-order nonlinear neutral difference equation Delta(III) (x(n) + c(n)x(n-k)) + Sigma(s=1) p(n)((s)) f(s)(x(n)-r(s)) = q(n), n greater than or equal to n(0), wherein, u greater than or equal to 1, k greater than or equal to 0, and r(s) greater than or equal to 0 are integers, {c(n)}, {p(n)((s))} (s = 1, 2,..., u) and {q(n)} are sequence of real numbers and f(s) is an element of C(R, R) (s = 1, 2,..., u). By using Krasnoselskii's fixed point theorem and some new techniques, we obtain sufficient conditions for the existence of nonoscillatory solutions for general {p(n)((s))} (s = 1, 2, . . . , u) and {q(n)} which means that we allow oscillatory {p(n)((s))} (s = 1, 2, . . . , n) {q(n)}. (C) 2003 Elsevier Science (USA). All rights reserved.