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 Δm(xn + cnxn−k )+ u s=1 p (s) n fs (xn−rs ) = qn, n n0, where m,u 1, k 0, and rs 0 are integers, {cn}, {p (s) n } (s = 1, 2, . . . ,u) and {qn} are sequence of real numbers and fs ∈ 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 (s) n } (s = 1, 2, . . . ,u) and {qn} which means that we allow oscillatory {p (s) n } (s = 1, 2, . . . ,u) and {qn}.  2003 Elsevier Science (USA). All rights reserved.