Electronic Journal of Differential Equations,
Vol. 2010(2010), No. 146, pp. 1-7.

Title: Existence of non-oscillatory solutions for a
       higher-order nonlinear neutral difference equation

Authors: Zhenyu Guo (Liaoning Shihua Univ., Fushun, China)
         Min Liu    (Liaoning Shihua Univ., Fushun, China)

Abstract:
 This  article concerns the solvability of the higher-order
 nonlinear neutral delay difference equation
 $$
 \Delta\Big(a_{kn}\dots\Delta\big(a_{2n}
 \Delta(a_{1n}\Delta(x_n+b_nx_{n-d}))\big)\Big)
 +\sum_{j=1}^s p_{jn}f_j(x_{n-r_{jn}})=q_n,
 $$
 where $n\geq n_0\ge0$, $d,k,j,s$ are positive integers,
 $f_j:\mathbb{R}\to \mathbb{R}$ and $xf_j(x)\geq 0$ for $x\ne 0$.
 Sufficient conditions for the existence of non-oscillatory solutions
 are established by using Krasnoselskii fixed point theorem.
 Five theorems are stated according to the range of
 the sequence $\{b_n\}$.

Submitted July 30, 2010. Published October 14, 2010.
Math Subject Classifications: 34K15, 34C10.
Key Words: Nonoscillatory solution; neutral difference equation;
           Krasnoselskii fixed point theorem.