Existence of Non-Oscillatory Solutions of Second-Order Neutral Delay Difference Equations

Abstract

In this paper, we consider the second-order neutral delay difference equation with positive and negative coefficients $$ \Delta r\_n \Delta (x\_n + cx\_{n–k} + p\_{n+1}x\_{n+1–l} = 0 $$ where $c \in \mathbb R, k ≥ 1$ and $m, l ≥ 0$ are integers, $\lbrace r\_n\rbrace^\infty \_{n=n0}, \lbrace p\_n\rbrace ^\infty \_{n=n0}$ and $\lbrace q\_n \rbrace ^\infty \_{n=n0}$ are sequences of non-negative real numbers. We obtain global results (with respect to $c$) which are some sufficient conditions for the existences of non-oscillatory solutions.