Oscillatory Behavior of First Order Neutral Delay Difference Equations

Abstract

In this paper, we establish some sufficient conditions for the oscillation of all solutions of first order neutral difference equation of the form \begin{equation}\Delta[r(n)(x(n)+p x(n-\tau))]+q(n)x(n-\sigma)=0, \quad n\geq n_0;\tag{$\ast$}\end{equation} where $\left\{r(n)\right\}$, $\left\{q(n)\right\}$ are sequences of positive real numbers, $p$ is a real number, and $\tau$ and $\sigma$ are positive integers. The results proved improve and generalize some of existing results in the literature. Some examples are inserted to illustrate our results.