On Oscillation of Solutions to Second Order Neutral Difference Equations

Abstract

In this paper, we consider a class of second order neutral difference equation of the form \begin{equation}\Delta\left(r(n)\Delta(x(n)-p(n)x(n+\tau))\right)+q(n)x(n+\sigma)=0,\quad n\geq n_0\tag{$\ast$}\end{equation} where $r(n)$ is a sequence of positive real numbers, $\left\{p(n)\right\}$ and $\left\{q(n)\right\}$ are sequence of nonnegative real numbers, and $\tau$ and $\sigma$ are integers. We discuss the oscillatory properties of the equation $(*)$ relating oscillation of these equations to existence of positive solutions to associated first order neutral inequalities.