Conditions for oscillation and convergence of solutions to second order neutral delay difference equations with variable coefficients

Abstract

In this paper, we deals with the second order neutral functional difference equation of the form
$$
\Delta(r(n) \Delta(x(n)-p(n) x(n-\tau)))+q(n) f(x(n-\sigma))=0 ; \quad n \geq n_0
$$
where $\{r(n)\},\{p(n)\}$ and $\{q(n)\}$ are sequences of real numbers, $\tau$ and $\sigma$ are positive integers and $f: R \rightarrow R$ is a real valued function. We determine sufficient conditions under which every solutions of $(*)$ is either oscillatory or tends to zero.