Asymptotic behavior in neutral difference equations with several retarded arguments

Abstract

In this paper, we study the asymptotic behavior of the solutions of a neutral type difference equation of the form% \[ \Delta \bigg[ x(n)+\sum_{j=1}^{w}c_{j}x(\tau _{j}(n))\bigg] +p(n)x(\sigma (n))=0,\quad n\geq 0 \] where $\tau _{j}(n)$, $j=1,\ldots,w$, are general retarded arguments, $\sigma (n) $ is a general deviated argument (retarded or advanced), $c_{j}\in \mathbb{R}$, $j=1,\ldots,w$, $(p(n))_{n\geq 0}$ is a sequence of positive real numbers such that $p(n)\geq p$, $p\in\mathbb{R}_{+}$, and $\Delta $ denotes the forward difference operator $\Delta x(n)=x(n+1)-x(n)$.

We also examine the convergence of the solutions when these are continuous and differentiable with respect to $c_{j}$, $j=1,\ldots,w$.