Convergence of the solutions for a neutral difference equation with negative coefficients

Abstract

Inthis paper, we investigate the asymptotic behavior of the solutions of a
neutral type difference equation of the form

\begin{equation*}
\Delta \left[ x(n)+\sum_{j=1}^{w}c_{j}x(\tau _{j}(n))\right] +\left(
-p(n)\right) x(\sigma (n))=0\text{, \ \ \ }n\geq 0
\end{equation*}

where $\tau _{j}(n),$ $j=1,...,w$ are general retarded arguments, $\sigma
(n) $ is a general deviated argument, $c_{j}\in
\mathbb{R}
$, $j=1,...,w$ , $\left( p(n)\right) _{n\geq 0}$ is a sequence of positive
real numbers such that $p(n)\geq p$, $p\in
\mathbb{R}
_{+}$, and $\Delta $ denotes the forward difference operator $\Delta
x(n)=x(n+1)-x(n)$.