Convergence and Divergence of the Solutions of a Neutral Difference Equation

Abstract

We investigate the asymptotic behavior of the solutions of a neutral type difference equation of the form $\Delta [x\left(n\right)+cx\left(\tau \right(n\left)\right)]+p\left(n\right)x\left(\sigma \right(n\left)\right)=0$, where $\tau \left(n\right)$ is a general retarded argument, $\sigma \left(n\right)$ is a general deviated argument (retarded or advanced), $c\in R$, ${\left(p\right(n\left)\right)}_{n\ge 0}$ is a sequence of positive real numbers such that $p\left(n\right)\ge p$, $p\in R_{+}$, and $\Delta$ denotes the forward difference operator $\Delta x\left(n\right)=x(n+1)-x\left(n\right)$. Also, we examine the asymptotic behavior of the solutions in case they are continuous and differentiable with respect to cc.