Asymptotic Behavior of HigherOrder Neutral Advanced Difference Equations

Abstract

In this paper, we study the asymptotic behavior of neutral advanced difference equations of the form \begin{equation*} \Delta ^{m}\left[ x(n)+cx(n+a)\right] +p(n)x(\sigma (n))=0,\qquad m\in \mathbb{N}, \qquad n\geq 0\text{,} \end{equation*} where $c\in \mathbb{R} $, $ \mathbb{N} \ni a\geq 2$, $(\sigma (n))$ is a sequence of positive integers such that $% (\sigma (n))\geq n+2$\ for all $n\geq 0$, $\left( p(n)\right) _{n\geq 0}$ is a sequence of real numbers, $\Delta $ denotes the forward difference operator $\Delta x(n)=x(n+1)x(n)$, and $\Delta ^{j}$ denotes the $j^{th}$ forward difference operator $\Delta ^{j}\left( x(n)\right) =\Delta \left( \Delta ^{j1}\left( x(n)\right) \right) $ for $j=2,3,...,m$. Examples illustrating the results are also given.