# Article

Keywords:
neutral type difference equation; third order difference equation; nonoscillatory solutions; asymptotic behavior
Summary:
In the paper we consider the difference equation of neutral type $$\Delta ^{3}[x(n)-p(n)x(\sigma (n))] + q(n)f(x(\tau (n)))=0, \quad n \in \Bbb N (n_0),$$ where $p,q\colon\Bbb N(n_0)\rightarrow \Bbb R_+$; $\sigma , \tau \colon\Bbb N\rightarrow \Bbb Z$, $\sigma$ is strictly increasing and $\lim \limits _{n \rightarrow \infty }\sigma (n)=\infty ;$ $\tau$ is nondecreasing and $\lim \limits _{n \rightarrow \infty }\tau (n)=\infty$, $f\colon\Bbb R\rightarrow {\Bbb R}$, $xf(x)>0$. We examine the following two cases: $0<p(n)\leq \lambda ^*< 1,\quad \sigma (n)=n-k,\quad \tau (n)=n-l,$ and $1<\lambda _*\leq p(n),\quad \sigma (n)=n+k,\quad \tau (n)=n+l,$ where $k$, $l$ are positive integers. We obtain sufficient conditions under which all nonoscillatory solutions of the above equation tend to zero as $n\rightarrow \infty$ with a weaker assumption on $q$ than the usual assumption $\sum \limits _{i=n_0}^{\infty }q(i)=\infty$ that is used in literature.
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