Asymptotic behaviour of solutions of third order nonlinear difference equations of neutral type

Abstract: In the paper we consider the difference equation of neutral type
\label{E} \Delta^{\!3}[x(n)-p(n)x(\sigma(n))] + q(n)f(x(\tau(n)))=0, \quad n \in\Bbb N (n_0), \tag{E}
where $p,q \Bbb N(n_0)\rightarrow\Bbb R_+$; $\sigma, \tau \Bbb N\rightarrow\Bbb Z$, $\sigma$ is strictly increasing and $\lim\limits_{n \rightarrow\infty}\sigma(n)=\infty;$ $\tau$ is nondecreasing and $\lim\limits_{n \rightarrow\infty}\tau(n)=\infty$, $f \Bbb R\rightarrow{\Bbb R}$, $xf(x)>0$. 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.

Keywords: neutral type difference equation, third order difference equation, nonoscillatory solutions, asymptotic behavior

Classification (MSC2000): 39A10

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