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Title: Asymptotic behaviour of solutions of third order nonlinear difference equations of neutral type (English)
Author: Andruch-Sobiło, Anna
Author: Drozdowicz, Andrzej
Language: English
Journal: Mathematica Bohemica
ISSN: 0862-7959 (print)
ISSN: 2464-7136 (online)
Volume: 133
Issue: 3
Year: 2008
Pages: 247-258
Summary lang: English
Category: math
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. (English)
Keyword: neutral type difference equation
Keyword: third order difference equation
Keyword: nonoscillatory solutions
Keyword: asymptotic behavior
MSC: 34K40
MSC: 39A10
MSC: 39A12
MSC: 39A21
MSC: 39A22
idZBL: Zbl 1199.39022
idMR: MR2494779
DOI: 10.21136/MB.2008.140615
Date available: 2010-07-20T17:28:03Z
Last updated: 2020-07-29
Stable URL:
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