| 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:
|
http://hdl.handle.net/10338.dmlcz/140615 |
| . |
| Reference:
|
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| Reference:
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| Reference:
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