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

Andruch-Sobiło, Anna, and Drozdowicz, Andrzej. "Asymptotic behaviour of solutions of third order nonlinear difference equations of neutral type." Mathematica Bohemica 133.3 (2008): 247-258. <http://eudml.org/doc/250534>.

@article{Andruch2008,
abstract = {In the paper we consider the difference equation of neutral type \[ \Delta ^\{3\}[x(n)-p(n)x(\sigma (n))] + q(n)f(x(\tau (n)))=0, \quad n \in \mathbb \{N\} (n\_0), \] where $p,q\colon \mathbb \{N\}(n_0)\rightarrow \mathbb \{R\}_+$; $\sigma , \tau \colon \mathbb \{N\}\rightarrow \mathbb \{Z\}$, $\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 $, $f\colon \mathbb \{R\}\rightarrow \{\mathbb \{R\}\}$, $xf(x)>0$. We examine the following two cases: \[ 0<p(n)\le \lambda ^*< 1,\quad \sigma (n)=n-k,\quad \tau (n)=n-l, \] and \[1<\lambda \_*\le p(n),\quad \sigma (n)=n+k,\quad \tau (n)=n+l,\] 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.},
author = {Andruch-Sobiło, Anna, Drozdowicz, Andrzej},
journal = {Mathematica Bohemica},
keywords = {neutral type difference equation; third order difference equation; nonoscillatory solutions; asymptotic behavior; neutral type difference equation; third order difference equation; nonoscillatory solutions; asymptotic behavior},
language = {eng},
number = {3},
pages = {247-258},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Asymptotic behaviour of solutions of third order nonlinear difference equations of neutral type},
url = {http://eudml.org/doc/250534},
volume = {133},
year = {2008},
}

TY - JOUR
AU - Andruch-Sobiło, Anna
AU - Drozdowicz, Andrzej
TI - Asymptotic behaviour of solutions of third order nonlinear difference equations of neutral type
JO - Mathematica Bohemica
PY - 2008
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 133
IS - 3
SP - 247
EP - 258
AB - In the paper we consider the difference equation of neutral type \[ \Delta ^{3}[x(n)-p(n)x(\sigma (n))] + q(n)f(x(\tau (n)))=0, \quad n \in \mathbb {N} (n_0), \] where $p,q\colon \mathbb {N}(n_0)\rightarrow \mathbb {R}_+$; $\sigma , \tau \colon \mathbb {N}\rightarrow \mathbb {Z}$, $\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 $, $f\colon \mathbb {R}\rightarrow {\mathbb {R}}$, $xf(x)>0$. We examine the following two cases: \[ 0<p(n)\le \lambda ^*< 1,\quad \sigma (n)=n-k,\quad \tau (n)=n-l, \] and \[1<\lambda _*\le p(n),\quad \sigma (n)=n+k,\quad \tau (n)=n+l,\] 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.
LA - eng
KW - neutral type difference equation; third order difference equation; nonoscillatory solutions; asymptotic behavior; neutral type difference equation; third order difference equation; nonoscillatory solutions; asymptotic behavior
UR - http://eudml.org/doc/250534
ER -