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

Anna Andruch-Sobiło; Andrzej Drozdowicz

Mathematica Bohemica (2008)

- Volume: 133, Issue: 3, page 247-258
- ISSN: 0862-7959

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topAndruch-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 -

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