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

Mathematica Bohemica (2008)

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

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## Abstract

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In the paper we consider the difference equation of neutral type ${\Delta }^{3}\left[x\left(n\right)-p\left(n\right)x\left(\sigma \left(n\right)\right)\right]+q\left(n\right)f\left(x\left(\tau \left(n\right)\right)\right)=0,\phantom{\rule{1.0em}{0ex}}n\in ℕ\left({n}_{0}\right),$ where $p,q:ℕ\left({n}_{0}\right)\to {ℝ}_{+}$; $\sigma ,\tau :ℕ\to ℤ$, $\sigma$ is strictly increasing and $\underset{n\to \infty }{lim}\sigma \left(n\right)=\infty ;$$\tau$ is nondecreasing and $\underset{n\to \infty }{lim}\tau \left(n\right)=\infty$, $f:ℝ\to ℝ$, $xf\left(x\right)>0$. We examine the following two cases: $0 and $1<{\lambda }_{*}\le p\left(n\right),\phantom{\rule{1.0em}{0ex}}\sigma \left(n\right)=n+k,\phantom{\rule{1.0em}{0ex}}\tau \left(n\right)=n+l,$ where $k$, $l$ are positive integers. We obtain sufficient conditions under which all nonoscillatory solutions of the above equation tend to zero as $n\to \infty$ with a weaker assumption on $q$ than the usual assumption $\sum _{i={n}_{0}}^{\infty }q\left(i\right)=\infty$ that is used in literature.

## How to cite

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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 -

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