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Zbl 1199.39022
Andruch-Sobi{ł}o, Anna; Drozdowicz, Andrzej
Asymptotic behaviour of solutions of third order nonlinear difference equations of neutral type.
(English)
[J] Math. Bohem. 133, No. 3, 247-258 (2008). ISSN 0862-7959

Summary: 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 $p,q\:\Bbb N(n_0)\rightarrow \Bbb R_+$; $\sigma , \tau \:\Bbb N\rightarrow \Bbb Z$, $\sigma$ is strictly increasing and $\lim \limits _{n \rightarrow \infty }\sigma (n)=\infty$; $\tau$ is nondecreasing and $\lim \limits _{n \rightarrow \infty }\tau (n)=\infty$, $f\:\Bbb R\rightarrow {\Bbb R}$, $xf(x)>0$. 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 the literature.
MSC 2000:
*39A22
39A10 Difference equations
39A12 Discrete version of topics in analysis
34K40 Neutral equations
39A21

Keywords: neutral type difference equation; third order difference equation; nonoscillatory solutions; asymptotic behavior

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