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Asymptotic behaviour of solutions of third order nonlinear difference equations of neutral type. (English) Zbl 1199.39022
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.

39A22Growth, boundedness, comparison of solutions (difference equations)
39A10Additive difference equations
39A12Discrete version of topics in analysis
34K40Neutral functional-differential equations
39A21Oscillation theory (difference equations)
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