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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 $\Delta ^{\!3}[x(n)-p(n)x(\sigma (n))] + q(n)f(x(\tau (n)))=0, \quad n \in \mathbb N (n_0),$ where $$p,q\:\mathbb N(n_0)\rightarrow \mathbb R_+$$; $$\sigma , \tau \:\mathbb N\rightarrow \mathbb Z$$, $$\sigma$$ is strictly increasing and $$\lim \limits _{n \rightarrow \infty }\sigma (n)=\infty$$; $$\tau$$ is nondecreasing and $$\lim \limits _{n \rightarrow \infty }\tau (n)=\infty$$, $$f\:\mathbb R\rightarrow {\mathbb R}$$, $$xf(x)>0$$. 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:
 39A22 Growth, boundedness, comparison of solutions to difference equations 39A10 Additive difference equations 39A12 Discrete version of topics in analysis 34K40 Neutral functional-differential equations 39A21 Oscillation theory for difference equations
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