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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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