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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.

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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