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Oscillatory and asymptotic behavior of second-order neutral difference equations with maxima. (English) Zbl 0984.39006

The authors investigate asymptotic and oscillatory properties of solutions of the neutral second order difference equation with maxima

${{\Delta }}^{2}\left({x}_{n}+{p}_{n}{x}_{n-k}\right)+{q}_{n}\underset{s\in \left[n-l,n\right]}{max}{x}_{s}=0,\phantom{\rule{2.em}{0ex}}\left(*\right)$

where $k,l$ are nonnegative integers and $\left[n-l,n\right]=\left\{n-l,n-l+1,\cdots ,n\right\}$, under some restrictions on the sequences $p,q$. A typical result is the following statement.

Suppose that ${q}_{n}\ne 0$, ${\sum }^{\infty }{q}_{n}=\infty$ and ${p}_{1}\le {p}_{n}\le {p}_{2}\le -1$. Then every bounded nonoscillatory solution ${x}_{n}$ of (*) satisfies ${lim}_{n\to \infty }{x}_{n}=0$.

Examples illustrating the general results of the paper are given. No comparison of the results and methods of the paper with those concerning the continuous counterpart of (*) ${\left(x\left(t\right)+p\left(t\right)x\left(t-\tau \right)\right)}^{\text{'}\text{'}}+q\left(t\right){max}_{s\in \left[t-\sigma ,t\right]}x\left(s\right)=0$ are presented.

MSC:
 39A11 Stability of difference equations (MSC2000)