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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 \[ \Delta ^2(x_n+p_nx_{n-k})+q_n\max_{s\in[n-l,n]}x_s=0, \tag{*} \] where \(k,l\) are nonnegative integers and \([n-l,n]=\{n-l,n-l+1,\dots,n\}\), under some restrictions on the sequences \(p,q\). A typical result is the following statement.
Suppose that \(q_n\neq 0\), \(\sum^\infty q_n=\infty\) and \(p_1\leq p_n\leq p_2\leq -1\). 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 (*) \((x(t)+p(t)x(t-\tau))''+q(t)\max_{s\in [t-\sigma,t]} x(s)=0\) are presented.

MSC:
39A11 Stability of difference equations (MSC2000)
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