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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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##### References:
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