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Zbl 0984.39006
Luo, J.W.; Bainov, D.D.
Oscillatory and asymptotic behavior of second-order neutral difference equations with maxima. (English)
[J] J. Comput. Appl. Math. 131, No.1-2, 333-341 (2001). ISSN 0377-0427

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. \par Suppose that $q_n\ne 0$, $\sum^\infty q_n=\infty$ and $p_1\leq p_n\leq p_2\leq -1$. Then every bounded nonoscillatory solution $x_n$ of (*) satisfies $\lim_{n\to \infty} x_n=0$. \par 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.
[Ondřej Došlý (Brno)]

MSC 2000:: *39A11 Stability of difference equations

Keywords: neutral second order difference equation with maxima; asymptotic behavior; oscillation

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