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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

Δ 2 (x n +p n x n-k )+q n max s[n-l,n] x s =0,(*)

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

Suppose that q n 0, q n = and p 1 p n p 2 -1. Then every bounded nonoscillatory solution x n of (*) satisfies lim n 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-τ)) '' +q(t)max s[t-σ,t] x(s)=0 are presented.

39A11Stability of difference equations (MSC2000)