Sharp conditions for the oscillation of delay difference equations. (English) Zbl 0685.39004

The authors are concerned with the difference equation \((*)\quad y_{n+1}-y_ n+p_ ny_{n-k}=0,\) \(n=0,1,2,...,\) where \(p_ n\geq 0\) for \(n\geq 0,\) k a positive integer. A sufficient condition for the oscillation of all solutions of (*) is presented in the form \[ \liminf_{n\to \infty}[\frac{1}{k}\sum^{n-1}_{i=n-k}p_ i]>\frac{k^ k}{(k+1)^{k+1}}. \] In the third part of the paper a sufficient condition for the existence of a positive solution y of (*) such that \(\lim_{n\to \infty}y_ n=0\) is given. For a similar problem see e.g. the reviewer and B. Szmanda [Demonstr. Math. 17, 153-164 (1984; Zbl 0557.39004)].
Reviewer: J.Popenda


39A12 Discrete version of topics in analysis
39A10 Additive difference equations


Zbl 0557.39004
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