Ladas, G.; Philos, Ch. G.; Sficas, Y. G. Sharp conditions for the oscillation of delay difference equations. (English) Zbl 0685.39004 J. Appl. Math. Simulation 2, No. 2, 101-112 (1989). 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 Cited in 4 ReviewsCited in 101 Documents MSC: 39A12 Discrete version of topics in analysis 39A10 Additive difference equations Keywords:delay difference equations; oscillatory behavior; oscillation; positive solution Citations:Zbl 0557.39004 PDFBibTeX XMLCite \textit{G. Ladas} et al., J. Appl. Math. Simulation 2, No. 2, 101--112 (1989; Zbl 0685.39004) Full Text: DOI