Oscillation of discrete analogues of delay equations. (English) Zbl 0723.39004

Erbe, L. H.; Zhang, B. G.

Differ. Integral Equ. 2, No. 3, 300-309 (1989).

The authors are concerned with the oscillation and nonoscillation of the solutions of first order linear and nonlinear difference equations with delay of the form \(y_{n+1}-y_ n+p_ ny_{n-m}=0,\) \(y_{n+1}-y_ n+p_ nf(y_{n-m})=0,\) \(y_{n+1}-y_ n+p_ n(1+y_ n)y_{n-m}=0,\) \(y_{n+1}-y_ n+p_ ny_{n-m}=f_ n\) and \(y_{n+1}-y_ n+\sum^{k}_{i=1}p_{in}y_{n-m_ i}=0,\) where \(n=1,2,...\), and m is a positive integer. Oscillation and nonoscillation criteria are established. Here a nontrivial solution is said to be oscillatory if for every \(N>0\) there exists an \(n\geq N\) such that \(y_ ny_{n+1}\leq 0\). Otherwise it is nonoscillatory.

Reviewer: Cui Baotong (Binzhou)

MSC:

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

Keywords:

oscillation; nonoscillation; first order linear and nonlinear difference equations with delay