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

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.


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