Characterization of oscillation of second order nonlinear difference equations. (English) Zbl 0598.39004

The second order nonlinear difference equation \(\Delta (r_{n-1}\Delta y_{n-1})+f(n,y_ n)=0\) is considered, where \(\Delta\) is the forward difference operator and yf(n,y) is assumed to be \(>0\) for \(y\neq 0\). Similar to differential equations the difference equation is classified as superlinear, strongly superlinear, sublinear and strongly sublinear, respectively.
For strongly superlinear and strongly sublinear equations necessary and sufficient conditions are derived that all solutions are oscillatory, and for superlinear and sublinear equations conditions for the existence of non-oscillatory solutions of a special kind are established. The obtained results are similar to those known from second order differential equations.
Reviewer: D.Dorninger


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