On the Oscillatory Behaviour of Solutions of Second Order Nonlinear Difference Equations

Abstract

By using simple discrete inqualities sufficient conditions are provided for the solution {$y_n$} of a difference equation of the form $\Delta (a_n\Delta y_n)+q_{n+1}f(y_{n+1}=r_n(n\in {\mathbb{N}}_0;a_n,q_n,r_{n}C\mathbb{R};f:\mathbb{R}\to \mathbb{R})$ to be oscillatory or to satisfy lim inf${}_{n\to \infty }|y_n|=0$. Also two other results are established for all solutions of this equation to be oscillatory when $r_n=0$ for all $n\in {\mathbb{N}}_0$.