Some oscillation theorems for second order nonlinear neutral type difference equations

Abstract

In this paper some new sufficient conditions for the oscillatory behavior of second order nonlinear neutral type difference equation of the form
$$\Delta\left(a_n \Delta\left(x_n+p_n x_{n-k}\right)\right)+q_n f\left(x_{\sigma(n+1)}\right)=0$$
where $\left\{a_n\right\},\left\{p_n\right\}$ and $\left\{q_n\right\}$ are real sequences, $\{\sigma(n)\}$ is a sequence of integers, $k$ is a positive integer and $f: \mathbb{R} \rightarrow \mathbb{R}$ is continuous with $u f(u)>0$ for $u \neq 0$ are established. Examples are provided to illustrate the main results.