## Oscillation criteria for second order nonlinear difference equations.(English)Zbl 0742.39003

Consider the nonlinear difference equation (1) $$\Delta^ 2x_ n+q_ nf(x_{n-\tau_ n})=0$$, $$n=0,1,2,\dots$$ where $$\Delta x_ n=x_{n+1}- x_ n$$, $$\{q_ n\}$$ is a sequence of nonnegative numbers, $$\tau_ n\in Z$$ with $$\lim_{n\to \infty}(n-\tau_ n)=\infty$$ and $$f: R\to R$$ is continuous with $$uf(u)>0$$ $$(u\neq 0)$$. The aim in this paper is to establish the following two theorems: Theorem 1. Assume that $$q_ n\geq 0$$ eventually with $$\sum^ \infty_{n=0}q_ n=\infty$$, and that $$\lim\inf_{| u| \to \infty}| f(u)| > 0$$. Then every solution of (1) oscillates. Theorem 2. Assume that $$n-\tau_ n$$ is nondecreasing, $$\tau_ n\in N=\{0,1,2,\dots\}$$, $$\sum^ \infty_{n=0} q_ n=\infty$$, and that $$f$$ is nondecreasing and there is a nonnegative constant $$M$$ such that $$\lim\sup_{u\to 0} u/f(u)=M$$. Then the difference $$\{\Delta x_ n\}$$ of every solution $$\{x_ n\}$$ of Eq. (1) oscillates.
Reviewer: W.Zhicheng

### MSC:

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