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


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