Oscillation and nonoscillation for second-order linear difference equations
The authors study the oscillatory properties of the equation \[ \Delta^{2}x_{n-1}+p_{n}x_{n}=0, \quad n=1,2,\ldots, \tag{1} \] where \(\Delta\) is the forward difference, \(\Delta x_{n}=x_{n+1}-x_{n}\), \(\Delta^{2}x_{n}=\Delta(\Delta x_{n})\), \(\{p_{n}\}^{\infty}_{n=1}\) is a real nonnegative sequence. Let \(\alpha_{0}=3-2\sqrt{2}\) and \(\mathbb N\) be the set of natural numbers. The following sufficient oscillatory conditions are given: if there exists \(n_{0}\in\mathbb{N}\) such that for every \(m\in\mathbb{N}\) \[ \sum_{i=2^{m}n_{0}}^{2^{m+1}n_{0}-1}p_{i}\leq \frac{\alpha_{0}}{2^{m+1}n_{0}}, \] then equation (1) is nonoscillatory; if there exists \(n_{0}\in\mathbb{N}\) and \(\alpha>\alpha_{0}\) such that for every \(m\in\mathbb{N}\) \[ \sum_{i=2^{m}n_{0}}^{2^{m+1}n_{0}-1}p_{i}\geq \frac{\alpha}{2^{m}n_{0}}, \] then equation (1) is oscillatory.
second-order linear difference equations
scientific article; zbMATH DE number 1428542