Oscillation criteria for difference equations with unbounded delay (Q1129479)

Consider the delay difference equation \[ x_{n+1}-x_n+p_nx_{\tau(n)}=0,\;n\in\mathbb{N}, \] \(p_n\geq 0\), where: (i) \(\tau: \mathbb{N}\mathbb{Z}\) is nondecreasing, (ii) \(\tau(n)<n\) for \(n\in \mathbb{N}\), (iii) \(\tau(n) \to\infty\) as \(n\to\infty\), (iv) there exists a monotone sequence \(\{n_k\}\) such that \(\tau\{n_k\} =n_{k-1}\), \(k=1,2, \dots, n_k\to \infty\) as \(n\to \infty\). The paper contains two criteria for oscillation of solution to the equation (*). Two illustrative examples are included.