Nonexistence and existence of positive solutions for difference equations with unbounded delay (Q1806560)

The paper deals with the delay difference equation with unbounded delay \[ x_{n+1}- x_n+ p_n x_{\tau(n)},=0\quad n = 0, 1,\dots,\tag{1} \] where \(p_n \in R\), \(\lim_{n \to \infty} \tau(n) = \infty\), \(\tau(n) < n\) for \(n \in N\), and \(\tau(n) \geq 0\) for \(n \geq \min \{m: \tau(m) \geq 0 \}\). The authors present sufficient conditions under which all solutions of (1) oscillate and sufficient conditions for the existence of positive solutions of (1).