An oscillation criterion for linear difference equations with general delay argument (Q1049757)

Consider the delay difference equation \[ x(n+1)-x(n)+p(n)x(\tau (n))=0,\tag{\(*\)} \] where \(\{p(n)\}_{n\geq 0}\) is a sequence of integers such that \(\tau (n)\leq n-1\) for all \(n\geq 0\) and \(\lim_{n\to \infty}\tau (n)=\infty\). The authors establish the following sufficient condition for the oscillation of all solutions of (\(*\)): Theorem. Assume that the sequence \(\{\tau (n)\}_{n\geq 0}\) is increasing, \(0<\alpha \leq -1+\sqrt{2}\), where \(\alpha =\lim \inf_{n\to \infty}\sum_{j=\tau (n)}^{n-1}p(j)\). If \(\lim \sup_{n\to \infty}\sum_{j=\tau (n)}^{n}p(j)>1-\frac{1}{2}(1-\alpha -\sqrt{1-2\alpha -\alpha ^{2}})\), then all solutions of (\(*\)) are oscillatory.