Oscillations of difference equations with several deviated arguments

Abstract

We establish sufficient conditions for the oscillation of all solutions to the retarded difference equation $$\Delta x(n) + \sum_{i=1}^{m}p_{i}(n)x(\tau _{i}(n))=0, \quad n\geq 0, $$ and the (dual) advanced difference equation $$\nabla x(n)-\sum_{i=1}^{m}p_{i}(n)x(\sigma _{i}(n))=0, \quad n \geq 1,$$ where \({(p_{i}(n)), 1\leq i\leq m}\) are sequences of nonnegative real numbers, \({(\tau _{i}(n)), 1 \leq i\leq m}\) are sequences of integers such that $$\tau _{i}(n)\leq n - 1 \quad \forall n\geq 0, \quad \text{and} \quad \lim\limits_{n\rightarrow \infty }\tau _{i}(n)=\infty, \quad 1 \leq i \leq m,$$ \({(\sigma _{i}(n)), 1\leq i \leq m}\) are sequences of integers such that $$\sigma _{i}(n)\geq n + 1 \quad \forall n\geq 1, \quad 1\leq i \leq m,$$ \({\Delta }\) denotes the forward difference operator \({\Delta x(n) = x(n + 1) - x(n)}\) and \({\nabla}\) denotes the backward difference operator \({\nabla x(n) = x(n) - x(n - 1)}\). Examples illustrating the results are also given.