Oscillation and Nonoscillation of Difference Equations with Several Delays

Consider the delay difference equation $$\begin{aligned} \Delta {}x(n)+\sum _{k=1}^{m}p_{k}(n)x(n-\tau _{k})=0 \quad \text {for}\ n=0,1,\ldots , \end{aligned}$$ where $$\Delta $$ is the forward difference operator, i.e., $$\Delta {}x(n):=x(n+1)-x(n)$$ , $$\tau _{k}$$ is a nonnegative integer and $$\{p_{k}(n)\}_{n=0}^{\infty }$$ is a nonnegative sequence of reals for $$k=1,2,\ldots ,m$$ . New oscillation and nonoscillation results, which essentially improve known results in the literature, are established. These results are extended to the more general difference equation $$\begin{aligned} \Delta {}x(n)+\sum _{k=1}^{m}p_{k}(n)x(\sigma _{k}(n))=0 \quad \text {for}\ n=0,1,\ldots . \end{aligned}$$ Examples illustrating the significance of the results are given.