Nonoscillation and oscillation criteria for class of higher - order difference equations involving generalized difference operator

Abstract

In this paper, sufficient conditions are obtained for nonoscillation/oscillation of all solutions of a class of higher-order difference equations involving the generalized difference operator of the form$\Delta _{a}^{k}(p_{n}\Delta _{a}^{2}y_{n})=f(n,y_{n},\Delta_{a}y_{n},\Delta _{a}^{2}y_{n},...,\Delta _{a}^{k+1}y_{n}),$where $\Delta _{a}$ is generalized difference operator which is defined as $\Delta _{a}y_{n}=y_{n+1}-ay_{n}, a\neq{0}.$