The oscillation of an $m$th order perturbed nonlinear difference equation

summary:We offer sufficient conditions for the oscillation of all solutions of the perturbed difference equation $$|\Delta^{m} y(k)|^{\alpha-1}\Delta^{m} y(k)+Q(k,y(k-\sigma_{k}), \Delta y(k-\sigma_{k}),\cdots, \Delta^{m-2}y(k-\sigma_{k}))$$ \hfill $=P(k,y(k-\sigma_{k}),\Delta y(k-\sigma_{k}),\cdots, \Delta^{m-1}y(k-\sigma_{k})),~k\geq k_{0}$ where $\alpha>0.$ Examples which dwell upon the importance of our results are also included