Oscillation for nth Order Nonlinear Difference Equations

Here, we shall provide sufficient conditions for the oscillation of all solutions of the nth order perturbed difference equation 18.1 $$|{\Delta ^n}y(k){|^{\sigma - 1}}{\Delta ^n}y(k) + Q(k,y(k - {\eta _k}),\Delta y(k - {\eta _k}), \cdots ,{\Delta ^{n - 2}}(k - {\eta _k})) = P(k,y(k - {\eta _k}),\Delta y(k - {\eta _k}), \cdots ,{\Delta ^{n - 1}}y(k - {\eta _k})),k \in N({k_0})$$ where σ > 0, and η _k ∈ Z = {⋯, −1, 0, 1, ⋯} with lim _k→∞(k − η _k ) = ∞.