Oscillation theorems and existence of positive monotone solutions for second order nonlinear difference equations

We offer sufficient conditions for the oscillation of all solutions of the perturbed difference equation Δ(an−1(Δyn−1)σ) + F(n, yn) = G(n, yn,Δyn), n ≥ 1, as well as for the existence of a positive monotone solution of the damped difference equation Δ(an(Δyn)σ) + bn(Δyn)σ + H(n, yn, Δyn) = 0, n ≥ 0, where 0 < σ = pq with p, q odd integers, or p even and q odd integers. Examples which dwell upon the importance of our results are also included.