Oscillations Generated by Deviating Arguments

Here, we shall consider the half-linear difference equation 16.1 $$\Delta [|\Delta y(k){|^{\sigma - 1}}\Delta y(k)] = \sum\limits_{i = 1}^n {{p_i}(k)|y({g_i}(k)){|^{\sigma - 1}}y({g_i}(k)),k \in N(a)} $$ where σ > 0. For each 1 ≤ i ≤ n we shall assume that (I)p _i (k) ≥ 0, max _k∈N(J)p _i (k) > 0 for any a ≤ J ∈ N, and(II)g _i : N(a) → Z is such that Δg _i (k) > 0 eventually, and lim _k→∞g _i (k) = ∞. For the difference equation (16.1) we shall provide sufficient conditions for the oscillation of all solutions, as well as necessary and sufficient conditions for the existence of both bounded and unbounded nonoscillatory solutions.