Necessary Conditions for the Solutions of Second Order Non-linear Neutral Delay Difference Equations to Be Oscillatory or Tend to Zero

We find necessary conditions for every solution of the neutral delay difference equation Δ(rnΔ(yn − pnyn−m)) + qnG(yn−k)= fn to oscillate or to tend to zero as n→∞, where Δ is the forward difference operator Δxn = xn+1 − xn, and pn, qn, rn are sequences of real numbers with qn ≥ 0, rn > 0. Different ranges of {pn}, including pn =±1, are considered in this paper. We do not assume that G is Lipschitzian nor nondecreasing with xG(x) > 0 for x = 0. In this way, the results of this paper improve, generalize, and extend recent results. Also, we provide illustrative examples for our results.