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

Abstract

We find necessary conditions for every solution of the neutral delay difference equation Δ(r n Δ(y n − p n y n−m )) + q n G(y n−k ) = f n to oscillate or to tend to zero as n → ∞, where Δ is the forward difference operator Δx n = x n+1 − x n , and p n , q n , r n are sequences of real numbers with q n ≥ 0, r n > 0. Different ranges of {p n }, including p n = ±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.

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