ABSTRACT. Sufficient conditions are obtained so that every solution of the neutral functional difference equationoscillates or tends to zero or ±∞ as n → ∞, where ∆ is the forward difference operator given by ∆x n = x n+1 − x n , p n , q n , u n , f n are infinite sequences of real numbers with q n > 0, u n ≥ 0, G, H ∈ C(R, R) and m ≥ 2 is any positive integer. Various ranges of {p n } are considered. The results hold for G(u) ≡ u, and f n ≡ 0. This paper corrects, improves and generalizes some recent results.

Abstract. In this paper, we obtain sufficient conditions so that every solution of neutral functional difference equationoscillates or tends to zero as n → ∞ , where the sequence {q n } may change sign. Here Δ is the forward difference operator given by Δx n = x n+1 − x n , {τ n } and {σ n } are increasing sequences, which are less than n and approaches ∞ as n approaches ∞ . This paper generalizes and extends some recent results.Mathematics subject classification (2010): 39A10, 39A12.