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.

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This article, is concerned with finding sufficient conditions for the oscillatory and asymptotic behaviour of solutions of the higher order neutral delay differential equationwhere, n ≥ 2, is any positive integer,The functional delays r j (t) ≤ t for each j and g(t) ≤ t, h(t) ≤ t and all of them approach ∞ as t → ∞.The results hold when u ≡ 0 and f (t) ≡ 0. This paper extends, generalizes and improves some known results and answers an open problem.