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. Sufficient conditions are obtained so that every solution ofoscillates or tends to zero as t → ∞. Various ranges of p(t) are considered. In order to accommodate sublinear cases, it is assumed thatThrough examples it is shown that if the condition on Q is weakened, then there are sublinear equations whose solutions tend to ±∞ as t → ∞.
ABSTRACT. In this paper, necessary and sufficient conditions for the oscillation and asymptotic behaviour of solutions of the second order neutral delay differential equation (NDDE)Since the results of this paper hold when r(t) ≡ 1 and G(u) ≡ u, therefore it extends, generalizes and improves some known results.
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