Oscillation of higher order neutral functional difference equations with positive and negative coefficients

Abstract

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.

Necessary and sufficient conditions on asymptotic behaviour of solutions of forced neutral delay dynamic equations

“…Our results improve, extend and complete the works done in [12,14,15,19] by dropping lim sup t→∞ |t − α(t)| < ∞ while dealing with unbounded solutions, and extends and improves the results in [11] by removing lim t→∞ A(t) ∈ (−1, 1) R . One can easily see that Theorems 1-3 remain valid for mixed type equations too, since in the proofs of the mentioned theorems, we indeed do not need the functions α, β and γ to be delays, we only require α and γ to satisfy the delay property in Lemmas 1 and 2.…”

“…Further, there are very few papers that consider the existence of nonoscillatory solutions (see [7,9,11,14,15,21,22]). In this work, we try to fill this gap in the literature, not only by generalizing some previously given results, but also by correcting and improving.…”

“…The motivation for this paper originates from the papers [12][13][14][15]; but unfortunately, the proofs in [14,15](T = Z, T = R) seem to be incomplete because the companion function defined in the mentioned papers is not always useful, i.e. for unbounded solutions.…”

“…of which oscillatory and asymptotic nature of solutions are also investigated by many authors; for instance, see [14,21,22], where the results are employed for linear delays of slope 1.…”

Oscillations of First Order Neutral Differential Equations with Positive and Negative Coefficients

“…The authors in [1] investigated the first order delay differential equations with positive and negative coefficients . While in [2], [5] and [6] the authors gave some sufficient conditions for the oscillation of neutral differential equation with positive and negative coefficients and constant delays . In this paper we give a generalization tosome results in [4] and [5]where we have used a variable delays.…”