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a b s t r a c tThe aim of this paper is to study the oscillation of the second order neutral differential equationsThe obtained results are based on the new comparison theorems, that enable us to reduce the problem of the oscillation of the second order equation to the oscillation of the first order equation. The obtained comparison principles essentially simplify the examination of the studied equations.

The aim of this paper is to study the asymptotic properties and oscillation of the n-th order delay differential equationThe results obtained are based on some new comparison theorems that reduce the problem of oscillation of an n-th order equation to that of the oscillation of one or more first order equations. We handle both the cases ∞ r −1/γ (t) dt = ∞ and ∞ r −1/γ (t) dt < ∞. The comparison principles simplify the analysis of equation (E).Вивчено асимптотичнi властивостi та осциляцiю диференцiального рiвняння n-го порядку з запiзненнямОтриманi результати базуються на деяких нових теоремах порiвняння, якi зводять задачу про осциляцiю рiвняння n-го порядку до такої ж задачi для одного або кiлькох рiвнянь першого порядку. Розглянуто обидва випадки:порiвняння дозволяють спростити аналiз рiвняння (E). Introduction.In this paper, we examine the asymptotic and oscillatory behavior of solutions of the n-th order (n ≥ 3) delay differential equation), and (H 1 ) γ is the ratio of two odd positive integers; (H 2 ) r(t) > 0, r (t) > 0, and q(t) > 0;(H 3 ) τ (t) ≤ t, lim t→∞ τ (t) = ∞, and τ (t) is nondecreasing;

We study oscillatory behavior of a class of fourth-order neutral differential equations with a p-Laplacian like operator using the Riccati transformation and integral averaging technique. A Kamenev-type oscillation criterion is presented assuming that the noncanonical case is satisfied. This new theorem complements and improves a number of results reported in the literature. An illustrative example is provided. MSC: 34C10; 34K11