New oscillation criteria for the second-order Emden-Fowler delay differential equation with a sublinear neutral term are presented. An essential feature of our results is that oscillation of the studied equation is ensured via only one condition. Furthermore, as opposed to the results by Agarwal et al. criteria can be applied to Emden-Fowler delay differential equations with noncanonical operators and a sublinear neutral term. Our results essentially improve, extend, and simplify some known ones reported in the literature. The results are illustrated with examples. K E Y W O R D S delayed argument, Emden-Fowler differential equation, oscillation, second-order, sublinear neutral term M S C ( 2 0 1 0 ) 34C10, 34K11
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;
The paper is devoted to the study of oscillation of solutions to a class of second-order half-linear neutral differential equations with delayed arguments. New oscillation criteria are established, and they essentially improve the well-known results reported in the literature, including those for non-neutral differential equations. The adopted approach refines the classical Riccati transformation technique by taking into account such part of the overall impact of the delay that has been neglected in the earlier results. The effectiveness of the obtained criteria is illustrated via examples.
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
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