An improved approach for studying oscillation of second-order neutral delay differential equations

Abstract

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.

Qualitative Properties of Solutions of Second-Order Neutral Differential Equations

“…In [43], Tripathy et al studied (3) and established several conditions of the solutions of (3) when considering the assumptions lim y→∞ A(y) = ∞ and lim y→∞ A(y) < ∞ for different values of the neutral coefficient p. In [44], Bohner et al obtained sufficient conditions for the oscillation of the solutions of (3) when γ = β, lim y→∞ A(y) < ∞ and 0 ≤ p(y) < 1. Grace et al [15] studied the oscillation of (3) when γ = β j , considering the assumptions lim y→∞ A(y) < ∞, lim y→∞ A(y) = ∞ and 0 ≤ p(y) < 1. In [45], Li et al established sufficient conditions for the oscillation of the solutions of (3), under the assumptions lim y→∞ A(y) < ∞ and p(y) ≥ 0.…”

“…Using the last inequality in (15), dividing by a(ϑ 0 (y))w δ 2 (ϑ 0 (y)) > 0, and then operating the power 1/γ on both sides, we obtain…”

On Oscillation of Second-Order Linear Neutral Differential Equations With Damping Term

“…In recent years, there has been a great interest in investigating the oscillatory behavior of solutions of various classes of second order neutral differential equations without damping term, and we refer the reader to the papers [2,3,4,5,8,11,12,13,14,16,17,18,19,21] and the references therein as examples of recent results on this topic. However, determining oscillation criteria for second-order neutral differential equations with damping term has not received a great deal of attention in the literature; moreover, the results obtained are for the cases 0 < p(t) ≤ p 0 < 1 or −1 < p 0 ≤ p(t) < 0, see the papers [7,9,20] as example.…”

Oscillatory Behavior of Second-Order Half-Linear Neutral Differential Equations with Damping

“…The oscillatory behavior of solutions to various classes of second order functional differential equations has been the object of research of a number of authors and many interesting results have been obtained. For some typical results, we refer the reader to [2][3][4]7,8,[10][11][12][15][16][17][18][19][20]23] and the references cited therein as examples of recent results on this topic. However, results on the oscillatory behavior of solutions of second-order neutral differential equations with damping term are relatively scarce in the literature; some results can be found, for example, in [5,6,21,22].…”