Oscillation theorems for second-order nonlinear delay differential equations of neutral type

Year 2019, Volume: 48 Issue: 3, 633 - 643, 15.06.2019

Başak Karpuz , Shyam S. Santra

Abstract

In this paper, new sufficient conditions are obtained for oscillation of second-order neutral delay differential equations of the form
\[\frac{d}{dt}\bigg[r(t)\frac{d}{dt}[x(t)+p(t)x(\tau(t))]\bigg]+q(t)G\bigl(x(\sigma(t))\bigr)=0\: for\: t\geq t_{0},\]
under the assumptions $\int^{\infty}\frac{1}{r(\eta)}d\eta=\infty$ and $\int^{\infty}\frac{1}{r(\eta)}d\eta<\infty$ for various ranges of the bounded neutral coefficient $p$. Unlike most of the previous results, $\tau^{\prime}$ is allowed to be oscillatory. Further, some illustrative examples showing applicability of the new results are included.

Keywords

Oscillation, nonoscillation, nonlinear, delay argument, second-order neutral differential equation

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