New criteria for the oscillation and asymptotic behavior of second-order neutral differential equations with several delays

Authors

BAŞAK KARPUZ
SHYAM SUNDAR SANTRA

DOI

10.3906/mat-2006-103

Abstract

In this paper, necessary and sufficient conditions for asymptotic behavior are established of the solutions to second-order neutral delay differential equations of the form \begin{equation} \frac{d}{d{}t}\Biggl(r(t)\biggl(\frac{d}{d{}t}[x(t)-p(t)x(\tau(t))]\biggr)^{\gamma}\Biggr)+\sum_{i=1}^{m}q_{i}(t)f_{i}\bigl(x(\sigma_{i}(t))\bigr)=0 \quad\text{for}\ t\geq{}t_{0}.\nonumber \end{equation} We consider two cases when $f_{i}(u)/u^{\beta}$ is nonincreasing for $\gamma>\beta$, and nondecreasing for $\beta>\gamma$, where $\beta$ and $\gamma$ are quotients of two positive odd integers. Our main tool is Lebesgue's dominated convergence theorem. Examples illustrating the applicability of the results are also given, and state an open problem.

Keywords

Oscillation, nonoscillation, nonlinear, delay argument, second-order neutral differential equations, Lebesgue's dominated convergence theorem

First Page

1990

Last Page

2003

Recommended Citation

KARPUZ, BAŞAK and SANTRA, SHYAM SUNDAR (2020) "New criteria for the oscillation and asymptotic behavior of second-order neutral differential equations with several delays," Turkish Journal of Mathematics: Vol. 44: No. 6, Article 3. https://doi.org/10.3906/mat-2006-103
Available at: https://journals.tubitak.gov.tr/math/vol44/iss6/3