## Turkish Journal of Mathematics

#### Article Title

#### 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