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Oscillation criteria for second order nonlinear neutral differential equations. (English) Zbl 1195.34098

The paper deals with the oscillatory behavior of second order nonlinear neutral delay differential equations of the form

${\left(r\left(t\right){\left(x\left(t\right)+p\left(t\right)x\left(t-\tau \right)\right)}^{\text{'}}\right)}^{\text{'}}+q\left(t\right)f\left(x\left(t\right),x\left(\sigma \left(t\right)\right)\right)=0,\phantom{\rule{2.em}{0ex}}\left(1\right)$

where $t\ge {t}_{0}>0$, $\tau \ge 0$ is a constant, and moreover $r,\sigma \in {C}^{1}\left(\left[{t}_{0},\infty \right),\left(0,\infty \right)\right)$, $p,q\in C\left(\left[{t}_{0},\infty \right),ℝ\right)$, and $f\in C\left({ℝ}^{2},ℝ\right)$. The purpose of the paper is to improve oscillation results obtained for equation (1) by W. Shi and P. Wang [Appl. Math. Comput. 146, No. 1, 211–226 (2003; Zbl 1037.34059)] using a generalized Riccati transformation and developing ideas exploited by Yu. V. Rogovchenko and F. Tuncay [Nonlinear Anal., Theory Methods Appl. 69, No. 1 (A), 208–221 (2008; Zbl 1147.34026)]. Two examples are provided to illustrate the relevance of obtained results.

##### MSC:
 34K11 Oscillation theory of functional-differential equations 34K40 Neutral functional-differential equations
##### References:
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