# On the oscillation of third-order quasi-linear neutral functional differential equations

## E. Thandapani and Tongxing Li

Address:

Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai, India

School of Control Science and Engineering, Shandong University, Jinan, Shandong 250061, P. R. China

E-mail:

ethandapani@yahoo.co.in

litongx2007@163.com

Abstract: The aim of this paper is to study asymptotic properties of the third-order quasi-linear neutral functional differential equation
\begin{equation*} \big [a(t)\big ([x(t)+p(t)x(\delta (t))]^{\prime \prime }\big )^\alpha \big ]^{\prime }+q(t)x^\alpha (\tau (t))=0\,, E \end{equation*}
where $\alpha >0$, $0\le p(t)\le p_0<\infty $ and $\delta (t)\le t$. By using Riccati transformation, we establish some sufficient conditions which ensure that every solution of () is either oscillatory or converges to zero. These results improve some known results in the literature. Two examples are given to illustrate the main results.

AMSclassification: primary 34K11; secondary 34C10.

Keywords: third-order, neutral functional differential equations, oscillation and asymptotic behavior.