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Oscillation theorems for certain even order neutral differential equations

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Qigui Yang and Sui Sun Cheng
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** Address.**

Q. Yang,
School of Mathematical Science, South China University of Technology,
Guangzhou, 510640 P. R. China

S.S. Cheng,
Department of Mathematics, Tsinghua University, Hsinchu, Taiwan 30043, R. O. China

** E-mail.**

qgyang@scut.edu.cn

sscheng@math.nthu.edu.tw

**Abstract.**

This paper is concerned with a class of even order nonlinear
differential equations of the form
\begin{multline*}
\frac{d}{dt}\Big( \Big|\left( x(t)+p(t)x(\tau (t))\right)
^{(n-1)}\Big| ^{\alpha -1}(x(t)+p(t)x(\tau (t)))^{(n-1)}\Big)\\
+F\big( t,x(g(t))\big) =0\,,
\end{multline*}
where $n$ is even and
$t\geq t_{0}$. By using the generalized Riccati transformation and
the averaging technique, new oscillation criteria are obtained which
are either extensions of or complementary to a number of existing
results. Our results are more general and sharper than some previous
results even for second order equations.

**AMSclassification. **

34A30, 34K11.

**Keywords. **

Neutral differential equation, oscillation criterion,
Riccati transform, averaging method.