\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small 
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 167, pp. 1--7.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or
http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2010/167\hfil Comparison theorems]
{Comparison theorems for second-order neutral
differential equations of mixed type}

\author[T. Li\hfil EJDE-2010/167\hfilneg]
{Tongxing Li}

\address{Tongxing Li \newline
 School of Control Science and Engineering,
Shandong University, Jinan, Shandong 250061, China}
\email{litongx2007@163.com}

\thanks{Submitted October 22, 2010. Published November 24, 2010.}
\subjclass[2000]{34K11}
\keywords{Oscillation; neutral functional differential
equations; \hfill\break\indent
mixed type; second-order; comparison theorem}

\begin{abstract}
 Three comparison theorems are established for the  oscillation
 of the second-order neutral differential equations of mixed type
 $$
 \big(r(t)[x(t)+p_1(t)x(t-\sigma_1)+p_2(t)x(t+\sigma_2)]'\big)'
 +q_1(t)x(t-\sigma_3) +q_2(t)x(t+\sigma_4)=0.
 $$
 Our results are new even when $p_2(t)=q_2(t)=0$. An
 example is provided to illustrate the main results.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

This article concerns the oscillatory behavior of the second-order
linear neutral differential equation of mixed type
\begin{equation}\label{lh1.1}
\left(r(t)[x(t)+p_1(t)x(t-\sigma_1)+p_2(t)x(t+\sigma_2)]'\right)'
+q_1(t)x(t-\sigma_3)
+q_2(t)x(t+\sigma_4)=0,
\end{equation}
for $t\geq t_0$.

We will use the following conditions:
\begin{itemize}
\item[(H1)] $r\in C^1([t_0,\infty),\mathbb{R})$,
$r(t)>0$ for $t\geq t_0$;

\item[(H2)] $p_i\in C([t_0,\infty),[0,a_i])$, where $a_i$ are constants
for $i=1$, $2$;


\item[(H3)] $q_j\in C([t_0,\infty),[0,\infty))$, and $q_j$ are not
eventually zero on any half line $[t_*,\infty)$ for $t_*\geq t_0$,
$j=1$, $2$;


\item[(H4)] $\sigma_i\geq0$ are constants, for $i=1, 2, 3, 4$.

\end{itemize}

We put $z(t)=x(t)+p_1(t)x(t-\sigma_1)+p_2(t)x(t+\sigma_2)$. By a
solution of  \eqref{lh1.1}, we mean a function $x\in
C([T_x,\infty), \mathbb{R})$ for some $T_x\geq t_0$ which has the
properties that $z\in C^1([T_x,\infty), \mathbb{R})$ and $rz'\in
C^1([T_x,\infty), \mathbb{R})$ and satisfying \eqref{lh1.1} on
$[T_x,\infty)$. We consider only those solutions $x$ of \eqref{lh1.1} which satisfy $\sup\{|x(t)|:t\geq T\}>0$ for all
$T\geq T_x$. We assume that \eqref{lh1.1} possesses such a
solution. As is customary, a solution of \eqref{lh1.1} is called
oscillatory if it has arbitrarily large zeros on $[t_0,\infty)$;
otherwise, it is called non-oscillatory. Equation \eqref{lh1.1}
is said to be oscillatory if all its solutions are oscillatory.

Recently, there has been much research activity concerning the
oscillation and non-oscillation of solutions of varietal types of
differential equations. We refer the reader to
\cite{ag2,bacu1,bacu2,dzurina,dzurina1,erbz,hasan,ladde,xu,zafer,zhang}
and the references cited therein.

D\v{z}urina \cite{dzurina1} presented  sufficient conditions for
the oscillation of the second-order differential equation with mixed
argument
$$
\big(\frac{1}{r(t)}u'(t)\big)'+p(t)u(\tau(t))+q(t)u(\sigma(t))=0,\quad
t\geq t_0.
$$


Some oscillation results for the  second-order neutral
differential equation
$$
(r(t)|z'(t)|^{\gamma-1}z'(t))'+q(t)|x(\sigma(t))|^{\gamma-1}x(\sigma(t))
=0,
$$
where $z(t)=x(t)+p(t)x(\tau(t))$ and $t\geq t_0$ were obtained by
\cite{dzurina2,dong,han2,liu}.

Regarding the oscillatory behavior of neutral differential equations
with mixed arguments; see e.g., the papers
\cite{AgG,dzurina3,gracei,grace1,grace2,yan1,yan}. Agarwal and Grace
\cite{AgG} studied the oscillation of the
 even-order equation
$$
(x(t)+ax(t-\tau)-bx(t+\tau))^{(n)}+q(t)x(t-g)+p(t)x(t+h)=0.
$$
D\v{z}urina et al. \cite{dzurina3} established some oscillation
criteria for the  mixed neutral equation
$$
\left(x(t)+p_1x(t-\tau_1)+p_2x(t+\tau_2)\right)''=q_1(t)x(t-\sigma_1)
+q_2(t)x(t+\sigma_2).
$$
Grace and Lalli \cite{gracei} examined the oscillatory behavior for
the second-order equation
$$
(x(t)+\lambda x(t-\tau))''=q(t)x(t-\sigma)+p(t)x(t+\beta).
$$
Grace \cite{grace1} obtained some oscillation theorems for the
odd-order neutral differential equation
$$
\left(x(t)+p_1x(t-\tau_1)+p_2x(t+\tau_2)\right)^{(n)}=q_1x(t-\sigma_1)
+q_2x(t+\sigma_2).
$$
Grace \cite{grace2} and Yan \cite{yan1} established several
sufficient conditions for the oscillation of solutions of odd-order
neutral functional differential equation
$$
(x(t)+cx(t-h)+Cx(t+H))^{(n)}+qx(t-g)+Qx(t+G)=0.
$$
Yan \cite{yan} considered the oscillation of even-order mixed
neutral differential equation
$$
\left(x(t)-c_1x(t-h_1)-c_2x(t+h_2)\right)^{(n)}+qx(t-g_1)+px(t+g_2)=0.
$$

To the best of our knowledge, there are only few results on the
oscillation of \eqref{lh1.1}. It is interesting to study
\eqref{lh1.1} since it has some applications in the study of
vibrating masses attached to an elastic bar (see \cite{hale}). The
aim of this paper is to establish some oscillation results
for \eqref{lh1.1}. The organization of this paper is as follows: In
Section 2, we reduce the problem of the oscillation of \eqref{lh1.1}
to the oscillation of the first-order inequalities
under the case when
\begin{equation}\label{lh1.3}
\int_{t_0}^\infty\frac{1}{r(t)}\,{\rm d}t=\infty.
\end{equation}
In Section 3, we give an example and a remark to illustrate our
results.

Below, when we write a functional inequality without specifying its
domain of validity we assume that it holds for all sufficiently
large $t$.

\section{Main results}
In the following, we will establish some oscillation criteria for
\eqref{lh1.1}.

Throughout this paper, we denote
\begin{gather*}
Q(t)=Q_1(t)+Q_2(t),\\
Q_1(t)=\min\{q_1(t),q_1(t-\sigma_1),q_1(t+\sigma_2)\},\\
Q_2(t)=\min\{q_2(t),q_2(t-\sigma_1),q_2(t+\sigma_2)\}.
\end{gather*}

\begin{theorem}\label{lth3.1}
Assume that \eqref{lh1.3} holds. Further, assume that
\begin{equation}\label{mh1}
[y(t)+a_1y(t-\sigma_1)+a_2y(t+\sigma_2)]'+Q(t)
\Big(\int_{t_1}^{t-\sigma_3}\frac{1}{r(s)}\,{\rm
d}s\Big)y(t-\sigma_3) \leq 0
\end{equation}
has no eventually positive solution for all sufficiently large
$t_1$, $t_1\geq t_0$. Then \eqref{lh1.1} is oscillatory.
\end{theorem}

\begin{proof}
 Let $x$ be a non-oscillatory solution of
\eqref{lh1.1}. Without loss of generality, we assume that there
exists $t_1\geq t_0$ such that $x(t)>0$, $x(t-\sigma_1)>0$,
$x(t+\sigma_2)>0$, $x(t-\sigma_3)>0$ and $x(t+\sigma_4)>0$ for all
$t\geq t_1$. Then $z(t)>0$ for $t\geq t_1$. In view of
\eqref{lh1.1}, we obtain
\begin{equation}\label{lcj}
(r(t)z'(t))'=-q_1(t)x(t-\sigma_3)-q_2(t)x(t+\sigma_4)\leq0,\quad t\geq
t_1.
\end{equation}
Thus, $r(t)z'(t)$ is non-increasing function. Consequently, it is
easy to conclude that there exist two possible cases of the sign of
$z'(t)$, that is, $z'(t)>0$ or $z'(t)<0$ eventually. If there exists
$t_2\geq t_1$ such that $z'(t_2)<0$, then from \eqref{lcj}, we see
that
$$
r(t)z'(t)\leq r(t_2)z'(t_2)<0,\quad t\geq t_2.
$$
Integrating the above inequality from $t_2$ to $t$, we obtain
$$
z(t)\leq z(t_2)+ r(t_2)z'(t_2)\int_{t_2}^t \frac{1}{r(s)}\,{\rm d}s.
$$
Letting $t\to\infty$, we obtain
$\lim_{t\to\infty}z(t)=-\infty$ due to \eqref{lh1.3}, which
is a contradiction. Thus, there exists a $t_2\geq t_1$ such that
\begin{equation}\label{118}
z'(t)>0
\end{equation}
for $t\geq t_2$.
Using \eqref{lh1.1}, for all sufficiently large $t$, we have
\begin{align*}
&(r(t)z'(t))'+q_1(t)x(t-\sigma_3)+q_2(t)x(t+\sigma_4)
+a_1(r(t-\sigma_1)z'(t-\sigma_1))'\\
&+a_1q_1(t-\sigma_1)x(t-\sigma_1-\sigma_3)
+a_1q_2(t-\sigma_1)x(t+\sigma_4-\sigma_1)\\
&+a_2(r(t+\sigma_2)z'(t+\sigma_2))'
+a_2q_1(t+\sigma_2)x(t+\sigma_2-\sigma_3)\\
&+a_2q_2(t+\sigma_2)x(t+\sigma_2+\sigma_4)
=0.
\end{align*}
Thus
\begin{equation}\label{259}
\begin{aligned}
&(r(t)z'(t))'+a_1(r(t-\sigma_1)z'(t-\sigma_1))'
+a_2(r(t+\sigma_2)z'(t+\sigma_2))'\\
&+Q_1(t)z(t-\sigma_3) +Q_2(t)z(t+\sigma_4)\leq 0.
\end{aligned}
\end{equation}
By \eqref{118}, we have $z(t+\sigma_4)\geq z(t-\sigma_3)$. Then,
from \eqref{259}, we obtain
\begin{equation}\label{jl2}
(r(t)z'(t))'+a_1(r(t-\sigma_1)z'(t-\sigma_1))'+a_2(r(t+\sigma_2)z'(t+\sigma_2))'+Q(t)z(t-\sigma_3)\leq0.
\end{equation}
It follows from \eqref{lcj} that
\begin{equation}\label{xj1}
z(t)=z(t_2)+\int_{t_2}^t\frac{r(s)z'(s)}{r(s)}\,{\rm d}s\geq
r(t)z'(t)\int_{t_2}^t\frac{1}{r(s)}\,{\rm d}s.
\end{equation}
Set $y(t)=r(t)z'(t)>0$. From \eqref{jl2} and \eqref{xj1}, we see
that $y$ is an eventually positive solution of
$$
[y(t)+a_1y(t-\sigma_1)+a_2y(t+\sigma_2)]'+Q(t)y(t-\sigma_3)\int_{t_2}^{t-\sigma_3}\frac{1}{r(s)}\,{\rm
d}s\leq 0.
$$
This completes the proof.
\end{proof}

\begin{theorem}\label{lth3.2}
Assume that \eqref{lh1.3} holds and
\begin{equation}\label{ssmh1}
u'(t)+Q(t)\frac{\int_{t_1}^{t-\sigma_3} \frac{1}{r(s)}\,{\rm
d}s}{1+a_1+a_2}u(t+\sigma_1-\sigma_3)\leq0
\end{equation}
has no eventually positive solution for all sufficiently large
$t_1$, $t_1\geq t_0$. Then  \eqref{lh1.1} is oscillatory.
\end{theorem}

\begin{proof}
 Let $x$ be a non-oscillatory solution of
\eqref{lh1.1}. Without loss of generality, we assume that there
exists $t_1\geq t_0$ such that $x(t)>0$, $x(t-\sigma_1)>0$,
$x(t+\sigma_2)>0$, $x(t-\sigma_3)>0$ and $x(t+\sigma_4)>0$ for all
$t\geq t_1$. Then $z(t)>0$ for $t\geq t_1$. Proceeding as in the
proof of Theorem \ref{lth3.1}, we obtain that $y(t)=r(t)z'(t)>0$ is
non-increasing and satisfies inequality \eqref{mh1}. Define
$$
u(t)=y(t)+a_1y(t-\sigma_1)+a_2y(t+\sigma_2)>0.
$$
Then
$$
u(t)\leq (1+a_1+a_2)y(t-\sigma_1).
$$
Substituting the above formulas into \eqref{mh1}, we find $u$ is an
eventually positive solution of
$$
u'(t)+Q(t)\frac{\int_{t_1}^{t-\sigma_3} \frac{1}{r(s)}\,{\rm
d}s}{1+a_1+a_2}u(t+\sigma_1-\sigma_3)\leq0.
$$
The proof is complete.
\end{proof}

 From Theorem \ref{lth3.2} and \cite[Theorem 2.1.1]{ladde}, we
establish the following corollary.

\begin{corollary}\label{xlth3.2}
Assume that \eqref{lh1.3} holds, $\sigma_1-\sigma_3<0$ and
\begin{equation}\label{1mh1}
\liminf_{t\to\infty}\int_{t+\sigma_1-\sigma_3}^tQ(u)
\Big(\int_{t_1}^{u-\sigma_3}\frac{1}{r(s)}\,{\rm d}s\Big) \,{\rm
d}u>\frac{1+a_1+a_2}{{\rm e}}
\end{equation}
for all sufficiently large $t_1$, $t_1\geq t_0$. Then  \eqref{lh1.1}
is oscillatory.
\end{corollary}


\begin{theorem}\label{lth3.3}
Assume that \eqref{lh1.3} holds and
\begin{equation}\label{xxmh1}
w'(t)-\frac{Q(t+\sigma_1)}{1+a_1+a_2}\Big(\int_{t_1}^{t+\sigma_1}
\,\frac{{\rm d}u}{r(u-\sigma_1)}\Big) w(t+\sigma_1-\sigma_3)\geq0
\end{equation}
has no eventually positive solution for all sufficiently large
$t_1$, $t_1\geq t_0$. Then  \eqref{lh1.1} is oscillatory.
\end{theorem}

\begin{proof} Let $x$ be a non-oscillatory solution of
\eqref{lh1.1}. Without loss of generality, we assume that there
exists $t_1\geq t_0$ such that $x(t)>0$, $x(t-\sigma_1)>0$,
$x(t+\sigma_2)>0$, $x(t-\sigma_3)>0$ and $x(t+\sigma_4)>0$ for all
$t\geq t_1$. Then $z(t)>0$ for $t\geq t_1$. Proceeding as in the
proof of Theorem \ref{lth3.1}, we obtain  \eqref{lcj}--\eqref{jl2}
for $t\geq t_2\geq t_1$. Integrating \eqref{jl2} from $t$ to
$\infty$ yields
\begin{equation}\label{123}
r(t)z'(t)+a_1r(t-\sigma_1)z'(t-\sigma_1)+a_2r(t+\sigma_2)z'(t+\sigma_2)\geq
\int_t^\infty Q(s)z(s-\sigma_3)\,{\rm d}s.
\end{equation}
Since $r(t)z'(t)$ is non-increasing, we get
\begin{equation}\label{234}
r(t)z'(t)+a_1r(t-\sigma_1)z'(t-\sigma_1)+a_2r(t+\sigma_2)z'(t+\sigma_2)
\leq (1+a_1+a_2)r(t-\sigma_1)z'(t-\sigma_1).
\end{equation}
In view of \eqref{123} and \eqref{234}, we have
\begin{equation}\label{345}
z'(t-\sigma_1)\geq \frac{1}{(1+a_1+a_2)r(t-\sigma_1)}\int_t^\infty
Q(s)z(s-\sigma_3)\,{\rm d}s.
\end{equation}
Integrating \eqref{345} from $t_2$ to $t$, we see that
\begin{align*}
z(t-\sigma_1)
&\geq\int_{t_2}^t\frac{1}{(1+a_1+a_2)r(u-\sigma_1)}\int_u^\infty
Q(s)z(s-\sigma_3)\,{\rm d}s\,{\rm d}u \\
&\geq \int_{t_2}^t\frac{1}{1+a_1+a_2}Q(s)z(s-\sigma_3)\int_{t_2}^s
\frac{1}{r(u-\sigma_1)} \,{\rm d}u \,{\rm d}s.
\end{align*}
Thus
$$
z(t)\geq\frac{1}{1+a_1+a_2}\int_{t_2}^{t+\sigma_1}
Q(s)z(s-\sigma_3)\int_{t_2}^s \frac{1}{r(u-\sigma_1)} \,{\rm d}u
\,{\rm d}s.
$$
Let
$$
w(t)=\frac{1}{1+a_1+a_2}\int_{t_2}^{t+\sigma_1}Q(s)z(s-\sigma_3)\int_{t_2}^s
\frac{1}{r(u-\sigma_1)} \,{\rm d}u \,{\rm d}s>0.
$$
Then $z(t)\geq w(t)$ and
\begin{align*}
w'(t)&=\frac{1}{1+a_1+a_2}Q(t+\sigma_1)z(t+\sigma_1-\sigma_3)\int_{t_2}^{t+\sigma_1}
\frac{1}{r(u-\sigma_1)} \,{\rm d}u \\
& \geq \frac{1}{1+a_1+a_2}Q(t+\sigma_1)
w(t+\sigma_1-\sigma_3)\int_{t_2}^{t+\sigma_1}
\frac{1}{r(u-\sigma_1)} \,{\rm d}u.
\end{align*}
Hence, we find $w$ is an eventually positive solution of
$$
w'(t)-\frac{Q(t+\sigma_1)}{1+a_1+a_2}\Big(\int_{t_2}^{t+\sigma_1}
\,\frac{{\rm d}u}{r(u-\sigma_1)}\Big) w(t+\sigma_1-\sigma_3)\geq0.
$$
This completes the proof.
\end{proof}

Due to Theorem \ref{lth3.3} and \cite[Theorem 2.4.1]{ladde}, we
obtain the following corollary.

\begin{corollary}\label{jlth3.2}
Assume that \eqref{lh1.3} holds, $\sigma_1-\sigma_3>0$ and
\begin{equation}\label{891mh1}
\liminf_{t\to\infty}\int_t^{t+\sigma_1-\sigma_3}Q(u+\sigma_1)
\Big(\int_{t_1}^{u+\sigma_1}\frac{1}{r(s-\sigma_1)}\,{\rm
d}s\Big)\,{\rm d}u>\frac{1+a_1+a_2}{{\rm e}}
\end{equation}
for all sufficiently large $t_1$, $t_1\geq t_0$.
Then  \eqref{lh1.1} is oscillatory.
\end{corollary}

\section{Example and remark}

For an application of our results, we will give the following
example.
 Consider the equation
\begin{equation}\label{y1}
[x(t)+a_1x(t-\sigma_1)+a_2x(t+\sigma_2)]''
+\frac{\alpha}{t}x(t-\sigma_3) +\frac{\beta}{t}x(t+\sigma_4)=0,\quad
t\geq t_0,
\end{equation}
where $a_1$, $a_2$, $\alpha$ and $\beta$ are positive constants.

Let $r(t)=1$, $p_1(t)=a_1$, $q_1(t)=\alpha/t$ and $q_2(t)=\beta/t$.
Then $Q_1(t)=\alpha/(t+\sigma_2)$, $Q_1(t)=\beta/(t+\sigma_2)$ and
$Q(t)=(\alpha+\beta)/(t+\sigma_2)$. Assume that $\sigma_3>\sigma_1$.
Since
$$
\liminf_{t\to\infty}\int_{t+\sigma_1-\sigma_3}^tQ(u)
\Big(\int_{t_1}^{u-\sigma_3}\frac{1}{r(s)}\,{\rm d}s\Big)\,{\rm d}u
=(\alpha+\beta)(\sigma_3-\sigma_1),
$$
we conclude that \eqref{y1} is oscillatory if
$$
(\alpha+\beta)(\sigma_3-\sigma_1)>\frac{1+a_1+a_2}{{\rm e}}
$$
due to Corollary \ref{xlth3.2}.

Suppose that $\sigma_3<\sigma_1$. Since
$$
\liminf_{t\to\infty}\int_t^{t+\sigma_1-\sigma_3}Q(u+\sigma_1)
\Big(\int_{t_1}^{u+\sigma_1}\frac{1}{r(s-\sigma_1)}\,{\rm d}s\Big)
\,{\rm d}u\\
=(\alpha+\beta)(\sigma_1-\sigma_3),
$$
we conclude that \eqref{y1} is oscillatory if
$$
(\alpha+\beta)(\sigma_1-\sigma_3)>\frac{1+a_1+a_2}{{\rm e}}
$$
due to Corollary \ref{jlth3.2}.


\begin{remark} \label{rmk3.1} \rm
The equation
\begin{equation}\label{mmm1}
[x(t)+a_1x(t-\sigma_1)]''+q_1(t)x(t-\sigma_3)=0,\quad
\sigma_1<\sigma_3,\; t\geq t_0
\end{equation}
is a special case of \eqref{lh1.1}. Applying results of
\cite[Theorem 2]{zafer} and \cite[Corollary 1]{zhang}, we obtain a
sufficient condition for \eqref{mmm1} to be oscillatory, that
is, if $a_1<1$ and
\begin{equation}\label{cc1}
\liminf_{t\to\infty}\int_{t-\sigma_3}^t q_1(s)(s-\sigma_3)\,{\rm
d}s>\frac{1}{(1-a_1){\rm e}},
\end{equation}
then  \eqref{mmm1} is oscillatory.

Note that Corollary \ref{xlth3.2}  transforms
\eqref{cc1} into
\begin{equation}\label{kkk1}
\liminf_{t\to\infty}\int_{t+\sigma_1-\sigma_3}^tQ_1(s)(s-\sigma_3-t_1)\,{\rm
d}s>\frac{1+a_1}{{\rm e}},
\end{equation}
for all sufficiently large $t_1$, $t_1\geq t_0$, where
$Q_1(t)=\min\{q_1(t),q_1(t-\sigma_1)\}$. Since
$$
\frac{1}{(1-a_1){\rm e}}> \frac{1+a_1}{{\rm e}}
$$
for $a_1>0$, our results improve their results in some sense.
Moreover, our results can be applied to \eqref{mmm1} when
$a_1\geq1.$
\end{remark}

\begin{thebibliography}{00}

\bibitem{AgG}  R. P. Agarwal, S. R. Grace,
Oscillation theorems for certain neutral functional differential
equations, Comput. Math. Appl. 38 (1999) 1--11.

\bibitem{ag2} R. P. Agarwal, M. Bohner, W. T. Li, Nonoscillation and
Oscillation: Theory of Functional Differential Equations, Marcel
Dekker, New York, 2004.

\bibitem{bacu1} B. Bacul\'{i}kov\'{a}, Oscillation criteria for second order
nonlinear differential equations, Arch. Math. 42 (2006) 141--149.

\bibitem{bacu2} B. Bacul\'{i}kov\'{a}, J. D\v{z}urina,
Oscillation theorems for second order neutral differential
equations, Comput. Math. Appl. (2010) DOI:
10.1016/j.camwa.2010.10.035.

\bibitem{tc} T. Candan, R. S. Dahiya, On the oscillation of certain mixed neutral equations, Appl.
Math. Lett. 21  (2008) 222--226.

\bibitem{dzurina}J. D\v{z}urina, I. P. Stavroulakis, Oscillation
criteria for second-order delay differential equations, Appl. Math.
Comput. 140 (2003) 445--453.

\bibitem{dzurina1} J. D\v{z}urina, Oscillation of second-order differential equations with mixed
argument, J. Math. Anal. Appl. 190 (1995) 821--828.

\bibitem{dzurina2} J. D\v{z}urina, D. Hud\'{a}kov\'{a}, Oscillation of second order neutral delay differential
equations, Math. Bohem. 134 (2009) 31--38.

\bibitem{dzurina3} J. D\v{z}urina, J. Busa, E. A. Airyan,
Oscillation criteria for second-order differential equations of
neutral type with mixed arguments, Differ. Equ. 38 (2002) 137--140.

\bibitem{dong} J. G. Dong, Oscillation behavior of second order
nonlinear neutral differential equations with deviating arguments,
Comput. Math. Appl. 59 (2010) 3710--3717.

\bibitem{erbz} L. Erbe, Q. Kong, B. G. Zhang, Oscillation Theory for
Functional Differential Eqautions, Marcel Dekker, New York, 1995.

\bibitem{gracei} S. R. Grace, B. S. Lalli,  On the oscillation of certain neutral
functional differential equations,  Funcialaj. Ekvacioj 36 (1993)
303--310.

\bibitem{grace1} S. R. Grace, On the oscillations of mixed neutral
equations, J. Math. Anal. Appl. 194 (1995) 377--388.

\bibitem{grace2} S. R. Grace, Oscillations of mixed neutral functional differential equations,
Appl. Math. Comput.  68 (1995) 1--13.

\bibitem{hale} J. K. Hale, Theory of Functional Differential
Equations, Spring-Verlag, New York, 1977.

\bibitem{han2} Z. Han, T. Li, S. Sun, Y. Sun, Remarks on the paper
[Appl. Math. Comput. 207 (2009) 388--396], Appl. Math. Comput. 215
(2010) 3998--4007.

\bibitem{hasan} M. Hasanbulli, Yu. V. Rogovchenko, Oscillation criteria for second order nonlinear neutral
differential equations, Appl. Math. Comput. 215 (2010) 4392--4399.


\bibitem{ladde}  G. S. Ladde, V. Lakshmikantham, B. G. Zhang, Oscillation Theory of
Differential Equations with Deviating Arguments, Marcel Dekker, New
York, 1987.

\bibitem{liu} L. H. Liu, Y. Z. Bai, New oscillation criteria for
second-order nonlinear neutral delay differential equations, J.
Comput. Math. Appl. 231 (2009) 657--663.

\bibitem{saker} S. H. Saker, Oscillation of second order neutral delay differential equations of Emden-Fowler
type, Acta Math. Hungar. 100 (2003) 37--62.

\bibitem{xu} Z. T. Xu,
Oscillation theorems related to averaging technique for second order
Emden-Fowler type neutral differential equations, Rocky Mountain J.
Math. 38 (2008) 649--667.

\bibitem{xu2} R. Xu, F. W. Meng, Oscillation criteria for second order
quasi-linear neutral delay differential equations, Appl. Math.
Comput. 192 (2007) 216--222.

\bibitem{yan1}J. R. Yan, Oscillations of higher order neutral differential equations of mixed
type, Israel. J. Math. 115 (2000) 125--136.

\bibitem{yan}J. R. Yan, Oscillation of higher order neutral
differential equations, J. Austral. Math. Soc. (Series A) 64 (1998)
73--81.

\bibitem{zafer} A. Zafer, Oscillation criteria for even order
neutral differential equations, Appl. Math. Lett. 11 (1998) 21--25.

\bibitem{zhang} Q. X. Zhang, J. R. Yan, L. Gao, Oscillation behavior
of even-order nonlinear neutral differential equations with variable
coefficients, Comput. Math. Appl. 59 (2010) 426--430.


\end{thebibliography}



\end{document}