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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 10, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2007 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2007/10\hfil Oscillation criteria]
{Oscillation criteria for second-order neutral differential
equations with distributed deviating arguments}

\author[G. Gui, Z. Xu\hfil EJDE-2007/10\hfilneg]
{Gaihua Gui, Zhiting Xu}

\address{Gaihua Gui \newline
School of Mathematical Sciences,
South China Normal University \\
Guangzhou, 510631,  China}
 \email{1981hgg@163.com}

\address{Zhiting Xu \newline
School of Mathematical Sciences,
South China Normal University \\
Guangzhou, 510631,  China}
 \email{xztxhyyj@pub.guangzhou.gd.cn}

\thanks{Submitted July 28, 2006. Published January 2, 2007.}
\subjclass[2000]{34K11, 34C10, 34K40}
\keywords{Oscillation; neutral differential equation; second order;
\hfill\break\indent
 distributed deviating argument; Riccati technique}

\begin{abstract}
 Using a class of test functions $\Phi(t,s,T)$ defined by Sun
 \cite{s2} and a generalized Riccati technique, we establish some new
 oscillation criteria for the second-order neutral differential
 equation with distributed deviating argument
 $$
 (r(t)\psi(x(t))Z'(t))'+\int^b_a
 q(t,\xi)f[x(g(t,\xi))]d\sigma(\xi)=0,\quad t\geq t_0,
 $$
 where $Z(t)=x(t)+p(t)x(t-\tau)$. The obtained results are different
 from most known ones and can be applied to many cases which are
 not covered by existing results.
\end{abstract}

%\def \p{\partial}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{example}[theorem]{Example}

\section{Introduction and Preliminaries}

Consider the second-order neutral differential equation with
distributed deviating argument
\begin{equation}
(r(t)\psi(x(t))Z'(t))'+\int^{b}_{a}q(t,\xi)f[x(g(t,\xi))]
d\sigma(\xi)=0,\quad  t\geq t_{0},
\label{e1.1}
\end{equation}
where $Z(t)=x(t)+p(t)x(t-\tau)$, $\tau\geq0$, and the following
 conditions are assumed to
hold without further mentioning:

\begin{itemize}
\item[(A1)] $r, p\in C(I,\mathbb{R})$ and $0\leq  p(t)\leq1$, $r(t)>0$ for $t\in I$,
$\int^{\infty}1/r(s)ds=\infty$, $I=[t_0,\infty)$;

\item[(A2)] $\psi\in C^{1}(\mathbb{R}, \mathbb{R})$, $\psi(x)>0$ for $x\neq0$;

\item[(A3)] $f\in C(\mathbb{R},\mathbb{R})$, $xf(x)>0$ for $x\neq0$;

\item[(A4)] $q\in C(I\times [a,b],[0,\infty))$ and $q(t,\xi)$ is not eventually zero
on any half-line $[t_{u},\infty)\times[a,b]$, $t_{u}\geq t_{0}$;

\item[(A5)] $g\in C(I\times [a,b],[0,\infty))$, $g(t,\xi)\leq t$ for $t\geq t_0$ and
$\xi \in [a,b]$, $g(t,\xi)$ has a continuous and positive partial derivative on
$I\times[a,b]$ with respect to the first variable $t$ and nondecreasing with respect to
the second variable $\xi$, respectively, and
$\liminf_{t\to\infty}g(t,\xi)=\infty$ for $\xi\in[a,b]$;

\item[(A6)] $\sigma\in C([a,b], \mathbb{R})$ is nondecreasing, and the
integral of \eqref{e1.1} is in the sense of Riemann-Stieltijes.
\end{itemize}

Let $\tau^*(t)=\min\{\tau(t)=t-\tau,\,\delta(t)=\min\,g(t,\xi)$
for $\xi\in[a,b]\}$ and let $T^*=\min\{\tau^*(t):t\geq 0\}$ and
$\tau^*_{-1}(t)=\sup\{s\geq 0: \tau^*(s)]\leq t\}$ for $t\geq
T^*$. Clearly $\tau^*_{-1}(t)\geq t$ for $t\geq T^*$,
$\tau^*_{-1}(t)$ is nondecreasing and coincides with the inverse
of $\tau^*(t)$ when latter exists. By a solution of \eqref{e1.1}
we means a nontrivial real-valued function $x(t)$ which has the
properties $Z(t)\in C^1([\tau^*_{-1}(t_0), \infty), \mathbb{R})$,
and $r(t)\psi(x(t)Z'(t)\in C^1([\tau^*_{-1}(t_0), \infty),
\mathbb{R})$. Our attention is restricted to those solutions
$x(t)$ of \eqref{e1.1} which exist on some half-line
$[t_{x},\infty)$ with $\sup\{|x(t)|: t\geq T\}>0$ for any $T\geq
t_{x}$, and satisfy \eqref{e1.1}. As usual, a solution $x(t)$ of
\eqref{e1.1} is called oscillatory if the set of its zeros is
unbounded from above, otherwise, it is called nonoscillatory.
 \eqref{e1.1} is called
oscillatory if all solutions are oscillatory.

We note that second order neutral delay differential equations have various applications
in problems dealing with vibrating masses attached to an elastic bar and some variational
problems, etc. For further applications and questions concerning existence and uniqueness
of solutions of neutral delay differential equations, see \cite{h1}.

In the last decades, there has been an increasing interest in obtaining sufficient
conditions for the oscillation and/or nonoscillation of solutions of second order linear
and nonlinear neutral delay differential equations with distributed deviating
arguments (see, for example, \cite{b2,l1,w1,x1,y1}
and the references therein). Very recently, in \cite{x1},
the results in \cite{c1,p1,w2} for second order differential equations have been
extended to \eqref{e1.1}.

For other oscillation results of various neutral functional differential equations
we refer the reader to the monographs \cite{a1,a2,b1,e1,g1}.

The purpose of this paper is to establish some new oscillation criteria for
\eqref{e1.1} by introducing a class of functions $\Phi(t,s,T)$ defined in the recent
paper \cite{s2} and a generalized Riccati technique. Our results are different from
most known ones in the sense that they are given in the form that
$\limsup_{t \to \infty}[\cdot]$ is greater than a constant, rather than in the form
$\limsup_{t \to\infty}[\cdot]=+\infty$. Thus, our results can be applied to many
cases, which are not covered by existing ones.

Following the idea of Sun \cite{s2}, we say that a function $\Phi=\Phi(t,s,T)$ belongs to a
function class $\mathrm{X} $, denoted by $\Phi\in \mathrm{X}$, if
$\Phi\in C(E,\mathbb{R})$, where $E=\{(t,s,T):t_0 \leq T\leq s\leq t<\infty\}$, which
satisfies $\Phi(t,t,T)=\Phi(t,T,T)=0$, $\Phi(t,s,T)\neq0$ for $T<s<t$, and has the
partial derivative $\partial \Phi/\partial s$ on $E$ such that $\partial \Phi/\partial s$
is locally integrable with respect to $s$ on $E$.

We now recall to introduce another class of functions defined by Philos
\cite{p1} which is
used extensively. Let $D_{0}=\{(t,s):t>s>t_{0}\}$ and $D=\{(t,s):t\geq s\geq t_{0}\}$.
Say that $H\in C(D,\mathbb{R})$ belongs to a function class $\mathrm{Y}$, denoted
by $H\in \mathrm{Y}$, if $H(t,t)=0$ for $t\geq t_0$, $H(t,s)>0$ on $D_0$, $H$ has
continuous partial derivatives on $D_{0}$ with respect to $t$ and $s$.

Let us state three sets of conditions commonly used as in literature;
see for example \cite{s1,x1}, which we rely on:
\begin{itemize}

\item[(S1)] $f'(x)$ exists, $f'(x)\geq k_{1}$ and $\psi(x)\leq L^{-1}$
for $x\neq0$;

\item[(S2)] $f'(x)$ exists, $f'(x)/\psi(x)\geq k_{2}$ for $x\neq0$;

\item[(S3)] $f(x)/x\geq k_3$ and $\psi(x)\leq L^{-1}$ for $x\neq0$,\\
where $k_1$, $k_2$, $k_3$ and $L$ are positive real numbers.
\end{itemize}

In addition, we will use the following conditions as in \cite{s1,x1}:
\begin{itemize}

\item[(N1)] There exists a positive real number $M$ such that
$\pm f(\pm uv)\geq Mf(u)f(v)$ for $uv\geq0$;

\item[(N2)] $u\psi'(u)>0$ for $u\neq0$.
\end{itemize}

In order to prove our theorems we need the following three Lemmas whose
proof can be found in \cite{x1}.



\begin{lemma}[\cite{x1}] \label{lem1.1}
 Suppose that (S1) and (N1) are satisfied. Let $x(t)$ be an
eventually positive solution of \eqref{e1.1}. Then there
exists a $T_0\geq t_{0}$ such that
\begin{equation}
Z(t)>0, \quad  Z'(t)>0  \quad
(r(t)\psi(x(t))Z'(t))'\leq0,\quad\text{for }
t\geq T_{0}.
\label{e1.2}
\end{equation}
Moreover, for $t\geq T_{0}$,
\begin{equation}
(r(t)\psi(x(t))Z'(t))'+Mf[Z(g(t,a))]\int^b_aq(t,\xi)
f[1-p(g(t,\xi))]d\sigma(\xi)\leq0.
\label{e1.3}
\end{equation}
\end{lemma}

\begin{lemma}[\cite{x1}] \label{lem1.2}
 Suppose that (S2), (N1) and (N2) are satisfied. Let $x(t)$ be an
eventually positive solution of \eqref{e1.1}. Then there exists
a $T_0\geq t_{0}$ such that
\eqref{e1.2} and \eqref{e1.3} hold.
\end{lemma}


\begin{lemma}[\cite{x1}] \label{lem1.3}
Suppose that (S3) is satisfied. Let $x(t)$ be an eventually
positive solution of \eqref{e1.1}. Then there exists a
$T_0\geq t_0$ such that \eqref{e1.2} holds.
Moreover,
\begin{equation}
(r(t)\psi(x(t))Z'(t))' + k_3 Z[g(t,a)]\int^b_aq(t,\xi)
[1-p(g(t,\xi))]d\sigma(\xi)\leq0, \quad t\geq T_{0}.
\label{e1.4}
\end{equation}
\end{lemma}

\section{Kamenev-type oscillation criteria}

\begin{theorem} \label{thm2.1}
Let (S1) and (N1) hold. If there exist functions
$\varphi\in C^{1}(I,\mathbb{R}^+)$, $\phi \in C^1(I,\mathbb{R})$
and $\Phi\in \mathrm{X}$
such that for each $T\geq t_0$,
\begin{equation}
\limsup_{t\to\infty}\int^t_T\Big\{\Phi^2(t,s,T)\Theta_1(s)-\gamma_1(s)
\big[\Phi'_s(t,s,T)+\frac{1}{2}l_1(s)\Phi(t,s,T)\big]^2\Big\}ds>0,
\label{e2.1}
\end{equation}
where
\begin{gather*}
l_{1}(t)=\frac{\varphi'(t)}{\varphi(t)} +
\frac{2k_{1}Lg'(t,a)\phi(t)}{r[g(t,a)]}, \quad
\gamma_1(t)=\frac{r[g(t,a)]\varphi(t)}{k_{1}Lg'(t,a)},\\
\Theta_{1}(t)=\varphi(t)\Big\{M\int^{b}_{a}q(t,\xi)
f[1-p(g(t,\xi))]d\sigma(\xi)+\frac{k_{1}Lg'(t,a)\phi^{2}(t)}{r[g(t,a)]}
 -\phi'(t)\Big\},
\end{gather*}
then \eqref{e1.1} is oscillatory.
\end{theorem}


\begin{proof}
 Let $x(t)$ be a nonoscillatory solution of \eqref{e1.1} on $I$. Without loss
of generality we assume that $x(t)>0$ for $t\geq t_0$, (The case of $x(t)<0$ can be
considered similarly). By Lemma \ref{lem1.1}, there exists
a $T_{0}\geq t_{0}$ such that \eqref{e1.2}
and \eqref{e1.3} hold. Define
\begin{equation}
v(t)=\varphi(t)\Big[\frac{r(t)\psi(x(t))Z'(t)}{f[Z(g(t,a))]}
+\phi(t)\Big]\quad \mbox{for all } t\geq T_{0}. \label{e2.2}
\end{equation}
Differentiating \eqref{e2.2} and using \eqref{e1.3}, it follows that
\begin{align*}
v't)&\leq \frac{\varphi'(t)}{\varphi(t)} v(t)
-\varphi(t)\Big[M\int^b_aq(t, \xi)f[1-p(g(t, \xi))]d{\sigma(\xi)}\\
&\quad +\frac{r(t)\psi(x(t))Z\,'(t)}{f^2[Z(g(t,a))]}
f'[Z(g(t,a))]Z'[g(t,a)]g'(t,a)-\phi'(t)\Big].
\end{align*}
Since $g(t, a)\leq t$ and $(r(t)\psi(x(t))Z\,'(t))'\leq 0$,
we have
$$
r(t)\psi(x(t))Z\,'(t)\leq r[g(t,a)]\psi[x(g(t,a))]Z\,'[g(t,a)].
$$
Thus,
\begin{align*}
v'(t)&\leq  \frac{\varphi'(t)}{\varphi(t)} v(t)
-\varphi(t)\Big[M\int^b_aq(t, \xi)f[1-p(g(t, \xi))]d{\sigma}(\xi)
-\phi'(t)\Big]\\
&\quad -\frac{k_1\varphi(t)g'(t,a)}{r[g(t,a)]\psi[x(g(t,a))]}
\Big(\frac{r(t)\psi(x(t))Z\,'(t)}{f[Z(g(t,a))]}\Big)^2\\
&\leq  \frac{\varphi'(t)}{\varphi(t)} v(t)
-\varphi(t)\Big[M\int^b_aq(t, \xi)f[1-p(g(t, \xi))]d{\sigma}(\xi)
-\phi'(t)\Big]\\
&\quad -\frac{k_1L\,\varphi(t)g'(t,a)}{r[g(t,a)]}
\big(\frac{v(t)}{\varphi(t)}-\phi(t)\big)^2.
\end{align*}
So that
\begin{equation}
v'(s)\leq-\Theta_1(s)+l_1(s)v(s)-\frac{1}{\gamma_1(s)}v^2(s).
\label{e2.3}
\end{equation}
Multiplying \eqref{e2.3} by $\Phi^2(t,s,T_0)$, and integrating from $T_0$ to
$t$, we get
\begin{align*}
&\int^t_{T_0}\Phi^2(t,s,T_0)\Theta_1(s)ds\\
&\leq\int^t_{T_0}\Phi^2(t,s,T_0)[-v'(s)+ l_1(s)v(s)]ds
-\int^t_{T_0}\frac{1}{\gamma_1(s)}\Phi^2(t,s,T_0)v^2(s)ds.
\end{align*}
Integrating by parts, we obtain
\begin{align*}
&\int^t_{T_0}\Phi^2(t,s,T_0)\Theta_1(s)ds\\
&\leq2\int^t_{T_0}\Phi(t,s,T_0)\big[\Phi'_s(t,s,T_0)
+\frac{1}{2}l_1(s) \Phi(t,s,T_0)\big]v(s)ds\\
&\quad -\int^t_{T_0}\frac{1}{\gamma_1(s)}\Phi^2(t,s,T_0)v^2(s)ds\\
&=\int^t_{T_0}\gamma_1(s)\big[\Phi'_s(t,s,T_0) +
\frac{1}{2}l_1(s)\Phi(t,s,T_0)\big]^2ds\\
&\quad -\int^t_{T_0}\frac{1}{\gamma_1(s)}\big\{\Phi(t,s,T_0)v(s)
-\gamma_1(s)\big[\Phi'_s(t,s,T_0)
+ \frac{1}{2}l_1(s)\Phi(t,s,T_0)\big]\big\}^2ds,
\end{align*}
which implies
$$
\int^t_{T_0}\big\{\Phi^2(t,s,T_0)\Theta_1(s)
-\gamma_1(s)\big[\Phi'_s(t,s,T_0)
+ \frac{1}{2}l_1(s)\Phi(t,s,T_0)\big]^2\big\}ds\leq0.
$$
This contradicts \eqref{e2.1} and completes the proof.
\end{proof}

\begin{theorem} \label{thm2.2}
Let (S2), (N1), (N2) hold. If there exist functions
$\varphi\in C^{1}(I,\mathbb{R}^+)$, $\phi \in C^{1}(I,\mathbb{R})$ and
$\Phi\in\mathrm{X}$ such that for each $T\geq t_0$,
\begin{equation}
\limsup_{t\to\infty}\int^t_T\Big\{\Phi^2(t,s,T)\Theta_2(s)-\gamma_2(s)
\big[\Phi'_s(t,s,T)+
\frac{1}{2}l_{2}(s)\Phi(t,s,T)\big]^2\Big\}ds>0, \label{e2.4}
\end{equation}
where
\begin{gather*}
l_{2}(t)=\frac{\varphi'(t)}{\varphi(t)} +
\frac{2k_{2}g'(t,a)\phi(t)}{r[g(t,a)]}, \quad
\gamma_2(t)=\frac{r[g(t,a)]\varphi(t)}{k_{2}g'(t,a)},\\
\Theta_2(t)=\varphi(t)\Big\{M\int^{b}_{a}q(t,\xi)
f[1-p(g(t,\xi))]d\sigma(\xi) +
\frac{k_{2}g'(t,a)\phi^{2}(t)}{r[g(t,a)]} -\phi'(t)\Big\},
\end{gather*}
then \eqref{e1.1} is oscillatory.
\end{theorem}


\begin{proof}
Let $x(t)$ be a nonoscillatory solution of
\eqref{e1.1} on $I$, say $x(t)>0$ for $t\geq t_{0}$. Then, by
Lemma \ref{lem1.2}, there exists a $T_{0}\geq t_{0}$ such
that \eqref{e1.2} and \eqref{e1.3} hold. We
consider the function $v(t)$ defined by \eqref{e2.2},
and proceeding as in the proof of
Theorem \ref{thm2.1} to get
\begin{align*}
v'(t) &\leq  \frac{\varphi'(t)}{\varphi(t)} v(t)
- \varphi(t)\Big[M \int^{b}_{a}q(t,\xi)f[1-p(g(t,\xi))]d\sigma(\xi)
-\phi'(t)\Big] \\
&\quad-\frac{\varphi(t)g'(t,a)}{r[g(t,a)]\psi[x(g(t,a))]}
\frac{f'[Z(g(t,a))]}{\psi[x(g(t,a))]}
\left(\frac{r(t)\psi(x(t))Z'(t)}{f[Z(g(t,a))]}\right)^{2}.
\end{align*}
Now, we use $x[g(t,a)]\leq Z[g(t,a)]$ and $(N2)$ to obtain
$$
\frac{f'[Z(g(t,a))]}{\psi[x(g(t,a))]}
\geq \frac{f'[Z(g(t,a))]}{\psi[Z(g(t,a))]}\geq k_{2}.
$$
Therefore,
\begin{align*}
v'(t)
&\leq  \frac{\varphi'(t)}{\varphi(t)}v(t)
- \varphi(t)\Big[M \int^{b}_{a}q(t,\xi)f[1-p(g(t,\xi))]d\sigma(\xi)
-\phi'(t)\Big]\\
&\quad -\frac{k_{2}\varphi(t)g'(t,a)}{r[g(t,a)]}
\left(\frac{r(t)\psi(x(t))Z'(t)}{f[Z(g(t,a))]}\right)^{2}\\
&=-\Theta_2(t) +l_2(t)v(t)-\frac{1}{\gamma_{2}(t)}v^{2}(t) .
\end{align*}
The rest of the proof is as in Theorem \ref{thm2.1}.
\end{proof}

\begin{theorem} \label{thm2.3}
 Let (S3) holds. If there exist functions
$\varphi\in C^{1}(I,\mathbb{R}^+)$, $\phi \in C^{1}(I,\mathbb{R})$ and
$\Phi\in \mathrm{X}$ such that for each $T\geq t_0$,
\begin{equation}
\limsup_{t\to\infty}\int^t_T\Big\{\Phi^{2}(t,s,T)\Theta_3(s)
-\gamma_3(s)\big[\Phi'_s(t,s,T) +
\frac{1}{2}l_3(s)\Phi(t,s,T)\big]^2\Big\}ds>0, \label{e2.5}
\end{equation}
where
\begin{gather*}
l_3(t)=\frac{\varphi'(t)}{\varphi(t)} +
\frac{2Lg'(t,a)\phi(t)}{r[g(t,a)]}, \quad
\gamma_3(t)=\frac{r[g(t,a)]\varphi(t)}{Lg'(t,a)},\\
\Theta_3(t)=\varphi(t)\Big\{k_{3}\int^{b}_{a}q(t,\xi)
[1-p(g(t,\xi))]d\sigma(\xi) +\frac{Lg'(t,a)\phi^{2}(t)}{r[g(t,a)]}
-\phi'(t)\Big\},
\end{gather*}
then \eqref{e1.1} is oscillatory.
\end{theorem}


\begin{proof}
Let $x(t)$ be a nonoscillatory solution of
\eqref{e1.1} on $I$, say $x(t)>0$ for $t\geq t_{0}$. Then,
 by Lemma \ref{lem1.3},
there exists a $T_{0}\geq t_{0}$ such that \eqref{e1.2}
and \eqref{e1.4} hold. We
define the function
\begin{equation}
v(t)=\varphi(t)\Big[\frac{r(t)\psi(x(t))Z'(t)}{Z[g(t,a)] }+
\phi(t)\Big] \quad \mbox{for } t\geq T_{0}.
 \label{e2.6}
\end{equation}
Differentiating \eqref{e2.6} and using \eqref{e1.4}, we obtain
\begin{align*}
v'(t) &\leq
\frac{\varphi'(t)}{\varphi(t)}v(t) - \varphi(t)\Big\{k_{3}
\int^b_aq(t,\xi)[1-p(g(t,\xi))]d\sigma(\xi)
-\phi'(t)\Big\}\\
&\quad - \frac{\varphi(t)g'(t,a)}{r[g(t,a)]\psi[x(g(t,a))]}
\left(\frac{r(t)\psi(x(t))Z'(t)}{Z(g(t,a))}\right)^2\\
&\leq  \frac{\varphi'(t)}{\varphi(t)}v(t)
-\varphi(t)\Big\{k_{3}\int^b_aq(t,\xi)[1-p(g(t,\xi))]d\sigma(\xi)
-\phi'(t)\Big\}\\
&\quad -\frac{L\varphi(t)g'(t,a)}{r[g(t,a)]}
\left(\frac{v(t)}{\varphi(t)}-\phi(t)\right)^2\\
&=-\Theta_3(t) +l_{3}(t)v(t)-\frac{1}{\gamma_{3}(t)}v^{2}(t).
\end{align*}
The rest of the proof follows the same lines as that
of Theorem \ref{thm2.1}.
\end{proof}

Let $\Phi(t,s,T)=\sqrt{H_{1}(t,s)H_{2}(s,T)}$, where $H_1, H_2\in Y$.
By Theorems \ref{thm2.1}-\ref{thm2.3}, we have the following interesting results.

\begin{corollary} \label{coro2.1}
 Let (S1), (N1) hold. If there exist functions
$\varphi\in C^1(I,\mathbb{R}^+)$, $\phi\in C^1(I,\mathbb{R})$,
$\Phi\in \mathrm{X}$ and
$H_1, H_2\in \mathrm{Y}$ such that for each $T\geq t_{0}$,
\begin{equation}
\limsup_{t\to\infty}\int^t_TH_1(t,s)H_2(s,T)\Big\{\Theta_1(s)-\frac{1}{4}\gamma_1(s)
[h_1(t,s)+h_2(s,T)+l_1(s)]^2\Big\}ds > 0, \label{e2.7}
\end{equation}
where $h_1(t,s)$ and $ h_2(s,T)$ are defined by
\begin{equation}
\frac{\partial H_1(t,s)}{\partial s}=h_1(t,s)H_1(t,s) \quad
\mbox{and} \quad \frac{\partial H_2(s,T)} {\partial s}=h_2(s,T)H_2(s,T),
\label{e2.8}
\end{equation}
then \eqref{e1.1} is oscillatory.
\end{corollary}


\begin{corollary} \label{coro2.2}
 Let (S2), (N1), (N2) hold. If there exist functions
$\varphi\in C^{1}(I,\mathbb{R}^+)$, $\phi\in C^{1}(I,\mathbb{R})$, $\Phi\in\mathrm{X}$
and $H_1, H_2\in \mathrm{Y}$ such that for each $T\geq t_0$,
\begin{equation}
\limsup_{t\to\infty}\int^t_TH_1(t,s)H_2(s,T)\Big\{\Theta_2(s)
-\frac{1}{4}\gamma_2(s)[h_1(t,s) +h_2(s,T) + l_2(s)]^2\Big\}ds >0,
\label{e2.9}
\end{equation}
where $h_1(t,s)$ and $h_2(s,T)$ are defined by \eqref{e2.8},
then \eqref{e1.1} is oscillatory.
\end{corollary}


\begin{corollary} \label{coro2.3}
 Let (S3) hold. If there are functions
$\varphi\in C^{1}(I,\mathbb{R}^+)$, $\phi \in C^{1}(I,\mathbb{R})$,
$\Phi\in \mathrm{X}$
and $H_1, H_2\in\mathrm{Y}$ such that for each $T\geq t_{0}$,
\begin{equation}
\limsup_{t\to\infty}\int^t_TH_1(t,s)H_2(s,T)\Big\{\Theta_3(s)
-\frac{1}{4}\gamma_3(s)[h_1(t,s) +h_2(s,T) + l_3(s)]^2\Big\}ds >0,
\label{e2.10}
\end{equation}
where $h_1(t,s)$ and $h_2(s,T)$ are defined by \eqref{e2.8}, then \eqref{e1.1}
is oscillatory.
\end{corollary}

Let $\Phi(t,s,T) = (t-s)(s-T)^{\alpha}$ for $\alpha>1/2$.
By Theorems \ref{thm2.1}-\ref{thm2.3},
we can establish the following important results.


\begin{corollary} \label{coro2.4}
 Let (S1) and (N1) hold. If there exist functions
$\varphi\in C^{2}(I,\mathbb{R}^+)$, $\phi\in C^1(I,\mathbb{R})$ and constants
$\alpha>1/2$, $m_1>0$ such that $\gamma_1(t)\leq m_1$ and for each $T\geq t_0$,
\begin{equation}
\limsup_{t\to\infty}\frac{1}{t^{2\alpha+1}}\int^t_T(t-s)^2(s-T)^{2\alpha}
\Big[\frac{1}{m_1}\Theta_1(s)+\frac{1}{2}l'_1(s)
-\frac{1}{4}l^2_{1}(s)\Big]ds>\frac{\alpha}{(2\alpha
-1)(2\alpha+1)}, \label{e2.11}
\end{equation}
then \eqref{e1.1} is oscillatory.
\end{corollary}

\begin{proof}
 Suppose that \eqref{e1.1} has a nonoscillatory solution $x(t)>0$. By using the
same arguments as in the proof of Theorem \ref{thm2.1} and denoting
$(t-s)(s-T_0)^{\alpha}$ by $\Phi(t,s,T_0)$, we have
\begin{align*}
&\int^t_{T_0}\Phi^{2}(t,s,T_0)\Theta_1(s)ds \\
&\leq\int^t_{T_0}\gamma_1(s) \big[\Phi'_{s}(t,s,T_0) +
\frac{1}{2}l_1(s)\Phi(t,s,T_0)\big]^2ds \\
&\leq m_1\int^t_{T_0}\Big\{\Phi'\,^2_{s}(t,s,T_0)
+ \Phi'_{s}(t,s,T_0)\Phi(t,s,T_0)l_1(s)
+\frac{1}{4}\Phi^2(t,s,T_0)l^2_{1}(s)\Big\}ds.
\end{align*}
Noting that
$$
\int^t_{T_0}\Phi'_{s}(t,s,T_0)\Phi(t,s,T_0)l_1(s)ds
=-\frac{1}{2} \int^t_{T_0}\Phi^2(t,s,T_0)l'_1(s)ds,
$$
we get
\begin{align*}
&\frac{1}{m_1}\int^t_{T_0} \Phi^2(t,s,T_0)\Theta_1(s)ds\\
&\leq \int^t_{T_0}\Big\{\Phi'\,^2_{s}(t,s,T_0)
-\frac{1}{2}\Phi^2(t,s,T_0)l'_1(s)
+ \frac{1}{4}\Phi^2(t,s,T_0)l^2_1(s)\Big\}ds \\
& =\int^t_{T_0}\Phi^2(t,s,T_0)\big[-\frac{1}{2}l'_1(s)
+\frac{1}{4}l^2_1(s)\big]ds
+\int^t_{T_0}[\alpha(t-s)(s-T_0)^{\alpha-1}-(s-T_0)^{\alpha}]^2ds\\
& =\int^t_{T_0}\Phi^2(t,s,T_0)\big[-\frac{1}{2}l'_1(s)
+\frac{1}{4}l^2_1(s)\big]ds
+\frac{\alpha}{(2\alpha-1)(2\alpha+1)}(t-T_0)^{2\alpha+1}.
\end{align*}
Therefore,
\begin{align*}
&\limsup_{t\to\infty}\frac{1}{t^{2\alpha+1}}
\int^t_{T_0}(t-s)^2(s-T_0)^{2\alpha}\Big[\frac{1}{m_1}\Theta_1(s)+
\frac{1}{2}l'_{1}(s)-\frac{1}{4}l^2_1(s)\Big]ds\\
&\leq\frac{\alpha}{(2\alpha -1)(2\alpha+1)},
\end{align*}
which contradicts \eqref{e2.11} and completest the proof.
\end{proof}

Similar to the proof of Corollary \ref{coro2.4}, by Theorems \ref{thm2.2}
 and \ref{thm2.3}, we can easily obtain the following results.

\begin{corollary} \label{coro2.5}
 Let (S2), (N1), (N2) hold. If there exist functions
$\varphi\in C^{2}(I,\mathbb{R}^+)$, $\phi\in C^1(I,\mathbb{R})$
and constants
$\alpha>1/2$, $m_2>0$ such that $\gamma_2(t)\leq m_2$ and for
each $T\geq t_0$,
\begin{equation}
\limsup_{t\to\infty}\frac{1}{t^{2\alpha+1}}\int^t_T(t-s)^2(s-T)^{2\alpha}
\Big[\frac{1}{m_2}\Theta_2(s)+\frac{1}{2}l'_2(s)
-\frac{1}{4}l^2_2(s)\Big]ds
>\frac{\alpha}{(2\alpha
-1)(2\alpha+1)},
\label{e2.12}
\end{equation}
then \eqref{e1.1} is oscillatory.
\end{corollary}


\begin{corollary} \label{coro2.6}
 Let  (S3) holds. If there exist functions
$\varphi\in C^{2}(I,\mathbb{R}^+)$, $\phi\in C^1(I,\mathbb{R})$ and constants
$\alpha>1/2$, $m_3>0$ such that $\gamma_3(t)\leq m_3$ and for each $T\geq t_0$,
\begin{equation}
\limsup_{t\to\infty}\frac{1}{t^{2\alpha+1}}\int^t_T(t-s)^2(s-T)^{2\alpha}
\Big[\frac{1}{m_3}\Theta_3(s)+\frac{1}{2}l'_3(s)
-\frac{1}{4}l^2_3(s)\Big]ds
>\frac{\alpha}{(2\alpha
-1)(2\alpha+1)},
\label{e2.13}
\end{equation}
then \eqref{e1.1} is oscillatory.
\end{corollary}

\section{Interval oscillation criteria}

We can easily see that the results in Section 2 involve the integral
of the coefficients $p$, $q$ and $r$, and hence, requires the
information of the coefficients on the entire
half-line $[t_0, \infty)$. In this section, we will
establish several interval
oscillation criteria for \eqref{e1.1}.

\begin{theorem} \label{thm3.1}
Let (S1) and (N1) hold. If for each $T\geq t_0$, there exist
functions $\varphi\in C^{1}(I,\mathbb{R}^+)$, $\phi \in C^1(I,\mathbb{R})$, $\Phi\in
\mathrm{X}$ and two constants $d>c\geq T$ such that
\begin{equation}
\int^d_c\Big\{\Phi^2(d,s,c)\Theta_1(s)-\gamma_1(s)
\big[\Phi'_s(d,s,c)+\frac{1}{2}l_1(s)\Phi(d,s,c)\big]^2\Big\}ds>0,
\label{e3.1}
\end{equation}
where $\Theta_1$, $\gamma_1$, $l_1$ are defined as in Theorem \ref{thm2.1},
then \eqref{e1.1} is oscillatory.
\end{theorem}


\begin{proof}
With the proof of Theorem \ref{thm2.1}, where $t$ and $T$ are replaced by
$d$ and $c$, respectively, we can easily see that every solution
of \eqref{e1.1} has at least one zero in $(c, d)$;
 i.e., every solution of \eqref{e1.1} has arbitrarily
large zeros on $[t_0, \infty)$. This completes the proof of
Theorem \ref{thm3.1}.
\end{proof}

Similar to the proof of Theorem \ref{thm3.1}, we can establish
the following theorems.

\begin{theorem} \label{thm3.2}
 Let (S2), (N1) and (N2) hold. If for each $T\geq t_0$,
there exist functions $\varphi\in C^{1}(I,\mathbb{R}^+)$,
$\phi \in C^{1}(I,\mathbb{R})$, $\Phi\in\mathrm{X}$ and two constants
$d>c\geq T$ such that
\begin{equation}
\int^d_c\Big\{\Phi^2(d,s,c)\Theta_2(s)-\gamma_2(s)
\big[\Phi'_s(d,s,c)+ \frac{1}{2}l_{2}(s)\Phi(d,s,c)\big]^2\Big\}ds>0,
\label{e3.2}
\end{equation}
where $\Theta_2$, $\gamma_2$, $l_2$ are defined as in Theorem \ref{thm2.2},
 then \eqref{e1.1} is oscillatory.
\end{theorem}


\begin{theorem} \label{thm3.3}
 Let (S3) holds. If for each $T\geq t_0$, there exist
functions $\varphi\in C^{1}(I,\mathbb{R}^+)$, $\phi \in C^{1}(I,\mathbb{R})$,
$\Phi\in\mathrm{X}$ and two constants $d>c\geq T$ such that
\begin{equation}
\int^d_c\Big\{\Phi^2(d,s,c)\Theta_3(s)-\gamma_3(s)
\big[\Phi'_s(d,s,c)+ \frac{1}{2}l_{3}(s)\Phi(d,s,c)\big]^2\Big\}ds>0,
\label{e3.3}
\end{equation}
where $\Theta_3$, $\gamma_3$, $l_3$ are defined as in Theorem \ref{thm2.3},
 then \eqref{e1.1} is oscillatory.
\end{theorem}


\begin{corollary} \label{coro3.1}
 Let (S1) and (N1) hold. If for each $T\geq t_0$, there exist
functions $H_1, H_2\in Y$ and two constants $d>c\geq T$ such that
\begin{equation}
\int^d_cH_1(d,s)H_2(s,c)\Big\{\Theta_1(s)-\frac{1}{4}\gamma_1(s)
\big[h_1(d,s)+h_2(s,c)+l_1(s)\big]^2\Big\}ds > 0, \label{e3.4}
\end{equation}
where $h_1(d,s)$ and $h_2(s,c)$ are defined by \eqref{e2.8}, then \eqref{e1.1}
is oscillatory.
\end{corollary}


\begin{corollary} \label{coro3.2}
 Let (S2), (N1) and (N2) hold.
If for each $T\geq t_0$, there exist functions $H_1, H_2\in Y$ and
two constants $d>c\geq T$ such that
\begin{equation}
\int^d_cH_1(d,s)H_2(s,c)\Big\{\Theta_2(s)-\frac{1}{4}\gamma_2(s)[h_1(d,s)
+h_2(s,c)+l_2(s)]^2\Big\}ds > 0,
\label{e3.5}
\end{equation}
where $h_1(d,s)$ and $h_2(s,c)$ are defined by \eqref{e2.8}, then \eqref{e1.1}
is oscillatory.
\end{corollary}


\begin{corollary} \label{coro3.3}
 Let (S3) holds. If for each
$T\geq t_0$, there exist functions $H_1, H_2\in Y$ and two constants
$d>c\geq T$ such that
\begin{equation}
\int^d_cH_1(d,s)H_2(s,c)\Big\{\Theta_3(s)-\frac{1}{4}\gamma_3(s)[h_1(d,s)
+h_2(s,c)+l_3(s)]^2\Big\}ds > 0,
\label{e3.6}
\end{equation}
where $h_1(d,s)$ and $h_2(s,c)$ are defined by \eqref{e2.8}, then \eqref{e1.1}
is oscillatory.
\end{corollary}

\section{Examples}

\begin{example} \label{exa4.1} \rm
 Consider the equation
\begin{equation}
\Big(\frac{1}{e^t(1+x^2(t))}\big(x(t)+(1-e^{-t})x(t-1)\big)'
\Big)'
+\mu\int^1_{\frac{1}{2}}\frac{\xi x(\ln t\xi)}{t}d\xi =0,\quad t\geq2,
\label{e4.1}
\end{equation}
where $r(t)=e^{-t}$, $\psi(x)= 1/(1+x^{2})$, $p(t)=1-e^{-t}$, $\mu>1$,
$g(t,\xi)=\ln t \xi$, $f(x)=x$ and $q(t,\xi)=\mu \xi/ t$.

If we take $k_1=L=M=1$, $m_1=2$, $\phi(t)=t$ and $\varphi(t)=1$,
then
$$
\Theta_1(t) =\frac{\mu}{2t^2} + \frac{1}{2}t^2 - 1, \quad l_1(t) = t,
\quad  \gamma_1(t)=2.
$$
Thus, the left-hand side of \eqref{e2.11} takes the following from
$$
\frac{\mu}{4}\limsup_{t\to\infty}\frac{1}{t^{2\alpha+1}}
\int^t_T(t-s)^{2}(s-T)^{2\alpha}\frac{1}{s^{2}}ds
=\frac{\mu}{4}\frac{1}{\alpha(2\alpha-1)(2\alpha + 1)}.
$$
For any $\mu>1$, there exists a constant $\alpha> 1/2$ such that
$\mu/4>\alpha^2$, i.e.,
$$
\frac{\mu}{4}\frac{1}{\alpha(2\alpha-1)(2\alpha + 1)}
>\frac{\alpha}{(2\alpha -1)(2\alpha+1)},
$$
i.e., \eqref{e2.11} holds for $\mu>1$.
By Corollary \ref{coro2.4}, \eqref{e4.1} is oscillatory for $\mu>1$.
\end{example}


\begin{example} \label{exa4.2}\rm
 Consider the equation
\begin{equation}
\Big(x^2(t)\big(x(t)+(1-\frac{1}{t})x(t-1)\big)'
\Big)'+\int^1_{\frac{1}{2}}t^4\xi^3x^3(t\xi)d\xi=0,
\quad t\geq2,
\label{e4.2}
\end{equation}
where $r(t)=1$, $\psi(x) = x^2$, $p(t)=1-1/t$,  $g(t,\xi)=t \xi$, $f(x)=x^3$
and $q(t,\xi)=t^4\xi^3$.

If we take $\varphi(t)=1$, $\phi(t)=0$, $m_2=2/3$, $k_2 =3$ and $M=1$, then
$$
\Theta_2(t)=\frac{t}{2}, \quad \gamma_2(t)=\frac{2}{3}, \quad l_2(t)=0.
$$
For any constant $T\geq 2$, there exists $n\in
\mathbb{N}_0=\{1,2,\cdots\}$ such that $2n \pi \geq T$. Let
$d=(2n+1)\pi$, $c=2n\pi \geq T$ and
$H_1(t,s)H_2(s,T)=|\sin(t-s)\sin(s-T)|$, we have
$$
H_1((2n+1)\pi,s)H_{2}(s,2n\pi)=\sin^2s\quad \mbox{for}\quad c\leq s\leq d.
$$
Thus, the left-hand side of \eqref{e3.5} takes the  from
$$
\int^{(2n+1)\pi}_{2n\pi}\sin^2s\big(\frac{s}{2}-\frac{4}{3}
\cot^2s\big)ds=\frac{(4n+1)\pi^2}{8}-\frac{2\pi}{3}>0.
$$
Hence, by Corollary \ref{coro3.2},  \eqref{e4.2} is oscillatory.
\end{example}


\begin{example} \label{exa4.3} \rm
 Consider the equation
\begin{equation}
\Big(\frac{1}{t(1+x^2(t))}\big(x(t)+\frac{1}{2}x(t-1)
\big)'\Big)'+\mu\int^1_0\frac{\xi}{t^2}\frac{x^3(\sqrt{t+\xi})
+3x(\sqrt{t+\xi})}{1+x^2(\sqrt{t+\xi})}d\xi=0, \quad  t\geq1,
\label{e4.3}
\end{equation}
where $r(t)=1/t$, $\psi(x) = 1/(1+x^2)$, $p(t)=1/2$, $g(t,\xi)=\sqrt{t+\xi}$,
$f(x)=(x^3+3x)/(1+x^2)$, $q(t,\xi)=\mu\xi/t^2$ and $\mu>2$.

If we take $k_3=L=1$, $m_3=2$, $\varphi(t)=1$, $\phi(t)=1/t$, then
$$
\Theta_3(t)=\frac{\mu+6}{4t^2}, \quad l_{3}(t) =\frac{1}{t},
\quad \gamma_3(t)=2.
$$
Thus, the left-hand side of \eqref{e2.13} takes the following from
$$
\frac{\mu}{8}\limsup_{t\to\infty}\frac{1}{t^{2\alpha+1}}
\int^t_T(t-s)^2(s-T)^{2\alpha}\frac{1}{s^2}ds
=\frac{\mu}{8} \frac{1}{\alpha(2\alpha-1)(2\alpha + 1)}.
$$
So, for any $\mu>2$, there exists a constant $\alpha>1/2$ such that
$$
\frac{\mu}{8}\frac{1}{\alpha(2\alpha-1)(2\alpha+1)}
>\frac{\alpha}{(2\alpha -1)(2\alpha+1)}.
$$
Therefore, by Corollary \ref{coro2.6},  Equation \eqref{e4.3} is oscillatory for $\mu>2$.
\end{example}


\subsection*{Acknowledgments}
The authors are  grateful to the anonymous referee for their valuable
 comments and suggestions which  led to an improvement of the
original manuscript.


\begin{thebibliography}{00}

\bibitem{a1} R. P. Agarwal, S. R. Grace, D. O'Regan,
\emph{Oscillation Theory of Difference and Functional Differential Equations},
 Kluwer. Dordreche, 2000.

\bibitem{a2} R. P. Agarwal, S. R. Grace, D. O'Regan,
\emph{Oscillation Theory for Second Order Dynamic Equations},
Taylor \& Francis Group, London, 2003.

\bibitem{b1} D. D. Bainov, D. P. Misher,
\emph{Oscillation Theory for Neutral Equations with Delay},
 Adam Hilger IOP Publishing Ltd, 1991.

\bibitem{b2} D. D. Bainov, V. Petrov,
\emph{Asymptotic properties of the nonoscillatory
solutions of second order neutral equations with a deviating argument},
 J. Math. Anal. Appl., \textbf{190} (1995) 645-653.

\bibitem{c1} W. J. Coles,
\emph{An oscillation criterion for second order differential
equations},  Proc. Amer. Math. Soc., \textbf{19} (1968) 755-759.

\bibitem{e1} L. H. Erbe, Q. Kong, B. G. Zhang,
\emph{Oscillation Theory for Functional Differential Equations},
 Marcel Dekker, New York, 1995.

\bibitem{g1} I. Gyori, G. Ladas,
\emph{Oscillation Theory of Delay Differential Equations
with Applications}, Clarendon Press, Oxford, 1991.

\bibitem{h1} J. K. Hale, \emph{Theory of Functional Differential Equations},
Springer-Verlag, New York, 1977.

\bibitem{k1} I. V. Kamenev,
\emph{An integral criterion for oscillation of linear
differential equation of second order},  Mat. Zametki., \textbf{23} (1978)
248-251.

\bibitem{l1} X. Z. Liu, X. L. Fu,
\emph{Nonlinear differential inequalities with
distritbuted deviating argument and applications},
Nonlin. World, \textbf{4} (1994) 409-427.

\bibitem{p1} Ch. G. Philos,
\emph{Oscillation theorems for linear differential equations
of second order},  Arch. Math., (Besel), \textbf{53} (1989) 482-492.

\bibitem{s1} Y. Sahiner,
\emph{On oscillation of second order neutral type delay
differential equations},  Appl. Math. Comput., \textbf{150} (2004) 697-706.

\bibitem{s2} Y. G. Sun,
\emph{New Kamenev-type oscillation criteria for second order
nonlinear differential equations with damping}, J. Math. Anal. Appl.,
\textbf{291} (2004) 341-351.

\bibitem{w1} P. Wang,
\emph{Oscillation criteria for second order neutral equations with
distributed deviating arguments},
Comput. Math. Appl., \textbf{47} (2004) 1935-1946.

\bibitem{w2} J. S. W. Wong,
\emph{On Kamenev-type oscillation theorems for second order
differential equations with damping},
J. Math. Anal. Appl.,  \textbf{258} (2001) 244-257.

\bibitem{x1} Z. Xu, P. Weng,
\emph{Oscillation of second order neutral equations with
distributed deviating argument}, J. Comput. Appl. Math.,
doi:10.1016/j.cam.2006.03.001.

\bibitem{y1} Y. H. Yu, X. L. Fu,
\emph{Oscillation of second order  neutral
equation with continuous distributed deviating argument},  Rad. Math.,
\textbf{7} (1991) 167-176.

\end{thebibliography}
\end{document}