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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 58, pp. 1--9.\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-2007/58\hfil Interval oscillation]
{Interval oscillation of second-order Emden-Fowler neutral delay
differential equations}

\author[M. Chen, Z. Xu\hfil EJDE-2007/58\hfilneg]
{Mu Chen, Zhiting Xu}  % in alphabetical order

\address{Mu Chen \newline
School of Mathematical Sciences, South China Normal University \\
Guangzhou, 510631,  China. \newline
Department of Mathematics, Zhanjiang Education College \\
Zhanjiang, Guangdong 524037, China}
\email{zjchenmu03@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 January 5, 2007. Published April 22, 2007.}
\subjclass[2000]{34K40, 34K11, 34C10}
\keywords{Interval oscillation; neutral delay differential equation;
\hfill\break\indent Emden-Fowler; Riccati technique; integral averaging method}

\begin{abstract}
 Employing  Riccati techniques and the integral averaging method,
 we establish interval oscillation criteria  for the second-order
 Emden-Fowler neutral delay differential equation
 \[
 [|x'(t)|^{\gamma-1}x'(t)]'+q_1(t)|y(t-\sigma)|^{\alpha-1}y(t-\sigma)
 +q_2(t)|y(t-\sigma)|^{\beta-1} y(t-\sigma)=0,
 \]
 where $t\geq t_0$ and $x(t)=y(t)+p(t)y(t-\tau)$.  The criteria
 obtained here are different from most known criteria in the sense
 that they are based on information only on a sequence of subintervals
 of $[t_0,\infty)$, rather than on the whole half-line. In particular,
 two interesting examples that illustrate the importance of our results
 are included.
\end{abstract}

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

\section{Introduction}

Consider the second-order Emden-Fowler neutral delay differential equation
\begin{equation} \label{e1.1}
[|x'(t)|^{\gamma-1}x'(t)]'+q_1(t)|y(t-\sigma)|^{\alpha-1}y(t-\sigma)
+q_2(t)|y(t-\sigma)|^{\beta-1} y(t-\sigma)=0,
\end{equation}
where $t\geq t_0$ and $x(t)=y(t)+p(t)y(t-\tau)$. In what follows we
assume that
\begin{itemize}
\item[(A1)] $\tau$ and $\sigma$ are nonnegative constants, $\alpha$,
$\beta$ and $\gamma$ are positive constants with $0<\alpha<\gamma<\beta$;

\item[(A2)]  $q_1,\, q_2\in \mathbf{C}([t_0,\infty), \mathbb{R}^+)$, $\mathbb{R}^+=(0,\infty)$;

\item[(A3)]  $p\in \mathbf{C}([t_0,\infty), \mathbb{R})$, and $-1<p_0\leq p(t)\leq 1$,
 $p_0$ is a constant.
\end{itemize}

For any $\varphi\in\mathbf{C}([t_0-\theta, t_0], \mathbb{R})$,
$\theta=\max\{\tau,\sigma\}$,  \eqref{e1.1} has a solution $y(t)$
extendable on $[t_0,\infty)$ satisfying the initial condition
$y(t)\equiv \varphi(t)$ for $[t_0-\theta, t_0]$; see, e.g., Hale
\cite{h1}. Our attention is restricted to those solutions $y=y(t)$ of
\eqref{e1.1} which exist on some half-line $[t_y,\infty)$ with
$\sup\{|y(t)|:t\geq T \}>0$ for any $T\geq t_y$, and satisfy
\eqref{e1.1}. As usual, a nontrivial solution of \eqref{e1.1} is
called oscillatory if it has arbitrarily large zeros, otherwise it
is called nonoscillatory. Finally, \eqref{e1.1} is called
oscillatory if all its solutions are oscillatory. We say that
\eqref{e1.1} satisfies the superlinear condition if $q_1(t)\equiv
0$ and it satisfies the sublinear condition if $q_2(t)\equiv 0$.

We note that second order neutral delay differential equation are
used in  many fields such as vibrating masses attached to an
elastic bar and some variational problems, etc., see Hale \cite{h1}.

In the last decades, there has been an increasing interest in
obtaining  sufficient conditions for the oscillation and/or
nonoscillation of second order linear and nonlinear neutral delay
differential equations (see, for example, the monographs \cite{a1,b1,e1}
and the references therein). Recently, the results of Atkinson \cite{a2}
and Belohorec \cite{b2} for the Emden-Fowler equation
\begin{equation} \label{e1.2}
y''(t)+q(t)|y(t)|^{\gamma-1}y(t)=0, \quad q\in \mathbf{C}([t_0, \infty), \mathbb{R})
\,\,\,\mbox{and}\,\,\, \gamma>0
\end{equation}
have been extended to the second order neutral delay differential equation
\begin{equation} \label{e1.3}
 [y(t)+p(t)y(t-\tau)]''+q(t)f((y-\sigma))=0
\end{equation}
by Wong \cite{w1} under the assumption that the nonlinear function $f$ satisfies the
sublinear condition
\begin{equation*}
0<\int_{0^+}^{\varepsilon}\frac{du}{f(u)},\quad \int_{0^-}^{-\varepsilon}\frac{du}{f(u)}
<\infty \quad \mbox{for all}\,\,\varepsilon>0,
\end{equation*}
as well as the superlinear condition
\begin{equation*}
0<\int_\varepsilon^\infty\frac{du}{f(u)},\quad \int_{-\varepsilon}^{-\infty}
\frac{du}{f(u)}<\infty \quad \mbox{for all}\,\,\varepsilon>0.
\end{equation*}
Also it will be of great interest to find some oscillation criteria for special case for
\eqref{e1.3}, even for the Emden-Fowler neutral delay differential  equation
\begin{equation} \label{e1.4}
[y(t)+p(t)y(t-\tau)]''+q(t)|y(t-\sigma)|^{\gamma-1}y(t-\sigma)=0,
\quad \gamma>0.
\end{equation}
This problem was posed by [13, Remark d]. As an positive answer to
it, Saker \cite{s1}, Saker and Manojlovi\`{c} \cite{s2}, and Xu
and Liu \cite{x1} have given some oscillation criteria for
\eqref{e1.1}, \eqref{e1.3} and \eqref{e1.4}. As we know, the
results obtained in \cite{s1,s2,w1,x1} involve the integral of the
functions $q,\, q_1,\, q_2$ and hence require the information of
those functions on the the entire half-linear $[t_0, \infty)$. As
pointed out in Kong \cite{k1}, oscillation is an interval
property, that is, it is more reasonable to investigate solutions
on an infinite set of bounded intervals. Therefore, the problem is
to find oscillation criteria which use only the information about
the involved functions on these intervals; outside of these
intervals the behavior of the functions is irrelevant. Such type
of criteria are referred to as interval oscillation criteria. The
first beautiful interval criteria in this direction was due to
Kong \cite{k1}, who gave some interval criteria for the
oscillation of the second order linear ordinary differentia
equation
\begin{equation} \label{e1.5}
(r(t)y'(t))'(t)+q(t)y(t)=0.
\end{equation}
Recently, Yang et al \cite{y1} extended Kong-type interval criteria to
certain neutral differential equations.

Motivated by the ideas of Kong \cite{k1} and Philos \cite{p1}, in this paper,
by using Riccati technique and the integral averaging method, we
will establish some interval oscillation criteria for
\eqref{e1.1}, that is, criteria given by the behavior of
\eqref{e1.1} only on a sequence of subintervals of $[t_0, \infty)$
(see Theorems \ref{thm2.1}--\ref{thm2.5} for details) rather than the whole
half-line. Our theorems essentially improve some known results in
\cite{s1,x1}. In particular, two interesting examples that illustrate
the importance of our results are also included.

\section{Main results}

In this section, we shall establish Kong-type interval oscillation
criteria for \eqref{e1.1} under the cases when $0\leq p(t)\leq 1$
and $-1<p_0\leq p(t)\leq 0$. It will be convenient to make the
following notations in the remainder of this paper. Define
\begin{gather*}
\mu=\min\big\{\frac{\beta-\alpha}{\beta-\gamma},
\frac{\beta-\alpha}{\gamma-\alpha}\big\},
\quad k=\frac{1}{(1+\gamma)^{1+\gamma}}\\
Q_1(t)=\mu[1-p(t-\sigma)]^\gamma[q_1^{\beta-\gamma}(t)
q_2^{\gamma-\alpha}(t)]^{1/{(\beta-\alpha)}},\\
Q_2(t)=\mu[q_1^{\beta-\gamma}(t)q_2^{\gamma-\alpha}(t)]^{1/{(\beta-\alpha)}}.
\end{gather*}

In the sequel, we say the a function $H=H(t,s)$  belongs to a
function class $\mathcal{H}$, denoted by $H\in \mathcal{H}$, if
$H\in \mathbf{C}(D,[0,\infty))$, where $D=\{(t,s):-\infty<s\leq
t<\infty\}$, and $H$  satisfies
\begin{itemize}

\item[(H1)] $H(t,t)=0$, $H(t,s)>0$ for $t>s$;

\item[(H2)] $H$ has partial derivatives $\partial H/\partial t$ and $\partial H/\partial s$ on $D$
 such that
\begin{equation*}
\frac{\partial H}{\partial t}(t,s)=h_1(t,s)\sqrt{H(t,s)} \quad \mbox{and}\quad
\frac{\partial H}{\partial s}(t,s)=-h_2(t,s)\sqrt{H(t,s)},
\end{equation*}
where $h_1, h_2\in L_{loc}(D, \mathbb{R})$.
\end{itemize}

The following Lemma will be useful for establishing oscillation
criteria for \eqref{e1.1} whose proof can be found in \cite{t1}.

\begin{lemma} \label{lem2.1}
Let $A_0$, $A_1$, $A_2\in \mathbf{C}([t_0, \infty), \mathbb{R})$ with $A_2>0$, and
$w\in \mathbf{C}^1([t_0, \infty), \mathbb{R})$. If there exist interval
$(a,b)\subset [t_0, \infty)$  such that
\begin{equation} \label{e2.1}
w'(s)\leq -A_0(s)+A_1(s)w(s)-A_2(s)|w(s)|^{(\gamma+1)/\gamma}, \quad s\in(a,b),
\end{equation}
then for any $c\in (a, b)$,
 \begin{equation} \label{e2.2}
\begin{aligned}
&\frac{1}{H(c,a)}\int^c_a\Big[H(s,a)A_0(s)-\frac{k\,\gamma^{\gamma}}{(A_2(s))^{\gamma}}
|\phi_1(s,a)|^{\gamma+1}\Big]ds\\
&+\frac{1}{H(b,c)}\int^b_c\Big[H(b,s)A_0(s)
-\frac{k\,\gamma^{\gamma}}{(A_2(s))^{\gamma}}|\phi_2(b,s)|^{\gamma+1}\Big]ds
\leq 0
\end{aligned}
\end{equation}
for every $H\in\mathcal{H}$, where
\begin{gather*}
\phi_1(s,a)=\frac{h_1(s,a)\sqrt{H(s,a)}+A_1(s)H(s,a)}{(H(s,a))^{\gamma/(\gamma+1)}},\\
\phi_2(b,s)=\frac{-h_2(b,s)\sqrt{H(b,s)}+A_1(s)H(b,s)}{(H(b,s))^{\gamma/(\gamma+1)}}.
\end{gather*}
\end{lemma}

\begin{theorem} \label{thm2.1}
Suppose that there exist interval $(a, b)\subset [t_0, \infty)$,
constant $c\in(a,b)$, and functions  $H\in \mathcal{H}$, $\rho\in
\mathbf{C}^1([t_0,\infty), \mathbb{R}^+)$, such that one of the following two
conditions is satisfied:
\begin{itemize}

\item[(C1)] $0\leq p(t)\leq 1$, and
\begin{equation} \label{e2.3}
\begin{aligned}
&\frac{1}{H(c,a)}\int^c_a\rho(s)
\big[H(s,a)Q_1(s)-k|\phi_1(s,a)|^{\gamma+1}\big]ds \\
&+\frac{1}{H(b,c)}\int^b_c\rho(s)\big[H(b,s)Q_1(s)
-k|\phi_2(b,s)|^{\gamma+1}\big]ds> 0;
\end{aligned}
\end{equation}

\item[(C2)] $-1<p_0\leq p(t)\leq 0$, and
\begin{equation} \label{e2.4}
\begin{aligned}
&\frac{1}{H(c,a)}\int^c_a\rho(s)\Big[H(s,a)Q_2(s)-k|\phi_1(s,a)|^{\gamma+1}\Big]ds
\\
& +\frac{1}{H(b,c)}\int^b_c\rho(s)\Big[H(b,s)Q_2(s)-k|\phi_2(b,s)|^{\gamma+1}
\Big]ds> 0,
\end{aligned}
\end{equation}
\end{itemize}
where
\begin{gather*}
\phi_1(s,a)=\frac{h_1(s,a)\sqrt{H(s,a)}+(\rho'(s)/\rho(s))H(s,a)}
{(H(s,a))^{\gamma/(\gamma+1)}},\\
\phi_2(b,s)=\frac{-h_2(b,s)\sqrt{H(b,s)}+(\rho'(s)/\rho(s))H(b,s)}
{(H(b,s))^{\gamma/(\gamma+1)}}.
\end{gather*}
Then every solution of \eqref{e1.1} has at least one zero in $(a,b)$ .
\end{theorem}

\begin{proof}
{\sl Case (C1)}: Let $y(t)$ be a nonoscillatory solution of \eqref{e1.1}.
Without loss of generality we may assume that $y(t)>0$ for $t\geq t_0$.
(The case of $y(t)<0$ can be considered similarly).
Furthermore, we may suppose that exists a $t_1\geq t_0$ such that
\begin{equation} \label{e2.5}
y(t)>0,\quad y(t-\tau)>0 , \quad  y(t-\sigma)>0 \quad
\mbox{for } t\geq t_1.
\end{equation}
As in \cite[Lemma 1 (1)]{y1}, for some
$T_0\geq t_1+\tau +\sigma$, we have immediately that
\begin{equation} \label{e2.6}
x(t)>0, \quad x' (t)>0, \quad  x''(t)>0 \quad \mbox{for } t\geq
T_0-\tau -\sigma.
\end{equation}
Using these inequalities and noting that $x(t)\geq y(t)$, we obtain
\begin{equation*}
 y(t)=x(t)-p(t)x(t-\tau)\geq [1-p(t)]x(t).
\end{equation*}
Thus, for all $t\geq T_0$,
\begin{equation*}
y(t-\sigma)\geq [1-p(t-\sigma)]x(t-\sigma).
\end{equation*}
Then \eqref{e1.1} implies that for $t\geq T_0$,
\begin{equation} \label{e2.7}
[|x'(t)|^{\gamma-1}x'(t)]' +q_1(t)[1-p(t-\sigma)]^\alpha
x^\alpha(t-\sigma)+q_2(t)[1-p(t-\sigma)]^\beta x^\beta(t-\sigma)\leq 0.
\end{equation}
Define
\begin{equation} \label{e2.8}
w(t)=\rho(t)\frac{(x' (t))^{\gamma}}{x^{\gamma}(t-\sigma)}.
\end{equation}
Differentiating $w(t)$, and using \eqref{e2.7}, we get
\begin{equation} \label{e2.9}
\begin{aligned}
w'(t)&\leq -\rho(t)\{q_1(t)[1-p(t-\sigma)]^\alpha x^{\alpha-\gamma}(t-\sigma)\\
&\quad +q_2(t)[1-p(t-\sigma)]^\beta x^{\beta-\alpha}(t-\sigma)\}
 +\frac{\rho'(t)}{\rho(t)}w(t)
-\gamma \rho(t)\Big(\frac{x'(t)}{x(t-\sigma)}\Big)^{\gamma+1},
\end{aligned}
\end{equation}
since $x'(t)<x'(t-\sigma)$. By Young's inequality  \cite[Theorem 61]{h2},
we have
\begin{align*}
&\frac{\beta-\gamma}{\beta-\alpha}q_1(t)[1-p(t-\sigma)]^\alpha x^{\alpha-\gamma}(t-\sigma)
+\frac{\gamma-\alpha}{\beta-\alpha}q_2(t)[1-p(t-\sigma)]^\beta
x^{\beta-\alpha}(t-\sigma)\\
& \geq [1-p(t-\sigma)]^\gamma[q_1^{\beta-\gamma}(t)
q_2^{\gamma-\alpha}(t)]^{1/(\beta-\alpha)},
\end{align*}
and consequently,
\begin{equation} \label{e2.10}
q_1(t)[1-p(t-\sigma)]^\alpha x^{\alpha-\gamma}(t-\sigma)+
q_2(t)[1-p(t-\sigma)]^\beta x^{\beta-\alpha}(t-\sigma)\geq Q_1(t),
\end{equation}
Combining this inequality with \eqref{e2.9},  we get
\begin{equation} \label{e2.11}
w'(t)\leq -\rho(t)Q_1(t)+\frac{\rho'(t)}{\rho(t)}w(t)
-\frac{\gamma}{\rho^{1/\gamma}(t)}|w(t)|^{(\gamma+1)/\gamma}\quad
\mbox{for }t\geq T_0.
\end{equation}
Comparing the above inequality and  \eqref{e2.1}, we find that
\begin{equation*}
A_0(t)=\rho(t)Q_1(t), \quad A_1(t)=\frac{\rho'(t)}{\rho(t)},\quad
A_2(t)=\frac{\gamma}{\rho^{1/\gamma}(t)}.
\end{equation*}
Applying Lemma \ref{lem2.1} to \eqref{e2.11}, we see that
inequality \eqref{e2.3}
 fails to hold, hence $y(t)$ has at least one zero in $(a,b)$.

{\sl Case (C2)}: Let $y(t)$ be a nonoscillatory solution of
\eqref{e1.1}. Without loss of generality we may assume that
$y(t)>0$ for $t\geq t_0$.  Furthermore, as in
\cite[Lemma 1(2)]{y1}, we suppose that there exists a $t_1>t_0$  such that
\eqref{e2.5} holds. Then, for some $T_0\geq t_1+\tau+\sigma$, we still
have \eqref{e2.6} holds for $t\geq T_0$. Note that
$y(t-\sigma)\geq x(t-\sigma)$ for $t\geq T_0$, \eqref{e1.1}
changes into
\begin{equation} \label{e2.12}
[|x'(t)|^{\gamma-1}x'(t)]' +q_1(t)x^\alpha(t-\sigma)
+q_2(t)x^\beta(t-\sigma)\leq 0,\quad t\geq T_0.
\end{equation}
Consider the function $w(t)$ also defined by \eqref{e2.8},
as in the proof of \eqref{e2.11}, we obtain
\begin{equation} \label{e2.13}
w'(t)\leq -\rho(t) Q_2(t)+\frac{\rho'(t)}{\rho(t)}w(t)
-\frac{\gamma}{\rho^{1/\gamma}(t)}|w(t)|^{(\gamma+1)/\gamma}.
\end{equation}
The rest of proof is similar to that of case $(C1)$ and hence is omitted.
\end{proof}

If the conditions of Theorem \ref{thm2.1} hold for a sequence $\{(a_n, b_n)\}$
 of intervals such that $\lim_{n\to\infty}a_n=\infty$, then we may
conclude that \eqref{e1.1} is oscillatory.
That is,  the following theorem is established.

\begin{theorem} \label{thm2.2}
For each $T\geq t_0$, if there exist  $H\in \mathcal{H}$,
$\rho\in C^1([t_0,\infty),\mathbb{R}^+)$, and constants $a,\,b,\,c\in \mathbb{R}$
with $T\leq a<c<b$, such that the conditions of Theorem \ref{thm2.1} are
 satisfied, then \eqref{e1.1} is oscillatory.
\end{theorem}

\begin{theorem} \label{thm2.3}
Suppose that there exist  $H\in \mathcal{H}$,
$\rho\in C^1([t_0,\infty), \mathbb{R}^+)$, and for each $\tau \geq t_0$,
such that one of the following conditions is
satisfied:
\begin{itemize}
\item[(C3)] $0\leq p(t)\leq 1$, and
\begin{gather} \label{e2.14}
\limsup_{t\to\infty}\int_\tau^t\rho(s)\Big[H(s,\tau)Q_1(s)
-k|\phi_1(s,\tau)|^{\gamma+1}\Big]ds>0 ,\\
\label{e2.15}
\limsup_{t\to\infty}\int_\tau^t\rho(s)\Big[H(t,s)Q_1(s)
-k|\phi_2(t,s)|^{\gamma+1}\Big]ds>0;
\end{gather}

\item[(C4)] $-1<p_0\leq p(t)\leq 0$, and
\begin{gather} \label{e2.16}
\limsup_{t\to \infty}\int_\tau^t\rho(s)\Big[H(s,\tau)Q_2(s)
-k|\phi_1(s,\tau)|^{\gamma+1}\Big]ds>0, \\
\label{e2.17}
\limsup_{t\to\infty}\int_\tau^t\rho(s)\Big[H(t,s)Q_2(s)
-k|\phi_2(t,s)|^{\gamma+1}\Big]ds>0,
\end{gather}
where $\phi_1$ and $\phi_2$ are defined in Theorem \ref{thm2.1}.
\end{itemize}
Then \eqref{e1.1} is oscillatory.
\end{theorem}

\begin{proof}
We only prove case (C3). The proof of case (C4) is similar.
For any $T\geq t_0$,
let $a=T$. In \eqref{e2.14} we choose $\tau=a$, then there exists $c>a$
such that
\begin{equation} \label{e2.18}
\int_a^c\rho(s)\Big[H(s,a)Q_1(s)-k|\phi_1(s,a)|^{\gamma+1}\Big]ds>0.
\end{equation}
In \eqref{e2.15} we choose $\tau=c$. Then there exists $b>c$ such that
\begin{equation} \label{e2.19}
\int_c^b\rho(s)\Big[H(b,s)Q_1(s)-k|\phi_2(b,s)|^{\gamma+1}\Big]ds>0.
\end{equation}
 From this inequality and \eqref{e2.18}, we obtain \eqref{e2.3}.
The conclusion thus comes from Theorem \ref{thm2.2} (C1). The proof is complete.
\end{proof}

With an appropriate choice of functions $H$, one can derive from
Theorem \ref{thm2.2} a number of oscillation criteria for \eqref{e1.1}.
For the case where $H:=H(t-s)\in \mathcal{H}$, we have
that $h_1(t-s)\equiv h_2(t-s)$ and denote them by $h(t-s)$.
The subclass of $\mathcal{H}$
containing such $H(t-s)$ is denoted by $\mathcal{H}_0$.
Applying Theorem \ref{thm2.2} to
$\mathcal{H}_0$, we get the following results.

\begin{theorem} \label{thm2.4}
For each $T\geq t_0$, if there exist  $H\in \mathcal{H}_0$,
$\rho\in C^1([t_0,\infty), \mathbb{R}^+)$, and constants $a,\,c\in \mathbb{R}$ with
$T\leq a<c$,  such that one of the following conditions is satisfied:
\begin{itemize}
\item[(C5)]  $0\leq p(t)\leq 1$, and
\begin{equation} \label{e2.20}
\int_a^c H(s-a)[\rho(s)Q_1(s)+\rho(2c-s)Q_1(2c-s)]ds>k\,\Theta(a,c);
\end{equation}

\item[(C6)]  $-1<p_0\leq p(t)\leq 0$, and
\begin{equation} \label{e2.21}
\int_a^c H(s-a)[\rho(s)Q_2(s)+\rho(2c-s)Q_1(2c-s)]ds>k\,\Theta(a,c),
\end{equation}
where
\begin{align*}
\Theta(a,c)=&\int_a^c\frac{1}{(H(s-a))^{\gamma}}\Big\{\rho(s)\Big|h(s-a)\sqrt{H(s-a)}
+\frac{\rho'(s)}{\rho(s)}H(s-a)\Big|^{\gamma+1}\\
&+\rho(2c-s)\Big|-h(s-a)\sqrt{H(s-a)}+\frac{\rho'(2c-s)}{\rho(2c-s)}
H(s-a)\Big|^{\gamma+1}\Big\}ds.
\end{align*}
\end{itemize}
Then \eqref{e1.1} is oscillatory.
\end{theorem}

Let
\begin{equation} \label{e2.22}
H(t,s)=(t-s)^\lambda,\quad (t,s)\in D,
\end{equation}
where $\lambda>\max\{1,\gamma\}$ is a constant. Then $H\in\mathcal{H}_0$ and
\begin{equation*}
h_1(t-s)=h_2(t-s)=h(t-s)=\lambda(t-s)^{\frac{\lambda-2}{2}}.
\end{equation*}
By Theorem \ref{thm2.3}, we obtain the following oscillation criteria
of Kamenev's type.

\begin{theorem} \label{thm2.5}
For each $\tau\geq t_0$ and for some $\lambda>\max\{1, \gamma\}$.
Then \eqref{e1.1} is oscillatory provided that one of the
following conditions is satisfied:
\begin{itemize}
\item[(C7)] $0\leq p(t)\leq 1$, and
\begin{gather} \label{e2.23}
\limsup_{t\to\infty}\frac{1}{t^{\lambda-\gamma}}\int_\tau^t(s-\tau)^\lambda Q_1(s)ds
>\frac{k\,\lambda^{\gamma+1}}{\lambda-\gamma}, \\
\label{e2.24}
\limsup_{t\to\infty}\frac{1}{t^{\lambda-\gamma}}\int_\tau^t(t-s)^\lambda Q_1(s)ds
>\frac{k\,\lambda^{\gamma+1}}{\lambda-\gamma};
\end{gather}

\item[(C8)]  $-1<p_0\leq p(t)\leq 0$, and
\begin{gather} \label{e2.25}
\limsup_{t\to\infty}\frac{1}{t^{\lambda-\gamma}}\int_\tau^t(s-\tau)^\lambda Q_2(s)ds
>\frac{k\,\lambda^{\gamma+1}}{\lambda-\gamma}, \\
 \label{e2.26}
\limsup_{t\to\infty}\frac{1}{t^{\lambda-\gamma}}\int_\tau^t(t-s)^\lambda Q_2(s)ds
>\frac{k\,\lambda^{\gamma+1}}{\lambda-\gamma}.
\end{gather}
\end{itemize}
\end{theorem}

\begin{proof}
We only prove Case (C7). The proof of Case (C8) is similar.
 Taking $H(t,s)$ as in \eqref{e2.22}
and $\rho(t)\equiv 1$ for $t\geq t_0$, we get
\begin{gather*}
\int_\tau^t\rho(s)|\phi_1(s,\tau)|^{\gamma+1}ds
=\lambda^{\gamma+1}\int_\tau^t(s-\tau)^{\lambda-(\gamma+1)}ds
=\frac{\lambda^{\gamma+1}}{\lambda-\gamma}(t-\tau)^{\lambda-\gamma},
\\
\int_\tau^t\rho(s)|\phi_2(t,s)|^{\gamma+1}ds
=\lambda^{\gamma+1}\int_\tau^t(t-s)^{\lambda-(\gamma+1)}ds
=\frac{\lambda^{\gamma+1}}{\lambda-\gamma}(t-\tau)^{\lambda-\gamma}.
\end{gather*}
In view of the fact $\lambda>\gamma$, it follows that
\begin{gather}\label{e2.27}
\lim_{t\to\infty}\frac{1}{t^{\lambda-\gamma}}\int_\tau^t\rho(s)
|\phi_1(s,\tau)|^{\gamma+1}ds=\frac{\lambda^{\gamma+1}}{\lambda-\gamma},\\
\label{e2.28}
\lim_{t\to\infty}\frac{1}{t^{\lambda-\gamma}}\int_\tau^t\rho(s)
|\phi_2(t,s)|^{\gamma+1}ds=\frac{\lambda^{\gamma+1}}{\lambda-\gamma},
\end{gather}
 From \eqref{e2.23} and \eqref{e2.27}, we have
\begin{align*}
&\limsup_{t\to\infty}\frac{1}{t^{\lambda-\gamma}}\int_\tau^t\rho(s)[H(s,\tau)Q_1(s)
-k|\phi_1(s,\tau)|^{\gamma+1}]ds\\
&=\limsup_{t\to\infty}\frac{1}{t^{\lambda-\gamma}}\int_\tau^t(s-\tau)^\lambda Q_1(s)ds
-\frac{k\,\lambda^{\gamma+1}}{\lambda-\gamma}>0.
\end{align*}
i.e., \eqref{e2.14} holds. Similarly,  \eqref{e2.24} and \eqref{e2.28}
 imply that \eqref{e2.15} holds,
By Theorem \ref{thm2.3} (C3),  \eqref{e1.1} is oscillatory. The proof is complete.
\end{proof}

\section{Examples}

In final section, we will show the application of our oscillation
criteria by two examples.


\begin{example} \label{exa3.1} \rm
 Consider the  equation
\begin{equation} \label{e3.1}
\big[|x'(t)|^{\gamma-1}x'(t)\big]'+q_1(t)|y(t-2)|^{\alpha-1}y(t-2)
+q_2(t)|y(t-2)|^{\beta-1} y(t-2)=0,
\end{equation}
where $t\geq 2$ and $x(t)=y(t)+p_0y(t-1)$, $-1<p_0<1$,
$0<\alpha<\gamma<\beta$ with
$\gamma=(\alpha+\beta)/2$ and $q_1,q_2\in \mathbf{C}([t_0,\infty), \mathbb{R}^+)$ with
\[
q_1(t)=q_2(t)=\begin{cases}
\eta(t-3n), & 3n< t\leq 3n+1,\\
\eta(-t+3n+2), & 3n+1<t< 3n+2,\\
q_0(t),& 3n+2\leq t\leq 3n+3,
\end{cases}
\]
for $n\in\{1,2,\dots\}$, where $q_0(t)$ is any positive continuous
function which makes $q_1(t)$ a continuous function, and
\begin{equation*}
\eta>\begin{cases}
 \dfrac{k(\lambda+2)\lambda^{\gamma+1}}{2(1-p_0)^\gamma(\lambda-\gamma)},
&  0\leq p_0<1,\\[3mm]
\dfrac{k(\lambda+2)\lambda^{\gamma+1}}{2(\lambda-\gamma)},
& -1<p_0\leq 0,
\end{cases}
\end{equation*}
is a constant for fixed $\lambda>\max{\{1,\gamma\}}$.
Now, we consider the following two cases:

{\bf Case 1:} $0\leq p_0<1$. Note that $ Q_1(t)=2(1-p_0)^\gamma q_1(t)$.
Let $a=3n$, $c=3n+1$, $H(t,s)=(t-s)^\lambda$ and $\rho(t)\equiv 1$. Then
\begin{align*}
&\int_a^c(s-a)^\lambda[Q_1(s)+Q_1(2c-s)]ds
-2k\lambda^{\gamma+1}\int_a^c(s-a)^{\lambda-\gamma-1}ds\\
& \geq 4\eta (1-p_0)^\gamma \int_{3k}^{3k+1}(s-3k)^{\gamma+1}ds
-2k\lambda^{\gamma+1}\int_{3k}^{3k+1}(s-3k)^{\lambda-(\gamma+1)}ds\\
& =\frac{4\eta (1-p_0)^\gamma }{\lambda+2}-\frac{2k\lambda^{\gamma+1}}
{\lambda-\gamma}>0,
\end{align*}
i.e., \eqref{e2.20} holds.

{\bf Case 2:} $-1<p_0\leq 0$. Note that $Q_2(t)\geq 2q_1(t)$. The rest
of proof is similar to that of Case 1. Thus, \eqref{e3.1} is oscillatory
 by Theorem \ref{thm2.4}.

However, the known results such as in \cite{s1,s2,w1,x1,y1} do not apply
to \eqref{e3.1}.
\end{example}


\begin{example} \label{exa3.2}\rm
 Consider the  neutral delay differential equation
\begin{equation} \label{e3.2}
[|x'(t)|^{\gamma-1}x'(t)]'+q_1(t)|y(t-2)|^{\alpha-1}y(t-2)
+q_2(t)|y(t-2)|^{\beta-1} y(t-2)=0,
\end{equation}
where $t\geq 2$ and $x(t)=y(t)+p_0y(t-1)$, $-1<p_0<1$,
$0<\alpha<\gamma<\beta$ with
$\gamma=(\alpha+\beta)/2$, and $q_1,\,q_2\in \mathbf{C}([t_0,\infty), \mathbb{R}^+)$ with
$q_1(t)q_2(t)\geq (\varepsilon/t^{\gamma+1})^2$ with
\begin{equation*}
\varepsilon>\begin{cases}
 \dfrac{k\gamma_0^{\gamma+1}}{2(1-p_0)^\gamma},
&  0\leq p_0<1,\\[3mm]
\dfrac{k\gamma_0^{\gamma+1}}{2},
& -1<p_0\leq 0,
\end{cases}
\end{equation*}
where $\gamma_0=\max\{1,\gamma\}$. Let $\rho(t)=t^{-(\gamma+1)}$.
The following two cases will be considered.

{\bf Case 1:} $0\leq p_0<1$. Note that
$ Q_1(t)\geq 2\varepsilon\,(1-p_0)^{\gamma}
t^{-(\gamma+1)}$. Let $\lambda>\gamma_0$. Then
\begin{align*}
\lim_{t\to\infty}\frac{1}{t^{\lambda-\gamma}}\int^t_{\tau}(s-\tau)^{\lambda}
Q_1(s)ds
&\geq 2\varepsilon(1-p_0)^{\gamma}\lim_{t\to\infty}\frac{1}{t^{\lambda-\gamma}}
\int^t_{\tau}\frac{(s-\tau)^{\lambda}}{s^{\gamma+1}}ds
\\
&=\frac{2\varepsilon(1-p_0)^{\gamma}}{\lambda-\gamma}\lim_{t\to\infty}
\frac{(t-\tau)^{\lambda}}{t^{\lambda}}\\
&=\frac{2\varepsilon(1-p_0)^{\gamma}}{\lambda-\gamma}.
\end{align*}
For any $\varepsilon>k$ $\gamma_0^{\gamma+1}/(2(1-p_0)^{\gamma})$,
there exists $\lambda>\gamma_0$ such that
\begin{equation*}
\frac{2\varepsilon(1-p_0)^{\gamma}}{\lambda-\alpha}
>\frac{k\,\lambda^{\gamma+1}}{\lambda-\alpha}.
\end{equation*}
This means that \eqref{e2.23} holds. By \cite[Lemma 3.1]{k1},
 \eqref{e2.24} holds for the same $\lambda$.
Applying Theorem \ref{thm2.5} (C7), we find that \eqref{e3.2} is oscillatory.

{\bf Case 2:} $-1<p_0\leq 0$. Note that
$Q_2(t)\geq 2\varepsilon \,t^{-(\gamma+1)}$.
The rest of proof is similar to that of Case 1.
Hence, Theorem \ref{thm2.5} (C8) holds.
Thus, \eqref{e3.2} is oscillatory .
\end{example}

\subsection*{Acknowledgments}
 The authors would like to express his great
appreciation to Professor Qingkai Kong for many valuable suggestions
and useful comments.

\begin{thebibliography}{00}

\bibitem{a1} R. P. Agarwal, S. R. Grace, D. O'Regan;
\emph{Oscillation Theory for Second Order Dynamic Equations},
Taylor \& Francis, London, 2003.

\bibitem{a2} F. V. Atkinson;
\emph{On second order  nonlinear oscillation},
Pacific J. Math. 5 (1955) 643-647.

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

\bibitem{b2} S. Belohorec;
\emph{Oscillatory solution of certain nonlinear differential
equations of the second order},
Math. Fyz. Casopis Sloven. Akad. Vied. 11 (1961) 250-255.

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

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

\bibitem{h2} G. H. Hardy, J. E. Littlewood, G. Polya;
\emph{Inequalities}, second ed., Cambridge University Press, Cambridge, 1988.

\bibitem{k1} Q. Kong;
 \emph{Interval criteria for oscillation of second order linear ordinary
differential equations}, J. Math. Anal. Appl. 229 (1999) 258-270.

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

\bibitem{s1} S. H. Saker;
\emph{Oscillation for second order neutral delay differential
equations of Emden-Fowler type}, Acta. Math. Hungar. 100 (1-2) (2003) 37-62.

\bibitem{s2} S. H. Saker, J. V. Manojlivi\`{c};
\emph{Oscillation criteria for second order superlinear neutral
delay differential equations}, Electron. J. Qual. Theory Differ.
Equ. 10 (2004) 1-22.

\bibitem{t1} A. Tiryaki, Y. Basci, I. G\"{u}lec;
\emph{Interval criteria for oscillation of  second order functional
differential equations}, Comput. Math. Appl. 59 (2005) 1487-1498.

\bibitem{w1} J. S. W. Wong;
\emph{Necessary and sufficient conditions for oscillation for
second order neutral differential equations},
J. Math. Anal. Appl. 252 (2000) 342-352.

\bibitem{x1}Z. Xu, X. Liu;
\emph{Philos-type oscillation criteria for Emden-Fowler neutral
delay differential equations},
J. Comput. Appl. Math. doi:10.1016/j.cam.2006. 09.012.

\bibitem{y1} Q. Yang, L. Yang, S. Zhu;
\emph{Interval criteria for oscillation of second order
nonlinear neutral differential equations},
Comput. Math. Appl. 46 (2003) 903-918.

\end{thebibliography}

\end{document}