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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 23, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2011/23\hfil Oscillation of solutions]
{Oscillation of solutions to odd-order nonlinear neutral
functional differential equations}

\author[T. Li, E. Thandapani\hfil EJDE-2011/23\hfilneg]
{Tongxing Li, Ethiraju Thandapani}  % in alphabetical order

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

\address{Ethiraju Thandapanii \newline
Ramanujan Institute for Advanced Study in Mathematics,
University of Madras, Chennai, India}
\email{ethandapani@yahoo.co.in}

\thanks{Submitted January 13, 2011. Published February 9, 2011.}
\subjclass[2000]{34K11, 34C10}
\keywords{Odd-order; neutral differential equation; oscillation;
\hfill\break\indent asymptotic behavior}

\begin{abstract}
 In this note,  we establish some new comparison theorems and
 Philos-type criteria for  oscillation of solutions to the
 odd-order nonlinear neutral functional differential equation
 \[
 [x(t)+p(t)x(\tau(t))]^{(n)}+q(t)x^\alpha(\sigma(t))=0,
 \]
 where $0\leq p(t)\leq p_0<\infty$ and $\alpha\geq1$.
\end{abstract}

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



\section{Introduction}

 This paper is concerned with the oscillation and
asymptotic behavior of  solutions to the odd-order nonlinear
neutral differential equation
\begin{equation}\label{E}
\big[x(t)+p(t)x(\tau(t))\big]^{(n)}+q(t)x^\alpha(\sigma(t))=0,
\end{equation}
where $n\geq3$ is an odd integer, $\alpha\geq1$ is the ratio of odd
positive integers, $p(t),q(t)\in C([t_{0},\infty))$ and
\begin{itemize}
\item[(H1)]   $q(t)>0$, $0\leq p(t)\leq p_0<\infty$;
\item[(H2)]  $\tau(t)= a+bt$, with  $b > 0$,
 $\sigma(t)\in C([t_0, \infty))$,
$\tau(t) \leq t$, $\tau \circ  \sigma = \sigma \circ \tau$,
$\lim_{t\to\infty}\sigma(t)=\infty$.
\end{itemize}

 We set $z(t)=x(t)+p(t)x(\tau(t))$. By a solution  of  \eqref{E},
we mean a function $x(t) \in C([T_{x},\infty ))$, $T_{x}\geq t_{0}$,
which has the property $z(t) \in C^n([T_{x},\infty ))$ and satisfies
\eqref{E} on $[T_{x},\infty)$. We consider only those solutions
$x(t)$ of \eqref{E} which satisfy $\sup \{|x(t)|:t \geq
T\}>0$ for all $T\geq T_{x}$.
 We assume that \eqref{E} possesses such a solution.
 A solution of \eqref{E} is called oscillatory if it has arbitrarily
large zeros  on $[T_{x},\infty)$ and otherwise, it is said to
be nonoscillatory.
 Equation \eqref{E} is said to
be almost oscillatory if all its solutions are oscillatory or
convergent to zero asymptotically.

Since the differential equations have important applications in the
natural sciences, technology and population dynamics, there is a
permanent interest in obtaining sufficient conditions for the
oscillation or nonoscillation of the solutions of various types of
even-order/odd-order differential equations; see references in this
article, and their references.

For the oscillation of odd-order neutral differential equations, see
e.g., \cite{bac,tca,pdas,pdas1,jd,kgo,han3,hl,bas,np1,np2,rnr,rnr1,ys,tang}.
They studied the oscillatory behavior of
odd-order neutral differential equations
\begin{gather*}
[x(t)+p(t)x(\tau(t))]^{(n)}+q(t)x(\sigma(t))=0,\\
[x(t)+p(t)x(t-\tau)]^{(n)}+q(t)h(x(t-\sigma))=0,
\end{gather*}
and established some oscillatory and asymptotic criteria for the
case when $-1\leq p(t)\leq1$.

To the best of our knowledge, the study of oscillatory behavior
of odd-order neutral differential equations has not been sufficient.
In this paper, we try to obtain some new oscillation results for
 \eqref{E}. To prove our results, we use the
following definition and remarks.

\begin{definition}\label{def1} \rm
Consider the sets $\mathbb{D}_0=\{(t,s):t>s\geq t_0\}$ and
$\mathbb{D}=\{(t,s):t\geq s\geq t_0\}$. Assume that $H\in
C(\mathbb{D},\mathbb{R})$ satisfies the following assumptions:

\item[(A1)]  $H(t,t)=0$, $t\geq t_0$; $H(t,s)>0$,
$(t,s)\in\mathbb{D}_0$;
\item[(A2)]  $H$ has a non-positive continuous partial
derivative with respect to the second variable in $\mathbb{D}_0$.

Then the function $H$ has the property $P$.
\end{definition}

\begin{remark}\label{R1} \rm
All functional inequalities considered in this paper are assumed
to hold eventually, that is, they are satisfied for all $t$
large enough.
\end{remark}

\begin{remark}\label{R2} \rm
Without loss of generality we can deal only
with the positive solutions of equation \eqref{E}.
\end{remark}

\section{Main results}

The Kiguradze's lemma is stated below, the readers may find this
result in \cite{kig, kigu}, which plays an important role in the
oscillation of higher-order differential equations.


\begin{lemma}[Kiguradze's lemma] \label{L1x}
 Let $f\in C^n([t_0,\infty),\mathbb{R})$ and its derivatives up to
order $(n-1)$ are of constant sign in $[t_0,\infty)$.
If $f^{(n)}$ is of constant sign and not identically zero on a
sub-ray of $[t_0,\infty)$, then
there exist $m\in \mathbb{Z}$ and $t_1\in[t_0,\infty)$ such that
$0\leq m\leq n-1$, and $(-1)^{n+m}ff^{(n)}\geq0$,
$$
ff^{(j)}>0\quad  \text{for } j=0,1,\dots,m-1 \text{ when } m\geq1
$$
and
$$
(-1)^{m+j}ff^{(j)}>0\quad  \text{for } j=m,m+1,\dots,n-1 \text{ when }
m\leq n-1
$$
hold on $[t_1,\infty)$.
\end{lemma}

\begin{lemma}[{\cite[Lemma 2.2.3]{rpa}}] \label{cxjL1x1}
Let $f$ be a function as in Lemma \ref{L1x}1. If
$\lim_{t\to\infty}f(t)\neq0$, then for every
$\lambda\in(0,1)$, there exists $t_\lambda\in[t_1,\infty)$ such that
$$
|f|\geq\frac{\lambda}{(n-1)!}t^{n-1}|f^{(n-1)}|
$$
holds on $[t_\lambda,\infty)$.
\end{lemma}

\begin{lemma}[\cite{chg}] \label{L1x1}
Let $f$ be a function as in Lemma \ref{L1x}1. If
$$
f^{(n-1)}(t)f^{(n)}(t)\leq 0,
$$
then for any constant $\theta\in(0,1)$ and sufficiently large $t$,
there exists a constant $M>0$, satisfying
$$
|f'(\theta t)|\geq Mt^{n-2}|f^{(n-1)}(t)|.
$$
\end{lemma}


\begin{lemma}\label{L1}
If $x$ is a positive solution of  \eqref{E}, then the corresponding
function $z(t)=x(t)+ p(t)x(\tau(t))$ satisfies
\begin{equation}\label{Can}
z(t)>0, \quad z^{(n-1)}(t)>0,\quad z^{(n)}(t)<0
\end{equation}
eventually.
\end{lemma}


Due to Lemma \ref{L1x}, the proof of the above lemma
is simple and so is omitted.


\begin{lemma}[{\cite[Lemma 3]{bas}}] \label{L1x3}
Let $f$ and $g\in C([t_0,\infty),\mathbb{R})$ and $\alpha\in
C([t_0,\infty),\mathbb{R})$ satisfies
$\lim_{t\to\infty}\alpha(t)=\infty$ and $\alpha(t)\leq t$
for all $t\in[t_0,\infty)$; further suppose that there exists $h\in
C([t_{-1},\infty),\mathbb{R}^{+})$, where
$t_{-1}:=\min_{t\in[t_0,\infty)}\{\alpha(t)\}$, such that
$f(t)=h(t)+g(t)h(\alpha(t))$ holds for all $t\in[t_0,\infty)$.
Suppose that $\lim_{t\to\infty}f(t)$ exists and
$\liminf_{t\to\infty}g(t)>-1$. Then
$\limsup_{t\to\infty}h(t)>0$ implies
$\lim_{t\to\infty}f(t)>0$.
\end{lemma}


\begin{lemma}\label{lle2.1}
Assume that $\alpha\geq1$, $c, d\in \mathbb{R}$. If
$c\geq0$ and $d\geq0$, then
\begin{equation}\label{li2.1}
c^\alpha+d^\alpha\geq\frac{1}{2^{\alpha-1}}(c+d)^\alpha.
\end{equation}
\end{lemma}


\begin{proof}
(i) Suppose that $c=0$ or $d=0$. Then we have \eqref{li2.1}.
(ii) Suppose that $c>0$ and $d>0$. Define the function
$f$ by $f(x)=x^\alpha,$ $x\in (0,\infty)$. Then
$f''(x)=\alpha(\alpha-1)x^{\alpha-2}\geq0$ for $x>0$. Thus, $f$ is a
convex function. By the definition of convex function, we have
$$
f\big(\frac{c+d}{2}\big)\leq\frac{f(c)+f(d)}{2};
$$
that is,
$$
c^\alpha+d^\alpha\geq\frac{1}{2^{\alpha-1}}(c+d)^\alpha.
$$
This completes the proof.
\end{proof}

Next, we  establish our main results. For the sake of
convenience, let
\begin{equation}\label{Q}
Q(t)=\min\{q(t),q(\tau(t))\}.
\end{equation}


\begin{theorem}\label{cxjT1}
Assume  that
\begin{equation}\label{cx}
\int_{t_0}^\infty t^{n-1}Q(t)\mathrm{d}t=\infty.
\end{equation}
Further, assume that the first-order neutral differential inequality
\begin{equation}
\Big(y(t)+\frac{{p_0}^\alpha}{b}\,y(\tau(t))\Big)'
+\frac{Q(t)}{2^{\alpha-1}}\Big(\frac{\lambda}{(n-1)!}\sigma^{n-1}(t)
\Big)^\alpha y^\alpha(\sigma(t))\leq0 \label{E2}
\end{equation}
has no positive solution for some $\lambda\in(0,1)$.
Then  \eqref{E} is almost oscillatory.
\end{theorem}

\begin{proof}
Assume that  $x$ is a positive solution of  \eqref{E},
which does not tend to zero asymptotically.  Then the corresponding
function $z$ satisfies
\begin{equation}
\begin{split}
z(\sigma(t))
&= x(\sigma(t))+ p(\sigma(t))x(\tau(\sigma(t)))\\
&\leq x(\sigma(t))+ p_0x(\sigma(\tau(t))),
\end{split}\label{zsig}
\end{equation}
where we have used the hypothesis (H1). On the other hand, it
follows from \eqref{E} that
\begin{equation}\label{EE}
z^{(n)}(t)+q(t)x^\alpha(\sigma(t))=0
\end{equation}
and moreover taking (H1) and  (H2) into account, we have
\begin{equation}
\begin{split}
0&=\frac{{p_0}^\alpha}{\tau'(t)}(z^{(n-1)}(\tau(t)))'
 +{p_0}^\alpha q(\tau(t))x^\alpha(\sigma(\tau(t)))\\
&=\frac{{p_0}^\alpha}{b}(z^{(n-1)}(\tau(t)))'+{p_0}^\alpha
q(\tau(t))x^\alpha(\sigma(\tau(t))).
\end{split}\label{Etau}
\end{equation}
Combining \eqref{EE} and \eqref{Etau}, we are led to
\begin{equation} \label{jx}
[z^{(n-1)}(t)+\frac{{p_0}^\alpha}{b}z^{(n-1)}(\tau(t))]'
+q(t)x^\alpha(\sigma(t))+{p_0}^\alpha
q(\tau(t))x^\alpha(\sigma(\tau(t)))\leq0,
\end{equation}
which in view of \eqref{li2.1}, \eqref{Q} and \eqref{zsig}
implies
\begin{equation}\label{Ez}
[z^{(n-1)}(t)+\frac{{p_0}^\alpha}{b}z^{(n-1)}(\tau(t))]'
+\frac{1}{2^{\alpha-1}}Q(t)z^\alpha(\sigma(t))\leq0.
\end{equation}

Next, we claim that
$z'(t)>0$ eventually. If not, then $\lim_{t\to\infty}z(t)=a>0$
($a$ is finite) due to Lemma \ref{L1x3}. From \eqref{Can}, we obtain
$\lim_{t\to\infty}z^{(k)}(t)=0$ for $k=1,2,\dots,n-1$.
Integrating \eqref{Ez} from $t$ to $\infty$ for a total of $(n-1)$
times and integrating the resulting inequality from $t_1$
($t_1$ is large enough) to $\infty$, we obtain
$$
\int_{t_1}^\infty\frac{(s-t_1)^{n-1}}{(n-1)!}Q(s)z^\alpha(\sigma(s))
\mathrm{d}s <\infty,
$$
which yields
$$
\int_{t_1}^\infty s^{n-1}Q(s)\mathrm{d}s<\infty.
$$
This contradicts \eqref{cx}. Hence by Lemma
\ref{cxjL1x1} and Lemma \ref{L1}, we obtain
$$
z(t)\geq\frac{\lambda}{(n-1)!}t^{n-1}z^{(n-1)}(t)\ \text{for every}\
\lambda \in(0,1).
$$
Thus, it follows from \eqref{Ez} that
\begin{equation}
[z^{(n-1)}(t)+\frac{{p_0}^\alpha}{b}z^{(n-1)}(\tau(t))]'
+\frac{Q(t)}{2^{\alpha-1}}\left(\frac{\lambda}{(n-1)!}
\sigma^{n-1}(t)z^{(n-1)}(\sigma(t))\right)^\alpha\leq0.
\label{tx}
\end{equation}
Therefore, setting $z^{(n-1)}(t)=y(t)$ in \eqref{tx}, one can see
that $y$ is a positive solution of \eqref{E2}. This contradicts our
assumptions and the proof is complete.
\end{proof}


\begin{remark}\label{R3} \rm
In the comparison principle in Theorem \ref{cxjT1} we do not
assume that the deviating arguments is either delay or
advanced type, and hence this result is applicable to all types of equations. Further, the comparison principle established in
Theorem \ref{cxjT1} reduces oscillation of equation \eqref{E} to find
conditions for the first-order neutral differential inequality
\eqref{E2} has no positive solution. Therefore, applying the
conditions for equation \eqref{E2} to have no positive solution, one can
immediately get oscillation criteria for equation \eqref{E}.
\end{remark}


\begin{theorem}\label{xjcT3}
Assume that \eqref{cx} holds. If the first-order differential
inequality
\begin{equation} \label{E7}
w'(t)+\frac{Q(t)}{2^{\alpha-1}
(1+\frac{{p_0}^\alpha}{b})}
\Big(\frac{\lambda}{(n-1)!}\sigma^{n-1}(t)\Big)^\alpha
w^\alpha(\tau^{-1}(\sigma(t)))\leq0
\end{equation}
has no positive solution for some $0<\lambda<1$,
then  \eqref{E} is almost oscillatory.
\end{theorem}

\begin{proof}
Assume that  $x$ is a positive solution of
\eqref{E}, which does not tend to zero asymptotically.  Then
$y(t)=z^{(n-1)}(t)>0$ is a decreasing solution of \eqref{E2}. We
 denote
$$
w(t)=y(t)+\frac{{p_0}^\alpha}{b}\,y(\tau(t)).
$$
It follows from $\tau(t)\leq t$ that
$$
w(t)\leq y(\tau(t))\Big(1+\frac{{p_0}^\alpha}{b}\Big).
$$
Substituting this into \eqref{E2}, we obtain that $w$ is a positive
solution of \eqref{E7}. A contradiction. This completes the proof.
\end{proof}


\begin{corollary}\label{Co3}
Assume that \eqref{cx}  holds,  and $\alpha=1$ and
$\sigma(t)<\tau(t)$.
 If
\begin{equation}\label{Con13}
\liminf_{t\to\infty} \int_{\tau^{-1}(\sigma(t))}^t \sigma^{n-1}(s)
Q(s)\,\mathrm{d}{}s>\frac{\left(1+\frac{p_0}{b}\right)(n-1)!}{\rm{e}},
\end{equation}
then  \eqref{E} is almost oscillatory.
\end{corollary}

\begin{proof}
 According to \cite[Theorem 2.1.1]{gsl}, the condition
\eqref{Con13} guarantees that \eqref{E7} with $\alpha=1$ has no
positive solution. Hence by Theorem \ref{xjcT3}, equation \eqref{E}
is almost oscillatory. This completes the proof of Corollary \ref{Co3}.
\end{proof}

Now, we shall establish some Philos-type oscillation criteria for
the oscillation of \eqref{E}.

\begin{theorem}\label{T1}
  Assume  that \eqref{cx} holds and $\sigma(t)\geq\tau(t)/2$.
Further, assume that the function $H\in C(\mathbb{D},\mathbb{R})$
has the property $P$ and there exist functions $h\in
C(\mathbb{D}_0,\mathbb{R})$ and
$\rho\in C^1([t_0,\infty),(0,\infty))$ such that
\begin{equation}\label{xingx}
-\frac{\partial}{\partial
s}H(t,s)-H(t,s)\frac{\rho'(s)}{\rho(s)}=h(t,s),\quad
(t,s)\in\mathbb{D}_0.
\end{equation}
If
\begin{equation}\label{th1}
\limsup_{t\to\infty} \frac{1}{H(t,t_0)}\int_{t_0}^t
K_1(t,s){\rm  d}s=\infty
\end{equation}
for all constants $M>0$, $L>0$ and for some $\beta\geq1$,
 where
 $$
K_1(t,s):=\big(\frac{L}{2}\big)^{\alpha-1}H(t,s)\rho(s)Q(s)
 -\Big(1+\frac{{p_0}^\alpha}{b}\Big)\frac{\beta\rho(s)h^2(t,s)}{2bMH(t,s)\tau^{n-2}(s)},
 $$
then \eqref{E} is almost oscillatory.
\end{theorem}

\begin{proof}
Assume that  $x$ is a positive solution of \eqref{E},
which does not tend to zero asymptotically.  Proceeding as in the
proof of Theorem \ref{cxjT1}, we obtain \eqref{Ez} and $z'(t)>0$.
Define
\begin{equation} \label{eth1}
w(t)=\rho(t)\frac{z^{(n-1)}(t)}{z\big(\frac{\tau(t)}{2}\big)},
\end{equation}
then $w(t)>0,$ and
\begin{equation}\label{eth2}
w'(t)=\rho'(t)\frac{z^{(n-1)}(t)}{z\big(\tau(t)/2\big)}
+\rho(t)\frac{z^{(n)}(t)z\big(\tau(t)/2\big)
-\frac{b}{2}z^{(n-1)}(t)z'\big(\tau(t)/2\big)}{z^2\big(\tau(t)/2\big)}.
\end{equation}
It follows from Lemma \ref{L1x1} and Lemma \ref{L1} that there
exists a constant $M>0$, such that
\begin{equation}\label{e}
z'\big(\tau(t)/2\big)\geq
M\tau^{n-2}(t)z^{(n-1)}(\tau(t))\geq M\tau^{n-2}(t)z^{(n-1)}(t),
\end{equation}
which in view of \eqref{eth1} and \eqref{eth2} yields
\begin{equation}\label{eth3}
w'(t)\leq\rho(t)\frac{z^{(n)}(t)}{z\big(\tau(t)/2\big)}+
\frac{\rho'(t)}{\rho(t)}w(t)
-\frac{bM}{2}\frac{\tau^{n-2}(t)}{\rho(t)}w^2(t),
\end{equation}
Define another function
\begin{eqnarray}\label{eth4}
v(t)=\rho(t)\frac{z^{(n-1)}(\tau(t))}{z\big(\tau(t)/2\big)},
\end{eqnarray}
then $v(t)>0,$ and
\begin{equation}\label{eth5}
\begin{split}
v'(t)&=\rho'(t)\frac{z^{(n-1)}(\tau(t))}{z\big(\tau(t)/2\big)} \\
&\quad +\rho(t)\frac{bz^{(n)}(\tau(t))z\big(\tau(t)/2\big)
-\frac{b}{2}z^{(n-1)}(\tau(t))z'\big(\tau(t)/2\big)}{z^2
\big(\tau(t)/2\big)}.
\end{split}
\end{equation}
It follows from \eqref{e}, \eqref{eth4} and \eqref{eth5} that
\begin{equation}\label{eth6}
v'(t)\leq \rho(t)\frac{z^{(n)}(\tau(t))}{z\big(\tau(t)/2\big)}+
\frac{\rho'(t)}{\rho(t)}v(t)-\frac{b M}{2}
\frac{\tau^{n-2}(t)}{\rho(t)}v^2(t).
\end{equation}
In view of \eqref{eth3} and \eqref{eth6}, we obtain
\begin{align*}
 w'(t)+\frac{{p_0}^\alpha}{b}v'(t)
&\leq \rho(t) \frac{z^{(n)}(t)+{p_0}^\alpha
 z^{(n)}(\tau(t))}{z\big(\tau(t)/2\big)}
 +\frac{\rho'(t)}{\rho(t)}w(t)\\
&\quad -\frac{bM}{2}\frac{\tau^{n-2}(t)}{\rho(t)}w^2(t)
 +\frac{{p_0}^\alpha}{b}[\frac{\rho'(t)}{\rho(t)}v(t)
 -\frac{bM}{2}\frac{\tau^{n-2}(t)}{\rho(t)}v^2(t)].
\end{align*}
It follows from \eqref{Ez} that there exists a constant $L>0$,
such that
\begin{equation}
\begin{split}
 w'(t)+\frac{{p_0}^\alpha}{b}v'(t)
&\leq -\big(\frac{L}{2}\big)^{\alpha-1}\rho(t)Q(t)
+\frac{\rho'(t)}{\rho(t)}w(t)
- \frac{bM}{2}\frac{\tau^{n-2}(t)}{\rho(t)}w^2(t)\\
&\quad +\frac{{p_0}^\alpha}{b}[ \frac{\rho'(t)}{\rho(t)}v(t)
-\frac{bM}{2}\frac{\tau^{n-2}(t)}{\rho(t)}v^2(t)].
\end{split}\label{eth8}
\end{equation}
Multiplying \eqref{eth8}, with $t$ replaced by $s$, by $H(t,s)$ and
integrating from $T$ to $t$ ,with $T\geq t_1$, we have
\begin{align*}
&\int_T^t\big(\frac{L}{2}\big)^{\alpha-1}H(t,s)\rho(s)Q(s)\mathrm{d}s\\
&\leq -\int_T^tH(t,s)w'(s){\rm  d}s
 +\int_T^tH(t,s)\frac{\rho'(s)}{\rho(s)}w(s)\mathrm{d}s
 - \int_T^t\frac{bM}{2}H(t,s)\frac{\tau^{n-2}(s)}{\rho(s)}w^2(s){\rm
d}s\\
&\quad -\frac{{p_0}^\alpha}{b}\int_T^tH(t,s)v'(s)\mathrm{d}s
 +\frac{{p_0}^\alpha}{b}\int_T^tH(t,s)\frac{\rho'(s)}{\rho(s)}v(s)
 \mathrm{d}s\\
&\quad - \frac{{p_0}^\alpha}{b}\int_T^t\frac{bM}{2}H(t,s)
 \frac{\tau^{n-2}(s)}{\rho(s)}v^2(s)\mathrm{d}s.
\end{align*}
It follows from the above inequality and \eqref{xingx} that
\begin{align*}
&\int_T^t\big(\frac{L}{2}\big)^{\alpha-1}H(t,s)\rho(s)Q(s)\mathrm{d}s\\
&\leq  H(t,T)w(T)-\int_T^th(t,s)w(s){\rm  d}s
 -\int_T^t\frac{bM}{2}H(t,s)\frac{\tau^{n-2}(s)}{\rho(s)}w^2(s)\mathrm{d}s\\
&\quad +\frac{{p_0}^\alpha}{b}H(t,T)v(T)
 -\frac{{p_0}^\alpha}{b}\int_T^th(t,s)v(s)\mathrm{d}s\\
&\quad -\frac{{p_0}^\alpha}{b}\int_T^t\frac{bM}{2}H(t,s)
 \frac{\tau^{n-2}(s)}{\rho(s)}v^2(s)\mathrm{d}s.
\end{align*}
Thus,  for any $\beta\geq1$,
\begin{equation} \label{eth10}
\begin{split}
&\int_T^t\big(\frac{L}{2}\big)^{\alpha-1}H(t,s)\rho(s)Q(s)\mathrm{d}s\\
&\leq  H(t,T)w(T)+\int_T^t\frac{\beta\rho(s)h^2(t,s)}{2bM\tau^{n-2}
 (s)H(t,s)}\mathrm{d}s \\
&\quad -\int_T^t\Big[\sqrt{\frac{bM\tau^{n-2}(s)H(t,s)}{2\beta\rho(s)}}w(s)+
 \sqrt{\frac{2\beta\rho(s)}{4bM\tau^{n-2}(s)H(t,s)}}h(t,s)\Big]^2
\mathrm{d}s \\
&\quad -\int_T^t\frac{(\beta-1)bM\tau^{n-2}(s)H(t,s)}{2\beta\rho(s)}
 w^2(s)\mathrm{d}s \\
&\quad +\frac{{p_0}^\alpha}{b}H(t,T)v(T)
 +\frac{{p_0}^\alpha}{b}\int_T^t\frac{\beta\rho(s)h^2(t,s)}
 {2bM\tau^{n-2}(s)H(t,s)}\mathrm{d}s \\
&\quad -\frac{{p_0}^\alpha}{b}\int_T^t\Big[\sqrt{\frac{bM\tau^{n-2}(s)
 H(t,s)}{2\beta\rho(s)}}v(s)
 + \sqrt{\frac{2\beta\rho(s)}{4bM\tau^{n-2}(s)H(t,s)}}h(t,s)\Big]^2
 \mathrm{d}s \\
&\quad -\frac{{p_0}^\alpha}{b}\int_T^t\frac{(\beta-1)bM\tau^{n-2}(s)
 H(t,s)}{2\beta\rho(s)}v^2(s)\mathrm{d}s.
\end{split}
\end{equation}
 From the above inequality, we obtain
\begin{align*}
&\int_T^t\Big[\big(\frac{L}{2}\big)^{\alpha-1}H(t,s)\rho(s)Q(s)
 -\big(1+\frac{{p_0}^\alpha}{b}\big)\frac{\beta\rho(s)h^2(t,s)}
{2bMH(t,s)\tau^{n-2}(s)}\Big]\mathrm{d}s \\
&\leq H(t,T)\Big(w(T)+\frac{{p_0}^\alpha}{b}v(T)\Big)\\
&\leq H(t,t_0)\Big(w(T)+\frac{{p_0}^\alpha}{b}v(T)\Big),
\end{align*}
which yields
$$
\frac{1}{H(t,t_0)}\int_{t_0}^t\Big[\big(\frac{L}{2}\big)^{\alpha-1}H(t,s)
\rho(s)Q(s)
 -\big(1+\frac{{p_0}^\alpha}{b}\big)
\frac{\beta\rho(s)h^2(t,s)}{2bMH(t,s)\tau^{n-2}(s)}\Big]\mathrm{d}s<\infty.
 $$
This contradicts condition \eqref{th1}. The proof is complete.
\end{proof}

As a consequence of Theorem \ref{T1}, we obtain the following
corollary.

\begin{corollary}\label{c1}
Let condition \eqref{th1} in Theorem \ref{T1}
be replaced by
\begin{gather*}
\limsup_{t\to\infty}
\frac{1}{H(t,t_0)}\int_{t_0}^tH(t,s)\rho(s)Q(s)\mathrm{d}s=\infty,\\
\limsup_{t\to\infty}
\frac{1}{H(t,t_0)}\int_{t_0}^t\frac{\rho(s)h^2(t,s)}{H(t,s)\tau^{n-2}(s)}{\rm
d}s<\infty.
\end{gather*}
Then  \eqref{E} is almost oscillatory.
\end{corollary}

It may happen that assumption \eqref{th1} in Theorem \ref{T1} fails
to hold.  The following result provide an essentially new
oscillation criterion for  \eqref{E}.


\begin{theorem}\label{T2}
  Assume  that \eqref{cx} holds  and
$\sigma(t)\geq\tau(t)/2$. Let $H,h,\rho$ be as in Theorem \ref{T1}
and
\begin{equation}\label{li10}
0<\inf_{s\geq t_0}\Big[\liminf_{t\to\infty}
\frac{H(t,s)}{H(t,t_0)}\Big]\leq\infty.
\end{equation}
Moreover, suppose that there exists a function
$m\in C([t_0,\infty),\mathbb{R})$ such that for all $T\geq t_0$ and for
some $\beta>1$, one has
\begin{equation}\label{li11}
\limsup_{t\to\infty}\frac{1}{H(t,T)}\int_T^tK_1(t,s)\mathrm{d}s\geq m(T)
\end{equation}
for all constants $M>0$ and $L>0$, where $K_1$ is defined as in
Theorem \ref{T1}. If
\begin{equation}\label{li112}
\limsup_{t\to\infty}\int_{t_0}^t\frac{\tau^{n-2}(s)m_+^2(s)}{\rho(s)}
\mathrm{d}s=\infty,
\end{equation}
where $m_+(t):=\max\{m(t),0\}$, then  \eqref{E} is almost
oscillatory.
\end{theorem}

\begin{proof}
 Assume that  $x$ is a positive solution of \eqref{E},
which does not tend to zero asymptotically. Proceeding as in the
proof of Theorem \ref{T1}, we obtain \eqref{eth10}, which implies
\begin{align*}
&\frac{1}{H(t,T)}\int_T^t\Big[\big(\frac{L}{2}\big)^{\alpha-1}
 H(t,s)\rho(s)Q(s)
 -\big(1+\frac{{p_0}^\alpha}{b}\big)
 \frac{\beta\rho(s)h^2(t,s)}{2bMH(t,s)\tau^{n-2}(s)}\Big]\mathrm{d}s\\
&\leq w(T)-\frac{1}{H(t,T)}\int_T^t\frac{(\beta-1)bM\tau^{n-2}(s)
H(t,s)}{2\beta\rho(s)}w^2(s)\mathrm{d}s\\
&\quad +\frac{{p_0}^\alpha}{b}v(T)-\frac{{p_0}^\alpha}{b}\frac{1}
 {H(t,T)}\int_T^t\frac{(\beta-1)bM\tau^{n-2}(s)
  H(t,s)}{2\beta\rho(s)}v^2(s)\mathrm{d}s.
\end{align*}
Therefore, for $t>T\geq t_1$, sufficiently large,
\begin{align*}
&\limsup_{t\to\infty}\frac{1}{H(t,T)}
\int_T^t\Big[\big(\frac{L}{2}\big)^{\alpha-1}H(t,s)\rho(s)Q(s)
 -\big(1+\frac{{p_0}^\alpha}{b}\big)
\frac{\beta\rho(s)h^2(t,s)}{2bMH(t,s)\tau^{n-2}(s)}\Big]\mathrm{d}s\\
&\leq w(T)+\frac{{p_0}^\alpha}{b}v(T)\\
&\quad -\liminf_{t\to\infty}\frac{1}
{H(t,T)}\int_T^t\frac{(\beta-1)bM\tau^{n-2}(s)H(t,s)}{2\beta\rho(s)}
\Big(w^2(s)+\frac{{p_0}^\alpha}{b}v^2(s)\Big)\mathrm{d}s.
\end{align*}
It follows from \eqref{li11} that
\begin{align*}
& w(T)+\frac{{p_0}^\alpha}{b}v(T) \\
&\geq  m(T) +\liminf_{t\to\infty}\frac{1}{H(t,T)}
\int_T^t\frac{(\beta-1)bM\tau^{n-2}(s)H(t,s)}{2\beta\rho(s)}
\Big(w^2(s)+\frac{{p_0}^\alpha}{b}v^2(s)\Big)\mathrm{d}s,
\end{align*}
for all $T\geq t_1$ and for any $\beta>1$. Consequently, for all
$T\geq t_1$, we obtain
\begin{equation}\label{li13}
w(T)+\frac{{p_0}^\alpha}{b}v(T)\geq m(T),
\end{equation}
and
\begin{equation}
\begin{split}
& \liminf_{t\to\infty}\frac{1}{H(t,t_1)}\int_{t_1}^t
 \frac{H(t,s)\tau^{n-2}(s)}{ \rho(s)}
 \Big(w^2(s)+\frac{{p_0}^\alpha}{b}v^2(s)\Big)\mathrm{d}s  \\
&\leq \frac{2\beta}{(\beta-1)bM}
 \Big(w(t_1)+\frac{{p_0}^\alpha}{b}v(t_1)-m(t_1)\Big)<\infty.
\end{split}\label{li12}
\end{equation}
Now we claim that
\begin{equation}\label{li14}
\int_{t_1}^\infty\frac{\tau^{n-2}(s)
\big(w^2(s)+\frac{{p_0}^\alpha}{b}v^2(s)\big)}{\rho(s)}\mathrm{d}s<\infty.
\end{equation}
Suppose to the contrary that
\begin{equation}\label{li15}
\int_{t_1}^\infty\frac{\tau^{n-2}(s)\left(w^2(s)+\frac{{p_0}^\alpha}{b}v^2(s)\right)}{
\rho(s)}\mathrm{d}s=\infty.
\end{equation}
By  \eqref{li15}, for any positive number $\kappa$, there
exists a $T_1\geq t_1$ such that, for all $t\geq T_1$,
$$
\int_{t_1}^t\frac{\tau^{n-2}(s)
\big(w^2(s)+\frac{{p_0}^\alpha}{b}v^2(s)\big)}{\rho(s)}\mathrm{d}s
\geq\frac{\kappa}{\rho}.
$$
Assumption \eqref{li10} implies the existence of a $\rho>0$ such
that
\begin{equation}\label{li16}
\inf_{s\geq
t_0}[\liminf_{t\to\infty}\frac{H(t,s)}{H(t,t_0)}]>\rho.
\end{equation}
From \eqref{li16}, we have
$$
\liminf_{t\to\infty}\frac{H(t,s)}{H(t,t_0)}>\rho>0,
$$
and there exists a $T_2\geq T_1$ such that
$H(t,T_1)/H(t,t_0)\geq\rho$, for all $t\geq T_2$. Using integration
by parts, we conclude that, for all $t\geq T_2$,
\begin{equation} \label{li17}
\begin{split}
&\frac{1}{H(t,t_1)}\int_{t_1}^t\frac{H(t,s)\tau^{n-2}(s)}{
\rho(s)}\Big(w^2(s)+\frac{{p_0}^\alpha}{b}v^2(s)\Big)\mathrm{d}s \\
&=\frac{1}{H(t,t_1)}\int_{t_1}^t[-\frac{\partial
H(t,s)}{\partial s}]\Big[\int_{t_1}^s\frac{\tau^{n-2}(u)
\big(w^2(u)+\frac{{p_0}^\alpha}{b}v^2(u)\big)}{\rho(u)}\mathrm{d}u\Big]
\mathrm{d}s \\
&\geq\frac{\kappa}{\rho}\frac{1}{H(t,t_1)}\int_{T_1}^t
[-\frac{\partial H(t,s)}{\partial s}]\mathrm{d}s
=\frac{\kappa H(t,T_1)}{\rho H(t,t_1)}.
\end{split}
\end{equation}
It follows from \eqref{li17} that, for all $t\geq T_2$,
$$
\frac{1}{H(t,t_1)}\int_{t_1}^t\frac{H(t,s)\tau^{n-2}(s)}{
\rho(s)}\Big(w^2(s)+\frac{{p_0}^\alpha}{b}v^2(s)\Big){\rm
d}s\geq \kappa.
$$
Since $\kappa$ is an arbitrary positive constant, we obtain
$$
\liminf_{t\to\infty}\frac{1}{H(t,t_1)}\int_{t_1}^t\frac{H(t,s)\tau^{n-2}(s)}{
\rho(s)}\Big(w^2(s)+\frac{{p_0}^\alpha}{b}v^2(s)\Big){\rm
d}s=\infty,
$$
which contradicts \eqref{li12}. Consequently, \eqref{li14} holds.
Thus, we obtain
$$
\int_{t_1}^\infty\frac{\tau^{n-2}(s)w^2(s)}{\rho(s)}{\rm
d}s<\infty,\quad
\int_{t_1}^\infty\frac{\tau^{n-2}(s)v^2(s)}{\rho(s)}\mathrm{d}s<\infty,
$$
and, by  \eqref{li13},
\begin{align*}
&\int_{t_1}^\infty\frac{\tau^{n-2}(s)m_+^2(s)}{\rho(s)}\mathrm{d}s \\
&\leq\int_{t_1}^\infty\frac{\tau^{n-2}(s)w^2(s)
+\big(\frac{{p_0}^\alpha}{b}\big)^2\tau^{n-2}(s)
v^2(s)+\frac{2{p_0}^\alpha}{b}\tau^{n-2}(s)w(s)v(s)}{\rho(s)}{\rm
d}s  \\
& \leq\int_{t_1}^\infty\frac{\tau^{n-2}(s)w^2(s)
 +\big(\frac{{p_0}^\alpha}{b}\big)^2\tau^{n-2}(s)v^2(s)+
\frac{{p_0}^\alpha}{b}\tau^{n-2}(s)[w^2(s)+v^2(s)]}{\rho(s)}{\rm
d}s<\infty, \\
\end{align*}
which contradicts \eqref{li112}. This completes the proof.
\end{proof}

Now, we establish some oscillation criteria for equation \eqref{E}
 when  $\sigma(t)\leq\tau(t)$.

\begin{theorem}\label{T3}
 Let $\sigma(t)\in C^1([t_{0},\infty))$ and $\sigma'(t)>0$.
Assume  that \eqref{cx} holds and
$\sigma(t)\leq\tau(t)$. Furthermore, assume that the function $H\in
C(\mathbb{D},\mathbb{R})$ has the property $P$ and there exist
functions $h\in C(\mathbb{D}_0,\mathbb{R})$ and $\rho\in
C^1([t_0,\infty),(0,\infty))$ such that \eqref{xingx} holds.  If
\begin{equation}\label{th2}
\limsup_{t\to\infty} \frac{1}{H(t,t_0)}\int_{t_0}^t
K_2(t,s){\rm
 d}s=\infty
\end{equation}
 for all constants $M>0$ and $L>0$ and for some $\beta\geq1$, where
 $$
K_2(t,s):=\big(\frac{L}{2}\big)^{\alpha-1}H(t,s)\rho(s)Q(s)
 -\big(1+\frac{{p_0}^\alpha}{b}\big)
\frac{\beta\rho(s)h^2(t,s)}{2\sigma'(s)MH(t,s)\sigma^{n-2}(s)},
 $$
then  \eqref{E} is almost oscillatory.
\end{theorem}

\begin{proof} Define $w$ and $v$ by
$$
w(t)=\rho(t)\frac{z^{(n-1)}(t)}{z\big(\sigma(t)/2\big)},\quad
v(t)=\rho(t)\frac{z^{(n-1)}(\tau(t))}{z\big(\sigma(t)/2\big)},
$$
respectively. The rest of the proof is similar to that of Theorem
\ref{T1} and so is omitted.
\end{proof}

From Theorem \ref{T3}, wiht a proof similar to the one of Theorem \ref{T2}, we
obtain the following result.


\begin{theorem}\label{T4}
 Let $\sigma(t)\in C^1([t_{0},\infty))$ and $\sigma'(t)>0$.
Assume  that \eqref{cx} holds and $\sigma(t)\leq\tau(t)$.
Let $H,h,\rho$ be as in Theorem \ref{T1}
such that \eqref{li10} holds. Further, suppose that there exists a
function $m\in C([t_0,\infty),\mathbb{R})$ such that for all $T\geq
t_0$ and for some $\beta>1$,
\begin{equation}\label{xyli11}
\limsup_{t\to\infty}\frac{1}{H(t,T)}\int_T^tK_2(t,s){\rm
d}s\geq m(T)
\end{equation}
for all constants $M>0$ and $L>0$, where $K_2$ is defined as
in Theorem \ref{T3}. If
\begin{equation}\label{txli112}
\limsup_{t\to\infty}\int_{t_0}^t\frac{\sigma'(s)\sigma^{n-2}(s)m_+^2(s)}{\rho(s)}
\mathrm{d}s=\infty,
\end{equation}
where $m_+(t):=\max\{m(t),0\}$, then \eqref{E} is almost
oscillatory.
\end{theorem}


\begin{remark}\label{T8} \rm
From Theorems \ref{T1}--\ref{T4}, we can derive different
 conditions for the oscillation of equation \eqref{E} with
different choices of  $\rho$, $H$ and $m$.
\end{remark}

For an application of our results,  we give the following example.

\begin{example} \label{exea1} \rm
 Consider the odd-order delay differential equation
\begin{equation} \label{exam}
[x(t)+p_0x\big(t/\tau\big)]^{(n)}
+\frac{q_0}{t^n}x \big(t/\sigma\big)=0,\quad t\geq1,
\end{equation}
where $p_0\in[0,\infty)$, $q_0\in(0,\infty)$ and $\sigma>\tau\geq 1$.

Let $q(t)=q_0/t^n$ and $v(t)=0$. Then $Q(t)=q_0/t^n$. Moreover, we
have
$$
\int_{t_0}^\infty s^{n-1}Q(s)\mathrm{d}s=q_0\int_1^\infty
\frac{1}{s}\mathrm{d}s=\infty.
$$
Hence by Corollary \ref{Co3}, equation \eqref{exam} is almost
oscillatory if
$$
q_0>\frac{(n-1)!(1+\tau p_0)\sigma^{n-1}}{\mathrm{e}
\ln(\sigma/\tau)}.
$$
If $p_0\in[0,1),$ then by \cite[Example 1]{bas}, equation
\eqref{exam} is almost oscillatory provided that
$$
q_0>\frac{(n-1)!\sigma^{n-1}}{\mathrm{e}(1-p_0)\ln \sigma}.
$$
\end{example}

We find that our results improve that of in  \cite{bas} for some cases.
For example, we let $\sigma=\mathrm{e}^2$ and $\tau=\mathrm{e}$. If we set
$p_0=7/8$ or $15/16$, we see that
$$
\frac{1}{2(1-p_0)}>1+\mathrm{e}p_0.
$$
Further our results hold for $p_0\geq1$.

One can construct examples easily to illustrate other results,
and the details are left to the reader.


\subsection*{Summary}
We have established criteria for the oscillation
of solutions to \eqref{E}.  Our technique permits us to
relax restrictions usually imposed on the coefficients of equation
\eqref{E}. So our results are of high generality, and are easily applicable 
as illustrated with a suitable example.

\subsection*{Acknowledgements}
The authors thank the anonymous referres for their suggestions which 
improve the content of this article.


\begin{thebibliography}{00}

\bibitem{rpa}  R. P. Agarwal, S. R. Grace and D. O'Regan;
\emph{Oscillation Theory for Difference and Functional Differential
Equations}, Kluwer Academic, Dordrecht, 2000.

\bibitem{bace} B. Bacul\'{i}kov\'{a},  E. M. Elabbasy,
S. H. Saker and J. D\v{z}urina;
\emph{Oscillation criteria for third-order nonlinear
differential equations}, Math. Slovaca 58 (2008) 1--20.

\bibitem{bac} B. Bacul\'{i}kov\'{a} and J. D\v{z}urina;
\emph{Oscillation theorems for second order neutral differential
equations}, Comput. Math. Appl. 61 (2011) 94--99.


\bibitem{tca} T. Candan and R. S. Dahiya;
\emph{Oscillatory and asymptotic behavior of odd order neutral
differential equations}, Dyn. Contin.
Discrete Impuls. Syst. Ser. A Math. Anal. 14 (2007) 767--774.


\bibitem{pdas} P. Das;
\emph{Oscillation in odd-order neutral delay differential equations},
Proc. Indian Acad. Sci. Math. Sci. 105 (1995)
219--225.

\bibitem{pdas1} P. Das, B. B. Mishra and C. R. Dash;
\emph{Oscillation theorems for neutral delay differential equations
of odd order}, Bull. Inst. Math. Acad. Sin. 1 (2007) 557--568.

\bibitem{jd} J. D\v{z}urina;
\emph{Oscillation theorems for neutral
differential equations of higher order}, Czechoslovak Math. J. 54
(2004) 185--195.

\bibitem{dzurrk} J. D\v{z}urina and R. Kotorova;
\emph{Properties of the third order
trinomial differential equations with delay argument}, Nonlinear
Anal. 71 (2009) 1995--2002.


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

\bibitem{kgo} K. Gopalsamy, B. S. Lalli and B. G. Zhang;
\emph{Oscillation of odd order neutral differential equations},
Czechoslovak Math. J. 42 (1992) 313--323.


\bibitem{han3} Z. Han, T. Li,  S. Sun and  C. Zhang;
\emph{Oscillation behavior of third-order neutral Emden-Fowler delay
dynamic equations on time scales}, Adv. Difference Equ. 2010 (2010)
1--23.

\bibitem{hl} Z. Han, T. Li, C. Zhang and S. Sun;
\emph{An oscillation criterion for third order neutral delay
differential equations}, J. Appl. Anal. (in press).

\bibitem{bas} B. Karpuz, \"{O}. \"{O}calan and S.  \"{O}zt\"{u}rk;
\emph{Comparison theorems on the oscillation and asymptotic
behavior of higher-order neutral differential equations},
 Glasgow Math. J. 52 (2010) 107--114.

\bibitem{kig} I. T. Kiguradze;
\emph{On the oscillation of solutions of
the equation $\frac{d^m u}{dt^m} + a(t)|u|^n \, \mbox{sgn}\,(u) =
0$}, Mat. Sb., 65 (1964), 172 - 187 (Russian).

\bibitem{kigu} I. T. Kiguradze;
\emph{The problem of oscillation of solutions of nonlinear
differential equations}, Diff. Urv., 1 (1965),
995 - 1006 (Russian).

\bibitem{ladas} G. Ladas and C. Qian;
\emph{Linearized oscillations for odd order neutral
delay differential equations}, J. Differential Equations 88 (1990)
238--247.

\bibitem{gsl} G. S. Ladde, V. Lakshmikantham and B. G. Zhang;
\emph{Oscillatory theory of differential equations with deviating
arguments}, Marcel Dekker, New York, 1987.

\bibitem{li2} T. Li;
\emph{Comparison theorems for second-order neutral differential
 equations of mixed type}, Electron. J. Diff. Eq. 167 (2010) 1--7.

\bibitem{lih} T. Li, Z. Han, C. Zhang and S. Sun;
\emph{On the oscillation of second-order Emden-Fowler neutral
differential equations}, J. Appl. Math. Comput. (in press).

\bibitem{np1} N. Parhi and R. N. Rath;
\emph{On oscillation of solutions
of forced nonlinear neutral differential equations of higher order},
Czechoslovak Math. J. 53 (2003) 805--825.

\bibitem{np2} N. Parhi and R. N. Rath;
\emph{On osillation of solutions of
forced nonlinear neutral differential equations of higher order II},
Ann. Polon. Math. 81 (2003) 101--110.

\bibitem{chg} Ch. G. Philos;
\emph{A new criterion for the oscillatory and
asymptotic behavior of delay differential equations}, Bull. Polish
Acad. Sci. S\'{e}r. Sci. Math. 29 (1981) 367--370.

\bibitem{chg2} Ch. G. Philos;
\emph{Oscillation theorems for linear
differential equations of second order}, Arch. Math. 53 (1989)
482--492.

\bibitem{rnr} R. N. Rath;
\emph{Oscillatory and asymptotic behavior of
solutions of higher order neutral equations}, Bull. Inst. Math. Acad.
Sinica 30 (2002) 219--228.

\bibitem{rnr1} R. N. Rath, L. N. Padhy and N. Misra;
\emph{Oscillation of solutions of non-linear neutral delay
differentialequations of higher order for $p(t)=1$},
 Arch. Math. (Brno) 40 (2004) 359--366.

\bibitem{yrf} Y. Rogovchenko and F. Tuncay;
\emph{Oscillation criteria for
second order nonlinear differential equations with damping},
Nonlinear Anal. TMA 69 (2008) 208--221.

\bibitem{ys} Y. \c{S}ahiner and A. Zafer;
\emph{Bounded oscillation of
nonlinear neutral differential equations of neutral type},
Czechoslovak Math. J. 51 (2001) 185--195.

\bibitem{saker} S. H. Saker and J. V. Manojlovi\'{c};
\emph{Oscillation criteria for second order
superlinear neutral delay differential equations}, E. J. Qualitative
Theory of Diff. Equ. 10 (2004)  1--22.

\bibitem{tang} X. H. Tang;
\emph{Linearized oscillation of odd order nonlinear neutral
delay differential equations (I)}, J. Math. Anal. Appl. 322 (2006)
864--872.

\bibitem{eth} E. Thandapani, S. Murugadass and S. Pinelas;
\emph{Oscillations criteria for second order nonlinear differential
equations with damping and mixed nonlinearities},
 Journal Functional Differential Equations,
17 (2010) 1--10.

\bibitem{zhangW} B. G. Zhang and W. T. Li;
\emph{On the oscillation of odd order neutral differential equations},
Fasc. Math. 29 (1999) 167--183.

\end{thebibliography}

\end{document}