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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 229, pp. 1--7.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/229\hfil Oscillation of solutions ]
{Oscillation of solutions to second-order half-linear differential
 equations with \\ neutral terms}

\author[T. Li, M. T. \c{S}enel, C. Zhang \hfil EJDE-2013/229\hfilneg]
{Tongxing Li, Mehmet Tamer \c{S}enel, Chenghui Zhang}  % in alphabetical order

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

\address{Mehmet Tamer \c{S}enel \newline
Department of Mathematics, Faculty of Sciences,
Erciyes University, Kayseri 38039, Turkey}
\email{senel@erciyes.edu.tr}

\address{Chenghui Zhang \newline
School of Control Science and Engineering,
Shandong University, Jinan, Shandong 250061, China}
\email{zchui@sdu.edu.cn}

\thanks{Submitted September 6, 2013. Published October 16, 2013.}
\subjclass[2000]{34C10, 34K11}
\keywords{Oscillation; neutral differential equation; comparison theorem}

\begin{abstract}
 This article studies the oscillatory behavior of  second-order
 half-linear differential equations with several neutral terms. Some
 criteria are presented that include those reported in \cite{TLi4}.
\end{abstract}

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

\section{Introduction}


Neutral differential equations are used for modeling many problems
arising in astrophysics, atomic physics, gas and fluid mechanics,
etc. Therefore, analysis of qualitative behavior of solutions to
such equations is important for applications. In particular,
oscillatory and nonoscillatory behavior of solutions to various
classes of neutral differential equations has always attracted
attention of researchers; see, e.g., the references in this article and
their references.
 In this article, we are concerned with the oscillation
of the second-order neutral differential equation
\begin{equation}\label{E}
\big(r(t)|z'(t)|^{\alpha-1}z'(t)\big)'
+q(t)|x(\sigma(t))|^{\alpha-1}x(\sigma(t))=0,
\end{equation}
where $z(t):=x(t)+\sum_{i=1}^mp_i(t)x(\tau_i(t))$, and
\begin{itemize}
\item[(H1)] $m>0$ is an integer, $q\in {\rm C}[t_{0},\infty)$, $r, p_i, \tau_i,
\sigma\in {\rm C}^{1}[t_{0},\infty)$;

\item[(H2)] $\alpha\geq1$, $r(t)> 0$,  $q(t)>0,$  $0\leq p_i(t)\leq
a_{i}<\infty$ for $i=1,2,\ldots,m$;

\item[(H3)]  $\lim_{t\to\infty}\sigma(t)=\infty$, 
$\tau_i\circ\sigma=\sigma\circ\tau_i$,  $\tau_i'(t)\geq \lambda_{i}>0$
for $i=1,2,\ldots,m$.
\end{itemize}
Also we assume that
\begin{equation}\label{R}
\lim_{t\to\infty}R(t)<\infty,
\quad R(t):=\int_{t_{0}}^{t}\frac{1}{r^{1/\alpha}(s)}\,\mathrm{d}s.
\end{equation}

By a solution to \eqref{E}, we mean a function $x\in {\rm
C}[T_{x},\infty),$ $T_{x}\geq t_{0},$ which has the property
$r|z'|^{\alpha-1}z'\in {\rm C}^{1}[T_{x},\infty)$ and satisfies
\eqref{E} on $[T_{x},\infty)$. We consider only those solutions $x$
of \eqref{E} which satisfy $\sup \{|x(t)|:t \geq T\}>0$
for all $T\geq T_{x}$ and tacitly assume that \eqref{E} possesses
such solutions. A solution of \eqref{E} is said to be oscillatory if
it does not have the largest zero on $[T_{x},\infty)$; otherwise, it
is called nonoscillatory. Equation \eqref{E} is said to be
oscillatory if all its solutions are oscillatory.

Below, we present some background details that motivate our study.
Bacul\'{i}kov\'{a} and D\v{z}urina \cite{BDz}, Li et al.
\cite{TLi,TLi4}, and Zhang et al. \cite{Zhang} studied the neutral
differential equation
\[
(r(t)(x(t)+p(t)x(\tau(t)))')'+q(t)x(\sigma(t))=0,
\]
and established some oscillation criteria under the conditions that
\[
0\leq p(t)\leq p_0<\infty    \quad   \text{and}   \quad
\tau\circ\sigma=\sigma\circ\tau.
\]
Agarwal et al. \cite{agarwal18}, Bacul\'{i}kov\'{a} and D\v{z}urina
\cite{BDz2}, Bacul\'{i}kov\'{a} et al. \cite{BDz3}, and Li et al.
\cite{TLi5, TLi7} considered oscillation of \eqref{E} in the case
where $m=1$. Assuming
\[
\lim_{t\to\infty} \int_{t_{0}}^{t}\frac{1}{r^{1/\alpha}(s)}\,\mathrm{d}s =\infty,
\]
Zhang et al. \cite{Zhang2} extended results of \cite{BDz} to
equation \eqref{E}.

We stress that results in
\cite{agarwal18,BDz2,BDz3,TLi5,TLi7,Zhang2} cannot be applied to
\eqref{E} in the case where \eqref{R} holds and $m\neq1$. 
Our objective in this work is to define a method for the analysis of
oscillatory properties of  \eqref{E} via the comparison
principles suggested by Zhang et al. \cite{Zhang2},
under assumption \eqref{R}.

In what follows, all functional inequalities are assumed to hold
eventually, that is, for all $t$ large enough.

\section{Oscillation criteria}

In what follows, we use the notation
\[
Q(t):=\min\{q(t),q(\tau_1(t)),q(\tau_2(t)),\ldots,q(\tau_m(t))\}
\]
and
\[
\tilde{Q}(t,t_1):=Q(t)(R(\eta(t))-R(t_1))^\alpha,
 \quad   \delta(t):=\int_{t}^{\infty}r^{-1/\alpha}(s)\,\mathrm{d}s,
\]
for $t\geq t_1$, $t_1\geq t_0$ is sufficiently large, where $\eta$
will be specified later.


\begin{theorem}\label{thm1}
Assume {\rm (H1)--(H3)} and \eqref{R}. Suppose that there exist two
functions $\eta,\xi\in {\rm C}[t_0,\infty)$ such that
$\eta(t)\leq\sigma(t)\leq\xi(t)$ and
$\lim_{t\to\infty}\eta(t)=\infty$. If the first-order
neutral differential inequalities
\begin{equation}\label{E1}
\Big(y(t)+\sum_{i=1}^m\frac{{a_i}^\alpha}{\lambda_i}\,y(\tau_i(t))\Big)'
+\frac{\tilde{Q}(t,t_1)}{(m+1)^{\alpha-1}}y(\eta(t)) \leq0
\end{equation}
and
\begin{equation}\label{e1}
\Big(u(t)+\sum_{i=1}^m\frac{{a_i}^\alpha}{\lambda_i}\,u(\tau_i(t))\Big)'
-\frac{Q(t)\delta^\alpha(\xi(t))}{(m+1)^{\alpha-1}}u(\xi(t))
\geq0
\end{equation}
have no positive solutions, then  \eqref{E} is oscillatory.
\end{theorem}

\begin{proof}
Let $x$ be an eventually positive solution of \eqref{E}. 
Using that $x\mapsto x^{\alpha}$ is convex for $\alpha\geq 1$ and $x>0$,
we obtain the following inequality that corresponds to (2.7) in 
\cite[Theorem 1]{Zhang2}, 
\begin{equation}\label{txe2}
\begin{aligned}
&(r(t)|z'(t)|^{\alpha-1}z'(t))'
+\sum_{i=1}^m\frac{{a_i}^\alpha}{\lambda_i}
(r(\tau_i(t))|z'(\tau_i(t))|^{\alpha-1}z'(\tau_i(t)))'\\
&+\frac{Q(t)}{(m+1)^{\alpha-1}}z^\alpha(\sigma(t))\leq0.
\end{aligned}
\end{equation}
Assuming that $x(t)>0$, Equation \eqref{E} implies that, for some $t_1$ 
large enough and for all $t\geq t_1$, either
\begin{equation}\label{case1}
z'(t)>0,   \quad   (r(t)|z'(t)|^{\alpha-1}z'(t))'<0,
\end{equation}
or
\begin{equation}\label{case2}
z'(t)<0,   \quad   (r(t)|z'(t)|^{\alpha-1}z'(t))'<0.
\end{equation}
Assume first that \eqref{case1} holds. Inequality \eqref{txe2}
reduces to
\begin{equation}\label{txe2i}
(r(t)(z'(t))^\alpha)'+\sum_{i=1}^m\frac{{a_i}^\alpha}{\lambda_i}(r(\tau_i(t))(z'(\tau_i(t)))^\alpha)'
+\frac{Q(t)}{(m+1)^{\alpha-1}}z^\alpha(\sigma(t))\leq0.
\end{equation}
The fact that $z'(t)>0$ and $z(t)>0$ imply that $z^{\alpha}$ is increasing.
Then, \eqref{txe2i} and $\eta(t)\leq\sigma(t)$ yield
\begin{equation}\label{txe2ii}
(r(t)(z'(t))^\alpha)'+\sum_{i=1}^m\frac{{a_i}^\alpha}{\lambda_i}(r(\tau_i(t))(z'(\tau_i(t)))^\alpha)'
+\frac{Q(t)}{(m+1)^{\alpha-1}}z^\alpha(\eta(t))\leq0.
\end{equation}
It follows from \eqref{case1} that $y:=(z')^\alpha r$ is positive and 
decreasing. Using that $z(t)>0$, we have
\begin{equation}\label{zy}
\begin{aligned}
z(t)&\geq \int_{t_1}^{t}\frac{(r(s)(z'(s))^\alpha)^{1/\alpha}}{r^{1/\alpha}(s)}
\,\mathrm{d}s\\
&\geq y^{1/\alpha}(t)\int_{t_1}^{t}\frac{\mathrm{d}s}{r^{1/\alpha}(s)}\\
&=y^{1/\alpha}(t)(R(t)-R(t_1)).
\end{aligned}
\end{equation}
Therefore, setting $y:=(z')^\alpha r$ in \eqref{txe2ii} and using
\eqref{zy}, one can see that $y$ is a positive solution of
\eqref{E1}. This contradicts our assumption that inequality
\eqref{E1} has no positive solutions.


Consider now the second case. It follows from \eqref{case2} that
$(r(t)(-z'(t))^\alpha)'>0$, and hence
\[
z'(s)\leq\frac{r^{1/\alpha}(t)z'(t)}{r^{1/\alpha}(s)} \quad
\text{for all} \quad  s\geq t,
\]
which, upon integration, leads to
\[
z(l)\leq z(t)+r^{1/\alpha}(t)z'(t)\int_{t}^{l}\frac{\mathrm{d}s}{r^{1/\alpha}(s)}.
\]
Since $z(t)> 0$, passing to the limit as $l\to\infty$, 
\[
0\leq z(t)+r^{1/\alpha}(t)z'(t)\int_{t}^{\infty}
\frac{\mathrm{d}s}{r^{1/\alpha}(s)}.
\]
Therefore,
\begin{equation}\label{litx123}
z(t)\geq-r^{1/\alpha}(t)z'(t)\delta(t).
\end{equation}
Since $z'(t)<0$, inequality \eqref{txe2} reduces to
\begin{equation}\label{txe2ik}
(r(t)(-z'(t))^\alpha)'+\sum_{i=1}^m
\frac{{a_i}^\alpha}{\lambda_i}(r(\tau_i(t))(-z'(\tau_i(t)))^\alpha)'
-\frac{Q(t)}{(m+1)^{\alpha-1}}z^\alpha(\sigma(t))\geq0.
\end{equation}
Note that $z(t)>0$ and $z'(t)<0$, thus $z^{\alpha}$ is decreasing.
It follows from $\sigma(t)\leq\xi(t)$ that
$z^{\alpha}(\sigma(t))\geq z^{\alpha}(\xi(t))$ and
\begin{equation}\label{txe2ik2}
(r(t)(-z'(t))^\alpha)'+\sum_{i=1}^m\frac{{a_i}^\alpha}{\lambda_i}
(r(\tau_i(t))(-z'(\tau_i(t)))^\alpha)'
-\frac{Q(t)}{(m+1)^{\alpha-1}}z^\alpha(\xi(t))\geq0.
\end{equation}
Therefore, setting $u:=(-z')^\alpha r$  and
using \eqref{litx123}, we have that $u$ is a positive solution
of \eqref{e1}. This contradicts our assumption and the proof is
complete.
\end{proof}

\begin{remark}\label{rmk1} \rm
When $m=1$ and $\alpha=1$, Theorem \ref{thm1} reduces to
\cite[Theorem 1]{TLi4}.
\end{remark}

Using additional assumptions on the coefficients of \eqref{E}, one
can deduce from Theorem~\ref{thm1} a number of oscillation criteria
applicable to different classes of equations. In what follows, we
use the notation $\tau_*(t):=\max\{\tau_i(t):i=1,2,\ldots,m\}$ and
$\tau(t):=\min\{\tau_i(t):i=1,2,\ldots,m\}$, the notation
$\tau_*^{-1}$ and $\tau^{-1}$ stand for the inverse of the functions
$\tau_*$ and $\tau$, respectively.


\begin{theorem}\label{thm2}
Assume {\rm (H1)--(H3)} and \eqref{R}. Suppose that there exist two
functions $\eta,\xi\in {\rm C}[t_0,\infty)$ such that
$\eta(t)\leq\sigma(t)\leq\xi(t)$ and
$\lim_{t\to\infty}\eta(t)=\infty$. Assume also that
\begin{equation}\label{assum1}
\tau_i(t)\geq t   \quad   \text{for }  i=1,2,\ldots,m.
\end{equation}
If the first-order functional differential inequalities
\begin{equation}\label{E2}
w'(t)+\frac{\tilde{Q}(t,t_1)}{(m+1)^{\alpha-1}(1+\sum_{i=1}^m\frac{{a_i}^\alpha}{\lambda_i})}\,w(\eta(t))\leq0
\end{equation}
and
\begin{equation}\label{e2}
h'(t)-\frac{Q(t)\delta^\alpha(\xi(t))}{(m+1)^{\alpha-1}(1+\sum_{i=1}^m\frac{{a_i}^\alpha}{\lambda_i})}\,h(\tau_*^{-1}(\xi(t)))\geq0
\end{equation}
have no positive solutions, then equation \eqref{E} is oscillatory.
\end{theorem}
\begin{proof}
Let $x$ be an eventually positive solution of \eqref{E}. As in
the proof of Theorem \ref{thm1}, there exist two possible cases
\eqref{case1} and \eqref{case2} for all $t\geq t_1$. Assume first
that \eqref{case1} holds. Along the same lines as in 
\cite[Theorem 2]{Zhang2}, we deduce that the inequality \eqref{E2} has a positive
solution, which contradicts our assumption. Consider now the second
case. It has been established in Theorem~\ref{thm1} that the function
$u:=(-z')^\alpha r$ is positive, increasing, and satisfies the
inequality \eqref{e1}. We now define
\begin{equation}\label{zeru}
h(t):=u(t)+\sum_{i=1}^m\frac{{a_i}^\alpha}{\lambda_i}\,u(\tau_i(t)).
\end{equation}
Then, we have
$$
h(t)\leq u(\tau_*(t))\Big(1+\sum_{i=1}^m\frac{{a_i}^\alpha}{\lambda_i}\Big).
$$
Substituting the latter inequality into \eqref{e1}, we see that $h$
is a positive solution of \eqref{e2}. This contradiction completes
the proof.
\end{proof}

\begin{remark}\label{rmk2} \rm
Theorem \ref{thm2} includes \cite[Theorem 2]{TLi4} in the case where
$m=1$ and $\alpha=1$.
\end{remark}

Combining Theorem \ref{thm2} with the oscillation criteria presented
in Ladde et al. \cite[Theorems 2.1.1 and 2.4.1]{Lad}, we obtain the
following result.

\begin{corollary}\label{Co2d}
Assume {\rm (H1)--(H3)}, \eqref{R}, and \eqref{assum1}. Suppose
further that there exist two functions 
$\eta,\xi\in {\rm C}[t_0,\infty)$ such that $\eta(t)<t$, $\xi(t)>\tau_*(t)$,
$\eta(t)\leq\sigma(t)\leq\xi(t)$, and
$\lim_{t\to\infty}\eta(t)=\infty$. If
\begin{equation}\label{Con1a}
\liminf_{t\to\infty} \int_{\eta(t)}^t
Q(s)R^\alpha(\eta(s))\,\mathrm{d}s>\frac{(m+1)^{\alpha-1}(1+\sum_{i=1}^m\frac{{a_i}^\alpha}{\lambda_i})}{\rm{e}}
\end{equation}
and
\begin{equation}\label{zhcon1}
\liminf_{t\to\infty} \int_{t}^{\tau_*^{-1}(\xi(t))}
Q(s)\delta^\alpha(\xi(s))\,\mathrm{d}s
>\frac{(m+1)^{\alpha-1}(1+\sum_{i=1}^m\frac{{a_i}^\alpha}{\lambda_i})}{\rm{e}},
\end{equation}
then  \eqref{E} is oscillatory.
\end{corollary}

\begin{proof}
By \cite[Theorem 2.1.1]{Lad}, assumption \eqref{Con1a} ensures that
the differential inequality \eqref{E2} has no positive solutions. On
the other hand, by \cite[Theorem 2.4.1]{Lad}, condition
\eqref{zhcon1} guarantees that the differential inequality
\eqref{e2} has no positive solutions. Application of Theorem
\ref{thm2} yields the result.
\end{proof}

\begin{remark}\label{rmrk3} \rm
When $\alpha=1$ and $m=1$, Corollary \ref{Co2d} includes
\cite[Corollary 3]{TLi4}.
\end{remark}

\begin{theorem}\label{thm3}
Assume {\rm (H1)--(H3)}, \eqref{R}, and let $\tau_*(t)\leq t$.
Suppose that there exist two functions 
$\eta,\xi\in {\rm C}[t_0,\infty)$ such that $\eta(t)\leq\sigma(t)\leq\xi(t)$ and
$\lim_{t\to\infty}\eta(t)=\infty$. If the first-order
functional differential inequalities
\begin{equation}\label{E71}
w'(t)+\frac{\tilde{Q}(t,t_1)}{(m+1)^{\alpha-1}(1+\sum_{i=1}^m\frac{{a_i}^\alpha}{\lambda_i})}\,
w(\tau^{-1}(\eta(t)))\leq0
\end{equation}
and
\begin{equation}\label{litxE71}
h'(t)-\frac{Q(t)\delta^\alpha(\xi(t))}{(m+1)^{\alpha-1}(1+\sum_{i=1}^m\frac{{a_i}^\alpha}{\lambda_i})}\,
h(\xi(t))\leq0
\end{equation}
have no positive solutions, then  \eqref{E} is oscillatory.
\end{theorem}

\begin{proof}
Let $x$ be an eventually positive solution of \eqref{E}. By the
proof of Theorem \ref{thm1}, there exist two possible cases
\eqref{case1} and \eqref{case2} for all $t\geq t_1$. Assume first
that \eqref{case1} holds. Following the same lines as in
\cite[Theorem 3]{Zhang2}, we conclude that the inequality
\eqref{E71} has a positive solution, which contradicts our
assumption. Consider now the second case. We have shown in
Theorem~\ref{thm1} that the function $u:=r(-z')^\alpha$ is positive,
increasing, and satisfies the inequality \eqref{e1}. By virtue of
$\tau_*(t)\leq t$, the inequality
$$
h(t)\leq
u(t)\Big(1+\sum_{i=1}^m\frac{{a_i}^\alpha}{\lambda_i}\Big)
$$
holds for the function $h$ defined by \eqref{zeru}. Substituting
this inequality into \eqref{e1}, we conclude that $h$ is a positive
solution of \eqref{litxE71}. This contradiction completes the proof.
\end{proof}


\begin{remark}\label{rmk4} \rm
Theorem \ref{thm3} includes \cite[Theorem 4]{TLi4} when
$\alpha=1$ and $m=1$.
\end{remark}

\begin{corollary}\label{Co3}
Assume {\rm (H1)--(H3)}, \eqref{R}, and let  $\tau_*(t)\leq t$.
Suppose also that there exist two functions 
$\eta,\xi\in {\rm C}[t_0,\infty)$ such that $\eta(t)<\tau(t)$, $\xi(t)>t$,
$\eta(t)\leq\sigma(t)\leq\xi(t)$, and
$\lim_{t\to\infty}\eta(t)=\infty$. If
\begin{equation}\label{Con1a3}
\liminf_{t\to\infty}
\int_{\tau^{-1}(\eta(t))}^tQ(s)R^\alpha(\eta(s))\,\mathrm{d}s
>\frac{(m+1)^{\alpha-1}(1+\sum_{i=1}^m\frac{{a_i}^\alpha}{\lambda_i})}{\rm{e}}
\end{equation}
and
\begin{equation}\label{litxCon1a3}
\liminf_{t\to\infty}
\int_{t}^{\xi(t)}Q(s)\delta^\alpha(\xi(s))\,\mathrm{d}s>\frac{(m+1)^{\alpha-1}(1+\sum_{i=1}^m\frac{{a_i}^\alpha}{\lambda_i})}{\rm{e}},
\end{equation}
then  \eqref{E} is oscillatory.
\end{corollary}

\begin{proof}
As in \cite[Theorem 2.1.1]{Lad}, condition \eqref{Con1a3}
ensures that the differential inequality \eqref{E71} has no positive
solutions. On the other hand, it follows from 
\cite[Theorem 2.4.1]{Lad} that condition \eqref{litxCon1a3} 
guarantees that the differential inequality \eqref{litxE71} 
has no positive solutions.
Application of Theorem \ref{thm3} completes the proof.
\end{proof}


\begin{remark}\label{rmrk5} \rm
When $m=1$ and $\alpha=1$,  Corollary \ref{Co3} reduces to
\cite[Corollary 5]{TLi4}.
\end{remark}

The following example illustrates possible applications of
the theoretical results obtained.

\begin{example}\label{examp1} \rm
For $t\geq1$, consider a second-order neutral differential equation
\begin{equation}\label{exe1}
\begin{aligned}
&\Big({\rm e}^t\Big(x(t)+\frac{1}{4}x\big(t-\frac{\pi}{4}\big)+\frac{1}{4}
x\big(t-\frac{\pi}{2}\big)\Big)'\Big)'\\
&+12\sqrt{65}{\rm e}^tx\Big(t-\frac{1}{8}\arcsin
\frac{\sqrt{65}}{65}\Big)=0.
\end{aligned}
\end{equation}
Let $\eta(t)=t-\pi$ and $\xi(t)=t+\pi/4$. It is not difficult to
verify that all assumptions of Corollary \ref{Co3} are satisfied.
Hence, equation \eqref{exe1} is oscillatory. As a matter of fact,
one such solution is $x(t)=\sin (8t)$.
\end{example}


\subsection*{Acknowledgements}
The authors want to express their sincere gratitude to Professor Julio G. Dix 
for the careful reading of the original manuscript and the useful
comments that helped us improve this article. 

This research is supported by National Key Basic Research Program of
 China (grant 2013CB035604) and NNSF of China (grants
61034007, 51277116, 51107069).


\begin{thebibliography}{99}

\bibitem{agarwal} R. P. Agarwal, M. Bohner, T. Li, C. Zhang;
\newblock Oscillation of second-order differential equations with a sublinear
 neutral term.
\newblock \emph{Carpathian J. Math.}, (2013, in press).

\bibitem{agarwal18} R. P. Agarwal, M. Bohner, T. Li, C. Zhang;
\newblock Oscillation of second-order Emden--Fowler neutral delay differential
 equations.
\newblock \emph{Ann. Math. Pura Appl.}, (2013, in press).

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

\bibitem{BDz2} B. Bacul\'{i}kov\'{a}, J. D\v{z}urina;
\newblock Oscillation theorems for second-order nonlinear neutral differential 
equations.
\newblock \emph{Comput. Math. Appl.}, 62 (2011) 4472--4478.

\bibitem{BDz3} B. Bacul\'{i}kov\'{a}, T. Li, J. D\v{z}urina;
\newblock Oscillation theorems for second-order superlinear neutral differential 
equations.
\newblock \emph{Math. Slovaca}, 63 (2013) 123--134.

\bibitem{candan} T. Candan;
\newblock The existence of nonoscillatory solutions of higher order nonlinear 
neutral equations.
\newblock \emph{Appl. Math. Lett.}, 25 (2012) 412--416.

\bibitem{dix2} J. G. Dix;
\newblock Oscillation of solutions to a neutral differential equation 
involving an $n$-order operator with variable coefficients and a forcing term.
\newblock \emph{Differ. Equ. Dyn. Syst.}, (2013, in press).

\bibitem{dix1} J. G. Dix, B. Karpuz, R. Rath;
\newblock Necessary and sufficient conditions for the oscillation of differential equations involving distributed arguments.
\newblock \emph{Electron. J. Qual. Theory Differ. Equ.}, 2011 (2011) 1--15.

\bibitem{dix} J. G. Dix, Ch. G. Philos, I. K. Purnaras;
\newblock Asymptotic properties of solutions to linear non-autonomous neutral 
differential equations.
\newblock \emph{J. Math. Anal. Appl.}, 318 (2006) 296--304.

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

\bibitem{GrL} S. R. Grace, B. S. Lalli;
\newblock Oscillation of nonlinear second order neutral delay 
differential equations.
\newblock \emph{Rat. Math.}, 3 (1987) 77--84.

\bibitem{Gra} M. K. Grammatikopoulos, G. Ladas, A. Meimaridou;
\newblock Oscillation of second order neutral delay differential equation.
\newblock \emph{Rat. Math.}, 1 (1985) 267--274.

\bibitem{Ro1} M. Hasanbulli, Yu. V. Rogovchenko;
\newblock Oscillation criteria for second order nonlinear neutral 
differential equations.
\newblock \emph{Appl. Math. Comput.}, 215 (2010) 4392--4399.

\bibitem{karpuz1} B. Karpuz;
\newblock Sufficient conditions for the oscillation and asymptotic 
behaviour of higher-order dynamic equations of neutral type.
\newblock \emph{Appl. Math. Comput.}, 221 (2013) 453--462.

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

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

\bibitem{TLi6} T. Li;
\newblock Comparison theorems for second-order neutral differential 
equations of mixed type.
\newblock \emph{Electron. J. Diff. Equ.}, 2010 (2010) no. 167,  1--7.

\bibitem{TLi5} T. Li, R. P. Agarwal, M. Bohner;
\newblock Some oscillation results for second-order neutral differential equations.
\newblock \emph{J. Indian Math. Soc.}, 79 (2012) 97--106.

\bibitem{TLi7} T. Li, Z. Han, C. Zhang, H. Li;
\newblock Oscillation criteria for second-order superlinear neutral 
differential equations.
\newblock \emph{Abstr. Appl. Anal.}, 2011 (2011), 1--17.

\bibitem{TLi3} T. Li, Z. Han, C. Zhang, S. Sun;
\newblock On the oscillation of second-order Emden--Fowler neutral 
differential equations. \newblock \emph{J. Appl. Math. Comput.}, 37 (2011),
 601--610.

\bibitem{TLi} T. Li, Z. Han, P. Zhao, S. Sun;
\newblock Oscillation of even-order neutral delay differential equations.
\newblock \emph{Adv. Difference Equ.}, 2010 (2010) 1--9.

\bibitem{TLi4} T. Li, Yu. V. Rogovchenko, C. Zhang;
\newblock Oscillation of second-order neutral differential equations.
\newblock \emph{Funkc. Ekvacioj}, 56 (2013) 111--120.

\bibitem{TLi2} T. Li, E. Thandapani, J. R. Graef, E. Tunc;
\newblock Oscillation of second-order Emden--Fowler neutral differential
 equations.
\newblock \emph{Nonlinear Stud.}, 20 (2013) 1--8.

\bibitem{Zhang} C. Zhang, R. P. Agarwal, M. Bohner, T. Li;
\newblock New oscillation results for second-order neutral delay dynamic
 equations.
\newblock \emph{Adv. Difference Equ.}, 2012 (2012) 1--14.

\bibitem{Zhang2} C. Zhang, M. T. \c{S}enel, T. Li;
\newblock Oscillation of second-order half-linear differential equations 
with several neutral terms.
\newblock \emph{J. Appl. Math. Comput.}, (2013, in press).

\end{thebibliography}

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