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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 186, 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 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/186\hfil Asymptotic behavior]
{Asymptotic behavior of solutions to higher order nonlinear delay 
differential equations}

\author[H. Liang \hfil EJDE-2014/186\hfilneg]
{Haihua Liang}  % in alphabetical order

\address{Haihua Liang   \newline
Department of Computer Science,
Guangdong Polytechnic Normal University, \newline 
Guangzhou, Guangdong 510665, China}
\email{haiihuaa@tom.com}

\thanks{Submitted June 27, 2014. Published September 3, 2014.}
\subjclass[2000]{34K11, 34K25}
\keywords{Higher order differential equation; delay differential equation,
\hfill\break\indent asymptotic behavior; oscillation}

\begin{abstract}
 In  this article, we study the oscillation and  asymptotic behavior of solutions
 to the nonlinear delay differential equation
 $$
 x^{(n+3)}(t)+p(t)x^{(n)}(t)+q(t)f(x(g(t)))=0.
 $$
 By using a generalized Riccati transformation and an integral averaging technique,
 we establish sufficient conditions for all solutions to oscillate, or to
 converge to zero.  Especially when the delay has the form $g(t)=at-\tau$,
 we provide two convenient oscillatory criteria.
 Some  examples are given to illustrate our results.
\end{abstract}

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

\section{Introduction}

In  this article, we study  the oscillation and the asymptotic behavior
of solutions to the $n+3$-order nonlinear delay
differential equation
\begin{equation}\label{me}
 x^{(n+3)}(t)+p(t)x^{(n)}(t)+q(t)f(x(g(t)))=0, \quad t\in I:=[a,+\infty)
\end{equation}
where  $q\in C(I,\mathbb{R}^+), p\in C^1(I,\mathbb{R})$ with
$p(t)\geq 0$ and it does not vanish identically on any $[T,\infty)\subset I$,
 $g\in C^1(I,\mathbb{R})$ with   $0<g(t)< t, g'(t)\geq 0$ and
$\lim_{t\to +\infty}g(t)=+\infty$,  $f\in C(\mathbb{R},\mathbb{R})$
and $f(u)/u\geq K (u\neq 0)$ for some positive constant $K$.

Our attention is restricted to those solutions of \eqref{me} which
exist on  $I$ and satisfy $\sup_{t\geq T}|x(t)|>0$ for any $T\geq a$.
We make a standing hypothesis that \eqref{me}  possess such solutions.
As usual,  a solution of \eqref{me} is called oscillatory if it has
arbitrarily large zeros, and non-oscillatory otherwise.
Equation \eqref{me} is called oscillatory if all its solutions are oscillatory.

The oscillation and asymptotic behavior have extensive applications 
in the real world.
See  the monographs \cite{Agarwalbook} for more details.
The problem of obtaining the   oscillation and asymptotic behavior of certain
higher-order nonlinear functional differential equations  has been studied by
a number of authors, see
\cite{Agarwalbook,AAT,AGW,ATZ,DoslyLomtatidze,Hartman,PN,Philo1989,Saker,Skerlik}
and the references cited therein.

In 1971 and 1977, \cite{Ladas,Mahfoud}  discussed  the oscillation of solutions 
of the  equation
$$
x^{(n)}(t)+a(t)f(x(g(t)))=0,
$$
where $0<g(t)<t, g(t)\to \infty$ as $t\to +\infty$ and $a(t)>0$.

Recently, the authors in \cite{AGW} studies the $2n$-order nonlinear
functional differential equation
$$
\frac{d^n}{dt^n}\Big(a(t)\big(\frac{d^nx(t)}{dt^n}\big)^\alpha\Big)+q(t)f(x(g(t))),
$$
where $\alpha$ is the ratio of two positive odd integers.
The  oscillation  theorems established here extend a number of existing results.

On the other hand, there are many publications about nonlinear functional
differential equations with damping. For example, the authors in
\cite{TA} investigated  the  third-order nonlinear functional differential equation
$$
(r_2(t)(r_1(t)y')')'+q(t)y'+f(y(g(t)))=0.
$$
Using a generalized Riccati transformation and integral averaging technique,
they  establish some new sufficient conditions which insure that any solution
 of this equation oscillates or converges to zero.

The authors in \cite{Houchengmin} studied the nonlinear functional differential
equation
\begin{equation}\label{houequation}
y^{(4)}(t)+p(t)y'(t)+q(t)f(y(g(t)))=0.
\end{equation}
By applying the generalized Riccati transformation,  it was shown that all
solutions of  \eqref{houequation}  oscillate or converge to zero
under some conditions.


The goal of the present paper is to  study the oscillation and asymptotic
behavior of solutions of the  nonlinear delay differential equation   \eqref{me}.
We note that   equation \eqref{me} with $n=1$ is exactly \eqref{houequation}.
The authors in \cite{Houchengmin} showed that
the oscillation and asymptotic behavior  of  \eqref{houequation} may
 yield useful information in real problems.
Therefore, we think that  it is  interesting to study the oscillation of
\eqref{me} since  it  extends the former studies. The main idea in the proof
of our results comes from  \cite{Houchengmin,TA}.
This  paper is organized as follows:
In Section 2, we   present some lemmas which are
useful in the proof of our main results.
Section 3 will provide  several  oscillatory and asymptotic  criteria
for system \eqref{me}. We note that, in many applications the delay $g(t)$
has the form $g(t)=t-\tau$ or the form $g(t)=at$.
As the corollary of our main results, we give two convenient oscillatory
and asymptotic  criterions  for system \eqref{me} having such a common delay;
see Corollaries  \ref{cor2} and   \ref{cor5}, respectively.
In Section 4, some examples illustrate our main results.

\section{Some preliminary lemmas}

In this section we state and prove some lemmas which we will use
in the proof of our main results.

\begin{lemma} \label{lem1}
Suppose the linear third-order differential equation
 \begin{equation}\label{3orderhomo}
u'''(t)+p(t)u(t)=0, \quad t \geq a
\end{equation}
has an eventually positive increasing solution.
If $x$ is a non-oscillatory solution of \eqref{me}, then  there exists
a constant $T$ such that
$|x^{(n)}(t)|>0$ for $t\geq T$.
\end{lemma}

\begin{proof}
Without loss of generality  that $x(t)>0$ for
$t\geq a$.  It is easy to see that $y=-x^{(n)}$ is a solution of
 \begin{equation}\label{lem1.eq1}
y'''(t)+p(t)y(t)=q(t)f(x(g(t))).
\end{equation}
By using a similar argument as that in the proof  of
\cite[Lemma 1.2]{Houchengmin},
 we conclude that  all solutions of \eqref{lem1.eq1} are non-oscillatory.
Thus $x^{(n)}(t)$ is eventually positive or eventually negative.
\end{proof}

\begin{lemma} \label{lem2}
Let $x$ be  a non-oscillatory solution of \eqref{me}.  If there exists
a constant $T_1\geq a$ such that $x(t)x^{(n)}(t)>0$  for $t\geq T_1$, then
$x(t)x^{(n+2)}(t)$ is eventually positive.
 \end{lemma}

\begin{proof}
Suppose firstly  that $x(t)>0$ for
$t\geq T_1$.   Since $g'(t)>0$ and $g(t)\to +\infty$ as $t\to +\infty$,
it follows that  $x(g(t))>0$ for $t>T_1'$ for  some constant $T_1'\geq T_1$. For
the sake of brevity we assume that $T_1=T_1'=a$ without loss of generality.

By  \eqref{me} we find that $x^{(n+3)}(t)<0$ for $t\geq a$. Thus
there exists a  $\mu\in \mathbb{R}\cup \{-\infty\}$ such that
$\lim_{t\to +\infty}x^{(n+2)}(t)=\mu$.
 In view of $x^{(n)}(t)>0, t\in I$, it turns out that $\mu\geq 0$.
Thus $x^{(n+2)}(t)$ is eventually positive.

The  case that $x(t)<0$ for $t\geq T_1$   can be discussed in a way completely
analogous to the previous one, and hence it is omitted.
This completes the proof.
\end{proof}


By a careful check of the proof of \cite[Lemma 1.1]{Houchengmin},
 we obtain the following result.

\begin{lemma} \label{lem3}
Assume that  $x\in C^{n}(I,\mathbb{R})$ such that $x(t)>0, x^{(n)}(t)\leq 0$
  for $t\in I$ and $x^{(n)}(t)$ does not vanish identically on any
$[T,\infty)\subset I$.
If $n$ is even (or odd), then there exists $l\in \{1,3, \dots, n-1\}$
(resp. $l\in \{0,2, \dots, n-1\}$) such that for all sufficiently large
$t$, $x(t)x^{(j)}(t)>0$ for $j=0,1, \dots, l$ and
$(-1)^{n+j-1}x(t)x^{(j)}(t)>0$ for $j=l+1, l+2, \dots, n-1$.
Furthermore,  if $l\geq 1$, then
 \begin{equation}\label{eq1lem3}
|x'(g(t))|\geq \frac{g^{l-1}(t)(t-g(t))^{n-l-1}}{2^{l-1}(l-1)!(n-l-1)!}|x^{(n-1)}(t)|
\end{equation}
 for all sufficiently large $t$.
 \end{lemma}

\begin{remark}  \rm
Lemma \ref{lem3} is different from \cite[Lemma 1.1]{Houchengmin}
 by pointing out that  inequality
\eqref{lem3}  is invalid for $l=0$.
 And the case $l=0$ needs  a  separate  treatment in the proof
of our main results.
\end{remark}

\section{Asymptotic Dichotomy}

In this section we  present  some sufficient conditions which
 guarantee that every solution of \eqref{me} oscillates or converges to
 zero.  Throughout this section we will  impose the following condition:
\begin{equation}\label{thstaequ1'}
\lim_{t\to \infty} \int_a^{t}[q(\tau)-Mp_+'(\tau)|d\tau =+\infty,
\end{equation}
for  any   $M>0$, where
$$
p_+'(t)=\begin{cases}
p'(t), &\text{if } p'(t)>0, \\
0,     &\text{if } p'(t)\leq 0.
\end{cases}
$$

\begin{theorem} \label{theorem1}
Suppose that  \eqref{3orderhomo} has an eventually positive increasing solution
and that  \eqref{thstaequ1'} holds.  Assume further that there exists
a $\rho\in C^1(I, \mathbb{R}^+)$ such that
\begin{equation}\label{eq1ofthe1}
\limsup_{t\to \infty} \int_T^{t}
\Big[K\rho(s)q(s)- \frac{2^{l-3}(l-1)!(n-l+2)!(\rho'(s))^2}{g^{l-1}(s)
(s-g(s))^{n-l+2}g'(s)\rho(s)}\Big]ds =+\infty
\end{equation}
holds for every $T\geq a$ and for all $l=2,4,\dots, n+2$ when $n $ is even and
for all $l=1,3,\dots, n+2$ when $n $ is odd.
Then every solution  $x$ of \eqref{me} is
oscillatory, or satisfies $x(t)\to 0$ as $t\to \infty$.
\end{theorem}

\begin{proof} Let  $x$  be a   non-oscillatory solution of \eqref{me}.
Without loss of generality, we may assume  that $x(t)>0$ and $x(g(t))>0$
for $t\geq a$.  By Lemma \ref{lem1},  there exists a constant $T\geq a$ such that
$x^{(n)}(t)>0$ or $x^{(n)}(t)<0$ for $t\geq T$.

  Consider firstly the case that $x^{(n)}(t)>0, t\geq T$. By \eqref{me}
we know that $x^{(n+3)}(t)<0, t\geq T$.  Therefore, it follows form Lemma \ref{lem3}
(it  worth mentioning  here that  $n$ is replaced with  $n+3$)
that there exists $l\in \{1,3, \dots, n+2\}$ (resp. $l\in \{0,2, \dots, n+2\}$)
when  $n$ is odd (resp. $n$ is even)  such that for all sufficiently
large $t$, $x^{(j)}(t)>0$ for $j=0,1, \dots, l$ and $(-1)^{n+j}x^{(j)}(t)>0$
for $j=l+1, l+2, \dots, n+2$.

 If $l\geq 1$, then we consider the function $w$ defined by
\begin{equation}\label{rict}
w(t)=\frac{\rho(t)x^{(n+2)}(t)}{x(g(t))}, \quad t\in I.
\end{equation}
According to Lemma \ref{lem2},  $w(t)$ is eventually positive.
It follows from \eqref{me} and Lemma \ref{lem3} that
\begin{equation} \label{rictiinequality2}
\begin{aligned}
&w'(t)\\
&= \frac{\rho'(t)}{\rho(t)}w(t)-\frac{\rho(t)q(t)f(x(g(t)))}{x(g(t))}
-\frac{\rho(t)p(t)x^{(n)}(t)}{x(g(t))}
 -\frac{\rho(t)x^{(n+2)}(t)x'(g(t))g'(t)}{x^2(g(t))}\\
&\leq  \frac{\rho'(t)}{\rho(t)}w(t)-K\rho(t)q(t)
-\frac{g^{l-1}(t)(t-g(t))^{n-l+2}g'(t)\rho(t)(x^{(n+2)}(t))^2}
 {2^{l-1}(l-1)!(n-l+2)!x^2(g(t))}\\
&=  \frac{\rho'(t)}{\rho(t)}w(t)-K\rho(t)q(t)
-w^2(t)\frac{g^{l-1}(t)(t-g(t))^{n-l+2}g'(t)}{2^{l-1}(l-1)!(n-l+2)!\rho(t)}\\
&=  -K\rho(t)q(t)
-\frac{g^{l-1}(t)(t-g(t))^{n-l+2}g'(t)}
{2^{l-1}(l-1)!(n-l+2)!\rho(t)}\Big(w(t)\\
&\quad -\frac{2^{l-1}(l-1)!(n-l+2)!\rho(t)\rho'(t)}{2\rho(t)g^{l-1}(t)(t-g(t))^{n-l+2}
 g'(t)}\Big)^2
 +\frac{2^{l-3}(l-1)!(n-l+2)!\rho'^2(t)}{\rho(t)g^{l-1}(t)
 (t-g(t))^{n-l+2}g'(t)}.
 \end{aligned}
\end{equation}
Thus
\begin{equation*}
w'(t)\leq  -K\rho(t)q(t)
+\frac{2^{l-3}(l-1)!(n-l+2)!\rho'^2(t)}{\rho(t)g^{l-1}(t)(t-g(t))^{n-l+2}g'(t)}.
\end{equation*}
Integration yields
 \begin{equation*}
\int_T^t\Big(K\rho(s)q(s) -\frac{2^{l-3}(l-1)!(n-l+2)!\rho'^2(s)}
 {\rho(s)g^{l-1}(s)(s-g(s))^{n-l+2}g'(s)}\Big)ds\leq w(T)-w(t),\quad t>T,
\end{equation*}
which contradicts  \eqref{eq1ofthe1}.

If $l=0$ (which means that $n$ is even), then
\begin{equation}\label{l=0,1}
\begin{gathered}
x'(t)<0, \quad x''(t)>0, \quad x'''(t)<0,\quad \dots, \\
x^{(n)}(t)>0, \quad x^{(n+1)}(t)<0, \quad x^{(n+2)}(t)>0
\end{gathered}
\end{equation}
for sufficiently large $t$, namely,  for $t\geq T_1$.
Let $\lim_{t\to \infty} x(t)=\mu$. If $\mu \neq 0$, then there exists a constant
$T_2\geq T_1$ such that
$x(g(t))\geq x(t)>\mu>0, t\geq T_2$.
From \eqref{me} we obtain
\begin{equation} \label{l=0,2}
 x^{(n+2)}(t)\leq x^{(n+2)}(T_2)-K\int_{T_2}^tx(g(u))q(u)du
 \leq x^{(n+2)}(T_2)-K\mu\int_{T_2}^tq(u)du, 
\end{equation}
for $t\geq T_2$.
By  \eqref{thstaequ1'} we know that $\int_{T_2}^{\infty}q(u)du=+\infty$.
Thus inequality \eqref{l=0,2} implies  that $x^{(n+2)}(t)$ is eventually
negative, a contradiction to \eqref{l=0,1}.

 Consider next  the case that $x^{(n)}(t)<0$ for  $t\geq T$.
 By Lemma  \ref{lem3}, $x(t)$ is eventually  monotonous and
$x^{(n-1)}(t)$ is eventually positive. Let
$$
\lim_{t\to +\infty}x(t)=\alpha_1, \quad
\lim_{t\to +\infty}x^{(n-1)}(t)=\alpha_2.
$$
We claim that  $\alpha_1=0$. If this is not true, then there exist
constants  $\beta_1, \beta_2>0$ such that
\begin{equation}\label{e2}
x(g(t))>\beta_1,\quad 0<x^{(n-1)}(t)<\beta_2,\quad   t\geq T_3
\end{equation}
for some constant $T_3>0$.

 Integrating \eqref{me} from $T_3$ to $t$ yields
\begin{align*}
&x^{(n+2)}(t)+\int_{T_3}^t[(p(u)x^{(n-1)}(u))'-p'(u)x^{(n-1)}(u)]du\\
&+\int_{T_3}^tx(g(u))q(u)\frac{f(x(g(u)))}{x(g(u))}du\\
&= x^{(n+2)}(T_3).
\end{align*}
Thus by \eqref{e2} we obtain 
\begin{equation} \label{arr2}
\begin{aligned}
& x^{(n+2)}(t)\\
&\leq x^{(n+2)}(T_3)+p(T_3)x^{(n-1)}(T_3)+\int_{T_3}^tp'(u)x^{(n-1)}(u)du
 -\int_{T_3}^t\beta_1 Kq(u)du\\
& \leq  x^{(n+2)}(T_3)+p(T_3)x^{(n-1)}(T_3)+\int_{T_3}^tx^{(n-1)}p_+'(u)du
 -\int_{T_3}^t\beta_1 Kq(u)du\\
& \leq  x^{(n+2)}(T_3)+p(T_3)x^{(n-1)}(T_3)+\int_{T_3}^t\beta_2p_+'(u)du
 -\int_{T_3}^t\beta_1 Kq(u)du\\
&=  x^{(n+2)}(T_3)+ p(T_3)x^{(n-1)}(T_3)-\beta_1 K\int_{T_3}^t[q(u)
 -\frac{\beta_2}{\beta_1 K}p_+'(u)]du.
\end{aligned}
\end{equation}
 By letting $t\to +\infty$, we get from \eqref{thstaequ1'} that
$x^{(n+2)}(t)\to -\infty$.
 Consequently, there is a constant $T_4\geq T_3$ such that
$x^{(n+2)}(t)\leq -1$ for $t\geq T_4$.
 Hence  $x^{(n+1)}(t)\leq x^{(n+1)}(T_4)-(t-T_4)\to -\infty$ as
$t\to +\infty$. By the same way, it follows that
 $x^{(n)}(t), x^{(n-1)}(t), \dots, x'(t), x(t)\to -\infty$ as  $t\to +\infty$.
 This contradict the assumption
 that $x(t)$ is eventually positive.
\end{proof}


\begin{remark} \rm
Conditions \eqref{thstaequ1'}  are not equivalent to
\cite[Condition (2.2)]{Houchengmin}.
We would like to  point out here  that, unfortunately,  the proof of the
main theorem in \cite{Houchengmin} contains an error. In fact, in
the first paragraph of Page 6,  under the assumption   $y(t)>0, y'(t)<0$,
the authors  conclude that $y'(t)\to 0$ as $t\to \infty$.
Obviously, this is not necessarily true.
\end{remark}

In what follows we give two interesting criteria for the oscillatory
and asymptotic  behavior of  the solutions to  \eqref{me}.

\begin{corollary} \label{cor1}
Suppose that \eqref{3orderhomo} has an eventually positive increasing solution
and that  \eqref{thstaequ1'} holds.  Assume further that
\begin{equation}\label{eq1ofcor1}
\limsup_{t\to \infty} \int_T^{t}
\Big[Kq(s)- \frac{2^{l-3}(l-1)!(n-l+2)!g'(s)}{g^{l+1}(s)(s-g(s))^{n-l+2}}\Big]g(s)ds
=+\infty
\end{equation}
holds for  all $l=2,4,\dots, n+2$ when $n $ is even and
for all $l=1,3,\dots, n+2$ when $n $ is odd.
Then every solution  $x$ of \eqref{me} is
oscillatory, or satisfies $x(t)\to 0$ as $t\to \infty$.
\end{corollary}

The conclusion of the above corollary follows from Theorem \ref{theorem1}
by letting $\rho(t)=g(t)$.

\begin{corollary} \label{cor2}
Suppose that  \eqref{3orderhomo} has an eventually positive increasing
solution and that   \eqref{thstaequ1'} holds. If
   $\lim_{t\to \infty}t/g(t)\geq \alpha>1$,
then every solution  $x$ of \eqref{me} is
oscillatory, or satisfies $x(t)\to 0$ as $t\to \infty$.
\end{corollary}

\begin{proof}
Since $\lim_{t\to \infty}t/g(t)\geq\alpha>1$, there exists a constant
$\bar{\alpha}>1$ such that
$t/g(t)>\bar{\alpha}$ for $t\geq T_1$, where $T_1\geq a$.
Hence
\begin{equation}  \label{coreq1}
\begin{aligned}
&\int_{T_1}^{t} \frac{g'(s)g(s)}{g^{l+1}(s)(s-g(s))^{n-l+2}}ds\\
&= \int_{T_1}^{t} \frac{g'(s)}{g^{n+2}(s)(s/g(s)-1)^{n-l+2}}ds \\
&\leq \frac{1}{(\bar{\alpha}-1)^{n-l+2}}\int_{T_1}^{t} \frac{g'(s)}{g^{n+2}(s)}ds\\
&= \frac{1}{(n+1)(\bar{\alpha}-1)^{n-l+2}}
 \Big(\frac{1}{g^{n+1}({T_1})}-\frac{1}{g^{n+1}(t)}\Big)\\
&< \frac{1}{g^{n+1}({T_1})(n+1)(\bar{\alpha}-1)^{n-l+2}},\quad
  \text{for all } t>{T_1}.
\end{aligned}
\end{equation}
By \eqref{thstaequ1'} we obtain that $\int_a^{\infty}q(t)dt=+\infty$. Note that
$\lim_{t\to \infty}g(t)=+\infty$, it turns out  that
$\int_a^{\infty}q(t)g(t)dt=+\infty$. Using this result  and
the inequality \eqref{coreq1}, the required conclusion follows
from  Corollary \ref{cor1}.
\end{proof}

In applications there are many models in which the delay $g(t)$ satisfies
the condition in Corollary \ref{cor2}.
As an example,  $g(t)=at-\tau$ for $a\in (0,1)$,
but not for $a=1$.  The  case $a=1$ will be discussed later.

Our next goal is to  present some new oscillation results for  \eqref{me},
by using the so-called  integral averages condition of Philos-type.
Following  the literature  \cite{Philo1989}, we introduce a class of functions $\Re$.
Let
$$
D_0=\{(t,s): t>s\geq a \}, \quad
D=\{(t,s): t\geq s\geq a \}.
$$
If  the function $H\in C(D, \mathbb{R})$ satisfies
\begin{itemize}
\item[(i)] $H(t,t)=0$ for $t\geq a$ and  $H(t,s)>0$ for $(t,s)\in D_0$,

\item[(ii)] $H$ has a continuous and non-positive partial derivative on
$D_0$ with respect to the second variable such that
  $$
\frac{\partial H(t,s)}{\partial s}=-h(t,s)\sqrt{H(t,s)}\quad  \text{for all }
  (t,s)\in D_0,
$$
\end{itemize}
then $H$ is said to belong to the class $\Re$.

\begin{theorem} \label{theorem2}
Suppose that equation \eqref{3orderhomo} has an eventually positive increasing
solution and that   \eqref{thstaequ1'} holds.
Assume further that there exist functions   $H\in \Re$ and
$\rho\in C^1(I, \mathbb{R}^+)$ such that
\begin{equation}\label{eq1ofthe2}
\limsup_{t\to \infty}\frac{1}{H(t,T)} \int_T^{t}
\Big[K\rho(s)H(t,s)q(s)- \frac{(\rho(s)h(t,s)
 -\sqrt{H(t,s)}\rho'(s))^2}{\rho^2(s)G_l(s)}\Big]ds
=+\infty,
\end{equation}
where
  $$
G_l(t)=\frac{g^{l-1}(t)(t-g(t))^{n-l+2}g'(t)}{a(l)\rho(t)}\quad
\text{with } a(l)=2^{l-3}(l-1)!(n-l+2)!,
$$
where $l=2,4,\dots, n+2$ when $n $ is even, and
 $l=1,3,\dots, n+2$ when $n $ is odd.
Then every solution  $x$ of \eqref{me} is
oscillatory, or satisfies $x(t)\to 0$ as $t\to \infty$.
\end{theorem}

\begin{proof}
Let $x$ be a non-oscillatory solution of \eqref{me}.
Without loss of generality, we may assume  that $x(t)>0$ and $x(g(t))>0$
for $t\geq a$.  By Lemma \ref{lem1},  there exists a constant $T\geq a$ such that
$x^{(n)}(t)>0$ or $x^{(n)}(t)<0$ for $t\geq T$.

Assume firstly that $x^{(n)}(t)>0$ for $t\geq T$.
It follows from  \eqref{me} that   $x^{(n+3)}(t)<0$
  and hence there exists $l\in \{1,3, \dots, n+2\}$
(resp. $l\in \{0,2, \dots, n+2\}$) when  $n$ is odd (resp. $n$ is even)
such that for all sufficiently large $t$, $x^{(j)}(t)>0$ for $j=0,1, \dots, l$
and $(-1)^{n+j}x^{(j)}(t)>0$ for $j=l+1, l+2, \dots, n+2$.

Defining again the function $w$ as in \eqref{rict}.
If $l\neq 0$, then we get from \eqref{rictiinequality2} that
\begin{equation}\label{eq2ofthe2}
K\rho(t)q(t)\leq -w'(t)+\frac{\rho'(t)}{\rho(t)}w(t)
-\frac{1}{4}w^2(t)G_l(t).
\end{equation}
Thus
\begin{align*}
&K\int_T^t H(t,s)\rho(s)q(s)ds\\
&\leq \int_T^t
\Big[-w'(s)H(t,s)+\Big(\frac{\rho'(s)}{\rho(s)}w(s)
-\frac{1}{4}w^2(s)G_l(s)\Big)H(t,s)\Big]ds.
\end{align*}
Using  integration by parts and noting that $H\in \Re$, we find
\begin{align*}
-\int_T^tw'(s)H(t,s)ds
&= w(T)H(t,T)+\int_T^tw(s)\frac{\partial H(t,s)}{\partial s}ds\\
&= w(T)H(t,T)-\int_T^tw(s)h(t,s)\sqrt{H(t,s)}ds.
\end{align*}
Let
$$
Q(t,s)=h(t,s)-\sqrt{H(t,s)}\frac{\rho'(s)}{\rho(s)},
$$
then
\begin{align*}
&K\int_T^t H(t,s)\rho(s)q(s)ds\\
&\leq w(T)H(t,T)  -\int_T^t\Big[w(s)\sqrt{H(t,s)}Q(t,s)
+\frac{1}{4}G_l(s)H(t,s)w^2(s)\Big]ds \\
&=  w(T)H(t,T)
 -\frac{1}{4}\int_T^tG_l(s)H(t,s)
\Big(w(s)+\frac{2Q(t,s)}{G_l(s)\sqrt{H(t,s)}}\Big)^2ds
 +\int_T^t\frac{Q^2(t,s)}{  G_l(s)}ds\\
&\leq  w(T)H(t,T)+\int_T^t\frac{Q^2(t,s)}{ G_l(s)}ds.
\end{align*}
It turns out that
\begin{equation} \label{rictiinequality3}
\frac{1}{H(t,T)}\int_T^t \Big[K H(t,s)\rho(s)q(s)-\frac{Q^2(t,s)}{  G_l(s)}\Big]ds
\leq w(T).
\end{equation}
This contradicts  \eqref{eq1ofthe2}.
The rest of the proof is the same as in Theorem \ref{theorem1},
and hence it is omitted.
\end{proof}

By letting $\rho(t)=g(t)$ in \eqref{eq1ofthe2},  from  Theorem  \ref{theorem2}
we obtain the following result.

\begin{corollary} \label{cor3}
Suppose that  \eqref{3orderhomo} has an eventually positive increasing solution
and that   \eqref{thstaequ1'} holds.  Assume further that there exists
function   $H\in \Re$ such that
\begin{equation}\label{cor3eq1}
\begin{aligned}
&\limsup_{t\to \infty}\frac{1}{H(t,T)} \int_T^{t}
\Big[Kg(s)H(t,s)q(s)\\
&- \frac{a(l)(g(s)h(t,s)
 -\sqrt{H(t,s)}g'(s))^2}{g^{l}(s)(s-g(s))^{n-l+2}g'(s)}\Big]ds
=+\infty,
\end{aligned}
\end{equation}
where  $l=2,4,\dots, n+2$ when $n $ is even and
  $l=1,3,\dots, n+2$ when $n $ is odd.
Then every solution $x$ of \eqref{me} is
oscillatory, or satisfies $x(t)\to 0$ as $t\to \infty$.
\end{corollary}


\begin{corollary} \label{cor4}
Suppose that \eqref{3orderhomo} has an eventually positive increasing solution
and that   \eqref{thstaequ1'} holds.  If  $g'(t)>0$ and there is a real
number $m\neq 0$ such that
\begin{equation}\label{cor4eq1}
\begin{aligned}
&\limsup_{t\to \infty}\frac{1}{[g(t)-g(T)]^m} \int_T^{t}
\Big[K\big(\frac{g(t)}{g(s)}-1\big)^mq(s)\\
&- \frac{m^2a(l)(g(t)-g(s))^{m-2}g^2(t)g'(s)}{g^{l+m+1}(s)(s-g(s))^{n-l+2}}\Big]ds
=+\infty,
\end{aligned}
\end{equation}
where  $l=2,4,\dots, n+2$ when $n $ is even and
  $l=1,3,\dots, n+2$ when $n $ is odd.
Then every solution   $x$ of \eqref{me} is
oscillatory, or satisfies $x(t)\to 0$ as $t\to \infty$.
\end{corollary}

\begin{proof}
Let $H(t,s)=[g(t)-g(s)]^m$, $\rho(t)=1/g^m(t)$, then
$H\in \Re,  \rho\in C^1(I,\mathbb{R}^+)$. Moreover,
$h(t,s)=mg'(s)(g(t)-g(s))^{m/2-1}$. Consequently,
\begin{equation} \label{cor4e2}
\begin{aligned}
&\frac{(\rho(s)h(t,s)-\sqrt{H(t,s)}\rho'(s))^2}{\rho^2(s)G_l(s)}\\
&=\frac{(g(t)-g(s))^{m-2}(m\rho(s)g'(s)-(g(t)-g(s))\rho'(s))^2}{\rho^2(s)G_l(s)}\\
&=\frac{m^2(g(t)-g(s))^{m-2}g^2(t)g'^2(s)}{G_l(s)g^2(s)} \\
&=\frac{m^2a(l)(g(t)-g(s))^{m-2}g^2(t)g'(s)}{g^{l+m+1}(s)(s-g(s))^{n-l+2}}.
\end{aligned}
\end{equation}
The required conclusion follows from \eqref{cor4eq1} and \eqref{cor4e2}.
\end{proof}


In some applications, the delay $g(t)$ has the form $g(t)=t-\tau$ with
$\tau>0$ which does not satisfy the condition
$\lim_{t\to \infty}t/g(t)\geq \alpha>1$ of Corollary \ref{cor2}.  Next we give a
convenient criterion for system \eqref{me} having such a delay.

\begin{corollary} \label{cor5}
Suppose the following conditions  hold:
\begin{itemize}
\item[(i)] Equation \eqref{3orderhomo} has an eventually positive increasing solution;

\item[(ii)]  Condition \eqref{thstaequ1'} holds and  there are  integer $m>1$ and
constant $\alpha>0$  such that $\lim_{t\to \infty}q(t)/t^{m-1}\geq \alpha$;

\item[(iii)]   $g(t)=at-\tau$  with   $0<a\leq 1$ and $\tau>0$.
\end{itemize}
Then every solution   $x$ of \eqref{me} is
oscillatory, or satisfies $x(t)\to 0$ as $t\to \infty$.
\end{corollary}

\begin{proof}
By  Corollary \ref{cor4}, it suffices to show that \eqref{cor4eq1} holds. 
For the sake of brevity, we
only give the proof of the case that $a=1$. The proof of the other cases is similar 
and hence is omitted.  Obviously,
condition (ii) implies that $q(t)/(t-\tau)^{m-1}>\alpha/2, t\geq T_1$ for some 
constant $T_1>a$. Hence
\begin{align*}
&\int_{T_1}^{t}\Big(\frac{g(t)}{g(s)}-1\Big)^mq(s)ds\\
&=\int_{T_1}^{t}\frac{(t-s)^m}{s-\tau}\cdot\frac{ q(s)}{(s-\tau)^{m-1}}ds\\
&\geq \frac{\alpha}{2}\int_{T_1}^{t}\frac{(t-s)^m}{s-\tau}ds\\
&=\frac{\alpha}{2}\int_{T_1}^{t}\frac{((t-\tau)-(s-\tau)))^m}{s-\tau}ds\\
&=\frac{\alpha}{2}\sum_{k=0}^{m}C_m^k(-1)^k(t-\tau)^{m-k}
 \int_{T_1}^{t}(s-\tau)^{k-1}ds\\
&=\frac{\alpha}{2}\Big((t-\tau)^mln\frac{t-\tau}{T_1-\tau}
+\sum_{k=1}^{m}C_m^k(-1)^k\frac{(t-\tau)^m-(t-\tau)^{m-k}(T_1-\tau)^k}{k}\Big),
\end{align*}
where $C_m^k=\frac{m!}{(m-k)!k!}$.
It turns out that
\begin{equation} \label{cor5eq1}
\begin{aligned}
&\lim_{t\to \infty}\frac{1}{(g(t)-g(T))^m} \int_T^{t}
\Big(\frac{g(t)}{g(s)}-1\Big)^mq(s)ds \\
&\geq   \lim_{t\to \infty}\frac{\alpha}{2}
\Big(\frac{(t-\tau)^m}{(t-T)^m}ln\frac{t-\tau}{T_1-\tau}
+\sum_{k=1}^{m}C_m^k(-1)^k\frac{(t-\tau)^m-(t-\tau)^{m-k}(T_1-\tau)^k}{k(t-T)^m}\Big)\\
& = +\infty.
\end{aligned}
\end{equation}
On the other hand,
\begin{equation} \label{cor5eq5}
\begin{aligned}
&\frac{1}{(g(t)-g(T))^m} \int_T^{t} \frac{(g(t)-g(s))^{m-2}g^2(t)g'(s)}
 {g^{l+m+1}(s)(s-g(s))^{n-l+2}}ds\\
&= \frac{(t-\tau)^2}{(t-T)^m} \int_T^{t} \frac{((t-\tau)-(s-\tau))^{m-2}}
 {(s-\tau)^{l+m+1}\tau^{n-l+2}}ds\\
&=\frac{1}{\tau^{n-l+2}}I_l(t),
\end{aligned}
\end{equation}
where
$$
I_l(t)=\frac{(t-\tau)^2}{(t-T)^m} \int_T^{t}
\frac{((t-\tau)-(s-\tau))^{m-2}}{(s-\tau)^{l+m+1}}ds.
$$

If $m=2$, then \begin{eqnarray}\label{cor5eq6'}
I_l(t)=\big(\frac{t-\tau}{t-T}\big)^2\int_T^{t}\frac{1}{(s-\tau)^{l+3}}ds<M_1, 
\end{eqnarray} 
where $M_1$ is a constant.

If $m>2$, then 
\begin{equation} \label{cor5eq6} 
\begin{aligned}
I_l(t)&= \frac{(t-\tau)^2}{(t-T)^m}\sum_{k=0}^{m-2}C_{m-2}^k(-1)^k(t-\tau)^{m-2-k}
\int_T^{t}(s-\tau)^{k-l-m-1}ds\\
&=  \big(\frac{t-\tau}{t-T}\big)^m
\sum_{k=0}^{m-2}C_{m-2}^k(-1)^k (t-\tau)^{-k}\frac{(T-\tau)^{k-l-m}-(t-\tau)^{k-l-m}}
{m+l-k}\\
&=  \big(\frac{t-\tau}{t-T}\big)^m
\sum_{k=0}^{m-2}C_{m-2}^k(-1)^k\frac{(T-\tau)^{k-l-m}(t-\tau)^{-k}-(t-\tau)^{-l-m}}
{m+l-k}\\
&<M_2,
\end{aligned}
\end{equation}
where $M_2$ is a constant.

By \eqref{cor5eq5}, \eqref{cor5eq6'} and \eqref{cor5eq6}, it is easy to see that
\begin{equation}\label{cor5eq7}
\limsup_{t\to \infty}\frac{1}{[g(t)-g(T)]^m} \int_T^{t} 
\frac{m^2a(l)(g(t)-g(s))^{m-2}g^2(t)g'(s)}{g^{l+m+1}(s)(s-g(s))^{n-l+2}}ds
<+\infty.
\end{equation}
Finally,   combining   \eqref{cor5eq1} with  \eqref{cor5eq7}, we find that 
\eqref{cor4eq1} holds. This completes the proof.
\end{proof}

\section{Examples}

In this section, we give examples that illustrate our main results.

\begin{example} \label{examp1} \rm 
Consider the  eighth-order delay differential equation
\begin{equation}\label{example1}
x^{(8)}(t)+\frac{1}{(1+2t)^2}\Big(\frac{t^2+t-2}{(1+t)^3\ln(1+t)}
+\frac{3}{(1+2t)}\Big)x^{(5)}(t)+\frac{3t+\sin t}{t^2-2}x(t-\ln t)=0, 
\end{equation}
for $t\geq 2$.
Here $n=5$, 
$$
p(t)=\frac{1}{(1+2t)^2}\Big(\frac{t^2+t-2}{(1+t)^3\ln(1+t)}+\frac{3}{(1+2t)}\Big), \quad 
q(t)=\frac{3t+\sin t}{t^2-2}
$$ 
 $\text{ and}\ f(x)=x$ with  $K=1$.

The equation $u'''+p(t)u=0$ has a positive and  strictly  increasing solution  
$u(t)=(2t+1)^{3/2}\ln(1+t)$.
It is easy to see that $\int_2^{\infty}q(t)dt=+\infty$, $p'(t)$  is 
eventually negative and hence that
\eqref{thstaequ1'} is true. Let $\rho(t)=t$, then it is easy to see that 
for $l=1,3,5,7$,
\begin{align*}
&\limsup_{t\to \infty} \int_2^{t}\Big[K\rho(s)q(s)
 - \frac{2^{l-3}(l-1)!(n-l+2)!(\rho'(s))^2}{g^{l-1}(s)(s-g(s))^{n-l+2}g'(s)\rho(s)}
 \Big]ds\\
&=\limsup_{t\to \infty} \int_2^{t}\Big[\frac{3s^2+s\sin s}{s^2-2}
-\frac{2^{l-3}(l-1)!(7-l)!}{(s-\ln s)^{l-1}(\ln s)^{7-l}(s-1)}\Big]ds
=+\infty.
\end{align*}
Consequently, by Theorem \ref{theorem1}, any solution of \eqref{example1} 
is oscillatory, or satisfies $x(t)\to 0$ as $t\to \infty$.
\end{example}

\begin{example} \label{examp2} \rm
Consider the  fourth-order delay differential equation
\begin{equation}\label{example2}
x^{(4)}(t)
+\frac{3(\ln^2t-2)}{t^3\ln^3t}x'(t)+\frac{t+1}{t^2+1}x
\Big((1+\sin\frac{1}{t^2+1})\frac{t}{2}\Big)=0,\quad t\geq 1.
\end{equation}
The delay  function $g(t)=(1+\sin\frac{1}{t^2+1})\frac{t}{2}$ satisfies    
$0<g(t)<t$, $\lim_{t\to +\infty}g(t)=+\infty$ and  $t/g(t)\geq 2/(1+\sin(1/2))>1$. 
It is not hard  to check that the equation $u'''+p(t)u=0$, with 
$p(t)=\frac{3(\ln^2t-2)}{t^3\ln^3t}$, has a positive and  strictly  increasing 
solution  $u(t)=t\ln^3t$. Moreover, since 
$$
p'(t)=\frac{3}{t^4\ln^4t}(6+6\ln t-ln^2t-3\ln^3t),
$$ 
$p'_+(t)=0$ for sufficiently  large $t$.  Clearly, 
$\int_1^{\infty}q(t)dt\geq \int_1^{\infty}\frac{t+1}{2t^2}dt=+\infty$,  which
 implies that \eqref{thstaequ1'} is true.
Thus, by Corollary \ref{cor2}, any solution of \eqref{example2} 
is oscillatory, or satisfies $x(t)\to 0$ as $t\to \infty$.
\end{example}



\begin{example} \label{examp3} \rm  
Consider the fifth-order delay differential equation
\begin{equation}\label{example3}
x^{(5)}(t) +\frac{2}{t^3(1+2\ln t)}x''(t)+(5+e^{-t}\cos t)tx(at-\tau)
(2+\exp[-x(at-\tau)])=0,
\end{equation}
for $t\geq 1$,
where $a\in (0,1], \tau >0$.
Obviously, the  function $f(x)=x(2+e^{-x})$ satisfies that  
$f(x)/x\geq 2$ for $x\neq 0$. It is easy to check that the equation $u'''+p(t)u=0$ 
has a positive and  strictly  increasing solution  $u(t)=t(2\ln t+1)$. 
Moreover, since $p'(t)\leq 0$ and 
$\int_1^{\infty}q(t)dt=\int_1^{\infty}(5+e^{-t}\cos t)tdt=+\infty$,  it
follows that \eqref{thstaequ1'} is satisfied. Clearly, $\lim_{t\to \infty}q(t)/t=5$.
Thus, by Corollary \ref{cor5}, any solution of \eqref{example3} 
is oscillatory, or satisfies
$x(t)\to 0$ as $t\to \infty$.
\end{example}

\subsection*{Acknowledgments}
This research was  supported  by the NSF of China (grant 11201086),
by the Foundation for Distinguished Young Talents in Higher Education of
Guangdong, China (grant 2012LYM\_0087), and by the Excellent Young Teachers 
Training Program for colleges and universities of Guangdong Province, China 
(grant Yq2013107).


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\end{document}