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

\AtBeginDocument{{\noindent\small {\em Electronic Journal of
Differential Equations}, Vol. 2005(2005), No. 65, pp. 1--14.\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 2005 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2005/65\hfil Asymptotic behaviour of solutions]
{Asymptotic behaviour of solutions to $n$-order functional
differential equations}

\author[S. Padhi\hfil EJDE-2005/65\hfilneg]
{Seshadev Padhi}

\address{Department of Applied Mathematics\\
Birla Institute of Technology\\
Mesra, Ranchi -835 215, India}

\curraddr{Department of Mathematics and Statistics, Mississippi State
University,  MS-39762, Mississippi State, USA}
\email{ses\_2312@yahoo.co.in}

\date{}
\thanks{Submitted September 4, 2004. Revised April 30, 2005. Published June 23, 2005.}
\subjclass[2000]{34C10, 34K15}
\keywords{Oscillatory solution; nonoscillatory solution; property A}

\begin{abstract}
We establish conditions for the linear differential equation
$$
    y^{(n)}(t)+p(t)y(g(t))=0
$$
to have property A. Explicit sufficient conditions for the
oscillation of the the equation is obtained while dealing with the
property A of the equations. A comparison theorem is obtained for
the oscillation of the equation with the oscillation of a third
order ordinary differential equation.
\end{abstract}

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

\section{Introduction}

 This paper concerns property A of the $n$-th order $(n\geq 2)$ delay
 differential equation
\begin{equation}\label{first}
y^{(n)}(t)+p(t)y(g(t))=0,
\end{equation}
under certain conditions on the coefficient function
 $p \in C([\sigma,\infty),[0,\infty)),\sigma\in R$, and
 $g \in C([\sigma,\infty),R)$ such that $g(t)\leq t$ and
$g(t)\to\infty$ as $t\to\infty$.

 It is interesting to note that we have obtained sufficient
 conditions for oscillation of all solutions of \eqref{first} while
 dealing with property A of the equation. These sufficient
  conditions are easily verifiable and different from earlier
  ones (See \cite{pdas,erbe,gopa,gyor,kopl,ladd}).
 Moreover, these sufficient conditions are consistent
 with the situation when $p(t)$ is a constant.

 A continuous function $y:[g(\sigma),\infty)\to R$ is said to
be a proper  solution of \eqref{first} if it is absolutely
continuous on $(t_{0},\infty),  t_{0}\geq \sigma$ along with its
derivatives up to the $(n-1)$th order and
  satisfies \eqref{first} almost everywhere on $(t_{0},\infty)$ and
  $\sup\{|y(s)|:s \geq t\}>0$ for $t\geq t_{0}$.
   A proper solution of \eqref{first} is called oscillatory if it has
  a sequence of zeros tending to infinity. Otherwise, it is called
  non-oscillatory. Equation \eqref{first} with $g(t)=t$ is said to be
disconjugate  on $[\sigma,\infty)$ if no nontrivial solution of
the equation has more than
 $(n-1)$ zeros, counting muntiplicities.

 A vast body of literature exist on the oscillation of \eqref{first}.
 One may see the monographs due to Lakshmikantham et al \cite{ladd},
  Gyori and Ladas \cite{gyor} and the references cited therein. Higher
  order differential equations with property A were studied by
  Parhi and Padhi \cite{pars} and  Koplatadze \cite{kopl}. We shall
see that    our results are different form their results.
 We observe that our results do not hold for the case $g(t)=t$
 (See Theorems \ref{thm2.1}-\ref{thm2.4} and \ref{thm2.25} and
 Corollaries \ref{coro2.5} and \ref{coro2.26}).

Let $y(t)$ be a positive solution of \eqref{first} for $t\geq
t_{0}\sigma$. Then there exists a $t_{1}>t_{0}$ such that
$y(g(t))>0$ for $t\geq t_{1}$. Then $y^{(n)}(t)\leq 0$ for $t\geq
t_{1}$,and so by a lemma due to Kiguradze \cite{kigu}, there
exists an integer $\emph{l}$, $0\leq \emph{l}\leq n-1$ such that
$n+\emph{l}$ odd and
\begin{equation}\label{second}
\begin{gathered}
y^{(i)}(t) >0 , \quad i=0,1,2,\dots ,l,\\
(-1)^{i+l}y^{(i)}(t) >0 , \quad i=l+1,\dots ,n.
\end{gathered}
\end{equation}
for large $t$. Again, for $l \in \{1,2,3,\dots ,n-1\}, n+l$ odd,
the following inequality holds for large $t$, say for
$t\geq t_{2}$.
\begin{equation}\label{third}
    |y(t)|
\geq\frac{(t-t_{2})^{(n-1)}}{(n-1)(n-2)\dots
(n-l)}|y^{(n-1)}(2^{n-l-1}t)|,     \quad t \geq t_{2}.
\end{equation}
Let $N$ denote the set of all nonoscillatory solutions of
\eqref{first} and $N_{l}$ denote the set of all nonoscillatory
solutions of \eqref{first} satisfying \eqref{second}. Then
\[
N = \begin{cases}
 N_{0}\cup N_{2}\cup\dots \cup N_{n-1}& \mbox{if $n$ is odd},\\
 N_{1}\cup N_{3}\cup\dots \cup N_{n-1}& \mbox{if $n$ is even.}
\end{cases}
\]

\subsection*{Definition} We say that \eqref{first} has property A if
any of its solution is oscillatory when $n$ is even and either is
oscillatory or satisfies $N_{0}$ when $n$ is odd.


The following conjecture is given in \cite[pp.29, Problem 1.14]{kigu},
which we state as a problem.

\begin{problem} \label{prob1.1}
Let $M_{n^{*}}=\max(\lambda(\lambda-1)(\lambda-2)\dots (\lambda-n+1))$.
If
\[
\int^{\infty}t^{n-1}\big[p(t)-\frac{M_{n^{*}}}{t^{n}}\big]\,dt=\infty,
\]
then \eqref{first} with $g(t)=t$ has property A.
\end{problem}

Our Theorem \ref{thm2.20} gives a partial answer to the above problem for
the case $n=2$ and $g(t)=t$ in \eqref{first}.

The following lemma, due to Kiguradze \cite{kigu}, is needed for our
use in the sequel.

\begin{lemma} \label{lem1.2}
Let for a certain $l \in \{1,2,3,\dots ,n-1\}$, the inequality
\eqref{second} hold. Then
\begin{equation}\label{fourth}
    \int^{\infty}_{t_{1}}s^{n-l-1}|y^{(n)}(s)|\,ds <\infty,
\end{equation}
\begin{equation}\label{fifth}
    y^{(i)}(t)\geq
y^{(i)}(t_{1})+\frac{1}{(l-i-1)!}\int^{t}_{t_{1}}(t-s)^{l-i-1}y^{(i)}(s)\,ds
\end{equation}
for $t\geq t_{1},i=0,1,2,\dots ,l-1$ and
\begin{equation}\label{sixth}
    y^{(l)}(t)\geq
    \frac{1}{(l-i-1)!}\int^{\infty}_{t}(s-t)^{n-l-1}|y^{(n)}(s)|\,ds
\end{equation}
for $t\geq t_{1}$. If in addition
\begin{equation}\label{seventh}
\int^{\infty}_{t_{1}}s^{n-l}|y^{(n)}(s)|\,ds = \infty,
\end{equation}
then there exists $t_{2}\geq t_{1}$ such that
\begin{equation}\label{eighth}
    y^{(l-1)}(t) \geq
    \frac{t}{(n-l)!}\int^{\infty}_{t}s^{n-l-1}|y^{(n)}(s)|\,ds
\end{equation}
for $t\geq t_{2}$ and
\begin{equation}\label{nineth}
    iy^{(l-1)}\geq ty^{(l-i+1)}(t)\geq (i-1)y^{(l-i)}(t)
\end{equation}
for $t\geq t_{2}$, $i\in \{1,2,\dots ,l\}$.
\end{lemma}

\section{Main Results}
\begin{theorem} \label{thm2.1}
Let $g(t)<t$ and for every $l\in \{1,2,3,\dots ,n-1\}$ such that
$n+l$ is odd,
\begin{equation}\label{tenth}
\limsup_{t\to\infty}(t-g(t))^{l}\int^{\infty}_{g^{-1}(t)}(s-t)^{n-l-1}p(s)\,ds
    > (n-l-1)!.l1
\end{equation}
hold. Then \eqref{first} has property A.
\end{theorem}

\begin{proof} Let $y(t)$ be a nonoscillatory solution of
\eqref{first}. Without loss of generality, we may assume that
$y(t)>0$ for $t\geq t_{0}>\sigma$. Thus there exists a $T_{1}\geq
t_{0}$ such that $y(g(t))>0$ for $t\geq T_{1}$. Consequently, from
\eqref{first}, it follows that $y^{(n)}(t)\leq 0$ for $t\geq
T_{1}$. Then, there exists a $l\in\{0,1,2,\dots ,n-1\}$ and $n+l$
odd such that \eqref{second} holds for some $t\geq t_{1}>T_{1}$. We
claim that $l=0$. If not, then $l\in\{1,2,\dots ,n-1\}$. Putting
$i=0$ in \eqref{fifth}, we get
\begin{equation}\label{one}
    y(t)\geq
    \frac{1}{(l-1)!}\int^{t}_{t_{1}}(t-s)^{l-1}y^{(l)}(s)\,ds, \quad
    t\geq t_{1}.
\end{equation}
We can find a $t_{2}\geq t_{1}$ such that $g(t)>t_{1}$ for
$t\geq t_{2}$. Hence, for $t\geq t_{2}$
\begin{equation*}
    y(t)\geq \frac{y^{(l)}(t)}{(l-1)!}\int^{t}_{g(t)}(t-s)^{l-1}\,ds
    \geq \frac{y^{(l)}(t)}{(l-1)!}.\frac{(t-g(t))^{l}}{l};
\end{equation*}
that is,
\begin{equation}\label{eleventh}
    y(t)\geq \frac{(t-g(t))^{l}}{l!}y^{(l)}(t).
\end{equation}
Using \eqref{sixth} in \eqref{eleventh}, we obtain
\begin{equation*}
\begin{split}
y(t) & \geq
\frac{(t-g(t))^{l}}{l!}.\frac{1}{(n-l-1)!}\int^{\infty}_{t}(s-t)^{n-l-1}|y^{(n)}(s)|\,ds\\
& \geq
\frac{(t-g(t))^{l}}{l!}.\frac{1}{(n-l-1)!}\int^{\infty}_{g^{-1}(t)}(s-t)^{n-l-1}|y^{(n)}(s)|\,ds\\
& \geq
\frac{(t-g(t))^{l}}{l!}.\frac{1}{(n-l-1)!}\int^{\infty}_{g^{-1}(t)}(s-t)^{n-l-1}p(s)y(g(s))\,ds\\
& \geq
\frac{(t-g(t))^{l}}{l!}.\frac{1}{(n-l-1)!}y(t)\int^{\infty}_{g^{-1}(t)}(s-t)^{n-l-1}p(s)\,ds
\end{split}
\end{equation*}
for $t\geq t_{2}$, which is a contradiction to the hypothesis of the
theorem. Hence \eqref{first} has property A. This completes the
proof of the theorem.
\end{proof}

\begin{theorem} \label{thm2.2}
Suppose that for every $l\in \{1,2,3,\dots ,n-1\}$ , $n+l$ is
odd,,
\begin{equation}\label{twelfth}
\limsup_{t\to\infty}t^{n-1}\int^{\infty}_{g^{-1}(t)}p(s)\,ds
> (n-1)\dots (n-l)2^{(n-1)(n-l)},
\end{equation}
holds. Then \eqref{first} has property A.
\end{theorem}

\begin{proof} Let $y(t)$ be a non-oscillatory solution of
\eqref{first}. Without any loss of generality, we may assume that
$y(t)>0$ for $t\geq t_{0}>\sigma$. Then there exists a $t_{1}\geq
t_{0}$ such that $y(g(t))>0$ for $t\geq t_{1}$. Consequently, it
follows from \eqref{first} that $y^{(n)}(t)\leq 0$ for
$t\geq t_{1}$ and \eqref{second} holds. If possible, suppose that
\eqref{first} has not property A. Then $l\in\{1,2,3,\dots ,n-1\}$.
Clearly \eqref{third} holds for some $t\geq t_{2}\geq t_{1}$.
Since $y'(t)>0$, then for $t>t.2^{l+1-n} \geq t_{2}$, we
have
\begin{equation}\label{thirteen}
    y(t)\geq y(2^{l+1-n}t) \geq
    \frac{1}{(n-1)\dots (n-l).2^{(n-1)(n-l)}}t^{n-1}y^{(n-1)}(t).
\end{equation}
On the other hand, integrating \eqref{first} from $t(\geq t_{2})$ to
$\infty$,we have
\begin{equation*}
    y^{(n-1)}(t) > \int^{\infty}_{t}p(s)y(g(s))\,ds >
    \int^{\infty}_{g^{-1}(t)}p(s)y(g(s))\,ds
    >y(t)\int^{\infty}_{g^{-1}(t)}p(s)\,ds.
\end{equation*}
Then \eqref{thirteen} gives
\begin{equation*}
1\geq \frac{1}{(n-1)\dots
(n-l).2^{(n-1)(n-l)}}t^{n-1}\int^{\infty}_{g^{-1}(t)}p(s)\,ds
\end{equation*}
for $t\geq t_{2}$, which contradicts \eqref{twelfth}. Hence
\eqref{first} has property A. The Theorem is proved.
\end{proof}

\begin{theorem} \label{thm2.3}
Suppose that $g(t)<t$ and for every $l\in\{1,2,3,\dots,n-1\}$
such that $n+l$ is odd, the following inequality
\begin{equation}\label{fourteen}
\limsup_{t\to\infty}\int^{t}_{g(t)}(t-s)^{l-1}\int^{\infty}_{g^{-1}(g^{-1}(s))}(u-s)^{n-l-1}p(u)\,du\,ds
    >(l-1)!.(n-l-1)!
\end{equation}
holds. Then \eqref{first} has property A.
\end{theorem}

\begin{proof} Let $y(t)$ be a nonoscillatory solution of
\eqref{first}. Without loss of generality, we may assume that
$y(t)>0$ and $y(g(t))>0$ for $t\geq t_{0}>\sigma$. Thus
\eqref{second} holds for some $t\geq t_{1}>t_{0}$. Suppose that
$l\in\{1,2,\dots ,n-1\}$. Putting $i=0$ in \eqref{fifth}, we get
\begin{equation}\label{fifteen}
    y(t)\geq
    \frac{1}{(l-1)!}\int^{t}_{t_{1}}(t-s)^{l-1}y^{(l)}(s)\,ds.
\end{equation}
 From \eqref{fifth}, we obtain
\begin{equation}\label{sixteen}
    y^{(l)}(t)\geq
    \frac{1}{(n-l-1)!}\int^{\infty}_{t}(s-t)^{n-l-1}p(s)y(g(s))\,ds.
\end{equation}
Then from \eqref{fifteen} and \eqref{sixteen}, we obtain
\begin{equation}\label{sixtin}
    y(t)\geq
\frac{1}{(n-l-1)!.(l-1)!}\int^{t}_{t_{1}}(t-s)^{l-1}
\int^{\infty}_{s}(u-s)^{n-l-1}p(u)y(g(u))\,du\,ds.
\end{equation}
We can find a $t_{2}\geq t_{1}$ such that $g(t)>t_{1}$ for
$t\geq t_{2}$. Thus, for $t\geq t_{2}$
\begin{equation*}
y(t)\geq
\frac{1}{(n-l-1)!.(l-1)!}\int^{t}_{g(t)}(t-s)^{l-1}
\int^{\infty}_{g^{-1}(g^{-1}(s))}(u-s)^{n-l-1}p(u)y(g(u))\,du\,ds
\end{equation*}
which in turn, yields
\begin{equation*}
1\geq \frac{1}{(n-l-1)!.(l-1)!}\int^{t}_{g(t)}(t-s)^{l-1}
\int^{\infty}_{g^{-1}(g^{-1}(s))}(u-s)^{n-l-1}p(u)\,du\,ds.
\end{equation*}
Taking limit sup., we obtain a contradiction. Consequently,
\eqref{first} has property A. Hence the theorem is proved.
\end{proof}

\begin{theorem} \label{thm2.4}
Let $g(t)<t$ and
\begin{equation}\label{seventeen}
    \limsup_{t\to\infty}\int^{t}_{g(t)}(s-g(t))^{n-1}p(s)\,ds
    > (n-1)!.
\end{equation}
Then \eqref{first} has no solution satisfying the property
$(-1)^{i}y^{(i)}(t)>0$ for large $t$.
\end{theorem}

\begin{proof} If possible, suppose that \eqref{first} has a
nonoscillatory solution $y(t)$ satisfying the property
$(-1)^{i}y^{(i)}(t)>0$ for large $t$. Then $l=0$ in \eqref{second}.
Suppose that $y(g(t))>0$ and $y(t)>0$ for some $t\geq t_{1}>\sigma$.
 From Lemma 1.2 due to Kiguradze and Chanturia \cite{kigu}, it
follows for $i=0$, that
\begin{equation*}
\begin{split}
y(t) &\geq
\frac{1}{(n-1)!}\int^{\infty}_{t}(s-t)^{n-1}p(s)y(g(s))\,ds\\
&\geq
\frac{1}{(n-1)!}\int^{g^{-1}(t)}_{t}(s-t)^{n-1}p(s)y(g(s))\,ds\\
&\geq \frac{y(t)}{(n-1)!}\int^{g^{-1}(t)}_{t}(s-t)^{n-1}p(s)\,ds,
\end{split}
\end{equation*}
that is,
\begin{equation*}
(n-1)! \geq \int^{g^{-1}(t)}_{t}(s-t)^{n-1}p(s)\,ds,
\end{equation*}
for some $t\geq t_{2} \geq t_{1}$. Then there exists a $t_{3}\geq
t_{2}$ such that $g(t)>t_{2}$ for $t\geq t_{3}$. Hence for $t\geq
t_{3}$, we have
\begin{equation*}
(n-1)! \geq \int^{t}_{g(t)}(s-g(t))^{n-1}p(s)\,ds.
\end{equation*}
Taking limit sup., we obtain a contradiction. Hence $l\neq 0$. The
theorem is proved.
\end{proof}

\begin{corollary} \label{coro2.5}
Suppose that $g(t)<t$, \eqref{seventeen} holds and either
\eqref{tenth} or \eqref{twelfth} or \eqref{fourteen}is satisfied.
Then every solution of \eqref{first} oscillates.
\end{corollary}

\begin{example} \label{exa2.6} \rm
 Consider
\begin{equation}\label{eighteen}
    y'''(t)+\frac{30}{t^{3}}y(t/2^{1/3})=0 ,
    \quad t\geq 2.
\end{equation}
By Theorem \ref{thm2.2}, \eqref{eighteen} has property A. In particular,
$y(t) = 1/t^{3}$ is a nonoscillatory solution of
\eqref{eighteen}.
\end{example}

\begin{example} \label{exa2.7} \rm
Consider
\begin{equation}\label{nineteen}
    y'''(t)+\frac{82}{t^{3}}y(t/3)=0 , \quad t\geq 1.
\end{equation}
\end{example}

Theorem \ref{thm2.1} can be applied to this example where as Theorem
\ref{thm2.3} fails to hold. On the other hand, \eqref{seventeen} is satisfied.
Hence by Corollary \ref{coro2.5}, all solutions of \eqref{nineteen} are
oscillatory.
\begin{example}
Inequality \eqref{fourteen} to the equation
\begin{equation}\label{twentith}
y'''(t)+\frac{63}{t^{3}}y(t/2)=0 , \quad t\geq 1.
\end{equation}
is satisfied, where as \eqref{tenth} fails to hold. Hence
Theorem \ref{thm2.3} can be applied to \eqref{twentith}. Further, since,
\eqref{seventeen} is satisfied, then all solutions of
\eqref{twentith} are oscillatory, by Corollary \ref{coro2.5}.
\end{example}

\textbf{Remark:} Let $p(t)=p >0$ be a constant and
$g(t)=t-\tau$, $\tau>0$ be a constant. Then \eqref{first} becomes
\begin{equation}\label{twentyfirst}
    y^{(n)}(t) +py(t-\tau)=0.
\end{equation}
Clearly, the conditions of \eqref{tenth},\eqref{twelfth} and
\eqref{fourteen} are consistent with $p(t)=p$ and $g(t)=t-\tau$.
Hence form Corollary \ref{coro2.5}, it follows that, if
\begin{equation}\label{twentysecond}
    p\tau^{n} >n!,
\end{equation}
then \eqref{twentyfirst} is oscillatory. \smallskip


The characteristic equation associated with \eqref{twentyfirst} is
given by
\begin{equation}\label{twentythird}
\lambda^{n}+pe^{-\tau\lambda}=0.
\end{equation}
Setting $F(\lambda)= \lambda^{n}+pe^{-\tau\lambda}$, we see that
$F(\lambda)>0$ for $\lambda \geq 0$. Suppose that $\lambda <0$. We
claim that $F(\lambda)>0$ for $\lambda < 0$. If possible suppose
that $F(\lambda)\leq 0$ for $\lambda<0$.
Then $\lambda^{n}\leq -pe^{-\tau\lambda}$.
Then $\lambda^{n}\tau^{n}\leq -n!.e^{-\tau\lambda}$. If $n$ is even,
then $\lambda^{n}\tau^{n}\leq 0$, a contradiction. Hence $n$ must be odd.
Let $\lambda=-\gamma,\gamma>0$. Then $\gamma^{n}\tau^{n}\geq n!.e^{\tau\gamma}$.
Setting $\tau\gamma = \beta$, we see that
$\beta^{n}\geq n!.e^{\beta}$, a contradiction. Hence our claim
holds, that is, $F(\lambda)>0$ for $\lambda<0$. Thus
\eqref{twentysecond} implies that all solutions of
\eqref{twentyfirst} are oscillatory.

 \textbf{Remark:} Although the conditions in Theorems \ref{thm2.1} and
 \ref{thm2.1} are legitimate,
 these are not efficient. When $g(t)$ is close to  $t$, the
 conditions \eqref{tenth} and \eqref{fourteen} fails to hold. This is
 evident from the following examples : If we replace
 $g(t)=\frac{t}{3}$ in \eqref{nineteen} by $g(t)=\frac{3t}{4}$,
 then the equation becomes
 \begin{equation}\label{twentithird}
 y'''(t)+\frac{82}{t^{3}}y(\frac{3t}{4})=0 ,
 t\geq 1.
 \end{equation}
Condition \eqref{tenth} fails to hold and hence  Theorem \ref{thm2.1} cannot
be applied to \eqref{twentithird}. Similarly,consider the equation
 \begin{equation}\label{twenthird}
 y'''(t)+\frac{46}{t^{3}}y(\frac{t}{2})=0,t\geq1.
 \end{equation}
Theorem \ref{thm2.3} can be applied to this example. On the other hand, if
$g(t)=\frac{t}{2}$ in \eqref{twenthird} is replaced by
$g(t)=\frac{10t}{11}$, then \eqref{twenthird} becomes
\begin{equation}\label{tenthird}
y'''(t)+\frac{46}{t^{3}}y(\frac{10t}{11})=0,t\geq1,
 \end{equation}
 then\eqref{fourteen} fails and hence
Theorem \ref{thm2.3}  cannot be applied. The following theorems provides
sufficient conditions for \eqref{first} to have property A  when
$g(t)$ is close to $t$.

\begin{theorem} \label{thm2.9}
Assume that $g(t)<t$ and $t-g(t)\to\infty$ as
$t\to\infty$.If, for every $l\in\{1,2,\dots ,n-1\}$ such that
$n+l$ is odd,
\begin{equation}\label{tenprime}
\limsup_{t\to\infty}(g(t))^{l}\int^{\infty}_{g^{-1}(t)}(s-t)^{n-l-1}p(s)\,ds
>(n-l-1)!.l!
\end{equation}
holds, then \eqref{first} has property A.
\end{theorem}

\begin{proof} We can find a $t_{2}>t_{1}$ such that $t-g(t)>t_{1}$
for $t\geq t_{2}$. Hence for $t\geq t_{2}$, \eqref{one} gives
\begin{equation*}
y(t)\geq \frac{y^{(l)}(t)}{(l-1)!}\int^{t}_{t-g(t)}(t-s)^{l-1}\,ds
\geq \frac{g^{l}(t)}{l!}y^{(l)}(t).
\end{equation*}
using \eqref{sixth}in the above inequality, we obtain a
contradiction. The proof is complete.
\end{proof}

\begin{corollary} \label{coro2.10}
Suppose that the conditions of Theorems \ref{thm2.4} and \ref{thm2.9} are
satisfied. then all solutions of \eqref{first} oscillates.
\end{corollary}

\begin{example} \label{exa2.11} \rm
By Theorem \ref{thm2.9}, \eqref{twentithird} has property A.
\end{example}

\begin{theorem} \label{thm2.12}
Let $g(t)<t$ and $t-g(t)\to\infty$ as
$t\to\infty$.If for every $l\in\{1,2,\dots ,n-1\}$ with $n+l$ odd,
\begin{equation}\label{fourtin}
\limsup_{t\to\infty}\int^{t}_{t-g(t)}(t-s)^{l-1}\int^{\infty}_{g^{-1}(g^{-1}(s))}(u-s)^{n-l-1}p(u)\,du\,ds
    >(l-1)!.(n-l-1)!
\end{equation}
holds, then \eqref{first} has property A.
\end{theorem}

\begin{proof}
Proceeding as in the proof of Theorem \ref{thm2.3}, we arrive
at \eqref{sixtin} for $t\geq t_{1}$. Then we can find a $t_{2}\geq
t_{1}$ such that $t-g(t)>t_{1}$ for $t\geq t_{2}$. Hence from
\eqref{sixtin}, we obtain
\begin{equation*}
y(t)\geq
\frac{1}{(n-l-1)!.(l-1)!}\int^{t}_{t-g(t)}(t-s)^{l-1}\int^{\infty}_{g^{-1}(g^{-1}(s))}(u-s)^{n-l-1}p(u)y(g(u))\,du\,ds
\end{equation*}
which further yields
\begin{equation*}
1\geq
\frac{1}{(n-l-1)!.(l-1)!}\int^{t}_{t-g(t)}(t-s)^{l-1}\int^{\infty}_{g^{-1}(g^{-1}(s))}(u-s)^{n-l-1}p(u)\,du\,ds.
\end{equation*}
Taking limit sup. both sides in the above inequality, we obtain a
contradiction. This completes the proof of the theorem.
\end{proof}

\begin{corollary} \label{coro2.13}
Suppose that the conditions of Theorem \ref{thm2.4} and \ref{thm2.12} are
satisfied. Then all solutions of \eqref{first} are oscillatory.
\end{corollary}

\begin{example} \label{exa2.14} \rm
By Theorem \ref{thm2.12}, \eqref{tenthird} has property A.
\end{example}

 Let $y(t)$ be a nonoscillatory solution of \eqref{first} such that
\eqref{one} holds for $t\geq t_{1}$. Then for $t>t_{2}\geq 2t_{1}$,
\eqref{one} gives
\begin{equation*}
    y(t)\geq
    \frac{1}{(l-1)!}\int^{t}_{t/2}(t-s)^{l-1}y^{(l)}(s)\,ds, \quad
    t\geq t_{1}.
\end{equation*}
Using \eqref{sixth} and the above inequality, we obtain the
following theorem.

\begin{theorem} \label{thm2.15}
Let $g(t)\leq t$. If for every $l\in \{1,2,\dots ,n-1\}$ such that $n+l$ is odd,
\begin{equation*}
\limsup_{t\to\infty}t^{l}\int^{\infty}_{g^{-1}(t)}(s-t)^{n-l-1}p(s)\,ds
>(n-l-1)!.l!.2^{l}
\end{equation*}
holds, then \eqref{first} has property A.
\end{theorem}

\begin{theorem} \label{thm2.16}
Let $g(t)\leq t$ and  for every $l\in\{1,2,\dots ,n-1\}$ such that $n+l$ is odd,
\begin{equation}
\limsup_{t\to\infty}\int^{t}_{t/2}(t-s)^{l-1}\int^{\infty}_{g^{-1}(g^{-1}(s))}(u-s)^{n-l-1}p(u)\,du\,ds
    >(l-1)!.(n-l-1)!
\end{equation}
holds, then \eqref{first} has property A.
\end{theorem}

\begin{proof}  Proceeding as in the proof of Theorem \ref{thm2.3}, we obtain
 \eqref{sixtin}. Then for $t\geq t_{2}>2t_{1}$, \eqref{sixtin} yields
 a contradiction. This completes the proof of the theorem.
 \end{proof}

 We note that when $g(t)=t/2$, then Theorem \ref{thm2.3}, \ref{thm2.12}
  and \ref{thm2.16} give
 same sufficient conditions to have property A of \eqref{first}.

\begin{corollary} \label{coro17}
Suppose that the conditions of Theorem \ref{thm2.4} are
satisfied. If either of the conditions of Theorem \ref{thm2.15} or
\ref{thm2.16} hold, then all solutions of \eqref{first} are oscillatory.
\end{corollary}

\begin{example}  \label{exa2.18} \rm
Consider
\begin{equation*}
y'''(t)+\frac{44}{t^{3}}y(\frac{3t}{5})=0, t\geq1.
\end{equation*}
Theorem \ref{thm2.1} and Theorem \ref{thm2.9} can be applied to this example,
whereas Theorem \ref{thm2.15} cannot be applied to this example.
\end{example}

\begin{example} \label{exa2.19} \rm
Consider
\begin{equation*}
y'''(t)+\frac{160}{t^{3}}y(\frac{t}{3})=0, t\geq1.
\end{equation*}
By Theorem \ref{thm2.15} this equation has property A, whereas
Theorem \ref{thm2.9} fails.
\end{example}

\begin{theorem} \label{thm2.20}
Let $g'(t)>0$. If for every
$l\in\{1,2,3,\dots ,n-1\}$ such that $n+l$ is odd,
\begin{equation}\label{twentyfourth}
    \int^{\infty}H_{l}(t)\,dt = \infty,
\end{equation}
then then for $n$ even every solution of \eqref{first} oscillates
and for $n$ odd every solution
 of \eqref{first} is either oscillates or tend to zero as $t\rightarrow\infty$,in particular,
 \eqref{first} has property A, where
\begin{equation}\label{twentyfifth}
H_{n-1}(t)=t^{n-1}p(t)-\frac{(n-1)!.(n-1)2^{n-4}t^{n-3}}{g'(t)g^{n-2}(t)}
\end{equation}
and
\begin{equation}\label{twentysixth}
    H_{l}(t) =
\frac{t^{l}}{(n-l-2)!}\int^{\infty}_{t}(s-t)^{n-l-2}p(s)\,ds-\frac{l!.l.2^{l-3}t^{l-2}}{g'(t)g^{l-1}(t)},
\end{equation}
for $l = 1,2,3,\dots ,n-2$.
\end{theorem}

\noindent \textbf{Remark:} Let $g(t)=t$ and $n=2$. From Theorem
\ref{thm2.20}, it follows that, if
\begin{equation}\label{twentyseventh}
    \int^{\infty}[tp(t)-\frac{1}{4t}]\,dt=\infty,
\end{equation}
then
\begin{equation}\label{twentyeighth}
    y''+p(t)y=0
\end{equation}
is oscillatory. This gives a partial answer to Problem \ref{prob1.1}. Further,
our result improves the results due to Kneser \cite[pp.45]{swan}
and Hille and Kneser \cite[Theorem 2.41]{swan}. We note that
Theorem \ref{thm2.20} holds for \eqref{first} with $g(t)=t$ for $n=2$ and
$n=3$. however, the theorem cannot be applied to higher order
ordinary differential equations, viz., \eqref{first} with
$g(t)=t$ and $n\geq 4$, because of the conditions
\eqref{twentyfourth} and \eqref{twentysixth}. Now, suppose that
$n=3$ and $g(t)=t$. then Theorem \ref{thm2.20} yields that, if
\begin{equation*}
    \int^{\infty}[t^{2}p(t)-\frac{2}{t}]\,dt=\infty,
\end{equation*}
then
\begin{equation}\label{twentynineth}
    y'''+p(t)y=0
\end{equation}
has property A. On the other hand, from  Hanan \cite[Theorem 5.7]{hana},
and  Kiguradze and Chanturia \cite[Theorem 1.1]{kigu}, it
follows that \eqref{twentynineth} has property A if
\begin{equation}\label{thirty}
    \int^{\infty}[t^{2}p(t)-\frac{2}{3\sqrt{3}t}]\,dt=\infty.
\end{equation}
hence  Theorem \ref{thm2.20} is yet to be improved.

\begin{proof}[Proof of Theorem \ref{thm2.20}]
If possible, suppose that \eqref{first} dose not have property A.
Then \eqref{first}  admits a nonoscillatory solution $y(t)$ such that
$y\in N_{l}$
 where $l\in \{1,2,3,\dots ,n-1\}$. We may assume, without any loss of
 generality, that $y(t)>0$ and $y(g(t))>0$ for $t\geq
 t_{1}>\sigma$. Clearly, \eqref{second} holds, where
 $l\in\{0,1,2,3,\dots ,n-1\}$ and $n+l$ odd.

Let $l=n-1$. Set $z(t)=\frac{t^{n-1}y^{(n-1)}(t)}{y(g(t))}$. Then
\begin{equation}\label{thirtyfirst}
    z'(t)=-t^{n-1}p(t)+\frac{n-1}{t}z(t)-g'(t)
    \frac{y'(g(t))}{y(g(t))}z(t).
\end{equation}
Putting $i=1$, $l=n-1$ in \eqref{fifth}, we obtain, for $t\geq t_{1}$
\begin{equation*}
y'(t)\geq \frac{1}{(n-2)!}(t-t_{1})^{n-2}y^{(n-1)}(t).
\end{equation*}
Hence for $t\geq 2t_{1}$, we get
\begin{equation*}
    y'(t)\geq
    \frac{t^{n-2}}{(n-2)!.2^{n-2}}y^{(n-1)}(t).
\end{equation*}
Thus, for $t\geq t_{2}>2t_{1}$,
\begin{equation*}
 y'(g(t))\geq\frac{(g(t))^{n-2}}{(n-2)!.2^{n-2}}y^{(n-1)}(t)
\end{equation*}
Using the above inequality, \eqref{thirtyfirst} yields
\begin{equation}\label{thirtysecond}
    z'(t)\leq -F_{n-1}(t),
\end{equation}
where
 \begin{equation*}
 F_{n-1}(t)=
t^{n-1}p(t)-\frac{n-1}{t}z(t)+\frac{g'(t)(g(t))^{n-2}}{(n-2)!2^{n-2}t^{n-1}}z^{2}(t),
\end{equation*}
which as a function of $z$, attains the minimum $H_{n-1}(t)$ given
in \eqref{twentyfifth}. Now, the integration of
\eqref{thirtysecond} from $t_{2}$ to $t$ yields $z(t)<0$  for
large $t$, a contradiction. Next, suppose that $l\in\{1,2,3,\dots
,n-2\}$. Setting $z_{1}(t)=\frac{t^{l}y^{(l)}(t)}{y(g(t))} , t\geq
t_{1}$, we see that $z_{1}(t)>0$ for $t\geq t_{1}$ and
\begin{equation}\label{thirtythird}
z'_{1}(t)=\frac{t^{l}y^{(l+1)}(t)}{y(g(t))}+\frac{l}{t}z_{1}(t)-
g'(t)\frac{y'(g(t))}{y(g(t))}z_{1}(t).
\end{equation}
Putting $i=1$ in \eqref{fifth}, we get
\begin{equation*}
    y'(t)\geq \frac{1}{(l-1)!}(t-t_{1})^{l-1}y^{(l)}(t).
\end{equation*}
Thus, for $t\geq t_{2}\geq 2t_{1}$,
\begin{equation*}
    y'(t)\geq \frac{1}{(l-1)!.2^{l-1}}t^{l-1}.y^{(l)}(t).
\end{equation*}
We can find a $t_{3}>t_{2}$ such that $g(t)>t_{2}$ for $t\geq
t_{3}$. Hence
\begin{equation}\label{thirtyfourth}
    y'(g(t))\geq
    \frac{1}{(l-1)!.2^{l-1}}(g(t))^{l-1}y^{(l)}(g(t))>
\frac{1}{(l-1)!.2^{l-1}}(g(t))^{l-1}y^{(l)}(t)
\end{equation}
for $t\geq t_{3}$. Putting $i=l+1, k=n$ and $s >t\geq t_{3}$ in the
inequality
\begin{equation}\label{thirtyfifth}
y^{(i)}(t)=\sum^{k-1}_{j=i}\frac{(t-s)^{j-i}}{(j-i)!}y^{(j)}(s)+\frac{1}
{(k-i-1)!}\int^{t}_{s}(t-u)^{k-i-1}y^{(k)}(u)\,du,
\end{equation}
and letting $s\rightarrow\infty$, we obtain
\begin{equation}\label{thirtysixth}
y^{(l+1)}(t)\leq
-\frac{y(g(t))}{(n-l-2)!}\int^{\infty}_{t}(s-t)^{n-l-2}p(s)\,ds.
\end{equation}
Making the use of \eqref{thirtyfourth} and \eqref{thirtysixth} in
\eqref{thirtythird}, we have
\begin{equation}\label{thirtyseventh}
    z'_{1}(t)\leq -F_{l}(t),
\end{equation}
where
\begin{equation*}
F_{l}(t)=\frac{g'(t).g^{l-1}(t)}{(l-1)!.2^{l-1}.t^{l}}z^{2}_{1}(t)
-\frac{l}{t}z_{1}(t)+\frac{t^{l}}{(n-l-2)!}\int^{\infty}_{t}(s-t)^{n-l-2}p(s)\,ds,
\end{equation*}
which as a function of $z_{1}$, attains the  minimum $H_{l}(t)$
given in \eqref{twentysixth}. In view of the conditions
\eqref{twentyfourth} and \eqref{twentysixth}, integration of
\eqref{thirtyseventh} yields a contradiction. Hence \eqref{first}
has property A, that is  $l=0$ for $t\geq t_{2}\geq t_{1}$.. Thus
the theorem is proved when $n$ is even. Now $l=0$ implies that $n$
is odd. Our theorem will be proved if we can show that
$y(t)\rightarrow 0 $ as $t\rightarrow\infty$. Since $l=0$ then $\lim
y(t)=\lambda, 0\leq\lambda <\infty$ exists. We claim that
$\lambda=0$. If not, them for $0<\epsilon<\lambda$,
 there exists a $t_{3}\geq t_{2}$ such that $y(g(t)) > \lambda-\epsilon$ for $t\geq t_{3}$.
 Now putting $i=0,k=n$ and  $s>t=t_{3}$ and letting $s\rightarrow\infty$ in \eqref{thirtyfifth}, we obtain

\begin{equation*}
y(t_{3})>(\lambda-\epsilon)\int^{\infty}_{t_{3}}(u-t_{3})^{n-1}p(u)\,du
\end{equation*}
which further gives
\begin{equation}\label{thirtiseveth}
\int^{\infty}_{t_{3}}(u-t_{3})^{n-1}p(u)\,du <\infty.
\end{equation}
On the other hand, the condition \eqref{twentyfourth} with $l=n-1$
yields that $\int^{\infty}_{t_{3}}t^{n-1}p(t)\,dt=\infty$ which
contradicts to \eqref{thirtiseveth}. Hence $\lambda=0$. This
completes the proof of the theorem.
\end{proof}

\begin{example} \label{exa2.21} \rm
Consider
\begin{equation}\label{thirtyeight}
    y'''(t)+\frac{24(t-1)^{2}}{t^{5}}y(t-1)=0 ,
    \quad t\geq 2.
\end{equation}
All the conditions of Theorem \ref{thm2.20} are satisfied. Hence
\eqref{thirtyeight} has property A. In particular, $y(t)=1/t^{2}$ is
a nonoscillatory solution of \eqref{thirtyeight}.
\end{example}

\begin{corollary} \label{coro2.22}
Suppose that the conditions of Theorems \ref{thm2.4} and \ref{thm2.20}
are satisfied. Then all solutions of \eqref{first} are oscillatory.
\end{corollary}

Now, we consider the following ordinary differential equations
associated with the delay differential equations
\eqref{eighteen}, \eqref{nineteen}, \eqref{twentith}, and
\eqref{thirtyeight}.
\begin{gather}\label{thirtinineth}
    y'''+\frac{30}{t^{3}}y=0, \quad t\geq 2. \\
\label{fourty}
y'''+\frac{82}{t^{3}}y=0, \quad t\geq 1. \\
\label{fortyfirst}
    y'''+\frac{63}{t^{3}}y=0, \quad t\geq 1. \\
\label{fortysecond}
    y'''+\frac{24(t-1)^{2}}{t^{3}}y=0, \quad t\geq 2.
\end{gather}
 From  Hanan \cite[Theorem 5.7]{hana}, it follows that
\eqref{thirtinineth}-\eqref{fortysecond} are oscillatory. We note
that a third order ordinary differential equation is said to be
oscillatory if it has an oscillatory solution ; otherwise,it is
called nonoscillatory. However, all solutions of
\eqref{thirtinineth}-\eqref{fortysecond}are not oscillatory. This is
because, \eqref{thirtinineth}-\eqref{fortysecond} are of Class I  or
$C_{I}$ and hence admits a nonoscillatory solution (see Lemma 2.2
and Theorem 3.1 in \cite{pari}). We may note that
Eq.\eqref{twentynineth} is said to be of Class I or $C_{I}$ if any
of its solution $y(t)$ for which $y(t_{0})=y'(t_{0})=0$ and
$y''(t_{0})>0, (\sigma<t_{0}<\infty)$ satisfies
$y(t)>0$ for $t\in[\sigma,t_{0})$. It seems that the presence of
delay in \eqref{nineteen} and \eqref{twentith} is responsible
for the change in the qualitative behaviour of solutions of the
equations. It is easy to construct an example of a third order delay
differential equation all solutions of which are oscillatory but it
is not difficult to construct such an example of a third order
ordinary differential equation. It is evident from the following
examples due to Dolan \cite{dola} and Parhi and the author
 \cite{parh} respectively.

\begin{example}Dolan \cite{dola}] \label{2.23}\rm
All solutions of
\begin{equation*}
    \{[z'-\frac{r'(t)}{r(t)}z]+r(t)z\}'=0
\end{equation*}
are oscillatory, where
$r(t)=[1+\sqrt{2}\epsilon\sin(t+\frac{\pi}{4})]^{-1}>0$, $t\geq 0,
0<\epsilon<\frac{1}{\sqrt{2}}$.
\end{example}

To the best of the authors knowledge, the following is the only
explicit example of which all solutions are oscillatory.

\begin{example}[Parhi and Padhi \cite{parh}] \label{exa2.24} \rm
All solutions of
\begin{equation*}
y'''-y''+\big(\frac{1}{1.0000004}+\frac{1}{t}\big)y'-\frac{k}{t^{2}}y=0,
    \quad t\geq 2
\end{equation*}
are oscillatory, where $k$ is a constant.
\end{example}

\begin{theorem} \label{thm2.25}
Let $n\geq3$. Suppose that for any $\mu\in(0,1/2)$, each of the the
third order ordinary differential equation
\begin{equation}\label{fortythird}
    u'''+G_{l}(t)u=0 , \quad i\in \{1,2,\dots ,n-1\}, \;  n+l \mbox{ odd}
\end{equation}
admits an oscillatory solution, where
\begin{equation}\label{fortyfourth}
G_{n-1}(t)=\frac{\mu}{(n-3)!}(g(t)-g(g(t)))^{n-3}\big(\frac{g(t)}{t}\big)^{2}p(t)
\end{equation}
and
\begin{equation}\label{fortyfifth}
\begin{aligned}
G_{l}(t)&=\frac{\mu}{(n-l-2)!.(l-1)!}
\Big(\int^{g^{-1}(t)}_{t}(s-t)^{n-l-2}p(s)\,ds\Big) \\
&\quad\times \big(g(t)-g(g(t))\big)^{l-2}\big(\frac{g(g(t))}{t}\big)^{2}
\end{aligned}
\end{equation}
for $l\in\{1,2,3,\dots ,n-2\}$. Then \eqref{first} has property A.
\end{theorem}

\begin{proof} If \eqref{first} has not property A, then it
admits a non-oscillatory solution $y(t)$ such that \eqref{second}
is satisfied for $l\in\{1,2,3,\dots ,n-1\}$. We may assume,
without any loss of generality, that $y(t)>0$ and $y(g(t))>0$ for
some $t\geq t_{1}>t_{0}>\sigma$. Let $l=n-1$. Setting
$x(t)=y^{(n-3)}(t)$, we see that
$x(t)>0,x'(t)>0,x''(t)>0$ and
$x'''(t)<0$ for $t\geq t_{2}\geq t_{1}$. For any
$\mu\in(0,1/2)$, there exists a $T\_{\mu}\geq>t_{2}$ such that
\begin{equation}\label{fortysixth}
    \frac{x(g(t))}{x(t)} \geq \mu \big(\frac{g(t)}{t}\big)^{2}
\end{equation}
for $t\geq T_{\mu}$ (See Theorem 2.2 in \cite{erbe}). Setting
$z(t)=x'(t)/x(t)$ for $t\geq T_{\mu}$, we get
\begin{equation}\label{fortyseventh}
z'(t)=\frac{x''(t)}{x(t)}-z^{2}(t).
\end{equation}
Further, assuming $u(t)=\exp\big(\int^{t}_{T_{\mu}}z(s)\,
ds\big)$ and using \eqref{fortysixth}, \eqref{fortyseventh} and
the inequality
\begin{equation*}
    y(t)\geq \frac{y^{(n-3)}(t)}{(n-3)!}(t-g(t))^{n-3},
\end{equation*}
we obtain
\begin{equation*}
u'''(t)+\frac{\mu}{(n-3)!}(g(t)-g(g(t)))^{n-3}\big(\frac{g(t)}{t}\big)^{2}
p(t)u(t)\leq     0
\end{equation*}
for $t\geq T_{\mu}$. From Lemma 4 in \cite{greg}, it follows that
\eqref{fortythird} with $l=n-1$ is disconjugate on
$[T_{\mu},\infty)$, a contradiction.

Next let $l\in\{1,2,3,\dots ,n-2\}$. Putting $i=l+1,k=n$ and
$s=g^{-1}(t)>t_{1}$ in \eqref{thirtyfifth}, we get
\begin{equation*}
y^{(l+1)}(t)+\frac{1}{(n-l-2)!}\Big(\int^{g^{-1}(t)}_{t}(s-t)^{n-l-2}p(s)\,ds
\Big)y(g(t))\leq     0.
\end{equation*}
for $t\geq T\geq t_{1}$, which further gives, for $t\geq T$
\begin{equation}\label{fortynineth}
\begin{aligned}
y^{(l+1)}(t)+\frac{1}{(n-l-2)!.(l-1)!}
\Big(\int^{g^{-1}(t)}_{t}(s-t)^{n-l-2}p(s)\,ds\Big)&\\
\times (g(t)-g(g(t)))^{l-2}y^{(l-2)}(w(t))&\leq 0
\end{aligned}
\end{equation}
where $g(g(t))=w(t)$. Let $x_{1}(t)=y^{(l-2)}(t)$. Then
$x_{1}(t)>0$, $x'_{1}(t)>0$, $x''_{1}(t)>0$ and
$x'''_{1}(t)<0$ for $t\geq T$ and hence we can
find a $t\geq T_{\mu}>T$ such that
\begin{equation*}
    \frac{x_{1}(w(t))}{x_{1}(t)}\geq \mu
    \big(\frac{w(t)}{t}\big)^{2};
\end{equation*}
that is,
\begin{equation}\label{fifty}
    \frac{y^{(l-2)}(w(t))}{y^{(l-2)}(t)}\geq \mu
    \big(\frac{w(t)}{t}\big)^{2},
\end{equation}
for $t\geq T_{\mu'}$. Then
$z'_{1}(t)=\frac{x''_{1}(t)}{x_{1}(t)}-z^{2}_{1}(t)$. Further,
setting $v(t)=e^{\left(\int^{t}_{T_{\mu'}}z_{1}(s)\,ds\right)}$ and
using \eqref{fifty},\eqref{fortynineth} gives
\begin{equation*}
\begin{aligned}
    v'''(t)+
\frac{\mu}{(n-l-2)!.(l-1)!}\Big(\int^{g^{-1}(t)}_{t}(s-t)^{n-l-2}p(s)\,ds\Big)&\\
\times (g(t)-g(g(t)))^{l-2}\big(\frac{w(t)}{t}\big)^{2}v(t)&\leq 0
\end{aligned}
\end{equation*}
for $t\geq T_{\mu'}$.This in turn implies that
\eqref{fortythird} is disconjugate, by in \cite[Lemma 4 ]{greg}, a
contradiction to the hypothesis of the theorem for the case
$l\in \{1,2,3,\dots ,n-2\}$. Hence \eqref{first} has property A. This
completes the proof of the theorem.
\end{proof}

\begin{corollary} \label{coro2.26}
Suppose that $g(t)<t$, $n\geq 3$. If all the conditions
of Theorems \ref{thm2.4} and \ref{thm2.25} are satisfied, then all solutions of
\eqref{first} are oscillatory.
\end{corollary}

\begin{example} \label{exa2.27} \rm
Consider
\begin{equation}\label{fiftyone}
    y'''(t)+e^{-1}y(t-1)=0, \quad t\geq 2.
\end{equation}
As $\liminf_{t\to\infty}\mu e^{-1}t(t-1)^{2}>\frac{2}{3\sqrt{3}}$,
then, for every $\mu\in(0,1/2)$, the equation
\begin{equation*}
    u'''+\mu e^{-1}\big(\frac{t-1}{t}\big)^{2}u=0
    , \quad t\geq 2
\end{equation*}
admits an oscillatory solution by Theorem 5.7 of
\cite{hana}. from Theorem \ref{thm2.25}, it follows that
\eqref{fiftyone} has property A. In particular, $y(t)=e^{-t}$ is a
solution of \eqref{fiftyone} for $t\geq 2$.
\end{example}

\noindent\textbf{Remark:} Consider  Equations \eqref{nineteen} and
\eqref{twentith}. For $0<\mu <\frac{\sqrt{3}}{82}$, it follows
that $\lim_{t\to\infty}t^{3}\mu
\frac{82}{9t^{3}}<\frac{2}{\sqrt{3}}$ and hence
$u'''+\mu\frac{82}{9t^{3}}u=0$ is nonoscillatory, by
\cite[Theorem 5.7]{hana}.
Similarly, for $0<\mu <\frac{4}{189\sqrt{3}}$, the equation
$u'''+\mu\frac{63}{4t^{3}}u=0$is nonoscillatory.
Hence Corollary \ref{coro2.26} cannot be applied to
\eqref{nineteen} and \eqref{twentith}.



\subsection*{Acknowledgements} 1. The author is thankful to the
anonymous referee for his/her helpful comments in revising the
manuscript to the present form.\\
2. This work is supported by Department of Science and technology,
New Delhi, Govt. of India ,BOYSCAST Programme under Sanction no.
100/IFD/5071/2004-2005 Dated 04.01.2005.


\begin{thebibliography}{99}

\bibitem{bain} D. D. Bainov and D. P. Mishev;
\emph{Oscillation Theory for Neutral Differential Equations with Delay},
Adam Hilger, New york, 1990.

\bibitem{pdas} P. Das; \emph{Oscillation criteria for odd order neutral
equations}, J. Math. Anal. Appl., 188(1994), 245-257.

\bibitem{dola} J. M. Dolan; \emph{On the relationship between the
oscillatory behaviour of a linear third order differential equation
and its adjoint}, J. Differential Equations, 7(1970), 367-388.

\bibitem{elia} U. Elias and A. Skerlik; \emph{On a conjecture about an
integral criterian for oscillation}, Arch. Math., 34(1998),
393-399.

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

\bibitem{gopa} K.Gopalsamy, B. S. Lalli and B. G. Zhang; \emph{Oscillation
of odd order neutral differential equations}, Czech. Math. J.
42(117)(1992), 313-323.

 \bibitem{greg} M. Gregus and M. Gera; \emph{Some results in the theory
of a third order linear differential equations}, Ann. Polonici Math.
XLII(1983), 93-102.

\bibitem{gyor} I. Gyori and G. Ladas; \emph{Oscillation Theory of Delay
Differential Equations}, Clarendon Press, Oxford, 1991.

\bibitem{hana} M.Hanan; \emph{Oscillation criteria for third order
linear differential equations}, Pacific J. math. 11(1961), 919-944.

\bibitem{kigu} I. T. Kiguradze and T. A. Chanturia; \emph{Asymptotic
Properties of Solutions of Nonautonomous Ordinary Differential
Equations}, Kluwer Academic Pub., London, 1993.

\bibitem{kopl} R. G. Koplatadze; \emph{On oscillatory properties of
solutions of functional differential equations}, Mem. Differential
Equations Math. Phys. 3(19940, 1-179.

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

 \bibitem{parh} N. Parhi and Seshadev Padhi;
\emph{On oscillatory linear differential
equations of third order}, Arch.Math.37(2001), No.3, 33-38.

\bibitem{pari} N. Parhi and Seshadev Padhi;
\emph{Asymptotic behaviour of solutions of third order delay differential
equations}, Indian J.Pure and Appl. Math. 33(10) 2002, 1609-1620.

\bibitem{pars} N. Parhi and Seshadev Padhi;
\emph{Asymptotic behaviour of solutions of delay differential equations
of n-th order}, Arch. Math. (BRNO), 37(2001),81-101.

\bibitem{swan} C. A. Swanson; \emph{Comparison and Oscillation Theory of
Linear Differential Equations}, Academic Press, New York and
London, 1968.
\end{thebibliography}

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