\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 $$\label{first} y^{(n)}(t)+p(t)y(g(t))=0,$$ 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 $$\label{second} \begin{gathered} y^{(i)}(t) >0 , \quad i=0,1,2,\dots ,l,\\ (-1)^{i+l}y^{(i)}(t) >0 , \quad i=l+1,\dots ,n. \end{gathered}$$ 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 $t\geq t_{2}$. $$\label{third} |y(t)| \geq\frac{(t-t_{2})^{(n-1)}}{(n-1)(n-2)\dots (n-l)}|y^{(n-1)}(2^{n-l-1}t)|, \quad t \geq t_{2}.$$ 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 $$\label{fourth} \int^{\infty}_{t_{1}}s^{n-l-1}|y^{(n)}(s)|\,ds <\infty,$$ $$\label{fifth} y^{(i)}(t)\geq y^{(i)}(t_{1})+\frac{1}{(l-i-1)!}\int^{t}_{t_{1}}(t-s)^{l-i-1}y^{(i)}(s)\,ds$$ for $t\geq t_{1},i=0,1,2,\dots ,l-1$ and $$\label{sixth} y^{(l)}(t)\geq \frac{1}{(l-i-1)!}\int^{\infty}_{t}(s-t)^{n-l-1}|y^{(n)}(s)|\,ds$$ for $t\geq t_{1}$. If in addition $$\label{seventh} \int^{\infty}_{t_{1}}s^{n-l}|y^{(n)}(s)|\,ds = \infty,$$ then there exists $t_{2}\geq t_{1}$ such that $$\label{eighth} y^{(l-1)}(t) \geq \frac{t}{(n-l)!}\int^{\infty}_{t}s^{n-l-1}|y^{(n)}(s)|\,ds$$ for $t\geq t_{2}$ and $$\label{nineth} iy^{(l-1)}\geq ty^{(l-i+1)}(t)\geq (i-1)y^{(l-i)}(t)$$ for $t\geq t_{2}$, $i\in \{1,2,\dots ,l\}$. \end{lemma} \section{Main Results} \begin{theorem} \label{thm2.1} Let $g(t) (n-l-1)!.l1 hold. Then \eqref{first} has property A. \end{theorem} \begin{proof} Let$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 $$\label{one} y(t)\geq \frac{1}{(l-1)!}\int^{t}_{t_{1}}(t-s)^{l-1}y^{(l)}(s)\,ds, \quad t\geq t_{1}.$$ 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, $$\label{eleventh} y(t)\geq \frac{(t-g(t))^{l}}{l!}y^{(l)}(t).$$ 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,, $$\label{twelfth} \limsup_{t\to\infty}t^{n-1}\int^{\infty}_{g^{-1}(t)}p(s)\,ds > (n-1)\dots (n-l)2^{(n-1)(n-l)},$$ 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 $$\label{thirteen} y(t)\geq y(2^{l+1-n}t) \geq \frac{1}{(n-1)\dots (n-l).2^{(n-1)(n-l)}}t^{n-1}y^{(n-1)}(t).$$ 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)(l-1)!.(n-l-1)! 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 $$\label{fifteen} y(t)\geq \frac{1}{(l-1)!}\int^{t}_{t_{1}}(t-s)^{l-1}y^{(l)}(s)\,ds.$$ From \eqref{fifth}, we obtain $$\label{sixteen} y^{(l)}(t)\geq \frac{1}{(n-l-1)!}\int^{\infty}_{t}(s-t)^{n-l-1}p(s)y(g(s))\,ds.$$ Then from \eqref{fifteen} and \eqref{sixteen}, we obtain $$\label{sixtin} y(t)\geq \frac{1}{(n-l-1)!.(l-1)!}\int^{t}_{t_{1}}(t-s)^{l-1} \int^{\infty}_{s}(u-s)^{n-l-1}p(u)y(g(u))\,du\,ds.$$ 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 $g(t) (n-1)!. Then \eqref{first} has no solution satisfying the property$(-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)0$be a constant and$g(t)=t-\tau$,$\tau>0$be a constant. Then \eqref{first} becomes $$\label{twentyfirst} y^{(n)}(t) +py(t-\tau)=0.$$ 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 $$\label{twentysecond} p\tau^{n} >n!,$$ then \eqref{twentyfirst} is oscillatory. \smallskip The characteristic equation associated with \eqref{twentyfirst} is given by $$\label{twentythird} \lambda^{n}+pe^{-\tau\lambda}=0.$$ 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 $$\label{twentithird} y'''(t)+\frac{82}{t^{3}}y(\frac{3t}{4})=0 , t\geq 1.$$ Condition \eqref{tenth} fails to hold and hence Theorem \ref{thm2.1} cannot be applied to \eqref{twentithird}. Similarly,consider the equation $$\label{twenthird} y'''(t)+\frac{46}{t^{3}}y(\frac{t}{2})=0,t\geq1.$$ 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 $$\label{tenthird} y'''(t)+\frac{46}{t^{3}}y(\frac{10t}{11})=0,t\geq1,$$ 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)(n-l-1)!.l! 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 $g(t)(l-1)!.(n-l-1)! holds, then \eqref{first} has property A. \end{theorem} \begin{proof} Proceeding as in the proof of Theorem \ref{thm2.3}, we arrive at \eqref{sixtin} for$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, $$\limsup_{t\to\infty}\int^{t}_{t/2}(t-s)^{l-1}\int^{\infty}_{g^{-1}(g^{-1}(s))}(u-s)^{n-l-1}p(u)\,du\,ds >(l-1)!.(n-l-1)!$$ 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, $$\label{twentyfourth} \int^{\infty}H_{l}(t)\,dt = \infty,$$ 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 $$\label{twentyfifth} H_{n-1}(t)=t^{n-1}p(t)-\frac{(n-1)!.(n-1)2^{n-4}t^{n-3}}{g'(t)g^{n-2}(t)}$$ and $$\label{twentysixth} H_{l}(t) = \frac{t^{l}}{(n-l-2)!}\int^{\infty}_{t}(s-t)^{n-l-2}p(s)\,ds-\frac{l!.l.2^{l-3}t^{l-2}}{g'(t)g^{l-1}(t)},$$ 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 $$\label{twentyseventh} \int^{\infty}[tp(t)-\frac{1}{4t}]\,dt=\infty,$$ then $$\label{twentyeighth} y''+p(t)y=0$$ 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 $$\label{twentynineth} y'''+p(t)y=0$$ 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 $$\label{thirty} \int^{\infty}[t^{2}p(t)-\frac{2}{3\sqrt{3}t}]\,dt=\infty.$$ 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 $$\label{thirtyfirst} z'(t)=-t^{n-1}p(t)+\frac{n-1}{t}z(t)-g'(t) \frac{y'(g(t))}{y(g(t))}z(t).$$ 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 $$\label{thirtysecond} z'(t)\leq -F_{n-1}(t),$$ 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 $$\label{thirtythird} z'_{1}(t)=\frac{t^{l}y^{(l+1)}(t)}{y(g(t))}+\frac{l}{t}z_{1}(t)- g'(t)\frac{y'(g(t))}{y(g(t))}z_{1}(t).$$ 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 $$\label{thirtyfourth} y'(g(t))\geq \frac{1}{(l-1)!.2^{l-1}}(g(t))^{l-1}y^{(l)}(g(t))> \frac{1}{(l-1)!.2^{l-1}}(g(t))^{l-1}y^{(l)}(t)$$ for$t\geq t_{3}$. Putting$i=l+1, k=n$and$s >t\geq t_{3}$in the inequality $$\label{thirtyfifth} y^{(i)}(t)=\sum^{k-1}_{j=i}\frac{(t-s)^{j-i}}{(j-i)!}y^{(j)}(s)+\frac{1} {(k-i-1)!}\int^{t}_{s}(t-u)^{k-i-1}y^{(k)}(u)\,du,$$ and letting$s\rightarrow\infty$, we obtain $$\label{thirtysixth} y^{(l+1)}(t)\leq -\frac{y(g(t))}{(n-l-2)!}\int^{\infty}_{t}(s-t)^{n-l-2}p(s)\,ds.$$ Making the use of \eqref{thirtyfourth} and \eqref{thirtysixth} in \eqref{thirtythird}, we have $$\label{thirtyseventh} z'_{1}(t)\leq -F_{l}(t),$$ 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 $$\label{thirtiseveth} \int^{\infty}_{t_{3}}(u-t_{3})^{n-1}p(u)\,du <\infty.$$ 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 $$\label{thirtyeight} y'''(t)+\frac{24(t-1)^{2}}{t^{5}}y(t-1)=0 , \quad t\geq 2.$$ 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$y''(t_{0})>0, (\sigma0$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 $$\label{fortythird} u'''+G_{l}(t)u=0 , \quad i\in \{1,2,\dots ,n-1\}, \; n+l \mbox{ odd}$$ admits an oscillatory solution, where $$\label{fortyfourth} G_{n-1}(t)=\frac{\mu}{(n-3)!}(g(t)-g(g(t)))^{n-3}\big(\frac{g(t)}{t}\big)^{2}p(t)$$ and \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} forl\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 $$\label{fortysixth} \frac{x(g(t))}{x(t)} \geq \mu \big(\frac{g(t)}{t}\big)^{2}$$ 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 $$\label{fortyseventh} z'(t)=\frac{x''(t)}{x(t)}-z^{2}(t).$$ 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$t\geq T\label{fortynineth} \begin{aligned} y^{(l+1)}(t)+\frac{1}{(n-l-2)!.(l-1)!} \Big(\int^{g^{-1}(t)}_{t}(s-t)^{n-l-2}p(s)\,ds\Big)&\\ \times (g(t)-g(g(t)))^{l-2}y^{(l-2)}(w(t))&\leq 0 \end{aligned} whereg(g(t))=w(t)$. 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, $$\label{fifty} \frac{y^{(l-2)}(w(t))}{y^{(l-2)}(t)}\geq \mu \big(\frac{w(t)}{t}\big)^{2},$$ for$t\geq T_{\mu'}$. Then$z'_{1}(t)=\frac{x''_{1}(t)}{x_{1}(t)}-z^{2}_{1}(t)$. Further, setting$v(t)=e^{\left(\int^{t}_{T_{\mu'}}z_{1}(s)\,ds\right)}and using \eqref{fifty},\eqref{fortynineth} gives \begin{equation*} \begin{aligned} v'''(t)+ \frac{\mu}{(n-l-2)!.(l-1)!}\Big(\int^{g^{-1}(t)}_{t}(s-t)^{n-l-2}p(s)\,ds\Big)&\\ \times (g(t)-g(g(t)))^{l-2}\big(\frac{w(t)}{t}\big)^{2}v(t)&\leq 0 \end{aligned} \end{equation*} fort\geq T_{\mu'}$.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$g(t)\frac{2}{3\sqrt{3}}$, 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. 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