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\AtBeginDocument{{\noindent\small {\em Electronic Journal of
Differential Equations}, Vol. 2007(2007), No. 22, pp. 1--5.\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 2007 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2007/22\hfil On asymptotic behaviour]
{On asymptotic behaviour of oscillatory solutions for fourth order
differential equations}

\author[S. Padhi, C. Qian\hfil EJDE-2007/22\hfilneg]
{Seshadev Padhi, Chuanxi Qian}  % in alphabetical order

\address{Seshadev Padhi\newline
Department of Applied Mathematics, Birla Institute of Technology,
Mesra, Ranchi-835 215, India} 
\email{ses\_2312@yahoo.co.in}

\address{Chuanxi Qian \newline
Department of Mathematics and Statistics \\
Mississippi State University \\
Mississippi state, MS 39762, USA} 
\email{qian@math.msstate.edu}

\thanks{Submitted December 2, 2006. Published February 4, 2007.}
\thanks{Supported by the Department of Science and
 Technology, New Delhi, Govt. of India, under
\hfill\break\indent BOYSCAST Programme
 vide Sanc. No. 100/IFD/5071/2004-2005 Dated   04.01.2005}
\subjclass[2000]{34C10} 
\keywords{Oscillatory solution; asymptotic behaviour}

\begin{abstract}
 We establish sufficient conditions for the linear differential
 equations of fourth order
 $$
 (r(t)y'''(t))' =a(t)y(t)+b(t)y'(t)+c(t)y''(t)+f(t)
 $$
 so that all oscillatory solutions of the equation satisfy
 $$
 \lim_{t\to\infty}y(t)=\lim_{t\to\infty}y'(t)=\lim_{t\to\infty}y''(t)=
 \lim_{t\to\infty}r(t)y'''(t)=0,
 $$
 where
 $r:[0,\infty)\to(0,\infty),a,b,c$ and $f:[0,\infty)\to R$
 are continuous functions. A suitable Green's function and its
 estimates are used in this paper.
\end{abstract}

\maketitle \numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{example}[theorem]{Example}
\newtheorem{remark}[theorem]{Remak}


\section{Introduction}

  Bainov and Dimitrova \cite{dimi} proved the following
 result (See also Theorem 3.1.1 with $n=4$ in \cite{bain}).

 \begin{theorem}\label{thm1.1}
Assume
\begin{gather}\label{e1.1}
\lim_{t\to\infty}\int^{t}_{t_{0}}\frac{1}{r_{1}(s_{1})}
\int^{s_{1}}_{t_{0}}\frac{1}{r_{2}(s_{2})}
\int^{s_{2}}_{t_{0}}\frac{1}{r_{3}(s_{3})}\,ds_{3}\,ds_{2}\,ds_{1}\,<\infty,
\\
 \label{e1.2}
\int^{\infty}_{t_{0}}|a(t)|\,dt<\infty, \\
\label{e1.3} \int^{\infty}_{t_{0}}|f(t)|\,dt<\infty,
\end{gather}
Then all solutions of
\begin{equation}\label{e1.4}
    (r_{3}(t)(r_{2}(t)(r_{1}(t)y'(t))')')'+a(t)F((Ay)(t))=f(t)
\end{equation}
are bounded and all oscillatory solutions of \eqref{e1.4} tend to
zero as $t\to\infty$, where $F\in C(R,R)$ and $F(u)$ is a bounded
function on $R, r_{i}\in C^{4-i}([t_{0},\infty);(0,\infty)),1 \leq
i \leq 3$, $a,f \in C([t_{0},\infty);R)$ and $A$ is an operator with
certain properties.
\end{theorem}

The motivation of the present work has come from Theorem
\ref{thm1.1}. Since $F(u)=u$ is not bounded, then Theorem
\ref{thm1.1} cannot be applied to its corresponding linear equation.
Our purpose is to show that under the conditions of Theorem
\ref{thm1.1}, every oscillatory solution of the considered equation
along with their first and second order derivatives tend to zero as
$t\to\infty$. In fact, we consider the more general fourth order
linear differential equations of the form
\begin{equation}\label{e1.5}
    (r(t)y'''(t))'=a(t)y(t)+b(t)y'(t)+c(t)y''(t)+f(t)
\end{equation}
where $r:[0,\infty)\to(0,\infty),a,b,c$ and $f:[0,\infty)\to R$ are
continuous functions. We shall show that all oscillatory solutions
of \eqref{e1.5} along with their first and second order derivatives
tend to zero as $t\to\infty$.

Asymptotic behaviour of oscillatory solutions of second order
differential equations have been studied by many authors, see
(\cite{sing,bsin, bsig, bsih}). For higher order differential
equations, one may see the paper due to Chen and Yeh \cite{chen} and
the recent work due to Padhi \cite{spad} and the references cited
therein. The monograph due to \cite{bain} gives a survey on the
asymptotic decay of oscillatory solutions of differential equations.
In \cite{spad}, Padhi considered a more general forced differential
equation where he obtained a new sufficient condition under which
all oscillatory solutions of the equation tend to zero as
$t\to\infty$. The result improve all earlier existing results. In a
recent note \cite{sesh}, Padhi studied the asymptotic behaviour of
oscillatory solutions of third order linear differential equations.
It seems that asymptotic behaviour of oscillatory solutions of
fourth order differential equations of the form \eqref{e1.5} has not
been studied in the literature. Motivated by the result in
\cite{sesh}, this work pays an attention for the asymptotic
behaviour of oscillatory solutions of the equations of the form
\eqref{e1.5}. The technique used in the work is the help of a
Green's function and its estimates. This technique was used by Padhi
\cite{sesh}. The sufficient conditions given in this paper may be
treated as a different set of condition given in \cite{spad}.
Sufficient conditions for oscillations of equations of the form
\eqref{e1.5} with $b(t)=0$ and $c(t)=0$ are given in \cite{bain}.

The work is organized as follows: Section 1 is introductory where as
the main result of the paper is given in Section 2 and an open
problem is left to the reader.

We note that a solution of the above mentioned equations  is said to
be oscillatory if it has arbitrarily large zeros.

\section{Main Results}

 The main result of the paper is the following.

\begin{theorem} \label{thm2.1}
Let \eqref{e1.2} and \eqref{e1.3} hold. Further suppose that
\begin{equation}\label{e1.6}
\int^{\infty}_{0}\frac{t^{2}}{r(t)}\,dt<\infty,\quad
\int^{\infty}_{0}|b(t)|\,dt<\infty, \quad
\int^{\infty}_{0}|c(t)|\,dt<\infty.
\end{equation}
 Then every oscillatory solution of the equation
\eqref{e1.5} satisfies
\begin{equation}\label{e1.7}
\lim_{t\to\infty}y(t)=\lim_{t\to\infty}y'(t)=\lim_{t\to\infty}y''(t)=
\lim_{t\to\infty}r(t)y'''(t)=0.
\end{equation}
\end{theorem}

\begin{proof}
 Let $\{t_{k}\}^{\infty}_{k=1}$, $1<t_{k}<t_{k+1}$, ($k=1,2,3,\dots)$
be such that $y(t_{k})=0$. Then for each natural $k$, there exists
$t_{k+1}'''\in(t_{k+1},t_{k+3})$ such that $y'''(t_{k+1}''')=0$.
Hence \eqref{e1.5} implies
\begin{equation}\label{e1.8}
    y'''(t)=\frac{1}{r(t)}\int^{t}_{t_{k+1}'''}[a(s)y(s)+b(s)y'(s)
+c(s)y''(s)+f(s)]\,ds.
\end{equation}
We can find $t'_{k+1}$ and $t''_{k+1}(t_{k+1}<t'_{k+1}<t''_{k+1})$
such that $y(t_{k+1})=0,y'(t'_{k+1})=0$ and $y''(t''_{k+1})=0$. We
note that $t'''_{k+1}\leq t''_{k+1}$.

If now we set
\begin{gather*}
\rho_{ik}=\max\{|y^{(i)}(t)|:\,t_{k+1}\leq t \leq
t''_{k+1}\},\quad i=0,1,2,\\
\rho_{3k}=\max\{r(t)|y'''(t)|: t_{k+1}\leq t \leq t''_{k+1}\}, \\
\epsilon_{k}=\int^{t''_{k+1}}_{t_{k}}
\big(\frac{t^{2}}{r(t)}+|a(t)|+|b(t)|+|c(t)|+|f(t)|\big)\,dt,
\end{gather*}
then it follows that
\begin{gather}\label{e1.9}
|y'''(t)|\leq
\frac{1}{r(t)}\epsilon_{k}(\rho_{0k}+\rho_{1k}+\rho_{2k}+1),\quad
t_{k+1}\leq t \leq t''_{k+1}, \\
\label{e1.10}
 \rho_{3k}\leq
\epsilon_{k}(\rho_{0k}+\rho_{1k}+\rho_{2k}+1),\,\,t_{k+1}\leq t \leq
t''_{k+1}.
\end{gather}
On the other hand, by conditions \eqref{e1.2},\eqref{e1.3} and
(\ref{e1.6}), we have
\begin{equation}\label{e1.11}
    \lim_{k\to\infty}\epsilon_{k}=0.
\end{equation}
Therefore, without any loss of generality, it can be assumed that
\begin{equation}\label{e1.12}
    \epsilon_{k}<\frac{1}{\sqrt{7}},\quad (k=1,2,3,\dots).
\end{equation}
By Green's formula, for each natural $k$, we have
\begin{equation}\label{e1.13}
  \begin{gathered}
y(t)=\int^{t''_{k+1}}_{t_{k+1}}G_{k}(t,s)y'''(s)\,ds,\\
y'(t)=\int^{t''_{k+1}}_{t_{k+1}}\frac{\delta G_{k}(t,s)}{\delta
s}y'''(s)
   \,ds,\\
y''(t)=\int^{t''_{k+1}}_{t_{k+1}}\frac{\delta^{2}
 G_{k}(t,s)}{\delta s^{2}}y'''(s)\,ds
\end{gathered}
\end{equation}
 where
\[
  G_{k}(t,s)=  \begin{cases}
 s \in [t_{k+1},t'_{k+1}]: \begin{cases}
 \frac{(t-t_{k})}{2}(2s-t-t_{k}),&t \leq s\\
 \frac{(s-t_{k+1})^{2}}{2},      & s \leq t.
 \end{cases} \\[6pt]
 s \in [t'_{k+1},t''_{k+1}]: \begin{cases}
 \frac{(t-t_{k+1})}{2}(2t'_{k+1}-t-t_{k}), & t \leq s\\
 \frac{(t-s)^{2}}{2}+\frac{(t-t_{k+1})}{2}(2t'_{k+1}-t-t_{k+1}),
  & s \leq t
 \end{cases}
 \end{cases}
\]
is the Green's function for $y'''(t)=0,y(t_{k+1})=0$,
$y'(t'_{k+1})=0$, $y''(t''_{k+1})=0$. Moreover,
\[
|G_{k}(t,s)|<\frac{3s^{2}}{2},\quad |\frac{\delta G_{k}(t,s)}{\delta
t}|<s, \quad |\frac{\delta^{2} G_{k}(t,s)}{\delta t^{2}}|<1,
\]
 for $t_{k+1} \leq s \leq t''_{k+1}$.
By these estimates and inequalities (\ref{e1.9}) and (\ref{e1.13}),
we have
\begin{gather*}
 \rho_{0k} \leq
\frac{3}{2}\epsilon_{k}(\rho_{0k}+\rho_{1k}+\rho_{2k}+1)
\int^{t''_{k+1}}_{t_{k+1}}\frac{s^{2}}{r(s)}\,ds \leq
\frac{3}{2}\epsilon^{2}_{k}(\rho_{0k}+\rho_{1k}+\rho_{2k}+1),
\\
\rho_{1k} \leq
\epsilon_{k}(\rho_{0k}+\rho_{1k}+\rho_{2k}+1)\int^{t''_{k+1}}_{t_{k+1}}
\frac{s}{r(s)}\,ds \leq
\epsilon^{2}_{k}(\rho_{0k}+\rho_{1k}+\rho_{2k}+1),
\\
\rho_{2k} \leq \epsilon_{k}(\rho_{0k}+\rho_{1k}+\rho_{2k}+1)
\int^{t''_{k+1}}_{t_{k+1}}\frac{1}{r(s)}\,ds \leq
\epsilon^{2}_{k}(\rho_{0k}+\rho_{1k}+\rho_{2k}+1).
\end{gather*}
 Thus
\begin{align*}
\rho_{0k}+\rho_{1k}+\rho_{2k} & \leq
\frac{7}{2}\epsilon^{2}_{k}(\rho_{0k}+\rho_{1k}
  +\rho_{2k})+\frac{7}{2}\epsilon^{2}_{k}\\
& \leq  \frac{7}{2}.\frac{1}{7}(\rho_{0k}+\rho_{1k}+\rho_{2k})
  +\frac{7}{2}\epsilon^{2}_{k}\\
& \leq \frac{1}{2}(\rho_{0k}+\rho_{1k}+\rho_{2k})+\frac{7}{2}
  \epsilon^{2}_{k}
\end{align*}
which in turn implies
$$
\rho_{0k}+\rho_{1k}+\rho_{2k} \leq 7\epsilon^{2}_{k}
$$
and from (\ref{e1.10})
\[
\rho_{3k}  \leq  \epsilon_{k}(\rho_{0k}+\rho_{1k}+\rho_{2k}+1)
 \leq  \epsilon_{k}(7\epsilon^{2}_{k}+1).
\]
Since (\ref{e1.11}) holds, then the above inequality yields that
$\rho_{ik}\to 0$ as $k\to\infty, i=0,1,2,3$. This completes the
proof of the theorem.
\end{proof}

\begin{example}\label{exa2.2} \rm
Consider the equation
\begin{equation}\label{e1.14}
    (t^{4}y'''(t))'=\frac{1}{t^{2}}y(t)+\frac{1}{t^{2}}y'(t)
+\frac{1}{t^{2}}y''(t)+f(t),\quad t\geq     1,
\end{equation}
where
\begin{align*}
f(t)&=\frac{\sin t}{t^{2}}+\frac{20 \cos t}{t^{3}} -\frac{190\sin
t}{t^{4}}-\frac{896\cos t}{t^{5}}
+\frac{1760\sin t}{t^{6}}\\
&\quad -\frac{\cos t}{t^{8}}-\frac{6\sin t}{t^{9}}
 -\frac{12 \cos t}{t^{9}}+\frac{42\sin t}{t^{10}}.
\end{align*}
 All the conditions of Theorem \ref{thm2.1} are satisfied.
We note that $y(t)=\frac{\sin t}{t^{6}}$ is an oscillatory solution
of the equation (\ref{e1.14}) satisfying the property (\ref{e1.7}).
\end{example}

It is clear that the conclusion of Theorem \ref{thm2.1} holds for
the homogeneous equation
\begin{equation}\label{e1.15}
(r(t)y'''(t))'=a(t)y(t)+b(t)y'(t)+c(t)y''(t)
\end{equation}
However, from Example \ref{exa2.2}, it seems that the forcing term
$f(t)$ plays a crucial role in constructing the example. Thus, it
would be interesting to obtain an example for the homogeneous
equation (\ref{e1.15}) satisfying the conclusions of Theorem
\ref{thm2.1} under the conditions \eqref{e1.2} and (\ref{e1.6}).

\begin{remark}\label{rem2.3} \rm
It would be interesting   to obtain sufficient conditions on the
coefficient functions using the above technique so that any
arbitrary oscillatory solution $y(t)$ of the general $n$-th order
linear differential equations of the form
\begin{equation}\label{e1.16}
    (r(t)y^{(n-1)}(t))'=\sum^{n-2}_{i=0}p_{i}(t)y^{(i)}(t)+f(t)
\end{equation}
satisfies
\begin{equation}\label{e1.17}
\lim_{t\to\infty}y(t)=\lim_{t\to\infty}y'(t)=\lim_{t\to\infty}y''(t)
=\dots=\lim_{t\to\infty}y^{(n-2)}(t)=\lim_{t\to\infty}r(t)y^{(n-1)}(t)=0
\end{equation}
where $r$ and $f$ are as defined earlier and $p_{i}:[0,\infty)\to
R\,(i=0,1,2,\dots,n-2)$.
\end{remark}

It seems that the following sufficient conditions are needed to
prove the above remark.

\subsection*{Open problem}
Under condition \eqref{e1.3} and
\begin{equation}\label{e1.18}
    \int^{\infty}_{0}\frac{t^{n-2}}{r(t)}\,dt<\infty,\quad
\int^{\infty}_{0}|p_{i}(t)|\,dt<\infty,\,\, \quad i=0,1,\dots,n-2,
\end{equation}
does every oscillatory solution $y(t)$ of \eqref{e1.16} satisfy
property \eqref{e1.17}?

A suitable Green's function and their estimates maybe  needed to
answer the above open problem. However, we have not found that
Green's function yet.


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\end{thebibliography}

\end{document}