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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 06, pp. 1--6.\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 2006 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2006/06\hfil Boundedness of solutions]
{Boundedness of solutions to fourth order differential equations
 with oscillatory restoring and forcing terms}
\author[B. S. Ogundare\hfil EJDE-2006/06\hfilneg]
{Babatunde S. Ogundare}

\address{Babatunde S. Ogundare \hfill\break
Department of Mathematics\\
Obafemi Awolowo University\\
Ile-Ife, Nigeria}
\email{bogunda@oauife.edu.ng \quad ogundareb@yahoo.com}

\date{}
\thanks{Submitted September 7, 2005. Published January 11, 2006.}
\subjclass[2000]{34C10, 34C11}
\keywords{Fourth order  differential equation; bounded solution;
\hfill\break\indent oscillatory solution; restoring and forcing terms}

\begin{abstract}
 This article concerns the fourth order differential equation
 $$
 x^{(iv)}+ax'''+bx''+g(x')+h(x)=p(t).
 $$
 Using the Cauchy formula for the particular solution of
 non-homogeneous linear differential equations with constant
 coefficients, we prove that the solution and its derivatives
 up to order three are bounded.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}


\section{Introduction}

 In this article, we study the boundedness of solutions
to the fourth-order nonlinear differential equation
\begin{equation}
x^{(iv)}+ax'''+bx''+g(x')+h(x)=p(t)\label{e1.1}
\end{equation}
where $a>0$ and $ b>0$ are positive constants with
$a^{2}>4b$, $g, h,$ and $p$ and their first derivatives are
continuous functions depending on the arguments shown.
In addition,  $h$ and $p$ are oscillatory.

Several authors have investigated the boundedness of solutions of
certain differential equations of the fourth order. We can mention
in this direction, the works of Afuwape and Adesina \cite{a1} where
the frequency-domain approach was used. Other articles in this
connection include Tiryaki and Tunc \cite{t1}, Tunc \cite{t2,t3,t4},
Tunc and Tiryaki \cite{t5} where the second  Lyapunov method was used.
All these results generalize in one way or another some results
on third order nonlinear differential equations see for instance
\cite{a2,a3,a4,a5,c1,r1,s1,s2}.

The present work was motivated by a relatively recent
paper of Andres \cite{a2}, where the existence of a bounded solution for
a third order non-linear differential equation with oscillatory
 restoring and forcing terms was proved.
We shall use the Cauchy formula for the particular solution
of non homogeneous linear part of the equation \eqref{e1.1},
to prove that the solution $x(t)$ and its derivatives
$x'(t)$, $x''(t)$  and $x'''(t)$ are bounded.

\section{Assumptions and Main Result}

The basic assumptions on the functions which appear in
\eqref{e1.1} are the following:
\begin{itemize}
\item[(i)]  $h$ and $p$ are oscillatory in the following sense:
 For each argument there exist such numbers
 $\beta _{1}>\alpha _{1}>u>\beta _{-1}>\alpha _{-1}$
 that for $f(\alpha _{1})<0, f(\beta _{1})>0$,
 $f(\alpha _{-1})<0, f(\beta _{-1})>0 $  where $f$ is either
 $h(x)$ or $p(t)$, $u$ is either $x$ or $t$ and all the roots
 of the restoring term $h(x)$ are isolated.

\item[(ii)] (a) $|h(x)|\leq H$,\quad (b) $|h'(x)| \leq H'$;

\item[(iii)]  (a) $|g(x')|<cx'\leq G$,\quad (b) $|g'(x')|\leq G'$;

\item[(iv)]   (a) $|p(t)|\leq P$,\quad (b) $|p'(t)|\leq P'$,\quad
(c) $|\int_{0}^{t}{P(\tau )d\tau }|\leq P_{0}$,\\
(d) ${\limsup_{t\to \infty }}|P(t)|>0$.
\end{itemize}
The main result of this paper is as follows.

\begin{theorem} \label{thm2.1}
Assume there exist positive constants $H, H', G, G', P, P', P_{0}, R$
such that for $|x|>R$ and $t>0$ the conditions (ii) and (iii) hold.
If in addition,
$$
\min[d(\overline{x}_{k},\overline{x}_{k+1}), d(\overline{x}_{k},
\overline{ x}_{k-1})] > 2 \frac{G+H+P}{b}
\big( \frac{2}{a}+\frac{a}{b}\big)
+\frac{P_{0}}{b},
$$
where $\overline{x}_{k}$ are roots of $h(x)$, $h'(x_{k})>0$ and
$\overline{x}_{k-1}, \overline{x}_{k+1}$ denote the couple adjacent
roots of $\overline{x}_{k}(k=0,\pm 2,\pm 4,\dots )$;
 then all solutions $x(t)$ of equation \eqref{e1.1} are bounded and
for each of them there exists a root $\overline{x}$ of $h(x)$ such
that $(x(t)-\overline{x})$ oscillates.
\end{theorem}

\section{Preliminary Results}
To prove our main result, we shall need the following result.

\begin{lemma} \label{lem3.1}
 If there exist positive constants, $H,  G, P$ such that for
 all $x\in \Re ^{1}$ and $t\geq 0$ the assumptions (ii)(a), (iii)(a) and
 (iv)(a) hold, then each solution $x(t)$ of \eqref{e1.1}
 satisfies the inequalities
\begin{gather}
{\limsup_{t\to \infty }}|x'(t)|\leq \frac{G}{c}:=D',\label{e3.1a}
\\
{\limsup_{t\to \infty }}|x''(t)|\leq
\frac{ H+G+P}{b}:=D'',\label{e3.1b}
\\
{\limsup_{t\to \infty }}|x'''(t)|
\leq \frac{2(H+G+P)}{a}:=D'''. \label{e3.1c}
\end{gather}
\end{lemma}

\begin{proof}
Let $z=x''$ then the equation \eqref{e1.1} reduces to
\begin{equation}
 z''+az'+bz=P(t)-g(x'(t))-h(x).\label{e3.2}
\end{equation}
Equation \eqref{e3.2} can also be rewritten as
\begin{equation}
z''+az'+bz=B,\label{e3.3}
\end{equation}
where $B=P(t)-g(x'(t))-h(x)$.
Thus the general solution of the equation \eqref{e3.3} satisfies
$$
|x''(t)|=|z(t)|=C_{1}e^{a_{1}t}+C_{2}e^{a_{2}t}
+ \int_{0}^{t}{\frac{e^{a_{1}\tau}-e^{a_{2}\tau }} {a_{1}-a_{2}}(B)}d\tau,
$$
where $a_{1,2}=\big(-a\pm \sqrt{a^{2}-4b}\big)/2$ and constants
$C_{1}$ and $C_{2}$ are arbitrary. Hence by the virtue of assumptions
 (i)-(iv) for $t\geq 0$, we have not only
$$
\big|\int_{0}^{t}{\frac{e^{a_{1}\tau }-e^{a_{2}\tau }}{a_{1}-a_{2}}[
P(t)-g(x'(t))-h(x)]}d\tau \big|
\leq \frac{H+G+P}{b}(1+\frac{a_{2}e^{a_{1}t}-a_{1}e^{a_{2}t}}{a_{1}-a_{2}}),
$$
but also
$$
\limsup_{t\to \infty }|x''(t)|\leq \frac{H+G+P}{b}=:D''.
$$
Furthermore on substituting $w=z'$ in \eqref{e3.2}, we have
$$
w'+aw=P(t)-bx''(t)-g(x'(t))-h(x).
$$
Following the same argument used in obtaining the general solution
for the equation \eqref{e3.3}, we have
$$
|x'''(t)|=|w(t)|= Ce^{-at}+\int_{0}^{t}{e^{-a\tau }
[P(t)-bx'' (t)-g(x'(t))-h(x)]}d\tau,
$$
and by assumptions (i)-(iii) for $t\geq T_{x}$, we have not only
\begin{align*}
|\int_{0}^{t}{e^{-a\tau }[P(t)-bx''(t)-g(x'(t))-h(x)]}d\tau |
&\leq 2\frac{H+G+P}{a}\int_{0}^{t}{e^{-a\tau }} d\tau \\
&\leq 2\frac{H+G+P}{a}(1-e^{-a(t-T_{x})}),
\end{align*}
but also
$$
\limsup_{t\to \infty }|x'''(t)|\leq 2\frac{H+G+P}{a}=:D'''.
$$
To establish the inequality \eqref{e3.1a}, we use the assumption
(iii)(a); i.e.,
given that $|g(x')|< cx'\leq G$, we have
$$
|cx'(t)|\leq c|x'(t)|\leq G;
$$
i.e., $|x'(t)|\leq G/c$.
Hence
$$
\limsup_{t\to\infty} |x'(t)|\leq \frac{G}{c}:=D'.
$$
This completes the proof of the lemma \ref{lem3.1}.
\end{proof}

\begin{lemma} \label{lem3.2}
Under the assumptions of Lemma \ref{lem3.1}. If (ii)(b) and (v)(d) hold
for $x\in \Re ^{1}$, then every solution $x(t)$ of \eqref{e1.1} either
satisfies the relation
\begin{equation}
\lim_{t\to \infty }x(t)=\overline{x},
\quad \lim_{t\to \infty} x'(t)
= \lim_{t\to \infty }x''(t)=\lim_{t\to \infty }x'''(t)=0
\quad (h(\overline{x} )=0,)\label{e*}
\end{equation}
or there exists a root $\overline{x}$ of $h(x)$ such that
$(x(t)-\overline{x}) $ oscillates.
\end{lemma}

\begin{proof}
Substituting a fixed bounded solution $x(t)$ of \eqref{e1.1} into itself
and integrating the result from $T_{x}$ to $t$, we have
\begin{equation}
\begin{aligned}
&\int_{T_{x}}^{t}{h(x(\tau )}d\tau \\
&=-\{ b[x'(t)-x'(T_{x})]+a[x''(t)-x''(T_{x})]
+\int_{T_{x}}^{t}{g(x'(\tau ))}d\tau \}
+\int_{T_{x}}^{t}{P(\tau )}d\tau.
\end{aligned}\label{e3.4}
\end{equation}
By condition (iii)(a), we have that
\begin{equation}
\begin{aligned}
&\int_{T_{x}}^{t}{h(x(\tau )}d\tau \\
&<-\{ b[x'(t)-x'(T_{x})]+a[x''(t)-x''(T_{x})]+c[x(t)-x(T_{x})]\}
+\int_{T_{x}}^{t}{P(\tau )}d\tau.
\end{aligned}\label{e3.5}
\end{equation}
Define $ I(t)\equiv{\int_{T_{x}}^{t}}{P(\tau ))}d\tau $.
By the virtue of the above condition, the assertion of the Lemma \ref{lem3.1}
 and
the boundedness of $x(t)$, there exists a constant $M_{x}$ such that
for $t\geq T_{x}$,
$$ |I(t)|\leq M_{x}; $$
i.e.,
\begin{equation}
\big|\int_{T_{x}}^{t}{h(x(\tau ))}d\tau \big|\leq|I(t)|
\leq M_{x}.\label{e3.6}
\end{equation}
Now let us assume that $x(t)$ does not converge to any root $\overline{x}$
of $h(x)$; i.e.,
\begin{equation}
\limsup_{t\to \infty }|x(t)-\overline{x}|>0 \label{e3.7}
\end{equation}
and simultaneously, for $t\geq T_{x}$,
\begin{equation}
h(x(t))\geq 0 \quad\text{or}\quad h(t)\leq 0.\label{e3.8}
\end{equation}
Then
$$
H(t):\equiv \int_{T_{x}}^{t}{h(x(\tau )})d\tau
$$
for $t\geq T_{x}$ which is a composed monotone function with a finite
or infinite limit for $t\to \infty $.
Since \eqref{e3.6} implies that divergent case can be disregarded,
it follows from  \eqref{e3.7} that not only
\begin{equation}
\lim_{t\to \infty }\int_{T_{x}}^{t}{|h(x(\tau ))|}d\tau
=\lim_{t\to \infty }|\int_{T_{x}}^{t}{|h(x(\tau
))|}d\tau |\leq M_{x}, \label{e3.9}
\end{equation}
but also
\begin{equation}
\lim_{t\to \infty }|x(t)-\overline{x}|
=0\,.\label{e3.10}
\end{equation}
Otherwise (i.e., if ${ \limsup_{t\to \infty} }|x(t)
-\overline{x}|>0$) the inequality
\eqref{e3.7} together with the fact that the roots of $h(x)$ are isolated
yields
$$
\liminf_{t\to \infty} |h(x(t))|
={\liminf_{t\to \infty} }|h(x(t))-h(\overline{x})|>0
$$
which is a contradiction to \eqref{e3.9}. Thus  \eqref{e3.6} and
\eqref{e3.8} imply
$$
{\limsup_{t\to \infty }}|h(x(t))|
={\limsup_{t\to \infty }}|h(x(t))-h(\overline{x})|>0
={\liminf_{t\to \infty}}|h(x(t))|.
$$
In what follows, $d(x,y)$ denotes
the distance between $x$ and $y$.
Consequently, there exists such a
sequence ${t_{i}}\geq T_{x}$ and a
constant $\widetilde{H}>0$, such that
$$
   \liminf_{t\to \infty \Rightarrow
t_{i}\to \infty } d(t_{i},t_{i-1})>0, \quad
|h(x(t_{i})|\geq \widetilde{H},
$$
and such that
$$
M_{x}\geq \lim_{t\to \infty }\int_{t_{i-1}}^{t_{i}}{|h(x(\tau
))|}d\tau =\sum_{i=2}^{\infty }{\int_{t_{i-1}}^{t_{i}}{|
h(x(\tau ))|}d\tau }
$$
implies
$$
\lim_{t\to \infty \Rightarrow
t_{i}\to \infty }{\int_{t_{i-1}}^{t_{i}}{|h(x(\tau ))|} d\tau }=0,
$$
or
$$
H'\limsup_{t\to \infty }|x'(t)|\geq
\lim_{t\to \infty }|\frac{dh(x(t))}{dx(t)}x'(t)|
=\limsup_{t\to \infty }|\frac{dh(t)}{dt}|=\infty.
$$
According to the assertion of the Lemma \ref{lem3.1}, this is
impossible and that is why $ (x(t)-\overline{x})$ necessarily
oscillates. The remaining part of the lemma follows from the
assertion
\begin{equation}
x(t)\in C^{n}[0, \infty ), \quad
\lim_{t\to \infty} |x^{n}(t)|< \infty, \label{e3.11}
\end{equation}
${\lim_{t\to \infty }}|x(t)|< \infty$ implies
$\lim_{t\to \infty }x^{k}(t)=0$,
where $n\geq 2$ is a natural numbers and $k=1,\dots (n-1)$.
This completes the proof.
\end{proof}

\begin{lemma} \label{lem3.3}
 Under the assumptions of the Lemma \ref{lem3.2}, suppose that
 (iv)(b) holds for all $t\geq 0$, and
${\limsup_{t\to \infty}}|p(t)|>0$ hold, where $P'$ is a suitable constant,
then for every bounded solution $x(t)$
of \eqref{e1.1} there exists  a root $\overline{x}$ of $h(x)$ such that
$(x(t)-\overline{x})$ oscillates.
\end{lemma}

\begin{proof}
If Lemma \ref{lem3.3} does not hold, then according to Lemma \ref{lem3.2}, equations
\eqref{e*} hold and the
fifth derivative of $x(t)$ satisfies
$$
x^{v}(t)=p'(t)-ax^{^{iv}}(t)-bx'''(t)-g'(x'(t))x''(t)-h'(x(t))x'(t).
$$
But by the ultimate boundedness of $x'(t), x''(t), x'''(t)$ and
$x^{iv}(t)$, there exists a constant $D_{5}$ such that
$$
{\limsup_{t\to \infty}} |x^{v}(t)|\leq D_{5}
$$
which according to \eqref{e3.11} gives the relations
$$
{\lim_{t\to \infty}}x(t)=\overline{x}\Rightarrow
\lim_{t\to \infty }h(x(t))=h(\overline{x})=0,
{\lim_{t\to \infty }}x^{j}(t)=0,
$$
$j=1,2,3$,
or
$$
{\limsup_{t\to \infty} }|p(t) |=\big|
x^{iv}(t)+ax''' (t)+bx''(t)+g(x'(t))+h(x(t))\big|=0,
$$
which is a contradiction to
$\limsup_{t\to \infty}|p(t)|>0$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm2.1}]
Let us assume on the contrary, that $x(t)$ is an unbounded solution
of \eqref{e1.1}; i.e., $\limsup_{t\to \infty } x(t)= \infty $.
It will follow from Lemma \ref{lem3.1} that there exists a number $T_{0}\geq 0$
large enough such that for $t\geq T_{0}$,
\begin{gather*}
|x'(t)|\leq D'+\epsilon _{1},\\
|x''(t)|\leq D''+\epsilon _{2}\\
|x'''(t)|\leq D'''+\epsilon_{3}
\end{gather*}
with $\epsilon _{i}>0$, ($i=1,2,3$) small enough constants.
Let $T_{1}\geq T_{0}$ be the last point with
$x(T_{1})=\overline{x}_{k}$, ($k$ even) and $T_{2}>T_{1}$ be the
first point with $x(T_{2})=\overline{x}_{k+1}$.
 Integrating \eqref{e1.1} from $T_{1}$ to $t$, $T_{1}\leq t\leq T_{2}$,
 we have
\begin{equation}
\begin{aligned}
&[ x'''(t)-x'''(T_{1})]+a[x''(t)-x''(T_{1})]+b[x'(t)-x' (T_{1})]\\
&+ {\int_{T_{1}}^{t}}{g(x'(\tau ))}d\tau
 +\int_{T_{1}}^{t}{h(\tau)}d\tau \\
&= \int_{T_{1}}^{t}{P(\tau )}d\tau.
\end{aligned} \label{e3.12}
\end{equation}
However, on replacing
${\int_{T_{1}}^{t}}{g(x'(\tau ))}d\tau $ with
$c[x(t)-x(T_{1})]$, for $T_{1}\leq t\leq T_{2}$, we have
$h(x(t))sgn x(t)\geq 0$. Multiplying \eqref{e3.12} by
$\mathop{\rm sgn}x$, we obtain
$$
|x(t)|\leq|x(T_{1})|+\frac{1}{c}[D'''+aD''+bD' + P_{0}] + \epsilon,
$$
where $c>0$ and  $\epsilon >0$ is arbitrary small constant,
a contradiction to $x(T_{2})=\overline{x}_{k+1}$ with respect to
condition (ii) of Theorem \ref{thm2.1}.
The remaining part of the proof follows from the Lemma \ref{lem3.3}.
\end{proof}

\subsection*{Acknowledgments}
The Author wishes to express his profound gratitude to the anonymous
referees for his/her valuable suggestions and contribution that
saw this work to its present form.

\begin{thebibliography}{99}

\bibitem{a1} A .U. Afuwape and O. A. Adesina:
\textit{Frequency-Domain Approach to Stability and Periodic Solutions
of Certain Fourth-Order Nonlinear Differential Equations\/},
Nonlinear Studies Vol. 12 (2005), No. 3,  259-269.

\bibitem{a2} J. Andres:
\textit{Boundedness of Solutions of the Third Order Differential
Equation with Oscillatory Restoring and Forcing Terms\/},
Czech. Math. J., Vol. 1 (1986), 1-6.

\bibitem{a3}  J. Andres: \textit{Boundedness result of solutions to
the equation $x'''+ax''+g(x')+h(x)=p(t)$ without the hypothesis
$h(x)\mathop{\rm sgn}x\geq 0$ for $|x|>R$\/},
Atti Accad. Naz. Lincie, VIII. Ser., Cl. Sci. Fis. Mat. Nat. 80 (1986),
No 7-12, 532-539.

\bibitem{a4} J. Andres: \textit{Note to a certain Third-Order Nonlinear
Differential Equation related to the Problem of Littlewood \/}, Fasc. Math.
 302 (1991), Nr 23, 5-8.

\bibitem{a5} J. Andres and S. Stanek:
\textit{Note to the Langrange Stability of Excited Pendulum Type
Equations\/}, Math. Slovaca,  43(1993), No. 5, 617-630.

\bibitem{c1} W. A. Coppel: Stability and Asymptotic Behaviour of
Differential Equations, D.C. Heath Boston 1975.

\bibitem{r1} R. Reissig, G. Sansone and R. Conti:
Non Linear Differential Equations of Higher Order Nourdhoff International
Publishing Lyden (1974).

\bibitem{s1} S. Sedziwy: \textit{Boundedness of Solutions of an N-th
Order Nonlinear Differential Equation,\/}, Atti Accad. Naz. Lincie,
VIII. Ser., Cl. Sci. Fis. Mat. Nat. 80 (1978) no. 64, 363-366.

\bibitem{s2} K.E. Swick: \textit{Boundedness and Stability for
Nonlinear Third Order Differential Equation\/}, Atti Accad.
Naz. Lincie, VIII. Ser., Cl. Sci. Fis. Mat. Nat. 80 (1974), no. 56. 859-865.

\bibitem{t1} A. Tiryaki and C. Tunc: \textit{Boundedness and the
Stability Properties of Solutions of Certain Fourth Order Differential
Equations via the Intrinsic Method\/}, Analysis, 16 (1996), 325-334.

\bibitem{t2} C. Tunc: \textit{A Note on the Stability and Boundedness
Results of Certain Fourth Order Differential Equations,\/}
Applied Mathematics and Computation, 155 (2004), no. 3, 837-843.

\bibitem{t3} C. Tunc: \textit{Some Stability and Boundedness Results
for the Solutions of Certain Fourth Order Differential Equations,\/},
Acta Univ. Palacki Olomouc. Fac. Rerum Natur. Math. 44 (2005), 161-171.

\bibitem{t4} C. Tunc: \textit{An ultimate Boundedness Result for a
Certain System of Fourth Order Nonlinear Differential Equations,\/}
Differential Equations and Applications, Vol. 5 (2005), 163-174.

\bibitem{t5} C. Tunc and A.Tiryaki: \textit{On the Boundedness and the
Stability Results for the Solutions of Certain Fourth Order
Differential Equations via the Intrinsic Method\/},
Applied Mathematics and Mechanics, 17 (1996), No. 11, 1039-1049.

\end{thebibliography}

\end{document}