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\markboth{\hfil Almost periodic solutions \hfil}%
{\hfil Hushang Poorkarimi \& Joseph Wiener \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc 16th Conference on Applied Mathematics, Univ. of Central Oklahoma},
\newline
Electronic Journal of Differential Equations, Conf. 07, 2001, pp. 99--102.
\newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or
http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu (login: ftp)}
 \vspace{\bigskipamount} \\
%
Almost periodic solutions of nonlinear hyperbolic equations with
time delay
%
\thanks{ {\em Mathematics Subject Classifications:} 
35B10, 35B15, 35J60, 35L70. \hfil\break\indent 
{\em Key words:} Nonlinear hyperbolic equation, time delay, 
almost periodic solution.
\hfil\break\indent 
\copyright 2001 Southwest Texas State University. \hfil\break\indent
Published July 20, 2001.} }

\date{}
\author{ Hushang Poorkarimi \& Joseph Wiener }
\maketitle

\begin{abstract}
 The almost periodicity of bounded solutions is established for a nonlinear
 hyperbolic equation with piecewise continuous time delay. The equation
 represents a mathematical model for the dynamics of gas absorption.
\end{abstract}

\newtheorem{thm}{Theorem}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{defn}[thm]{Definition}



\section{Introduction}

In this paper we are interested in determining almost periodicity
for a unique bounded solution of nonlinear hyperbolic equations
with time delay. The initial value problem under investigation is
the following:
\begin{eqnarray}
&u_{xt}(x,t)+a(x,t)u_{x}(x,t)=C(x,t,u(x,[t])) & \label{E} \\
&u(0,t)=u_{0}(t),  \label{C}
\end{eqnarray}
where $a$ and $C$ are defined in the domain $D:(0,l) \times
\mathbb{R} \to \mathbb{R} $, and $[t]$ denotes the greatest
integer function: $[t]=n$ when $n \leq t < n+1$,  for an integer
$n$.  In this case the delay function is piecewise constant. The
existence of a unique bounded solution of problem
(\ref{E})--(\ref{C}) has been discussed earlier [1].


Equation (\ref{E}) with condition (\ref{C}) under assumption
\[
a(x,t) \geq m > 0 \quad\hbox{in } D
\]
has a unique bounded solution via Volterra integral equation
\begin{equation} \label{e1}
u(x,t)=u_{0}(t)+\int_{0}^{x}\int_{-\infty}^{t}
e^{-\int_{\tau}^{t}a(\xi,\theta)\,d\theta }
C(\xi,\tau,u(\xi,n))\,d\tau\, d\xi
\end{equation}

Let us notice that, in case of periodicity, the period has to be
the same for all the functions involved [2]. This result is based
on the equivalence of (\ref{E})--(\ref{C}) with integral equation
(\ref{e1}), and it can be stated as the following assertion.

\begin{thm}
If $u_{0}(t)$, $a(x,t)$, and $C(x,t,u(x,[t])$ are periodic in $t$
with period $T$, then the unique bounded solution of (\ref{e1})
is also periodic in $t$, with the same period $T$.
\end{thm}

\paragraph{Proof} From (\ref{e1}), one obtains
\[
u(x,t+T)=u_{0}(t+T)+\int_{0}^{x}\int_{-\infty}^{t+T}
e^{-\int_{\tau}^{t+T}a(\xi,\theta)\,d\theta}
C(\xi,\tau,u(\xi,n))\,d\tau\, d\xi .
\]
Making the substitution $\tau = \eta + T$ and taking into account
\[
\int_{\eta+T}^{t+T} a(\xi,\theta)\,d\theta = \int_{\eta+T}^{\eta}
a(\xi,\theta)\,d\theta + \int_{\eta}^{t}a(\xi,\theta)\,d\theta +
\int_{t}^{t+T} a(\xi,\theta)\,d\theta
\]
we have
\[
u(x,t+T)=u_{0}(t)+\int_{0}^{x}\int_{-\infty}^{t}e^{-\int_{\eta}^{t}
a(\xi,\theta)\,d\theta }C(\xi,\eta,u(\xi,n))\,d\eta d\xi=u(x,t)
\]
which proves the periodicity of $u$ in $t$ with period $T$.

\begin{defn}[Bohr's Definition of $\epsilon $--almost periodicity]
\rm For any $\epsilon >0$, there exists a number $l(\epsilon )>0$
with property that any interval of length $l(\epsilon )$ of the
real line contains at least one point with abscissa $\delta $,
such that $$\left| u(x,t+\delta )-u(x,t)\right| <\epsilon ,\
(x,t)\in D\,,$$ the number $\delta $ is called translation number
of $u(x,t)$ corresponding to $\epsilon $, or an $\epsilon
$-almost period of $u(x,t)$.
\end{defn}

The following lemma will be used to prove that the unique bounded
solution (in $D$) of equation (\ref{e1}) is almost periodic in
$t$.

\begin{lem}
Assume the following conditions hold true in regard to the
equation
\begin{equation} \label{e2}
V_{t}(x,t)+a(x,t)V(x,t)=f(x,t),\hbox{ in
}D:(0,l)\times\mathbb{R}\rightarrow\mathbb{R}
\end{equation}
\begin{enumerate}
\item $a(x,t)$, $f(x,t)$ are almost periodic in $t$,
uniformly with respect to $x$;
\item $a(x,t)\geq m>0$ in $D$.
\end{enumerate}
Then the unique bounded solution of  (\ref{e2}), given by
\begin{equation} \label{e3}
V(x,t)=\int_{-\infty }^{t}e^{-\int_{\tau }^{t}a(x,\theta
)\,d\theta }f(x,\tau )\,d\tau ,
\end{equation}
is almost periodic in $t$, uniformly with respect to $x$, and
\begin{equation} \label{e4}
\left| V(x,t)\right| \leq \frac{1}{m}\sup \left| f(x,t)\right|
,\quad (x,t)\in D.
\end{equation}
\end{lem}

\paragraph{Proof} We obtain from (\ref{e2}), changing $t$ to $t+\delta$:
\[
V_{t}(x,t+\delta)+a(x,t+\delta)V(x,t+\delta)=f(x,t+\delta),
\]
and subtracting (\ref{e2}) from it,
\begin{eqnarray*}
\lefteqn{ \left[V(x,t+\delta)-V(x,t)\right]_{t}+a(x,t+\delta)
\left[V(x,t+\delta)-V(x,t)\right] }\\
&=& f(x,t+\delta)-f(x,t)-\left[ a(x,t+\delta)-a(x,t)
\right]V(x,t).
\end{eqnarray*}
Taking into account the almost periodicity of $a(x,t)$, $f(x,t)$
and boundedness of $V(x,t)$ in $D$, one obtains in $D$, according
to (\ref{e4}):
\begin{eqnarray*}
\sup\left\vert V(x,t+\delta)-V(x,t)\right\vert & \leq &
\frac{1}{m}
\sup\left\vert f(x,t+\delta)-f(x,t)\right\vert \\
& &+ \frac{M}{m}\sup\left\vert a(x,t+\delta)-a(x,t)\right\vert,
\end{eqnarray*}
where $M=\sup\left\vert V(x,t)\right\vert$, $(x,t)\in D$. We
choose $\delta$ such that,
\[
\left\vert f(x,t+\delta)-f(x,t)\right\vert < \frac{m\epsilon}{2},
\quad \hbox{and}\quad \left\vert a(x,t+\delta)-a(x,t)\right\vert
< \frac{m\epsilon}{2M}
\]
for sufficiently large $t$, i.e., $f(x,t)$ must be an
$(m\epsilon)/2$-almost periodic and $a(x,t)$ is
$(m\epsilon)/2M$-almost periodic. Then
\begin{equation} \label{e5}
\sup\left\vert V(x,t+\delta)-V(x,t)\right\vert \leq
\frac{\epsilon}{2} + \frac{\epsilon}{2}=\epsilon \quad\hbox{for
all such } \delta \in \mathbb{R}.
\end{equation}
In other words, for any $\epsilon > 0$, there exists a number
$l(\epsilon) >0$ with the property that any interval
$(a,a+l)\in \mathbb{R}$ contains an $%
\epsilon$-almost period of $V(x,t)$. This means that $V(x,t)$ is
an almost periodic function in $t$, uniformly with respect to $x
\in [0,l]$ by Bohr's definition of almost periodicity. Let us
conclude now with the result on almost periodicity of the unique
bounded solution of  (\ref{e1}) in $D$.

\begin{thm}
Consider equation (\ref{E}) in $D$, and assume $u_{0}(t)$,
$a(x,t)$, and \break $C(x,t,u(x,[t]))$ are almost periodic in
$t$, uniformly with respect to $x \in [0,l]$, and $a(x,t) \geq m
> 0$. Also assume that $C(x,t,u(x,[t]))$ is continuous on $D
\times \mathbb{R}$, with $C(x,t,0)$ bounded on $D$, and satisfies
the Lipschitz condition
\[
\left\vert C(x,t,u(x,[t]))-C(x,t,V(x,[t]))\right\vert \leq
L\left\vert u(x,[t])-V(x,[t])\right\vert
\]
where $L$ is a positive constant. Then the unique bounded
solution of (\ref{E})--(\ref{C}) in $D$ is almost periodic in
$t$, uniformly with respect to $x \in [0,l]$.
\end{thm}

\paragraph{Proof} Let the first approximation be $u_{0}(x,t)\equiv 0$.
Next approximation is then
\[
u_{1}(x,t)=u_{0}(t)+\int_{0}^{x}\int_{-\infty}^{t}
e^{-\int_{\tau}^{t}a(\xi,\theta)\,d\theta }C(\xi,\tau,0)\,d\tau\,
d\xi\,.
\]
Since $V(x,t)=\frac{\partial}{\partial x}u_{1}(x,t)$, then from
the equation
\[
V_{t}(x,t)+a(x,t)V(x,t)=C(x,t,0)
\]
by Lemma 2 we obtain the almost periodicity of $V(x,t)$. But
\[
u_{1}(x,t)=u_{0}(t)+\int_{0}^{x}V(\xi,t)\,d\xi.
\]
This shows that $u_{1}(x,t)$ is almost periodic in $t$, uniformly
with respect to $x \in [0,l]$, and
\[
u_{2}(x,t)=u_{0}(t)+\int_{0}^{x}\int_{-\infty}^{t}
e^{-\int_{\tau}^{t}a(\xi,\theta)\,d\theta
}C(\xi,\tau,u_{1}(\xi,\tau))\,d\tau\, d\xi .
\]
The relation $\overline{V}(x,t)=\frac{\partial}{\partial
x}u_{2}(x,t)$ and equation
\[
\overline{V_{t}}(x,t)+a(x,t)\overline{V}(x,t)=C(x,t,u_{1}(x,t))
\]
implies almost periodicity of
\[
u_{2}(x,t)=u_{0}(t)+\int_{0}^{x}\overline{V}(\xi,t)\,d\xi ,
\]
by Lemma 2. Then $u_{3}(x,t)$ is almost periodic by a similar
argument. Consequently, all successive approximations
$u_{n}(x,t), \quad n=1,2,\ldots$ are almost periodic functions in
$t$, uniformly with respect to $x \in [0,l]$.
 Hence the solution
\[
u(x,t)=\lim_{n\to\infty} u_{n}(x,t)
\]
is also almost periodic in $t$, uniformly with respect to $x\in
[0,l]$.


\begin{thebibliography}{9} \frenchspacing

\bibitem{1}  Poorkarimi, H., and Wiener, J., (1986), ``Bounded Solutions of
Non-linear Hyperbolic Equations with Delay'', Proceedings of the
VII International Conference on Non-Linear Analysis, V.
Lakshmikantham, Ed., 471--478

\bibitem{2}  Poorkarimi, H., ``Asymptotically Periodic Solutions for Some
Hyperbolic Equations'', Libertas Mathematica, Vol.8, 1998,
117--122.

\bibitem{3}  Tikhonov, A. N., and Samarskii, A. A., \textit{Equations of
Mathematical Physics}, Pergamon Press, New York, 1963.

\bibitem{4}  Corduneanu, C., \textit{Almost Periodic Functions}, Wiley, New
York, 1968.
\end{thebibliography}

\noindent\textsc{Hushang Poorkarimi \& Joseph Wiener}\\
University of Texas-Pan American \\
Department of Mathematics \\
Edinburg, TX 78539, USA\\
\texttt{poorkar@panam.edu \& jwiener@panam.edu}


\end{document}