\pdfoutput=1\relax\pdfpagewidth=8.26in\pdfpageheight=11.69in\pdfcompresslevel=9 \documentclass[twoside]{article} \usepackage{amssymb} % used for R in Real numbers \pagestyle{myheadings} \setcounter{page}{99} \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 $$\label{e1} u(x,t)=u_{0}(t)+\int_{0}^{x}\int_{-\infty}^{t} e^{-\int_{\tau}^{t}a(\xi,\theta)\,d\theta } C(\xi,\tau,u(\xi,n))\,d\tau\, d\xi$$ 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 $$\label{e2} V_{t}(x,t)+a(x,t)V(x,t)=f(x,t),\hbox{ in }D:(0,l)\times\mathbb{R}\rightarrow\mathbb{R}$$ \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 $$\label{e3} V(x,t)=\int_{-\infty }^{t}e^{-\int_{\tau }^{t}a(x,\theta )\,d\theta }f(x,\tau )\,d\tau ,$$ is almost periodic in $t$, uniformly with respect to $x$, and $$\label{e4} \left| V(x,t)\right| \leq \frac{1}{m}\sup \left| f(x,t)\right| ,\quad (x,t)\in D.$$ \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 $$\label{e5} \sup\left\vert V(x,t+\delta)-V(x,t)\right\vert \leq \frac{\epsilon}{2} + \frac{\epsilon}{2}=\epsilon \quad\hbox{for all such } \delta \in \mathbb{R}.$$ 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}