Content-Type: multipart/mixed; boundary="-------------0505290414966" This is a multi-part message in MIME format. ---------------0505290414966 Content-Type: text/plain; name="05-191.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="05-191.keywords" Massera criterion, Periodic solution. ---------------0505290414966 Content-Type: application/x-tex; name="mas_1.TEX" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="mas_1.TEX" %\documentclass[reqno]{amsproc} \documentclass[reqno,11pt]{amsart} %\AtBeginDocument{\noindent\small %http://www.mathpreprints.com/math/Preprint/Oleg/20040212/2/ \newline} %\documentclass[reqno]{amsart} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newtheorem{theo}{Theorem} \newtheorem{rem}{Remark} \newtheorem{lem}{Lemma} \newtheorem{conj}{Conjecture} \newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem{df}{Definition} \newtheorem{hyp}{Hypothesis} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand\eps\varepsilon \newcommand\ph\varphi \newcommand\kap\varkappa \newcommand\R {\mathbb{R}} \newcommand\T {\mathbb{T}} \newcommand\Z {\mathbb{Z}} \newcommand\N {\mathbb{N}} \newcommand\bC {\mathbb{C}} \renewcommand{\refname}{References} \renewcommand{\Re}{\mbox{\rm Re}\,} \renewcommand{\Im}{\mbox{\rm Im}\,} %%%%%%%%%%%%%%%%%%%%%%% %\hoffset -35mm %\voffset -35mm %\overfullrule 0pt %\tolerance 1000 %\textwidth=510pt %\textheight=720pt %%%%%%%%%%%%%%%%%%%%%%% \begin{document} \title[A Note On A Theorem Of Massera] {A Note On A Theorem Of Massera} \author[Oleg Zubelevich]{Oleg Zubelevich\\ \\ \tt Department (\#803) of Differential Equations\\ Moscow State Aviation Institute\\ Volokolamskoe Shosse 4, 125993, Moscow, Russia\\ E-mail: ozubel@yandex.ru} \address{Department (\# 803) of Differential Equations Moscow State Aviation Institute Volokolamskoe Shosse 4, 125993, Moscow, Russia} \email{ozubel@yandex.ru} \curraddr{2-nd Krestovskii Pereulok 12-179, 129110, Moscow, Russia} %\date{} %\thanks{Partially supported by grants RFBR 02-01-00400.} \subjclass[2000]{34G10} \keywords{Massera criterion, Periodic solution.} \begin{abstract}In the present paper we consider a non autonomous inhomogeneous $\omega$-periodic linear differential equation on a reflexive Banach space and show that if it has a bounded solution then it has an $\omega$-periodic solution. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}{Definition}[section] \section{Introduction} In this paper we consider a non autonomous inhomogeneous $\omega$-periodic linear differential equation on a reflexive Banach space and show that if it has a bounded solution then it has an $\omega$-periodic solution. Theorems that deduce existence of periodic solutions from assumption of existence for bounded solutions have been studied by many authors. First Massera \cite{8} proved such a theorem for a linear inhomogeneous system of finite number of ODE and for two dimensional nonlinear ODE. A problem of existence of $m\omega$-periodic ($m>0$, integer) solution for ordinary differential system that possesses bounded solution with some stability property was considered in \cite{10}. Chow \cite{4} obtained Massera's type result for linear functional-differential equations with finite delay. In \cite{7} are shown such a sort results for functional-differential equations with infinite delay and for some class of integral equations. In \cite{Makay} different aspects of the Massera type results for non linear functional-differential equations are considered and examples of existence and non existence are given. Other results for periodic solutions to functional-differential equations with delay are presented in \cite{3},\cite{6}. \section{Setup of the Problem} Consider an inhomogeneous linear system of ordinary differential equations: \begin{equation} \label{lin_sys} \dot{x}=A(t)x+b(t),\quad x=(x_1,\ldots,x_n).\end{equation} The matrix $A$ and vector $b$ are continuous and $\omega$-periodic in $t\ge 0$. The celebrated Massera's theorem states that if system (\ref{lin_sys}) has a bounded solution then it has an $\omega$-periodic solution. Our aim is to generalize this theorem to the case of any reflexive Banach space $E$. Nevertheless, it is convenient to reformulate our problem in terms of Poincare's mappings. In such a terms our theorem can be applied to linear PDE. Give an informal description of transfer from the differential problem to mapping of the space $E$. Exact result is presented in the next sections, it is concerned to a stable point of some Poinrare's mapping. So assume that $b(t),x\in E$ and $A(t)$ is an $\omega$-periodic family of linear operators of $E$, $b(t)$ is also $\omega$-periodic. Introduce a linear homogeneous equation $$\dot{x}=A(t)x,\quad x(t_0)=\hat x.$$ Suppose that there exists a family of the Cauchy operators $\{K(t,t_0)\}_{t,t_0\in \mathbb{R}}$ such that the solution to this equation presents as follows $$x(t)=K(t,t_0)\hat{x}.$$ According to Du Hammel's principle, present system (\ref{lin_sys}) with given $\hat{x}=x(t_0)$ as an integral equation: \begin{equation} \label{int_eq} x(t)=K(t,t_0)\hat x+\int^t_{t_0}K(t,\tau)b(\tau)\,d\tau. \end{equation} The initial value $\hat x=x(0)=x(\omega)$ for an $\omega$-periodic solution to problem (\ref{int_eq}) is found from the equation: $$x(\omega)=K(\omega,0)x(\omega)+\int^\omega_{0}K(\omega,\tau)b(\tau)\,d\tau.$$ Thus, writing $g=\int^\omega_{0}K(\omega,\tau)b(\tau)\,d\tau$ and $Q=K(\omega,0)$ we look for a fixed point of a mapping \begin{equation} \label{main_P} Px=Qx+g. \end{equation} If equation (\ref{int_eq}) has a bounded solution with initial value $x(0)=x_0$ then sequence $\{P^n x_0\}_{n\in \mathbb{N}}$ is bounded in $E$. \section{Main theorem}Let $Q:E\to E$ be a linear bounded transformation of a reflexive Banach space $E$, let $g$ be an element of $E$ and the mapping $P$ be given by (\ref{main_P}). \begin{theo} \label{main_th}Assume that there exists an element $x_0\in E$ such that the sequence $\{P^n x_0\}_{n\in \mathbb{N}}$ is bounded. Then the mapping $P$ has a fixed point $\hat{x}\in E$ i.e. $P(\hat{x})=\hat{x}.$\end{theo} \section{Proof of Theorem \protect\ref{main_th}} Let $V$ be another reflexive Banach space. Consider a bounded linear operator $T:V\to V$ and construct operators $T_n,\quad n\in \mathbb{N}$ as follows: $$ T_n=\frac{1}{n}\sum_{k=1}^nT^k.$$ \begin{lem}[\'a la Yosida's ergodic theorem \protect\cite{Yosida}] \label{yo} If for some $z_0\in V$ the sequence $\{T^nz_0\}$ is bounded: $$\sup_{n\in \mathbb{N}}\{\|T^nz_0\|\}=c<\infty$$ then the sequence $\{T_nz_0\}$ contains a subsequence $\{T_{n'}z_0\}$ such that $\{T_{n'}z_0\}\to \hat{z}$ weakly as $n'\to\infty$ and \begin{equation} \label{nep}T\hat{z}=\hat{z}.\end{equation} \end{lem} \proof Note that the sequence $\{T_nz_0\}$ is also bounded. Indeed, $$\|T_nz_0\|\le \frac{1}{n}\sum_{k=1}^n\|T^kz_0\|\le c.$$ Thus, we can pick the announced subsequence. We shall prove that the element $\hat{z}$ is desired fixed point of $T$. Since the sequence $\{T^nz_0\}$ is bounded we obtain $$ TT_{n'}z_0-T_{n'}z_0=\frac{1}{n'}\Big(T^{n'+1}z_0-Tz_0\Big)\to 0\quad \mathrm{as}\quad n'\to\infty.$$ For any $ f\in V^*$ this implies: \begin{equation} \label{weal} (TT_{n'}z_0,f)-(T_{n'}z_0,f)\to 0.\end{equation} On the other hand the following formulas hold true: \begin{align} (T_{n'}z_0,f)&\to (\hat{z},f),\label{1}\\ (TT_{n'}z_0,f)=(T_{n'}z_0,T^*f)&\to (\hat{z},T^*f)=(T\hat{z},f)\label{2}. \end{align} Gathering formulas (\ref{weal}), (\ref{1}), (\ref{2}) we see (\ref{nep}). \endproof Now we are ready to prove Theorem \ref{main_th}. As the space $V$ take a space $E\times \mathbb{R}$. As the mapping $T:V\to V$ take a mapping $(x,y)\mapsto (Qx+gy,y)$. Applying Lemma \ref{yo} to the mapping $T$ and a point $z_0=(x_0,1)$ we obtain a fixed point $\hat{z}=(\hat{x},\hat{y})$ for the mapping $T$. Obviously, if we show that $\hat{y}=1$ then $\hat{x}$ is a fixed point of $P$ and the Proof is concluded. Let $z=(x,y)\in V$ then define $p\in V^*$ as follows: $(z,p)=y$. Note that $(T^nz,p)=(z,p)$ and $(T_nz,p)=(z,p)$. Therefore we have: $$1=(T_{n'}z_0,p)\to (\hat{z},p)$$ and this implies that $(\hat{z},p)=1.$ Theorem \ref{main_th} is proved. \begin{thebibliography}{99} \bibitem{3}T. Burton and L. Hatvani On the existence of periodic solutions of some non-linear functional differential equations with unbounded delay, Nonlinear Anal. 16 (1991), 389-398. \bibitem{4}S.-N. Chow, Remarks on one dimensional delay-differential equations, J. Math. Anal. Appl. 41 (1973), 426-429. \bibitem{6}L. Hatvani and T. Krisztin, On the existence of periodic solutions for linear inhomogeneous and quasi-linear functional differential equations, J. Differential Equations 97 (1992), 1-15. \bibitem{7}G. Makay, Periodic solutions of liear differential and integral equations, J. of Differential and Integral Equations 8 (1995), 2177-2187. \bibitem{Makay} G. Makay, On some possible extensions of Massera's theorem, EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 16. \bibitem{8} J. Massera, The existence of periodic solutions of systems of differential equations, Duke Math. J. (1950), 457-475. \bibitem{10} T. Yoshizawa, Stability theory and the existence of periodic solutions and almost periodic solutions, Springer-Verlag, (1975). \bibitem{Yosida} K. Yosida, Mean ergodic theorem in Banach spaces, Proc. Imp. Acad. Tokyo, 14 (1938), 292-294. \end{thebibliography} \end{document} ---------------0505290414966--