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Massera criterion, Periodic solution.
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\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}
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