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\markboth{\hfil stability  of periodic evolution families \hfil EJDE--2003/14}
{EJDE--2003/14\hfil Constantin  Bu\c se \&  Oprea Jitianu \hfil}

\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol. {\bf 2003}(2003), No. 14, pp. 1--10. \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} \\
 %
  A new theorem on exponential stability
  of periodic evolution families on Banach spaces
 %
\thanks{ {\em Mathematics Subject Classifications:}
26D10, 34A35, 34D05, 34B15, 45M10, 47A06.
\hfil\break\indent
{\em Key words:} Almost periodic functions, exponential stability,
 periodic evolution families \hfil\break\indent
 of operators, integral inequality, differential
 inequality on Banach spaces.
\hfil\break\indent
\copyright 2003 Southwest Texas State University. \hfil\break\indent
Submitted November 13, 2002. Published February 11, 2003.} }
\date{}
%
\author{Constantin  Bu\c se \&  Oprea Jitianu}
\maketitle

\begin{abstract}
  We consider a mild solution $v_f(\cdot, 0)$ of a well-posed
  inhomogeneous Cauchy problem $\dot v(t)=A(t)v(t)+f(t)$, $v(0)=0$
  on a complex Banach  space $X$, where $A(\cdot)$ is a $1$-periodic
  operator-valued function. We prove that if $v_f(\cdot, 0)$ belongs
  to $AP_0(\mathbb{R}_+, X)$ for each  $f\in AP_0(\mathbb{R}_+, X)$
  then for each $x\in X$ the solution of the well-posed Cauchy problem
  $\dot u(t)=A(t)v(t)$, $u(0)=x$ is uniformly exponentially stable.
  The converse statement is also true. Details about the space
  $AP_0(\mathbb{R}_+, X)$ are given in the section 1, below. Our approach
  is based on the spectral theory of evolution semigroups.
\end{abstract}

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

\section{Introduction}

 Let $X$ be a  complex Banach space and $\mathcal{L}(X)$ the Banach
algebra of all linear and bounded operators acting on $X$. The norms of
vectors in $X$ and of operators in $\mathcal{L}(X)$ will be denoted
 by $\|\cdot\|$. Let $\mathbb{R}_+$ the set of all non-negative real numbers
and let $\mathbb{J}$ be either $\mathbb{R}$ or $\mathbb{R}_+$. The
Banach space of all $X$-valued, bounded and uniformly continuous
functions on $\mathbb{J}$ will be denoted by $BUC(\mathbb{J}, X)$, and
the Banach space of all $X$-valued, almost periodic
 functions on $\mathbb{J}$ will be denoted by $AP(\mathbb{J}, X)$. It is known
 that $AP(\mathbb{J}, X)$
is the smallest closed subspace of $BUC(\mathbb{J}, X)$ containing functions
of the form
$$
t\mapsto f_{\mu, x}(t):=e^{i\mu t}x :J\to X,\quad \mu\in\mathbb{R},
\quad x\in X,
$$
see e.g. \cite{LZ}. The set of all $X$-valued functions on
$\mathbb{R}_+$ for which there exist $t_f\ge 0$ and $F_f\in
AP(\mathbb{R}_+, X)$ such that $f(t)=0$ if
 $t\in [0, t_f]$ and $f(t)=F_f(t)$ if $t\ge t_f$ will be denoted by $\mathcal{A}
_0(\mathbb{R}_+, X)$. Let $AP_0(\mathbb{R}_+, X)$ the smallest
closed subspace of $BUC(\mathbb{R}_+, X)$ which contains $\mathcal{A}_0(\mathbb{R}_+, X)$. The subspace of $BUC(\mathbb{J}, X)$
consisting of all $X$-valued, continuous, $1$-periodic functions
  such that $f(0)=0$ will be denoted by $P_1^0(\mathbb{J}, X)$. An $X$-valued, trigonometric polynomial function is given by
$$
t\mapsto p(t):=\sum_{k=-n}^n c_ke^{i\mu_k t}x_k :\mathbb{R}\to X,
\quad c_k\in\mathbb{C}, \mu_k\in\mathbb{R}, x_k\in X.
$$
The set of all functions $f$ on $\mathbb{R}_+$ for which there exist
$t_f\ge 0$ and a $X$-valued, trigonometric polynomial function $p_f$ such
that $f(t)=0$ if $t\in [0, t_f]$ and $f(t)=p_f(t)$ if $t\ge t_f$
will be denoted by
 $TP_0(\mathbb{R}_+, X)$. It is clear that $TP_0(\mathbb{R}_+, X)$ is a subset of
$\mathcal{A}_0(\mathbb{R}_+, X)$ and $P_1^0(\mathbb{R}_+, X)$ is the
closure in
 $BUC(\mathbb{R}_+, X)$ of a part of $TP_0(\mathbb{R}_+, X)$.
Let ${\bf T}=\{T(t): t\ge 0\}\subset \mathcal{L}(X)$ be a strongly continuous
semigroup on $X$ and $A: D(A)\subset X\to X$ its infinitesimal generator. It is
well known that the Cauchy problem
\begin{equation} \label{A0x}
\begin{gathered}
\dot u(t) = A u(t) \quad t\geq 0\\
 u(0) = x, \quad x\in X
 \end{gathered}
\end{equation}
is well-posed (see \cite{S1,S2,NaNi} and the references therein for the
 well-posednness of abstract differential equations) and the mild
 solution of \eqref{A0x} is given by
$u(t)=T(t)x, (t\ge 0)$. Moreover, for a locally Bochner integrable
function $f:\mathbb{R}_+\to X$, the mild solution of the
inhomogeneous Cauchy Problem
\begin{gather*}
\dot u(t)= Au(t)+f(t),\quad t\ge 0,\\
 u(0)=y, \quad y\in X
 \end{gather*}
is given by
$$ u_f(t, y)=T(t)y+\int_0^tT(t-\xi)f(\xi)d\xi,\quad t\ge 0.
$$
In particular the Cauchy problem
\begin{gather*}
\dot u(t) = Au(t)+e^{i \mu t}x \quad t\geq 0,\\
 u(0) =0\,,
\end{gather*}
 where $\mu\in\mathbb{R}$ and
$x\in X$, has the solution
$$
u_f(t, 0)=u_{\mu, x}(t)=\int_0^tT(t-\xi)e^{i\mu\xi}xd\xi,\quad t\ge 0.
$$
The Datko-Neerven's theorem (\cite{Da, Ne}) states that {\it a strongly continuous
semigroup  ${\bf T}= \{ T(t): t\geq 0\}\subset \mathcal{L}(X)$
 is exponentially stable},
 that is, there exist the constants
$N>0$ and $\nu >0$ such that
$$\|T(t)\|\leq Ne^{-\nu t} \quad \mbox{ for all } t\geq 0,
$$
{\it if and only if it acts
boundedly on one of the spaces $L^p(\mathbb{R}_+, X)$ or
$C_0(\mathbb{R}_+, X)$  by convolution.}
With other words if ${\cal X}$ is one of the spaces
$L^p(\mathbb{R}_+, X)$ or $C_0(\mathbb{R}_+, X)$ then the strongly
continuous semigroup ${\bf T}$ is
 exponentially stable if and only if for each function $f\in {\cal X}$ the solution
 $u_f(\cdot, 0)$ belongs to ${\cal X}$. Here $C_0(\mathbb{R}_+, X)$ is the space
consisting of all $X$-valued, continuous functions vanishing at
infinity, endowed with the sup-norm, and $L^p(\mathbb{R}_+, X)$, $1\le p<\infty$, denotes the usual Lebesgue-Bochner space of all
measurable functions $f:\mathbb{R}_+\to X$
 identifying functions which are equal almost everywhere and such that
$$
\|f\|_p:=\Big(\int_0^{\infty} \|f(s)\|^pds\Big)^{1/p}<\infty.
$$
When $X$ is a complex Hilbert space the Neerven-Vu's theorem
(\cite{Ne1,Ne2,Vu})
states that {\it the strongly continuous semigroup ${\bf T}$ on $X$
 is exponentially stable if and only if}
\begin{equation} \label{e1}
\sup_{\mu\in\mathbb{R}}\,\sup_{t\ge 0}\|u_{\mu, x}(t)\|=M(x)<\infty
\quad \mbox{for each } x\in X.
\end{equation}
In fact Neerven and Vu showed that if \eqref{e1} holds then the resolvent
 $R(\lambda, A):=(\lambda-A)^{-1}$ exists and is uniformly bounded in
$\{\lambda\in\mathbb{C}: \mathop{\rm Re}(\lambda)>0\}$. This result is valid for
semigroups defined on Banach spaces. The Gearhart-Pr\"uss-Herbst-Howland's
theorem (see \cite{Ge,He,Ho,Hu,P,W}) states that for semigroups
 on Hilbert spaces the uniform boundedness of the resolvent in
 $\{\mathop{\rm Re}(\lambda)>0\}$ implies
the exponential stability. A short history of these results and many more
 details about their relationships with abstract differential equations can
be found in \cite{B1,BJP,Vu}.

 For a well-posed, non-autonomous Cauchy problem
\begin{equation}
\begin{gathered}
\dot u(t) = A(t)u(t), \quad t\geq 0,\\
 u(0) = x, \quad x\in X
 \end{gathered} \label{e2}
\end{equation}
with  (possibly unbounded) linear operators $A(t)$, the mild solutions lead
to an evolution family on $\mathbb{R}_+$,
$\mathcal{U}=\{U(t,s): \ t\geq s\geq 0\} \subset \mathcal{L}(X)$, that is:
\begin{itemize}

\item[$(e_1)$] $U(t, r)=U(t, s)U(s, r)$ for all $t\ge s\ge r\ge 0$ and
   $U(t,t)=I$ for any $t\ge 0$, (I is the identity operator in $\mathcal{L}(X))$;

\item[$(e_2)$] the maps $(t,s)\mapsto U(t,s)x:\{(t,s):t\geq s\geq
 0\}\to X$ are continuous for each $x\in X$.
\end{itemize}
 An evolution family is {\it exponentially bounded} if there exist
 $\omega\in \mathbb{R}$ and $M_{\omega}>0$ such that
 \begin{equation}
 \|U(t,s)\|\leq M_{\omega}e^{\omega (t-s)},  \mbox{ forall }
 t\geq s\geq 0, \label{e3}
 \end{equation}
 and {\it exponentially stable} if \eqref{e3} holds with some negative
 $\omega$.
 If the evolution family $\mathcal{U}$ verifies the condition
\begin{itemize}
\item[$(e_3)$] $ U(t,s)=U(t-s,0)$ for  all $t\geq s\geq 0$,
\end{itemize}
then the family ${\bf T}=\{U(t,0): t\geq  0\}\subset\mathcal{L}(X)$ is
a strongly continuous semigroup on $X$. In this case the estimate \eqref{e3}
holds automatically.
 If the Cauchy problem \eqref{e2} is $1$-periodic, that is, $A(t+1)=A(t)$ for every
 $t\geq 0$, then the corresponding evolution family $\mathcal{U}$
 is $1$-periodic, that is,
\begin{itemize}
\item[$(e_4)$] $U(t+1, s+1) = U(t,s)$ for all $t\geq s\geq 0$.
\end{itemize}
 Every  $1$-periodic evolution family is exponentially bounded,
 see for example (\cite{BP}, Lemma 4.1). For a locally Bochner integrable function
 $f:\mathbb{R}_+\to X$, the mild solution of the well-posed, inhomogeneous
 Cauchy problem
 \begin{gather*}
\dot v(t)=A(t)v(t)+f(t),\quad t\geq 0,\\
u(0)=x
\end{gather*}
is given by
$$
v_{f}(t,x) : = U(t,0)x+\int^t_0 U(t, \tau)f(\tau)d\tau,\quad (t\ge 0).
$$
We also consider evolution families on the line. We  shall use the
same notations as in the case of evolution families on
$\mathbb{R}_+$ except that the
 variables $s$ and $t$ can take any value in $\mathbb{R}$.  For more details about the strongly continuous
semigroups and evolution families we refer to \cite{EN}.
The Datko-Neerven's theorem can be extended for evolution families in the
both cases, on the line and on the half-line, see the papers
(\cite[Theorem 6]{Da},
\cite[Theorem 2.2]{MRS}, \cite{CLMR}, or the monograph \cite{CL}.
It seems that the Neerven-Vu's theorem cannot be extended for periodic
evolution families, but some weaker results, which will be described  as
follows, hold.

We  recall the notion of evolution semigroup. For more details we refer
to \cite{CL,CLMR} and references therein.
 Let $\mathcal{U}=\{U(t, s): t\ge s\in\mathbb{R}\}$
 be a $1$-periodic evolution family, $t\ge 0$, and $G\in AP(\mathbb{R}, X)$.
The function given by
\begin{equation}
s\mapsto (\mathcal{S}(t)G)(s):=U(s, s-t)G(s-t):\mathbb{R}\to X,\label{e4}
\end{equation}
belongs to $AP(\mathbb{R}, X)$ and the one-parameter family $\mathcal{S}=\{\mathcal{S}(t): t\ge 0\}$ is a strongly continuous semigroup on
$AP(\mathbb{R}, X)$, see for example \cite{NM}.  $\mathcal{S}$ is
called  an evolution semigroup on  $AP(\mathbb{R}, X)$.

\section{Results}

\begin{lemma} \label{lm1}
Let $f\in AP_0(\mathbb{R}_+, X)$, $\tau\ge 0$
and $\mathcal{U}=\{U(t, s): t\ge s\in\mathbb{R}\}\subset \mathcal{L}(X)$
be a $1$-periodic evolution family of bounded linear operators on
$X$. Then the function $S(\tau)f$ given by
\begin{equation}
[S(\tau)f](s):=\begin{cases}
U(s, s-\tau)f(s-\tau), & \mbox{if } s\ge\tau\\
0, & \mbox{if }  0\le s<\tau
\end{cases}\label{e5}
\end{equation}
belongs to $AP_0(\mathbb{R}_+, X).$
\end{lemma}

\paragraph{Proof}
First we prove that $S(\tau)g\in\mathcal{A}_0(\mathbb{R}_+, X)$ for any $g$
in $\mathcal{A}_0(\mathbb{R}_+,X)$. Let $t_g$ and $F_g$
 (as in the definition of the set $\mathcal{A}_0(\mathbb{R}_+, X))$.
Let $t_{S(\tau)g}:=\tau+t_g$ and
$F_{S(\tau)g}:=U(\cdot, \cdot-\tau)F_g(\cdot-\tau)$. If $\tau\le
s\le \tau+t_g$ then $g(s-\tau)=0$ and $[S(\tau)g](s)=0$. Moreover,
if $s\ge \tau+t_g$ then $g(s-\tau)=F_g(s-\tau)$ and therefore
$[S(\tau)g](s)=F_{S(\tau)g}(s)$. Thus $S(\tau)g\in\mathcal{A}_0(\mathbb{R}_+, X)$. Let now $\varepsilon >0$ and $g\in\mathcal{A}_0(\mathbb{R}_+, X)$ such that $\|f-g\|_{BUC(\mathbb{R}_+,
X)}<\varepsilon$. Then $S(\tau)g$ belongs to
 $\mathcal{A}_0(\mathbb{R}_+, X)$, and
\begin{align*}
\|S(\tau)f-S(\tau)g\|_{BUC(\mathbb{R}_+, X)}
& =\sup_{s\ge\tau}\|U(s, s-\tau)[f(s-\tau)-g(s-\tau)]\|\\
& \le Me^{\omega\tau}\sup_{s\ge\tau}\|f(s-\tau)-g(s-\tau)\|
\le Me^{\omega\tau}\varepsilon.
\end{align*}
This completes the proof. \hfill$\square$ \smallskip

Now, it is easy to see, that the family $\{S(\tau): \tau\ge 0\}$
is a semigroup of linear and bounded operators on
$AP_0(\mathbb{R}_+, X)$.


\begin{lemma} \label{lm2}
Let $\mathcal{U}$ be an 1-periodic evolution
family of bounded linear operators on $X$. The semigroup
${\bf S}=\{S(t): t\ge 0\}$ associated to $\mathcal{U}$ on
$AP_0(\mathbb{R}_+, X))$, defined in \eqref{e5}   is strongly continuous.
\end{lemma}

\paragraph{Proof} For each $f\in AP_0(\mathbb{R}_+, X)$ and any
$\tau\ge 0$, we have
\begin{align*}
\|S(\tau)f-f\|_{AP_0  (\mathbb{R}_+, X)}
& \le\sup_{s\in [t_f, \tau+t_f]}\|f(s)\|
+\sup_{s\in [t+t_f, \infty)}\|(S(\tau)f)(s)-F_f(s)\|\\
&\le \|\mathcal{S}(\tau)F_f-F_f\|_{AP(\mathbb{R},X)}
+\sup_{s\in [t_f, \tau+t_f]}\|f(s)\|.
\end{align*}
 The last term tends to $0$ when $\tau\to 0$, because the semigroup $\mathcal{S}$
 (which is defined in \eqref{e4}) is strongly continuous,
 the function $f$ is uniformly continuous on $\mathbb{R}_+$, and $f(t_f)=0$.
\hfill$\square$\smallskip

 The semigroup ${\bf S}$
is called an {\it evolution semigroup associated to $\mathcal{U}$ on the space
 $AP_0(\mathbb{R}_+, X).$} The main result of the our paper is the following
 theorem.

\begin{theorem} \label{thm3}
Let $\mathcal{U}=\{U(t, s): t\ge
s\in\mathbb{R}\}$ be an
 1-periodic evolution family of bounded linear operators acting on $X$. The
following two statements are equivalent:
\begin{itemize}
\item[(i)] $\mathcal{U}$ is exponentially stable;
\item[(ii)] $v_f(\cdot, 0)$ belongs to $AP_0(\mathbb{R}_+, X)$ for each
$f\in AP_0(\mathbb{R}_+, X)$.
\end{itemize}
\end{theorem}

The following Lemma is the key tool in our proof of (i)
implies (ii) from Theorem \ref{thm3}.

\begin{lemma} \label{lm4}
Let $\mathcal{U}=\{U(t, s): t\ge s\in \mathbb{R}\}$ be a
 $1$-periodic evolution family of bounded linear operators on $X$,
${\bf S}=\{S(t): t\ge 0\}$
 the evolution semigroup associated to $\mathcal{U}$ on the space
 $AP_0(\mathbb{R}_+, X)$, defined in \eqref{e5},
$(G, D(G))$ the infinitesimal generator of ${\bf S}$ and
 $u$ and $f\in AP_0(\mathbb{R}_+, X)$.
The following two statements are equivalent:
\begin{itemize}
\item[(j)] $u\in D(G)$ and $Gu=-f$;
\item[(jj)] $u(t)=\int_0^t U(t, s)f(s)ds$ for all $t\ge 0$.
\end{itemize}
\end{lemma}

\paragraph{Proof} This Lemma can be shown as in \cite[Lemma 1.1]{MRS}.
For sake of  completness we present the details.
\\
(j) implies (jj). For each $t\ge 0$,
$S(t)u-u=\int_0^tS(\xi)Gud\xi$; therefore,
\begin{align*}
u(t) &=(S(t)u)(t)-(\int_0^tS(\xi)Gu\,d\xi)(t)\\
& = U(t, 0)u(0)-\int_0^tU(t, t-\xi)(Gu)(t-\xi)\,d\xi\\
& = \int_0^t U(t, t-\xi)f(t-\xi)\,d\xi
  = \int_0^t U(t, \tau)f(\tau)\,d\tau.
\end{align*}
(jj) implies (j). Let $t>0$ be fixed. We prove that
\begin{equation}
\frac{1}{t}(-S(t)u+u)=\frac{1}{t}\int_0^t S(r)fdr.\label{e6}
\end{equation}
If $s\ge t$, we have:
\begin{align*}
\frac{1}{t}(-S(t)u+u)(s) & =\frac{1}{t}[-U(s, s-t)u(s-t)+u(s)]\\
& =\frac{1}{t}[\int_0^s U(s, \tau)f(\tau)d\tau-
\int_0^{s-t}U(s, \tau)f(\tau)d\tau]\\
& =\frac{1}{t}\int_0^tU(s, s-r)f(s-r)dr\\
& =\frac{1}{t}(\int_0^tS(r)fdr)(s).
\end{align*}
If $0\le s<t$, we have
\begin{align*}
\frac{1}{t}(-S(t)u+u)(s) &=\frac{1}{t} u(s)
 =\frac{1}{t}\int_0^sU(s, \tau)f(\tau)d\tau\\
& =\frac{1}{t}\int_0^sU(s, s-r)f(s-r)dr\\
& =\frac{1}{t}(\int_0^s S(r)fdr)(s)\\
& =\frac{1}{t}(\int_0^t S(r)fdr)(s).
\end{align*}
Passing to the limit as $t\to 0$ in \eqref{e6} we get the conclusion (j).
\hfill$\square$ \smallskip

Recall that $\sigma(L)$ denotes the {\it spectrum} of the bounded linear operator
$L$ acting on $X$, and
$\rho(L):=\mathbb{C}\setminus\sigma(L)$ is the resolvent set of $L$. The
{\it spectral radius} of $L$ is $r(L):=\sup\{|\lambda|: \lambda\in\sigma(L)\}$
and the {\it spectral bound} is
 $s(L):=\sup\{\mathop{\rm Re}(\lambda): \lambda\in\sigma(L)\}$. For the
proof of the following result see for example
 (\cite{B}, Proof of Theorem 4 and Lemma 3).

\begin{theorem} \label{thm5}
Let $\mathcal{U}=\{U(t, s): t\ge s\}$ be a $1$-periodic
 evolution family on the Banach space $X$, $V:=U(1, 0)$ the monodromy operator
and $\mathcal{S}$ the evolution semigroup associated to $\mathcal{U}$ on
the space $AP(\mathbb{R}, X)$, given in \eqref{e4}. The following four
statements are equivalent:
\begin{itemize}
\item[(i)] $\mathcal{U}$ is  exponentially stable;
\item[(ii)] $r(V)<1$;
\item[(iii)] $\sup_{n\in\mathbb{N}}\|\sum_{k=0}^n e^{i\mu k}V^k\|=
M_{\mu}<\infty$, for all $\mu\in\mathbb{R}$;
\item[(iv)] for each $f\in P_1^0(\mathbb{R}_+, X)$ and each
$\mu\in\mathbb{R}$  the function\\
 $t\mapsto \int_0^t U(t, s)e^{-i\mu s}f(s)ds$ is bounded on
$\mathbb{R}_+$.
\end{itemize}
\end{theorem}

\paragraph{Proof of Theorem \ref{thm3}}
(i) implies (ii). Let ${\bf S}$ denote
 the evolution semigroup associated to $\mathcal{U}$ on the space $AP_0(\mathbb{R}_+, X)$, defined in (9) and $(G, D(G))$ its infinitesimal generator. $\mathcal{U}$ is
exponentially stable, that is, \eqref{e5} holds with some negative $\omega$ for
every pairs $(t, s)$ with $t\ge s$, so $\omega_0({\bf S})$ is negative
 and $0\in \rho(G)$. Then
$G$ is an invertible operator. It follows that for every
 $f\in AP_0(\mathbb{R}_+, X)$
there is $u\in D(G)$ such that $Gu=-f$. Using Lemma \ref{lm4} it results
that $u=v_f(\cdot, 0)$, so $v_f(\cdot, 0)$ belongs to
$AP_0(\mathbb{R}_+, X)$.
\\
(ii) implies (i). Let $\mu\in\mathbb{R}$ and
 $f\in P_1^0(\mathbb{R}_+, X)$. The function $t\mapsto e^{-i\mu t}f(t)$ belongs
to the space $AP_0(\mathbb{R}_+, X)$. Thus the function
 $t\mapsto \int_0^t U(t, s)e^{-i\mu s}f(s)ds$ is bounded on $\mathbb{R}_+$
because it belongs to the space $AP_0(\mathbb{R}_+, X)$, too.
Using Theorem \ref{thm5} ((iv) implies (i)), it follows that
$\mathcal{U}$ is exponentially stable.
\hfill$\square$

\begin{remark} \label{rmk6} \rm
Combining the equivalence between (i) and
(iv) from Theorem \ref{thm5} with the result from Theorem \ref{thm3} it is
easy to see that an evolution family $\mathcal{U}$, as in Theorem \ref{thm3},
is exponentially stable if and only if for each $f\in
AP_0(\mathbb{R}_+, X)$, the solution   $v_f(\cdot, 0)$
 is bounded on $\mathbb{R}_+$.
\end{remark}

\section{Applications}

An immediate consequence of Theorem \ref{thm3} is the spectral mapping
theorem for the evolution semigroup ${\bf S}$ on
$AP_0(\mathbb{R}_+, X)$. Similar results can be found in
(\cite{MRS}, Theorem 2.2) for evolution semigroups on
$C_{00}(\mathbb{R}_+, X)$ and in \cite[Theorem 5]{B1} for
evolution semigroups on $AAP_0(\mathbb{R}_+, X)$. Here
$C_{00}(\mathbb{R}_+, X)$ denotes the space of all $X$-valued
continuous functions on $\mathbb{R}_+$ such that
$f(0)=\lim_{t\to\infty}f(t)=0$ and $AAP_0(\mathbb{R}_+, X)$ is the
space of all $X$-valued functions $h$ on $\mathbb{R}_+$ such that
$h(0)=0$ and there exist $f\in C_0(\mathbb{R}_+, X)$ and $g\in
AP(\mathbb{R}_+, X)$ such that $h=f+g$.


\begin{theorem}\label{thm7}
Let $\mathcal{U}$ be a $1$-periodic evolution family of bounded
 linear operators on  $X$. The  evolution semigroup
 ${\bf S}$ associated to $\mathcal{U}$ on $AP_0(\mathbb{R}_+, X)$ satisfies
 the spectral mapping theorem, as follows
$$e^{t\sigma(G)}=\sigma(S(t))\setminus\{0\}, \quad t\ge 0.$$
Moreover,
$\sigma(G)=\{\lambda\in\mathbb{C}: \mathop{\rm Re}(\lambda)\le s(G)\}$,
and
$$ \sigma(S(t))=\{\lambda\in\mathbb{C}: |\lambda|\le r(S(t))\},\quad
 \mbox{ for all } t>0.
$$
\end{theorem}

 Another application of Theorem \ref{thm3} is the following inequality of
 Landau-Kallman-Rota's type.
For more details about the theorems of this form, see \cite{BD}
and \cite{B1}. Let $\mathcal{Y}$ one of the following spaces:
$C_{00}(\mathbb{R}_+, X)$, $AAP_0(\mathbb{R}_+, X)$, or
$AP_0(\mathbb{R}_+, X)$.

\begin{theorem} \label{thm8}
Let $\mathcal{U}=\{U(t, s): t\ge s\ge 0\}$ be a $1$-periodic
evolution family of bounded linear operators acting on $X$ and let
$f\in\mathcal{Y}$. Suppose that the following conditions are
fulfilled:
\begin{itemize}
\item[(i)] $v_f(\cdot, 0)=\int_0^{\cdot}U(\cdot, s)f(s)ds$ belongs to
$\mathcal{Y}$;
\item[(ii)] $w_f(\cdot):=\int_0^{\cdot}(\cdot-s)U(\cdot, s)f(s)ds$ belongs
to $\mathcal{Y}$.
 \end{itemize}
If\;  $\sup\{\|U(t, s)\|: t\ge s\ge 0\}=M<\infty$ then
$$
\|v_f(\cdot, 0)\|_\mathcal{Y}^2\le 4M^2\|f\|_\mathcal{Y}\cdot\|w_f(\cdot)\|_\mathcal{Y}.
$$
\end{theorem}
For the proof of Theorem \ref{thm8} in the cases $\mathcal{Y}=C_{00}(\mathbb{R}_+, X)$ or $\mathcal{Y}=AAP_0(\mathbb{R}_+, X)$
we refer the reder to (\cite{BD,B1}). The last case
can be obtained in a similar way.


The hypothesis from the Neerven-Vu's theorem can be formulated as follows:
\begin{quote}
There exist a positive constant $K$ such that
$$\sup_{t\ge 0}\|\int_0^tT(\xi)e^{-i\mu\xi}xd\xi\|
\le K\|e^{i\mu\cdot}x\|_{BUC(\mathbb{R}_+, X)},
$$
for all $x\in X$.
\end{quote}
Then the following result is  natural.

\begin{theorem} \label{thm9}
Let $\mathcal{U}=\{U(t, s): t\ge s\in\mathbb{R}\}$ be a 1-periodic
evolution family of bounded linear operators acting on $X$.
The following two statements are equivalent:
\begin{enumerate}
\item $\mathcal{U}$ is exponentially stable;
\item for each $p\in TP_0(\mathbb{R}_+, X)$ the solution $v_p(\cdot, 0)$
belongs to $AP_0(\mathbb{R}_+, X)$ and there exists a positive
constant $K$ such that
\begin{equation}
\|v_p(\cdot, 0)\|_{AP_0(\mathbb{R}_+, X)}\le K\|p\|_{AP_0(\mathbb{R}_+, X)}.
\label{e7}
\end{equation}
\end{enumerate}
\end{theorem}

\paragraph{Proof} The proof of $1\Rightarrow 2$ is obvious.
We will prove that 2 implies 1.
Let $f\in AP_0(\mathbb{R}_+, X)$ and $p_n\in TP_0(\mathbb{R}_+,X)$ be such
that the sequence $(p_n)$ converges to $f$ in
$AP_0(\mathbb{R}_+, X)$. From \eqref{e7} it follows that $(v_{p_n}(\cdot, 0))$
 converges in $AP_0(\mathbb{R}_+, X)$. On the other hand it is easy to see
that $(v_{p_n}(\cdot, 0))$ converges pointwise to $v_f(\cdot, 0)$. Thus $v_f(\cdot, o)$
lies in $AP_0(\mathbb{R}_+, X)$ and the assertion follows from
Theorem \ref{thm3}.


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\noindent\textsc{Constantin Bu\c se}\\
Department of Mathematics,
West University of Timi\c soara\\
Bd. V. P\^arvan No. 4,
1900-Timi\c soara, Rom\^ania\\
e-mail: buse@hilbert.math.uvt.ro \smallskip

\noindent\textsc{Oprea Jitianu}\\
Department of Applied Mathematics, University of Craiova\\
Bd. A. I. Cuza 13,
1100-Craiova, Rom\^ania\\
e-mail: jitianu@ucv.netmasters.ro

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