\documentclass[reqno]{amsart} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2003(2003), No. 125, pp. 1--7.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2003 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2003/125\hfil Exponential stability] {Exponential stability of linear and almost periodic systems on Banach spaces} \author[Constantin Bu\c se \& Vasile Lupulescu\hfil EJDE-2003/125\hfilneg] {Constantin Bu\c se \& Vasile Lupulescu} % in alphabetical order \address{Constantin Bu\c se \hfill\break Department of Mathematics, West University of Timi\c soara, Bd. V. P\^arvan, No. 4, Timi\c soara, Rom\^ania} \email{buse@hilbert.math.uvt.ro} \address{Vasile Lupulescu \hfill\break Department of Mathematics, "Constantin Br\^ancu\c si"- University of Tg. Jiu, Bd. Republicii, No. 1, Tg. Jiu, Rom\^ania} \email{vasile@utgjiu.ro} \date{} \thanks{Submitted November 13, 2003. Published December 16, 2003.} \subjclass[2000]{35B10, 35B15, 35B40, 47A10, 47D03} \keywords{Almost periodic functions, uniform exponential stability, \hfill\break\indent evolution semigroups} \begin{abstract} Let $v_f(\cdot, 0)$ the mild solution of the well-posed inhomogeneous Cauchy problem $$\dot v(t)=A(t)v(t)+f(t), \quad v(0)=0\quad t\ge 0$$ on a complex Banach space $X$, where $A(\cdot)$ is an almost periodic (possible unbounded) operator-valued function. We prove that $v_f(\cdot, 0)$ belongs to a suitable subspace of bounded and uniformly continuous functions if and only if for each $x\in X$ the solution of the homogeneous Cauchy problem $$\dot u(t)=A(t)u(t), \quad u(0)=x\quad t\ge 0$$ is uniformly exponentially stable. Our approach is based on the spectral theory of evolution semigroups. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \section{ Introduction} Let $X$ be a complex Banach space and $\mathcal{L}(X)$ the Banach algebra of all bounded linear operators on $X$. The norms on $X$ and $\mathcal{L}(X)$ will be denoted by $\|\cdot\|$. We recall that a family $\mathcal{U}=\{U(t, s)\}_{t\ge s}$ of bounded linear operators acting on $X$, is a {\it strongly continuous and exponentially bounded evolution family} (which we will call simply an evolution family), if $U(t, t)=\mathop{\rm Id}$ (Id is the identity operator on $X)$, $U(t, s)U(s, r)=U(t, r)$ for all $t\ge s\ge r$, for each $x\in X$ the map $(t, s)\mapsto U(t, s)x$ is continuous and there exist $\omega\in \mathbb{R}$ and $M_{\omega}\ge 1$ such that $$\|U(t, s)\|\le M_{\omega}e^{\omega(t-s)}\quad\mbox{for all } t\ge s.\eqno{(1.1)}$$ If $\mathcal{F}(\mathbb{R}, X)$ is a suitable Banach function space, then for each $t\ge 0$ the operator $\mathcal{T}(t)$ defined by $$(\mathcal{T}(t)f)(s)=U(s, s-t)f(s-t),\quad s\in\mathbb{R}\eqno{(1.2)}$$ acts on $\mathcal{F}(\mathbb{R}, X)$ and the family $\{\mathcal{T}(t)\}_{t\ge 0}$ is a strongly continuous semigroup which is called the {\it evolution semigroup} associated with the family $\mathcal{U}$ on the space $\mathcal{F}(\mathbb{R}, X)$. For example, $\mathcal{F}(\mathbb{R}, X)=C_{00}(\mathbb{R}, X)$ the Banach space of all continuous functions that vanish at infinities and $\mathcal{F}(\mathbb{R}, X)=L^p(\mathbb{R}, X)$ with $1\le p<\infty$, the usual Lebesgue-Bochner space, are suitable. Similar results were obtained when $\mathcal{F}(\mathbb{R}, X)$ are certain subspaces of $BUC(\mathbb{R}, X)$ the Banach space of all $X$-valued, bounded and uniformly continuous functions on $\mathbb{R}$, endowed with the sup-norm. Let $\mathbb{R}_+:=[0, \infty)$. The space $BUC(\mathbb{R}_+, X)$ can be defined in a similar way. We will use the following closed subspaces of $BUC(\mathbb{R}, X)$, see \cite{[C],[LZ],[Z85]}: $AP(\mathbb{R}, X)$ is the smallest closed subspace of $BUC(\mathbb{R}, X)$ which contains all functions of the form: $$t\mapsto e^{i\mu t}x:\mathbb{R}\to X, \quad\mu\in\mathbb{R}, \quad x\in X;$$ $C_0^+(\mathbb{R}, X)$ is the subspace of $BUC(\mathbb{R}, X)$ consisting by all functions vanishing at $\infty$; $AAP_r^+(\mathbb{R}, X)$ is the space consisting by all functions $f$ with relatively compact range for which there exist $g\in AP(\mathbb{R}, X)$ and $h\in C_0^+(\mathbb{R}, X)$ such that $f=g+h$. $P_q(\mathbb{R}, X)$, with strictly positive fixed $q$, is the space consisting by all continuous and $q$-periodic functions. The evolution family $\mathcal{U}$ is called $q$-periodic if the function $U(t+\cdot, s+\cdot)$ is $q$-periodic for every pair $(t, s)$ with $t\ge s$. Also we say that the family $\mathcal{U}$ is {\it asymptotically almost periodic with relatively compact range} (a.a.p.r.) if for each $x\in X$ and each pair $(t, s)$ with $t\ge s$, the map $U(t+\cdot, s+\cdot)x$ lies in the space $AAP_r^+(\mathbb{R}, X)$. If the evolution family $\mathcal{U}$ is $q$-periodic and $\mathcal{F}(\mathbb{R}, X)=P_q(\mathbb{R}, X)$ or $\mathcal{F}(\mathbb{R}, X) =AP(\mathbb{R}, X)$ then the semigroup $\mathcal{T}=\{\mathcal{T}(t)\}_{t\ge 0}$ defined in (1.2) acts on $P_q(\mathbb{R}, X)$ or $AP(\mathbb{R}, X)$ and it is strongly continuous. Moreover, if $\mathcal{U}$ is a.a.p.r. and for each $x\in X$, $\lim_{t\to 0+} U(s, s-t)x=x$, uniformly for $s\in \mathbb{R}$, then the evolution semigroup $\mathcal{T}$ is defined on $AAP_r^+(\mathbb{R}, X)$ and is strongly continuous. More details related to these results can be found in \cite{[ABHN],[BC],[CL],[D],[HR],[LM95],[MRS],[NM99]}. Interesting results on this subject in the general framework of dynamical systems have been obtained by D. N. Cheban \cite{[C99],[C01]}. \section{Almost periodic evolution families and evolution semigroups} An $X$-valued function $f$ defined on $\mathbb{R}$ is called almost periodic (a.p.) if it belongs to the space $AP(\mathbb{R}, X)$. Let $\mathcal{U}$ be a strongly continuous and exponentially bounded evolution family on the Banach space $X$ and let $f$ be a $X$-valued function on $\mathbb{R}$. We will consider the following hypotheses about $\mathcal{U}$ and $f$. \begin{itemize} \item[(H1)] The function $U(\cdot, \cdot-t)x$ is a.p. for every $t\ge 0$ and any $x\in X$. \item[(H2)] The function $U(\cdot, \cdot-t)x$ has relatively compact range for every $t\ge 0$ and any $x\in X$. \item[(H3)] For each $x\in X$ $\lim_{t\to 0}U(s, s-t)x=x$ uniformly for $s\in\mathbb{R}$. \item[(H4)] The function $f$ is a.p. \end{itemize} It is well-known that (H1) implies (H2). \begin{theorem} \label{thm2.1} \begin{itemize} \item[(i)] If the evolution family $\mathcal{U}$ satisfies (H1) and $f$ satisfies (H4) then for each $t\ge 0$, the function $\mathcal{T}(t)f$ is a.p. \item[(ii)] If $\mathcal{U}$ satisfies (H2) and $f$ satisfies (H4) then for each $t\ge 0$, the map $\mathcal{T}(t)f$ has relatively compact range. \item[(iii)] If $\mathcal{U}$ satisfies (H1) and (H3) then the semigroup $\mathcal{T}$ acts on $AP(\mathbb{R}, X)$ and is strongly continuous. \item[(iv)] If $\mathcal{U}$ satisfies (H1) and (H3) then the evolution semigroup $\mathcal{T}$ is defined on $AAP_r^+(\mathbb{R}, X)$ and is strongly continuous. \end{itemize} \end{theorem} \begin{proof} (i) Let $p_n(t):=\sum_{k=0}^nc_k e^{i\mu_kt}x_k$ with $c_k\in\mathbb{C}$, $\mu_k\in\mathbb{R}$, $t\in\mathbb{R}$ and $x_k\in X$ such that $p_n(s)$ converges uniformly at $f(s)$ for $s\in\mathbb{R}$. Then $U(s, s-t)p_n(s-t)$ converges uniformly at $U(s, s-t)f(s-t)$ for $s\in\mathbb{R}$. Since the map: $$s\mapsto U(s, s-t)p_n(s-t)=\sum_{k=0}^nc_ke^{i\mu_k(s-t)}U(s, s-t)x_k$$ is a. p. its limit $U(\cdot, \cdot-t)f(\cdot-t)$ is a.p. as well. \noindent (ii) Let $t\ge 0$ be fixed. First we prove that for each $x\in X$ and each $\mu\in\mathbb{R}$ the function $s\mapsto U(s, s-t)e^{i\mu(s-t)}x$ has relatively compact range. Let $(s_n)$ be a sequence of real numbers such that $(U(s_n, s_n-t)x)$ converges in $X$. Since the sequence $(e^{i\mu(s_n-t)})$, is bounded in $\mathbb{C}$, we can suppose that the sequence $(e^{i\mu(s_n-t)}U(s_n, s_n-t)x))$ converges in $X$. Let $p_N(s-t)=\sum_{k=0}^Nc_ke^{i\mu_k(s-t)}x_k$, as above, be such that $p_N(s-t)\to f(s-t)$ uniformly for $s\in\mathbb{R}$. Let $\varepsilon>0$ and $N_0\in\mathbb{N}$ be such that the inequality $$Me^{\omega t}\|f(s_n-t)-p_{N_0}(s_n-t)\|<\frac{\varepsilon}{2}$$ holds for $n$ sufficiently large. We denote by $y_t$ the limit in $X$ of the sequence $(U(s_n, s_n-t)p_{N_0}(s_n-t))$. Then, for $n$ sufficiently large, we have \begin{align*} &\|U(s_n, s_n-t)f(s_n-t)-y_t\|\\ &\le \|U(s_n, s_n-t)f(s_n-t)-U(s_n, s_n-t)p_{N_0}(s_n-t)\|\\ &\quad +\|U(s_n, s_n-t)p_{N_0}(s_n-t)\|\\ &\le Me^{\omega t}\|f(s_n-t)-p_{N_0}(s_n-t)\|+\|U(s_n, s_n-t)p_{N_0}(s_n-t)-y_t\| <\varepsilon. \end{align*} Hence the map $U(\cdot, \cdot-t)f(\cdot-t)$ has relatively compact range. \noindent (iii) Let $f\in AP(\mathbb{R}, X)$ and $\varepsilon>0$. We can choose $N_0\in\mathbb{N}$ and $\delta>0$ such that the following three inequalities \begin{gather*} \sup_{s\in\mathbb{R}}\|U(s, s-t) p_{N_0}(s-t)-p_{N_0}(s-t)\| \le \sum_{k=0}^{N_0}|c_k\||U(s, s-t)x_k-x_k\|<\frac{\varepsilon}{3},\\ \sup_{s\in\mathbb{R}}\|p_{N_0}(s-t)-f(s-t)\|<\frac{\varepsilon}{3},\\ \sup_{s\in\mathbb{R}}\|f(s-t)-f(s)\|<\frac{\varepsilon}{3} \end{gather*} hold for all $0\le t<\delta$. Now it is clear that $\lim_{t\to 0}\|\mathcal{T}(t)f-f\|_{\infty}=0$, hence the semigroup $\mathcal{T}$ is strongly continuous. \noindent (iv) Finally we show that the semigroup $\mathcal{T}$ given in (1.2) on $AAP_r^+(\mathbb{R}, X)$ is strongly continuous. Let $\varepsilon>0$ be fixed. We can choose $\delta_1>0$ such that the inequality $$\sup_{s\in \mathbb{R}}\|f(s-t)-f(s)\|<\frac{\varepsilon}{2}$$ holds for $0\le t<\delta_1$. Since $f$ has relatively compact range there exist $s_1, s_2, \dots, s_{\nu}$ in $\mathbb{R}$ such that: $$\overline{\mathop{\rm range }(f)}\subset \cup_{k=1}^\nu B\big(f(s_k), \frac{\varepsilon}{6Me^{\omega t}}\big), \quad \omega>0,\; t\ge 0.$$ Let $s\in\mathbb{R}, t\ge 0$ and $k\in\{1, \dots, \nu\}$ such that $f(s-t)\in B\left(f(s_k), \frac{\varepsilon}{6Me^{\omega t}}\right)$. From hypothesis it follows that there exists $\delta_2>0$ such that the inequality $$\|U(s, s-t)f(s_k)-f(s_k)\|< \varepsilon/6$$ holds for $0\le t<\delta_2$. Let $\delta=\min\{\delta_1, \delta_2\}$. Then for every $t$ in $[0, \delta)$, we have \begin{align*} &\|U(s, s-t)f(s-t)-f(s)\|\\ &\le \|U(s, s-t)f(s-t)-U(s, s-t)f(s_k)\|+\|U(s, s-t)f(s_k)-f(s_k)\|\\ &\quad +\|f(s_k)-f(s-t)\|+\|f(s-t)-f(s)\|\\ &\le Me^{\omega t}\|f(s-t)-f(s_k)\|+\frac{\varepsilon}{6} +\frac{\varepsilon}{6}+\frac{\varepsilon}{2}<\varepsilon; \end{align*} therefore, $\lim_{t\to 0}\|\mathcal{T}(t)f-f\|_{\infty}=0$. In the above considerations we supposed that $\mathcal{T}$ acts on $AAP_r^+(\mathbb{R}, X)$. Next, we show that this is true. Let $f\in AAP_r^+(\mathbb{R}, X)$ and $t\ge 0$ be fixed. From the hypothesis it results that there exist a sequence $(s_n)$ of real numbers and $y_t, z_t$ in $X$ such that $$f(s_n-t)\to y_t\quad\mbox{and}\quad U(s_n, s_n-t)y_t\to z_t\quad\mbox{as} n\to\infty.$$ Then $U(s_n, s_n-t)f(s_n-t)\to z_t$ as $n\to\infty$. Indeed, we have $$\|U(s_n, s_n-t)f(s_n-t)-z_t\|\le \|U(s_n, s_n-t)[f(s_n-t)-y_t]\|+ \|U(s_n, s_n-t)y_t-z_t\|\to 0$$ as $n\to\infty$. \end{proof} \section{Evolution semigroups and exponential stability} Let $\mathcal{F}_q(\mathbb{R}, X):=P_q(\mathbb{R}, X)\oplus C_{0}^+(\mathbb{R}, X)$ and $\mathcal{U}$ be a $q$-periodic evolution family of bounded linear operators on the Banach space $X$. It is easy to see that the evolution semigroup $\mathcal{T}$ defined in (1.2) acts on $\mathcal{F}_q(\mathbb{R}, X)$ and it is strongly continuous. By $\mathcal{F}_q^0(\mathbb{R}_+, X)$ we will denote the subspace of $BUC(\mathbb{R}_+, X)$ consisting of all functions $f$ on $\mathbb{R}_+$ for which $f(0)=0$ and there exists $F_f$ in $\mathcal{F}_q(\mathbb{R}, X)$ such that $F_f(t)=f(t)$ for all $t\ge 0$. For such $f$ we consider the map: $$(\mathcal{S}(t)f)(s):=\begin{cases} U(s, s-t)f(s-t)&\mbox{if } s\ge t\\ 0&\mbox{if } 0\le s0 then \tilde G_f(0)+\tilde H_f(0)=0, and if t=0 then$$ \tilde G_f(0)+\tilde H_f(0)=(\mathcal{T}(0)G_f)(0)+(\mathcal{T}(0)H_f)(0) = U(0, 0)G_f(0)+U(0, 0)H_f(0)=0. On the other hand it is clear that \tilde f=\tilde G_f+\tilde H_f on \mathbb{R}_+, hence \tilde f belongs to \mathcal{F}_q^0(\mathbb{R}_+, X). Using the strong continuity of \mathcal{T} and the uniform continuity of f, it follows that \begin{align*} \|\mathcal{S}(t)f-f\|_{\infty} &\le \sup_{s\ge t}\|(\mathcal{T}(t)F_f)(s)-F_f(s)\|+\sup_{s\in [0, t]}\|f(s)\|\\ &\le \|\mathcal{T}(t)F_f-F_f\|_{\mathcal{F}_q(\mathbb{R}, X)}+\sup_{s\in [0, t]} \|f(s)\|. \end{align*} The last term tends to 0 when t tends to 0. Therefore, the semigroup \mathcal{S} is strongly continuous. \end{proof} The following theorem seems to be a new characterization of the exponential stability for evolution families. \begin{theorem} \label{thm3.2} Let \mathcal{U} be a q-periodic evolution family of bounded linear operators on the Banach space X. The following two statements are equivalent. \begin{enumerate} \item The family \mathcal{U} is exponentially stable, that is, we can choose a negative \omega such that (1.1) holds. \item For each f in \mathcal{F}_q^0(\mathbb{R}_+, X) the map t\mapsto \int_0^t U(t, \tau)f(\tau)d\tau: \mathbb{R}_+\to X is an element of \mathcal{F}_q^0(\mathbb{R}_+, X). \end{enumerate} \end{theorem} \begin{proof} (2)\Rightarrow (1)\quad It is clear that \mathcal{F}_q^0(\mathbb{R}_+, X) contains C_{00}(\mathbb{R}_+, X). Then we can apply \cite[Theorem 3]{[B98]} which works with C_{00}(\mathbb{R}_+, X) instead of C_0(\mathbb{R}_+, X). Here C_{00}(\mathbb{R}_+, X) denotes the subspace of BUC(\mathbb{R}_+, X) consisting by all functions that vanish at 0 and \infty. (1)\Rightarrow (2)\quad \mathcal{U} is exponentially stable so the semigroup \mathcal{S} defined in (3.1) is exponentially stable as well. Then the generator G:D(G)\subset\mathcal{F}_q^0(\mathbb{R}_+, X)\to \mathcal{F}_q^0(\mathbb{R}_+, X) $$of \mathcal{S} is an invertible operator. The proof of Theorem \ref{thm3.2} will be complete using the following lemma. \end{proof} \begin{lemma} \label{lm3.3} Let \{u, f\} belong to \mathcal{F}_q^0(\mathbb{R}_+, X). The following statements are equivalent. \begin{enumerate} \item u\in D(G) and Gu=-f. \item u(t)=\int_0^tU(t, s)f(s)ds for all t\ge 0. \end{enumerate} \end{lemma} This Lemma is well-known for certain spaces instead of \mathcal{F}_q^0(\mathbb{R}_+, X). \smallskip Let \mathcal{A}_0(\mathbb{R}_+, X) be the set of all X-valued functions f on \mathbb{R}_+ for which there exist t_f\ge 0 and F_f\in AP(\mathbb{R}, X) such that F_f(t_f)=0 and$$ f(t)=\begin{cases} 0& \mbox{if } t\in [0, t_f]\\ F_f(t)&\mbox{if } t>t_f. \end{cases} $$The smallest closed subspaces of BUC(\mathbb{R}_+, X) which contains \mathcal{A}_0(\mathbb{R}_+, X) will be denoted by \mathcal{AP}_0(\mathbb{R}_+, X). By AAP_{r0}^+(\mathbb{R}_+, X) we will denote the space consisting by all functions f for which there exists F_f\in AAP_r^+(\mathbb{R}, X) such that F_f(0)=0 and F_f=f on \mathbb{R}_+. \begin{proposition} \label{prop3.3} \begin{enumerate} \item If the evolution family \mathcal{U} satisfies the hypothesis (H1) and (H3) then the semigroup \mathcal{S}, given in (3.1) acts on \mathcal{AP}_0(\mathbb{R}, X). Moreover the semigroup \mathcal{S} is strongly continuous. \item If the family \mathcal{U} satisfies {\bf h_1, h_2} and (H3) then the semigroup \mathcal{S} acts on AAP_{r0}^+(\mathbb{R}, X) and is strongly continuous. \end{enumerate} \end{proposition} The proof of (1) can be obtained as in \cite[Lemma 2.2]{[BJ03]}, and the proof on (2) as in \cite[Lemma 2.2]{[B02]}. Thus we omit their proof. For every real fixed T we consider the spaces BUC([T, \infty), X) and AP([T, \infty), X) Recall that AP([T, \infty)) is bounded locally dense in BUC([T, \infty), X); that is, for every \varepsilon>0, every bounded and closed interval I\subset [T, \infty) and every f\in C(I, X) there exist a function f_{\varepsilon, I}\in AP([T, \infty), X) and a positive constant L, independent of \varepsilon and I such that$$ \sup_{s\in I}\|f(s)-f_{\varepsilon, I}(s)\|\le \varepsilon $$and \|f_{\varepsilon, I}\|_{BUC([T, \infty), X)}\le L\|f\|_{C(I, X)} (see \cite {[Ne]}, page 335). Let BUC_0(\mathbb{R}_+, X) be the space of functions in BUC(\mathbb{R}_+, X) for which f(0)=0. It is clear that \mathcal{A}_0(\mathbb{R}_+, X) is bounded locally dense in BUC_0(\mathbb{R}_+, X) hence \mathcal{AP}_0(\mathbb{R}_+, X) is bounded locally dense in BUC_0(\mathbb{R}_+, X) as well. \begin{theorem} \label{thm3.4} Suppose that \mathcal{U} is an evolution family that satisfies hypotheses (H1) and (H3). The following statements are equivalent. \begin{enumerate} \item The family \mathcal{U} is exponentially stable. \item For each f\in\mathcal{AP}_0(\mathbb{R}_+, X) the map t\mapsto \int_0^t U(t, s)f(s)ds:\mathbb{R}_+\to X is in \mathcal{AP}_0(\mathbb{R}_+, X). \end{enumerate} \end{theorem} \begin{proof} The implication (1)\Rightarrow (2) follows as in \cite[Theorem 2.3]{[BJ03]}. Now we shoe that (2)\Rightarrow (1). By the uniform boundedness theorem there is a constant K>0 such that for every g\in\mathcal{AP}_0(\mathbb{R}_+, X),$$ \sup_{t>0}\Big\|\int_0^tU(t, s)g(s)ds\Big\|\le K\|g\|_{\infty}\,. For a given f\in C_0(\mathbb{R}_+, X) and t>0, let M_t=\sup_{0\le r\le s\le t}\|U(s, r)\| and let f_t\in\mathcal{AP}_0(\mathbb{R}_+, X) be a mapping such that \begin{gather*} \sup_{0\le s\le t}\|f(s)-f_t(s)\|\le\frac{1}{tM_t}\|f\|_{C_0(\mathbb{R}_+, X)},\\ \|f_t\|_{BUC_0(\mathbb{R}_+, X)}\le L\|f\|_{C_0(\mathbb{R}_+, X)}. \end{gather*} It follows that \begin{align*} \Big\|\int_0^tU(t, s)f(s)ds\Big\| &\le \Big\|\int_0^tU(t, s)[f(s)-f_t(s)]ds\Big\| +\Big\|\int_0^tU(t, s)f_t(s)ds\Big\|\\ &\le (1+KL)\cdot \|f\|_{C_0(\mathbb{R}_+, X)}\,. \end{align*} Then by \cite[Theorem 3]{[B98]}, \mathcal{U} is exponentially stable. \end{proof} Now we can write the spectral mapping theorem for the evolution semigroup \mathcal{S} on \mathcal{AP}_0(\mathbb{R}_+, X) corresponding to an evolution family \mathcal{U}. Of course similar results hold for the spaces \mathcal{F}_q^0(\mathbb{R}_+, X) and AAP_{r0}^+(\mathbb{R}_+, X). With (G, D(G)) we will denote the generator of \mathcal{S} with its maximal domain. By \sigma(G) we denote the spectrum of G. The spectral bound s(G) is defined by s(G)=\sup\{\mathop{\rm Re}(\lambda): \lambda\in\sigma(G)\}, $$and the spectral radius of \mathcal{S}(t) is defined by$$ r(\mathcal{S}(t))=\sup\{ |\lambda|: \lambda\in\sigma(\mathcal{S}(t))\}. $$\begin{theorem} \label{thm3.5} If \mathcal{U} is an evolution family that satisfies the hypothesis (H1) and (H3) then the evolution semigroup \mathcal{S} associated with \mathcal{U}, defined on \mathcal{AP}_0(\mathbb{R}_+, X), satisfies the spectral mapping theorem; that is,$$ \sigma(\mathcal{S}(t))\setminus\{0\}=e^{t\sigma(G)}, \quad t\ge 0. $$Moreover, \sigma(G)=\{\lambda\in\mathbb{C}: \mathop{\rm Re}(\lambda)\le s(G)\}, and for every t>0,$$ \sigma(\mathcal{S}(t))=\{\lambda\in\mathbb{C}: |\lambda|\le r(\mathcal{S}(t))\,.  \end{theorem} \subsection*{Acknowledgements} The authors would like to thank the anonymous referees for their comments and suggestions on a preliminary version of this article. \begin{thebibliography}{00} \bibitem{[ABHN]} W. Arendt, C. J. K. Batty, M. Hieber, and F. Neubrander, {\it Vector-valued Laplace Transforms and Cauchy Problems}, Monographs in Mathematics, vol. {\bf 96}, Birckha\"user, Basel, 2001. \bibitem{[BC]} C. J. K. Batty, R. Chill, {\it Bounded convolutions and solutions of inhomogeneous Cauchy problems}, Forum. Math. {\bf 11}, No.2, 253-277, (1999). \bibitem{[B98]} C. Bu\c se, {\it On the Perron-Bellman theorem for evolutionary processes with exponential growth in Banach spaces}, New Zealand Journal of Mathematics, {\bf 27}, (1998), 183-190. \bibitem{[BJ03]} C. Bu\c se, O. Jitianu, {\it A new theorem on exponential stability of periodic evolution families on Banach spaces,} Electron. J. Diff. Eqns., Vol. {\bf 2003}, (2003), No. 14, pp.1-10. \bibitem{[B02]} C. Bu\c se, {\it A spectral mapping theorem for evolution semigroups on assymptotically almost periodic functions defined on the half line}, Electron. J. Diff. Eqns., Vol. {\bf 2002} (2002), No. 70, pp. 1-11. \bibitem{[C99]} D. N. Cheban, {\it Relationship between different types of stability for linear almost periodic systems in Banach spaces}, Electron. J. Diff. Eqns., Vol. {\bf 1999} (1999), No. 46, pp.1-9. \bibitem{[C01]} D. N. Cheban, {\it Uniform exponential stability of linear periodic systems in a Banach spaces}, Electron. J. Diff. Eqns., Vol. {\bf 2001} (2001), No. 03, pp. 1-12. \bibitem{[CLMR]} S. Clark, Y. Latushkin, S. Montgomery-Smith, and T. Randolph, {\it Stability radius and internal versus, external stability: An evolution semigroup approach,} SIAM J. Control and Optimization, {\bf 38}, (2000),1757-1793. \bibitem{[C]} C. Corduneanu, {\it Almost-Periodic Functions}, Interscience Wiley, New-York-London, Sydney-Toronto, 1968. \bibitem{[CL]} C. Chicone, Y. Latushkin, {\it Evolution Semigroups in Dynamical Systems and Differential Equations}, Amer. Math. Soc., 1999. \bibitem{[D]} R. Datko, {\it Uniform asymptotic stability of evolutionary processes in a Banach space}, SIAM J. Math. Anal. {\bf 3} (1972), 428-445. \bibitem{[HR]} W. Hutter, F. R\"abiger, {\it Spectral mapping theorems for evolution semigroups on spaces of almost periodic functions}, Quaest. Math. {\bf 26}, No.2, 191-211(2003). \bibitem{[LM95]} Y. Latushkin, S. Montgomery-Smith, {\it Evolutionary semigroups and Lyapunov theorems in Banach spaces}, J. Funct. Anal. {\bf 127}(1995), 173-197. \bibitem{[LZ]} B. M. Levitan, V. V. Zhicov, {\it Almost Periodic Functions and Differential Equations}, Cambridge University Press, 1982. \bibitem{[MRS]} Nguyen Van Minh, F. R\"abiger and R. Schnaubelt, {\it Exponential stability, exponential expansiveness, and exponential dichotomy of evolution equations on the half-line}, Integral Equations Operator Theory, {\bf 32}, (1998), 332-353. \bibitem{[NM99]} S. Naito, Nguyen Van Minh, {\it Evolution semigroups and spectral criteria for almost periodic solutions of periodic evolution equations}, J. Diff. Equations {\bf 152} (1999), 338-376. \bibitem{[Ne]} J. van Neerven, {\it Characterization of Exponential Stability of a Semigroups of Operators in Terms of its Action by Convolution on Vector-Valued Function Spaces over $\mathbb{R}_+$,} J. Diff. Eq. {\bf 124}, No. 2 (1996), 324-342. \bibitem{[Z85]} S. D. Zaidman, {\it Almost-Periodic Functions in Abstract Spaces}, Research Notes in Math. {\bf 126}, Pitman, Boston-London-Melbourne, 1985. \end{thebibliography} \end{document}