\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. 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