\documentclass[reqno]{amsart} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2004(2004), No. 38, pp. 1--9.\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 2004 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2004/38\hfil New characterizations of asymptotic stability] {New characterizations of asymptotic stability for evolution families on Banach spaces} \author[S. B\^arz\u a, C. Bu\c se, \& J. Pe\v cari\'c \hfil EJDE-2004/38\hfilneg] {Sorina B\^arz\u a, Constantin Bu\c se, \& Josip Pe\v cari\'c} \address{Sorina B\^arz\u a \hfill\break Department of Mathematics, Karlstad University, Universitetgatan 2, 65188-Karlstad, Sweden} \email{Sorina.Barza@kau.se} \address{Constantin Bu\c se \hfill\break Department of Mathematics, West University of Timi\c soara, Bd. V. P\^arvan 4, 300223-Timi\c soara, Rom\^ania} \email{buse@hilbert.math.uvt.ro} \address{Josip Pe\v cari\'c \hfill\break Faculty of Textile Technology, University of Zagreb, Pierottijeva 6, 10000-Zagreb, Croatia} \email{pecaric@element.hr} \date{} \thanks{Submitted March 10, 2004. Published March 22, 2004.} \subjclass{47A30, 93B35, 35B40, 46A30} \keywords{Evolution family of bounded linear operators, \hfill\break\indent uniform exponential stability, Datko-Rolewicz theorem} \begin{abstract} We generalize the Datko - Rolewicz theorem on exponential stability in the non-autonomous case. Also, we extend the results obtained by Jan van Neerven \cite{[Ne02]}. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{remark}[theorem]{Remark} \section{Introduction} Let $\mathbb{R}_+$ be the set of non-negative real numbers, $\mathbf{T}=\{T(t)\}_{t\ge 0}$ be a strongly continuous semigroup on a Banach space $X$ and $\omega_0(\mathbf{T}):=\inf_{t>0}\frac{\ln\|T(t)\|}{t}$ its uniform exponential growth. It is well known the autonomous version of Datko theorem (\cite{[Da70]}) which says that \begin{quote} If for each $x\in X$ the map $t\mapsto \|T(t)x\|$ belongs to the space $L^2(\mathbb{R}_+)$ then the semigroup $\mathbf{T}$ is exponentially stable, that is $\omega_0(\mathbf{T})$ is strictly negative. \end{quote} This result was generalized by Pazy (\cite{[Pa83]}) who proved that the exponent $p=2$ from the autonomous version of Datko theorem may be replaced by every $1\le p<\infty$. Moreover, from the Pazy proof follows an interesting individual stability result. Namely {\it if a trajectory of the semigroup $\mathbf{T}$, (i.e. a map $t\mapsto T(t)x$ with $x\in X$), belongs to the space $L^p(\mathbb{R}_+)$, then it decay to $0$ at $\infty$.} On the other hand a classical result says that if a real valued function $f$ on $\mathbb{R}_+$ is uniformly continuous and $\int_0^\infty |f(t)|dt<\infty$ then it decay to $0$ at $\infty$, see for example \cite{[B59]}. Then we can say that each trajectory of a strongly continuous semigroup which belongs to the space $L^p(\mathbb{R}_+)$is uniformly continuous on $\mathbb{R}_+$ if and only if it decay to $0$ at $\infty$. In order to introduce the nonautonomous results of this type we recall the notion of solid space over $\mathbb{R}_+$. The set of all $\mathbb{R}$-valued functions $f$ defined on $\mathbb{R}_+$ will be denoted by $\mathcal{F}(\mathbb{R}_+,\mathbb{R})$. Let $\rho :\mathcal{F}(\mathbb{R}_+, \mathbb{R})\to [0,\infty ]$ be a map with the following properties: \begin{itemize} \item[(N1)] $\rho (f)=0$ if and only if $f=0$. \item[(N2)] $\rho (af)=|a|\rho (f)$ for every real scalar $a$ and every $f\in \mathcal{F}(\mathbb{R}_+, \mathbb{R})$ with $\rho (f)<\infty$. \item[(N3)] $\rho (f+g)\le \rho (f)+\rho (g)$ for all $f,g\in \mathcal{F}(\mathbb{R}_+, \mathbb{R})$. \end{itemize} We will denote by $F=F_{\rho}$ the set $\{f\in \mathcal{F}(\mathbb{R}_+, \mathbb{R}) : |f|_F :=\rho (f)<\infty\}$. It is clear that the pair $(F,|\cdot|_F)$ is a linear normed space. Every normed subspace $E$ of $F$ will be called {\it normed function space}. A normed function space is called {\it solid} if for each $f\in\mathcal{F}(\mathbb{R}_+, \mathbb{R})$ and each $g\in E$ for which $|f|\le |g|$ we have that $f\in E$ and $|f|_E\le |g|_E$. For more details about Banach functions spaces we refer to the books \cite{[KPS],[Rol85],[BS88],[Sch96]}. Let $X$ be a Banach space and $\mathcal{L}(X)$ the Banach algebra of all linear and bounded operators acting on $X$. The norm on $X$ and on $\mathcal{L}(X)$ will be denoted by $\|\cdot\|$. Recall that a family $\mathcal{U}=\{U(t, s): t\ge s\ge 0\}$ in $\mathcal{L}(X)$ is called {\it evolution family with exponential growth} if $U(t, t)=Id$, ($Id$ is the identity operator in $\mathcal{L}(X)$), $U(t, s)U(s, r)=U(t, r)$ for all $t\ge s\ge r\ge 0$ and there exist the real constants $\omega$ and $M$ such that \begin{equation} \|U(t, s)\|\le Me^{\omega(t-s)}\quad \mbox{for all } t\ge s\ge 0. \label{e1} \end{equation} We may suppose that $\omega>0$ and $M\ge 1$. The evolution family $\mathcal{U}$ is called {\it uniformly bounded} if we can choose $\omega=0$ in \eqref{e1} and {\it uniformly exponentially stable} if there exist a negative $\omega$ such that \eqref{e1} holds. Let $E$ be a solid space. For the moment we suppose that for every positive $T$ the space $E$ contains the characteristic function of the interval $[0, T]$. We will see that this is not a restriction. We will suppose that the space $E$ satisfies one or more of the following hypotheses: \begin{itemize} \item[(H1)] $\lim_{T\to\infty}|\chi_{[0, T]}|_E=\infty$. %\label{e2} \item[(H2)] For every positive $t$, the function $h\mapsto |\chi_{[h, t+h]}|_E$ is nondecreasing on $\mathbb{R}_+$. \item[(H3)] There exists a positive number $\delta$ such that \begin{equation} K_{\delta}:=\inf_{t\ge 0}|\chi_{[t, t+\delta]}|_E>0\,. \label{e3} \end{equation} \item[(H4)] There exists a positive function $h$, with $h(\infty)=\infty$, such that \begin{equation} 1+|\chi_{[s, t]}|_E\ge h(t-s)\mbox{ for all } t\ge s\ge 0. \label{e4} \end{equation} \end{itemize} It is easily to see that (H1) does not imply (H2), but (H2) implies (H3), and (H4) implies (H1). Moreover (H3) and (H4) do not imply (H2). To see this, let $a$ be a strictly decreasing function on $\mathbb{R}_+$ with $a(\infty)=1$, and $E$ be the solid space consisting by all real-valued and locally measurable functions $f$ (we identify every two functions which are equal almost everywhere) for which $$ |f|_E:=\int_0^\infty a(r)|f(r)|dr<\infty. $$ Then the space $E$ is solid, satisfies (H4) and (H3) (because the infimum from \eqref{e3} is equal to $\delta>0$), but it does not satisfy (H2). Let $\mathcal{U}$ be an evolution family with exponential growth and let $s\ge 0$ and $x\in X$, be fixed. When $0\le t0$ for all $t> 0$. The non-autonomous version of Datko theorem (\cite{[Da73],[DK74],[Rol87]}) says that the evolution family $\mathcal{U}$ is exponentially stable if and only if there exists a real number $1\le p<\infty$ such that for each $s\ge 0$ and each $x\in X$, the map $U_s^x$ belongs to $L^p(\mathbb{R}_+)$ and $\sup_{s\ge 0}|U_s^x|_p<\infty$. The non-autonomous version of Rolewicz theorem (\cite{[Rol86],[Rol87]}) says that if for each $s\ge 0$ and each $x\in X$, $\phi\circ U_s^x$ belongs to $L^1(\mathbb{R}_+)$ and for each $x\in X$ we have that $$ \sup_{s\ge 0}|\varphi\circ U_s^x|_1<\infty $$ then the evolution family $\mathcal{U}$ is exponentially stable. The reverse statement of the Rolewicz theorem is not true. We mention that Datko and Rolewicz used in their proofs the continuity of the map $t\mapsto U(t, s)x: [s, \infty)\to X$ for every $x\in X$. In the papers \cite{[B97],[BD02]} it is shown that the spaces $L^p$ and $L^1$ in the above theorems can be replaced by a solid space satisfying (H1) and (H2). Moreover by an example in \cite{[B97]} it is shown that (H1) and (H2) cannot be removed. However, in this paper we will prove that it is possible to put (H3) and (H4) instead of (H2). If $E$ is {\it rearrangement invariant} solid function space over $\mathbb{R}_+$ (see e. g. \cite{[KPS]} or \cite[page 222]{[Ne96]} for this class of spaces) then the hypotheses (H2) and (H3) are equivalent and these hypotheses are satisfied automatically. Moreover (H1) and (H4) are equivalent in this case. \section{The Datko theorem for weighted spaces} To prove the main results we need the following Lemma whose proof can be found in \cite[Lemma 4]{[B98]}. \begin{lemma} \label{lm2.1} Let $\mathcal{U}$ be an evolution family which has exponential growth. If there exist a function $g:\mathbb{R}_+\to (0, \infty)$ and a $t_0>0$ such that $g(t_0)<1$ and if in addition $$ \|U(t, s)\|\le g(t-s)\quad \mbox{for all } t\ge s\ge 0 $$ then $\mathcal{U}$ is uniformly exponentially stable. \end{lemma} \begin{theorem} \label{thm2.2} Let $\mathcal{U}$ be an evolution family with exponentially growth on a Banach space $X$. If for each $s\ge 0$ and each $x\in X$ the map $t\mapsto (U_ s^x)(t)$ belongs to a solid space $E$ which verifies the hypotheses (H3) and if $$ \sup_{s\ge 0}| U_ s^x |_E:=M(x)<\infty $$ then the evolution family $\mathcal{U}$ is uniformly bounded. If, in addition, the space $E$ satisfies (H4) then the evolution family $\mathcal{U}$ is uniformly exponentially stable. \end{theorem} \begin{proof} Let $s\ge 0, t\ge s+\delta, x\in X$ and $ t-\delta\le\tau0$ and $\lim_{t\to\infty}\int_t^{t+\mu}a(r)dr=\infty$ for some positive $\mu$. If, in addition, for each $x\in X$, $$ \sup_{s\ge 0}\Big[\sup_{t\ge s}\int_t^{t+\mu}a(r)U_s^x(r)dr\Big]<\infty $$ then the evolution family $\mathcal{U}$ is exponentially stable. \end{corollary} \begin{proof} It suffices to apply Theorem \ref{thm2.2} for the solid space $E$ consisting by all real valued, locally measurable functions $f$ defined on $\mathbb{R}_+$ for which $$ \rho(f):=\sup_{t\ge 0}\int_t^{t+\mu} a(r)|f(r)|dr<\infty. $$ \end{proof} With the above notation, let us consider the real-valued map $$ V_s^x(r):=\|U(r+s, s)x\|, \quad r\ge 0. $$ It is interesting to see what happens if we put $V_s^x$ instead of $U_s^x$ in Theorem \ref{thm2.2}. A result in this spirit was shown in \cite{[MSS01]}, where the exponential stability property of the evolution family $\mathcal{U}$ was obtained under the following two assumptions: \begin{enumerate} \item The normed solid space $E$ satisfies (H1). \item There exists a strictly increasing unbounded sequence $(t_n)$ of positive real numbers such that: $$ \sup_{n\in{\bf N}}(t_{n+1}-t_n)<\infty\mbox{ and } \inf_{n\in{\bf N}}| \chi_{[t_n, t_{n+1}]}|_E>0. $$ \end{enumerate} Next, we obtain same conclusion without using the second assumption above. Let $f$ be a $X$-valued function defined on $\mathbb{R}_+$. Then the map $$ t\mapsto \|f(t)\|: \mathbb{R}_+\to \mathbb{R}_+ $$ will be denoted by the symbol $\|f\|$. Let $E(\mathbb{R}_+, X)$ be the linear space of all $X$-valued functions defined on $\mathbb{R}_+$ for which $\|f\|$ lies in the space $E$. We will endow the space $E(\mathbb{R}_+, X)$ with the norm $|f|_{E(\mathbb{R}_+, X)}:=| \|f\| |_E$. \begin{theorem} \label{thm2.4} Let $\mathcal{U}$ be an evolution family with exponential growth and $E$ be a solid space over $\mathbb{R}_+$ which satisfies (H1). If for each $s\ge 0$ and each $x\in X$ the map $V_s^x$ belongs to the space $E$ and $$ \sup_{s\ge 0}|V_s^x|_E= K(x)<\infty $$ then $\mathcal{U}$ is uniformly exponentially stable. \end{theorem} Before proving this theorem, we recall the following known Lemma, see (\cite[Lemma 8.12.3']{[Rol87]}) or \cite{[DK74]} for the case of reversible evolution families. \begin{lemma} \label{lm2.5} Let $\mathcal{U}=\{U(t, s), t\ge s\ge 0\}$ be an evolution family with exponential growth. If $\mathcal{U}$ is not uniformly exponentially stable then for all $T>0$ and all $0q\|x\|\mbox{ for all } T\ge\tau\ge 0. \label{e7} \end{equation} \end{lemma} \begin{lemma} \label{lm2.6} Under the hypotheses of Theorem \ref{thm2.4}, it follows that there is a positive constant $K$ such that \begin{equation} \sup_{s\ge 0}|V_s^x|_E\le K\|x\|\mbox{ for all } x\in X. \label{e8} \end{equation} \end{lemma} \begin{proof} For each $s\ge 0$ let us consider the linear and bounded operator $V_s:X\to E(\mathbb{R}_+, X)$ given by $$ (V_sx)(t):=U(s+t, s)x, \quad t\in\mathbb{R}_+,\; x\in X. $$ Then for each $x\in X$, we have $$ |V_sx|_{E(\mathbb{R}_+, X)}=| \|U(s+\cdot, s)\| |_E=|V_s^x|_E\le K(x). $$ The assertion of Lemma \ref{lm2.6} follows by the Uniform Boundedness Principle applied to the family $\mathcal{V}:=\{V_s: s\ge 0\}$. \end{proof} \begin{proof}[Proof of Theorem \ref{thm2.4}] Suppose that $\mathcal{U}$ is not uniformly exponentially stable. Then from \eqref{e7} and \eqref{e8} follows that $$ K\ge q|\chi_{[0, T]}|_E $$ for all positive real number $T$, which is a contradiction. \end{proof} To the best of our knowledge the result in Theorem \ref{thm2.4} is new and generalizes to the non-autonomous case some recently obtained autonomous or periodic versions in literature; see (\cite[Theorem 4.2]{[Ne95]}) or (\cite[Theorem 4.5]{[BP01]}). Using the method developed by Schnaubelt (\cite{[Sch00]}), see also \cite{[CL99]}, we can prove the following generalization of the $L^1$-version of Datko theorem. \begin{theorem} \label{thm2.7} Let $\mathcal{U}:=\{U(t, s): t\ge s\ge 0\}$ be an evolution family with exponential growth on a Banach space $X$. We suppose that for each $x\in X$ the map $$ (t, s)\mapsto U(t, s)x:\{(t, s): t\ge s\ge 0\} $$ is measurable. Then $\mathcal{U}$ is uniformly exponentially stable if and only if \begin{equation} \sup_{s\ge 0}\int_s^\infty\|U(t, s)x\|dt<\infty \label{e9} \end{equation} for all $x\in X$. \end{theorem} \begin{proof} As in the proof of Lemma \ref{lm2.6}, there exists a positive constant $K$, (independent of $x$ and $s$), such that \begin{equation} \|U_s^x\|_{L^1(\mathbb{R}_+)}\le K\|x\|. \label{e10} \end{equation} Let us consider the evolution semigroup $\mathbf{T}=\{T(t)\}_{t\ge 0}$ associated with $\mathcal{U}$ on $L^1(\mathbb{R}_+, X)$. Recall that for each $t\ge 0$ and each $f\in L^1(\mathbb{R}_+, X)$ the map $T(t)f$ is given by $$ (T(t)f)(s)=\begin{cases} U(s, s-t)f(s-t),& s\ge t\\ 0,& 0\le sFrom the hypothesis on the measurability and using the fact that the evolution family $\mathcal{U}$ has exponential growth it follows that the map $T(t)f$ belongs to $L^1(\mathbb{R}_+, X)$ for all $t\ge 0$ and all $f\in L^1(\mathbb{R}_+, X)$. Moreover it is easy to see that the evolution semigroup $\mathbf{T}$ has exponential growth. Thus for each $f\in L^1(\mathbb{R}_+, X)$, the map $t\mapsto \|T(t)f\|_{L^1(\mathbb{R}_+, X)}$ is measurable, see e.g. (\cite[Remark 4.3]{[Ne95]}). From \eqref{e10} using the Fubini theorem follows \begin{align*} \int_0^{\infty} \|T(t)f\|_{L^1(\mathbb{R}_+, X)}dt &=\int_0^\infty\int_0^\infty\chi_{[t, \infty)}(s)\|U(s, s-t)f(s-t)\|\,ds\,dt\\ &=\int_0^\infty\int_0^s \|U(s, \xi)f(\xi)\|\,d\xi \,ds\\ &=\int_0^\infty\int_0^\infty\chi_{[0, s]}(\xi)\|U(s, \xi)f(\xi)\|\,ds\,d\xi\\ &=\int_0^\infty\int_\xi^\infty\|U(s, \xi)f(\xi)\|\,ds\,d\xi\\ &\le K\|f\|_{L^1(\mathbb{R}_+, X)}. \end{align*} Now we apply the Datko-Pazy theorem for $p=1$ (see the beginning of our paper) and use the well-known fact that if the semigoup $\mathbf{T}$ is exponentially stable then the evolution family $\mathcal{U}$ is uniformly exponentially stable as well, see \cite[Theorem 2.2]{[CLMR]}. \end{proof} \begin{remark} \label{rmk2.8} \rm (1) The result contained in the above theorem may be known. It follows, for example, from (\cite[Corollary 3.2]{[BD02]}), for $\phi(t)=t$, $t\ge 0$. However, the main hypothesis of this Corollary is the boundedness of the function $(s, x)\mapsto \int_s^\infty\phi(\|U(t, s)x\|)dt$ on $\mathbb{R}_+\times \overline{B}(0, 1)$, where $\overline{B}(0, 1)$ is the closed unit ball in $X$ and $\phi$ is a nondecreasing function such that $\phi(t)>0$ for every $t>0$, which seems to be a more strongly require than the similar one from Theorem \ref{thm2.7}. \noindent(2) The result stated in Theorem \ref{thm2.7} holds under the general hypothesis that for each $x\in X$ and some real-valued, strictly increasing (or nondecreasing and positive on $(0, \infty)$) and convex function $\Phi$ on $\mathbb{R}_+$, one has \begin{equation} \sup_{s\ge 0}\int_s^\infty \Phi(\|U(t, s)x\|)dt<\infty. \label{e10b} \end{equation} \end{remark} \begin{proof}[Proof of 2] For every $k=1, 2, 3, \cdots$ let us consider the set $$ X_k=\big\{ x\in X: \sup_{s\ge 0}\int_s^\infty \Phi(\|U(t, s)x\|)\le k. \big\} $$ By the assumption \eqref{e10} follows that $X=\cup_{k\ge 1}X_k$. Using the well-known Fatou Lemma it is easily to see that each $X_k$ is closed. Then there is a natural number $k_0$ such that $X_{k_0}$ has nonempty interior. Let $x_0\in X$ and $\delta>0$ such that $X_{k_0}$ contains the open ball with the centre in $x_0$ and radius $\delta$. We will prove that the open ball which the centre in origin and radius $\frac{\delta}{2}$ is also contained in $X_{k_0}$. Indeed for each positive $s$ and each $x\in X$ with $\|x\|\le \delta$, one has \begin{align*} \int_0^\infty \Phi(\|U(t, s)(\frac{1}{2}x)\|)dt &\le \int_s^\infty\Phi(\frac{\|U(t, s)(x+x_0)\|+\|U(t, s)x_0\|}{2})dt\\ &\le \frac{1}{2}(\int_s^\infty\Phi(\|U(t, s)(x+x_0)\|)dt +\int_s^\infty\Phi(\|U(t, s)x_0\|)dt)\\ &\le k_0. \end{align*} Now we can apply \cite[Corollary 3.2]{[BD02]}. We remark that in this proof only the strong measurability of the maps $t\mapsto U(t, s)$ $( s\ge 0, t\ge s)$ were used. The ``if" part can be obtained in the following way. Upon replacing $\Phi$ be a some multiple of itself we may assume that $\Phi(1)=1$. It is clear that $\Phi(0)=0$. Let $N$ and $\nu$ two positive constants such that $$ \|U(t, s)\|\le Ne^{-\nu(t-s)}\quad \mbox{for all } t\ge s\ge 0. $$ Then for a sufficiently large and positive $h$, (independent of s), we have $$ \int_s^\infty\Phi(\|U(t, s)x\|)\le \int_0^h\Phi(Ne^{-\nu u})du+\int_h^\infty Ne^{-\nu u}du<\infty. $$ Finally we remark that the result holds even if the set of all $x\in X$ for which \eqref{e10} holds is a second category in $X$. \end{proof} Another result of this type can be formulate as follows. \begin{theorem} \label{thm2.9} Let $E$ be a solid Banach function space over $\mathbb{R}_+$ which satisfies (H1) and $\mathcal{U}$ be an evolution family such that for each positive $s$ the map $t\mapsto U(t, s)$ is strongly measurable on $[s, \infty)$. If the norm of $E$ has the Fatou property \cite{[Ne02]} and if the set of all $x\in X$ for which \begin{equation} \sup_{s\ge 0}| \|U(\cdot+s, s)x\| |_E<\infty \label{e11} \end{equation} is of the second category then $\mathcal{U}$ is uniformly exponentially stable. \end{theorem} \begin{proof} As above, (see also \cite{[Ne02]} for the semigroup case), using the triangle inequality in the space $E$ instead of convexity it follows that \eqref{e11} holds for every $x\in X$. Then we apply Theorem \ref{thm2.4} above to complete the proof. \end{proof} The following result shows that the hypothesis on the convexity of $\Phi$ from Remark 2.8 may be removed. However the converse statement of the Theorem \ref{thm2.10} below does not hold without the convexity of $\Phi$, see \cite[Example 8.12.1]{[Rol87]}. \begin{theorem} \label{thm2.10} Let $\phi: \mathbb{R}_+\to \mathbb{R}_+$ be a nondecreasing function such that $\phi(t)>0$ for all $t>0$ and $\mathcal{U}=\{U(t, s)\}_{t\ge s}$ be an evolution family such that for each $s\ge 0$ the map $t\mapsto U(t, s)$ is strongly measurable. If the set of all $x\in X$ for which \begin{equation} M_{\phi}(x):=\sup_{s\ge 0}\int_s^\infty \phi(\|U(t, s)x\|)dt<\infty \label{e12} \end{equation} is of second category in $X$ then $\mathcal{U}$ is uniformly exponentially stable. \end{theorem} \begin{proof} First we prove that the family $\mathcal{U}$ is uniformly bounded. Indeed for each $x\in X$ satisfying \eqref{e12} there exists a real number $C(x)$ such that \begin{equation} \sup_{t\ge s\ge 0}\|U(t, s)x\|\le C(x), \label{e13} \end{equation} see \cite[Lemma1]{[BD01]}. It is clear that \eqref{e13} holds for every $x\in X$, because it holds for each $x$ in a set of second category in $X$. Then we apply the Uniform Boundedness Theorem to obtain the uniform boundedness of $\mathcal{U}$. On the other hand \eqref{e12} can be written as \begin{equation} M_{\phi}(x)=\sup_{s\ge 0}\int_0^\infty \phi(\|U(t+s, s)x\|)dt<\infty. \label{e14} \end{equation} From \cite[Lemma 3.2.1]{[Ne96]} follows that there exists an Orlicz's space $E$ which satisfies (H1) and such that for each $x$ which satisfies \eqref{e14}, the map $t\mapsto \|U(t+s, s)x\|$ belongs to $E$. Using \eqref{e14} we can derive \eqref{e11}. Now we apply Theorem \ref{thm2.9} to complete the proof. \end{proof} We conclude by stating another related result. \begin{proposition} \label{prop2.11} Let $\mathcal{U}=\{U(t, s): t\ge s\ge 0\}$ be an evolution family with exponential growth on a Banach space $X$ and $x\in X$ be fixed. If for each $s\ge 0$, the map $U_s^x$ (or the map $V_s^x$) belongs to a rearrangement invariant solid space $E$ which verifies the hypothesis (H1), then the trajectory $U(s+\cdot, s)x$ of the evolution family $\mathcal{U}$ is asymptotically stable, that is, for each $s\ge 0$, one has: $$ \lim_{t\to\infty} U(s+t, s)x=0.$$ \end{proposition} The proof of this proposition follows the arguments in \cite[Theorem 2.1]{[B96]}, and we omit it. \subsection*{Acknowlegment} The authors would like to thank Professor Yuri Latushkin for his idea in the proof of Lemma \ref{lm2.6}. \begin{thebibliography}{99} \bibitem{[B59]} I. Barb\u alat, Systems d'equations d'oscilations non-linearis, {\it Rev. Roumaine Math. Pures Appl.} iv, {\bf 2}(1959), 267--270. \bibitem{[BS88]} C. Bennett and R. Sharpley, {\it Interpolation of Operators}, Pure Appl. Math., Vol. {\bf 129}, (1988). \bibitem{[B96]} C. Bu\c se, Nonuniform exponential stability and Orlicz functions, {\it Comm. Math. 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