\documentclass[twoside]{article} \usepackage{amsmath, amssymb} % font used for R in Real numbers \pagestyle{myheadings} \markboth{\hfil Rolewicz's Theorem \hfil EJDE--2001/45} {EJDE--2001/45\hfil C. Bu\c{s}e \& S. S. Dragomir \hfil} \begin{document} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}, Vol. {\bf 2001}(2001), No. 45, pp. 1--5. \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 proof for a Rolewicz's type theorem:\\ An evolution semigroup approach % \thanks{ {\em Mathematics Subject Classifications:} 47A30, 93D05, 35B35, 35B40, 46A30. \hfil\break\indent {\em Key words:} Evolution family of bounded linear operators, evolution operator semigroup, \hfil\break\indent Rolewicz's theorem. \hfil\break\indent \copyright 2001 Southwest Texas State University. \hfil\break\indent Submitted May 14, 2001. Published June 20, 2001.} } \date{} % \author{ C. Bu\c{s}e \& S. S. Dragomir } \maketitle \begin{abstract} Let $\varphi$ be a positive and non-decreasing function defined on the real half-line and $\mathcal{U}$ be a strongly continuous and exponentially bounded evolution family of bounded linear operators acting on a Banach space. We prove that if $\varphi$ and $\mathcal{U}$ satisfy a certain integral condition (see the relation (\ref{1.2}) below) then $\mathcal{U}$ is uniformly exponentially stable. For $\varphi$ continuous, this result is due to S. Rolewicz. \end{abstract} \newtheorem{theorem}{Theorem} \newtheorem{lemma}{Lemma} \section{Introduction} Let $X$ be a real or complex Banach space and $L\left( X\right)$ the Banach algebra of all linear and bounded operators on $X$. Let $\mathbf{T}=\left\{ T\left( t\right) :t\geq 0\right\} \subset L\left( X\right)$ be a strongly continuous semigroup on $X$ and $\omega _{0}\left( \mathbf{T}\right) =\lim_{t\rightarrow \infty }\frac{\ln \left( \left\| T\left( t\right) \right\| \right) }{t}$ be its growth bound. The Datko-Pazy theorem (\cite{Da,Pa}) states that $\omega _{0}\left( \mathbf{T}\right) <0$ if and only if for all $x\in X$ the maps $t\longmapsto \left\| T\left( t\right) x\right\|$ belongs to $L^{p}\left( \mathbb{R}_{+}\right)$ for some $1\leq p<\infty$. A family $\mathcal{U}=\left\{ U\left( t,s\right) :t\geq s\geq 0\right\} \subset L\left( X\right)$ is called an \textit{evolution family }of bounded linear operators on $X$ if $U\left( t,t\right) =\mathbf{I}$ (the identity operator on $X$) and $U\left( t,\tau \right) U\left( \tau ,s\right) =U\left( t,s\right)$ for all $t\geq \tau \geq s\geq 0$. Such a family is said to be \textit{strongly continuous} if for every $x\in X$, the maps \begin{equation*} \left( t,s\right) \mapsto U\left( t,s\right) x:\left\{ \left( t,s\right) :t\geq s\geq 0\right\} \rightarrow X \end{equation*} are continuous, and \textit{exponentially bounded }if there are $\omega >0$ and $K_{\omega }>0$ such that $$\left\| U\left( t,s\right) \right\| \leq K_{\omega }e^{\omega \left( t-s\right) }\text{ \ for all }t\geq s\geq 0. \label{1.1}$$ The family $\mathcal{U}$ is called \textit{uniformly exponentially stable} if (\ref{1.1}) holds for some negative $\omega$. If $\mathbf{T}=\left\{ T\left( t\right) :t\geq 0\right\} \subset L\left( X\right)$ is a strongly continuous semigroup on $X$, then the family $\left\{ U\left( t,s\right) :t\geq s\geq 0\right\}$ given by $U\left( t,s\right) =T\left( t-s\right)$ is a strongly continuous and exponentially bounded evolution family on $X$. Conversely, if $\mathcal{U}$ is a strongly continuous evolution family on $X$ and $U\left( t,s\right) =U\left( t-s,0\right)$ then the family $\mathbf{T}% =\left\{ T\left( t\right) :t\geq 0\right\}$ given by $T\left( t\right) =U\left( t,0\right)$ is a strongly continuous semigroup on $X$. The Datko-Pazy theorem can be obtained from the following result given by S. Rolewicz (\cite{R1}, \cite{R2}). \textit{Let }$\varphi :\mathbb{R}_{+}\rightarrow \mathbb{R}_{+}$ \textit{be a continuous and nondecreasing function such that} $\varphi \left( 0\right) =0$ \textit{and} $\varphi \left( t\right) >0$ \textit{for all} $t>0$. \textit{If} $\mathcal{U=}\left\{ U\left( t,s\right) :t\geq s\geq 0\right\} \subset L\left( X\right)$ \textit{is a strongly continuous and exponentially bounded evolution family on the Banach space} $X$ \textit{such that} $$\sup_{s\geq 0}\int_{s}^{\infty }\varphi \left( \left\| U\left( t,s\right) x\right\| \right) dt=M_{\varphi }<\infty \text{,\ \ for all }x\in X,\;\left\| x\right\| \leq 1, \label{1.2}$$ \textit{then }$\mathcal{U}$ \textit{is uniformly exponentially stable.} A shorter proof of the Rolewicz theorem was given by Q. Zheng \cite{Zh} who removed the continuity assumption about $\varphi$. Other proofs of (the semigroup case) Rolewicz's theorem were offered by W. Littman \cite{Li} and J. van Neervan \cite[pp. 81-82]{Ne}. Some related results have been obtained by K.M. Przy\l uski \cite{P}, G. Weiss \cite{W} and J. Zabczyk \cite{Z}. In this note we prove the following: \begin{theorem} \label{t1}Let $\varphi :\mathbb{R}_{+}\rightarrow \mathbb{R}_{+}$ be a nondecreasing function such that $\varphi \left( t\right) >0$ for all $t>0$. If $\mathcal{U=}\left\{ U\left( t,s\right) :t\geq s\geq 0\right\} \subset L\left( X\right)$ \textit{is a strongly continuous and exponentially bounded evolution family of operators on }$X$ such that (\ref{1.2}) holds, then $\mathcal{U}$ is uniformly exponentially stable. \end{theorem} Our proof of Theorem \ref{t1} is very simple. In fact, we apply a result of Neerven (see below) for the evolution semigroup associated to $\mathcal{U}$ on $C_{00}\left( \mathbb{R}_{+},X\right)$, the space of all continuous, $X-$valued functions defined on $\mathbb{R}_{+}$ such that $f\left( 0\right) =\lim_{t\rightarrow \infty }f\left( t\right) =0$. \begin{lemma} \label{l1}Let $\mathcal{U}$ be a strongly continuous and exponentially bounded evolution family of operators on $X$ such that $$\sup_{s\geq 0}\int_{s}^{\infty }\varphi \left( \left\| U\left( t,s\right) x\right\| \right) dt=M_{\varphi }\left( x\right) <\infty \text{,\ \ for all }x\in X.\; \label{1.3a}$$ Then $\mathcal{U}$ is uniformly bounded, that is, \begin{equation*} \sup_{t\geq \xi \geq 0}\left\| U\left( t,\xi \right) \right\| <\infty. \end{equation*} \end{lemma} \paragraph{Proof of Lemma \ref{l1}} Let $x\in X$ and $N\left( x\right)$ be a positive integer such that $M_{\varphi }\left( x\right) 0$. If we choose $x=0$ in (\ref{1.3a}), then we get $\varphi \left( 0\right) =0,$ and thus from (\ref{1.3}) we obtain \begin{eqnarray} N\left( x\right) \varphi \left( \frac{\left\| U\left( t,s\right) x\right\| }{% K_{\omega }e^{\omega N}}\right) &=&\int_{s}^{\infty }\varphi \left( \frac{1_{% \left[ t-N,t\right] }\left( u\right) \left\| U\left( t,s\right) x\right\| }{% K_{\omega }e^{\omega N}}\right) du \label{1.4} \\ &\leq &\int_{s}^{\infty }\varphi \left( \left\| U\left( u,s\right) x\right\| \right) du\leq M_{\varphi }\left( x\right) . \notag \end{eqnarray} We assume that $\varphi \left( 1\right) =1$ (if not, we replace $\varphi$ be some multiple of itself). Moreover, we may assume that $\varphi$ is a strictly increasing map. Indeed if $\varphi \left( 1\right) =1$ and $a:=\int_{0}^{1}\varphi \left( t\right) dt,$ then the function given by \begin{equation*} \bar{\varphi}\left( t\right) =\left\{ \begin{array}{lll} \int_{0}^{t}\varphi \left( u\right) du, & \text{if} & 0\leq t\leq 1 \\[5pt] \dfrac{at}{at+1-a}, & \text{if} & t>1 \end{array} \right. \end{equation*} is strictly increasing and $\bar{\varphi}\leq \varphi$. Now $\varphi$ can be replaced by some multiple of $\bar{\varphi}$. From (\ref{1.4}) it follows that if $t\geq s+N\left( x\right)$ and $x\in X$, then \begin{equation*} \left\| U\left( t,s\right) \right\| \leq K_{\omega }e^{\omega N\left( x\right) },\;\;\;\text{for all }x\in X. \end{equation*} Using this inequality and the exponential boundedness of the evolution family, we have that $$\sup_{t\geq \xi \geq 0}\left\| U\left( t,\xi \right) x\right\| \leq K_{\omega }e^{\omega N\left( x\right) },\;\;\;\;\text{for each }x\in X. \label{1.5}$$ The conclusion of Lemma \ref{l1} follows from (\ref{1.5}) and the Uniform Boundedness Theorem. \quad$\Box$\medskip Let $\mathcal{U=}\left\{ U\left( t,s\right) :t\geq s\geq 0\right\}$ be a strongly continuous and exponentially bounded evolution family of bounded linear operators on $X$. We consider the strongly continuous evolution semigroup associated to $\mathcal{U}$ on $C_{00}\left( \mathbb{R}% _{+},X\right)$. This semigroup is defined by $$\left( \mathfrak{T}\left( t\right) f\right) \left( s\right) :=\left\{ \begin{array}{lll} U\left( s,s-t\right) f\left( s-t\right) , & \text{if} & s\geq t \\[5pt] 0, & \text{if} & 0\leq s\leq t \end{array} ,\right. t\geq 0 \label{1.6}$$ for all $f\in C_{00}\left( \mathbb{R}_{+},X\right)$. It is known that $\mathbf{\mathfrak{T}}=\left\{ \mathfrak{T}\left( t\right) :t\geq 0\right\}$ is a strongly continuous semigroup and in addition $\omega _{0}\left( \mathbf{% \mathfrak{T}}\right) <0$ if and only if $\mathcal{U}$ is uniformly exponentially stable (\cite{MRS}, \cite{CL}, \cite{CLMR}). \paragraph{Proof of Theorem \ref{t1}.} Let $\varphi$ be as in Theorem \ref{t1}. We assume that $\varphi \left( 1\right) =1$. Then \begin{equation*} \Phi \left( t\right) :=\int_{0}^{t}\varphi \left( u\right) du\leq \varphi \left( t\right) \text{ for all }t\in \left[ 0,1\right] . \end{equation*} Without loss of generality we may assume that \begin{equation*} \sup_{t\geq 0}\left\| \mathfrak{T}\left( t\right) \right\| \leq 1, \end{equation*} where $\mathbf{\mathfrak{T}}$ is the semigroup defined in (\ref{1.6}). Then for all $f\in C_{00}\left( \mathbb{R}_{+},X\right)$ with $\left\| f\right\| _{\infty }\leq 1$, one has \begin{eqnarray*} \lefteqn{\int_{0}^{\infty }\Phi \left( \left\| \mathfrak{T}\left( t\right) f\right\|_{C_{00}\left( \mathbb{R}_{+},X\right) }\right) dt }\\ &=&\int_{0}^{\infty }\Phi \left( \sup_{s\geq t}\left\| U\left( s,s-t\right) f\left( s-t\right) \right\| \right) dt\\ &=&\int_{0}^{\infty }\Phi \left( \sup_{\xi \geq 0}\left\| U\left( t+\xi ,\xi \right) f\left( \xi \right) \right\| \right) dt \\ &=&\int_{0}^{\infty }\left( \int_{0}^{\infty }1_{\left[ 0,\sup_{\xi \geq 0}\left\| U\left( t+\xi ,\xi \right) f\left( \xi \right) \right\| \right] }\left( u\right) \varphi \left( u\right) du\right) dt \\ &=&\sup_{\xi \geq 0}\int_{0}^{\infty }\left( \int_{0}^{\infty }1_{ \left[ 0,\left\| U\left( t+\xi ,\xi \right) f\left( \xi \right) \right\| % \right] }\left( u\right) \varphi \left( u\right) du\right) dt \\ &=&\sup_{\xi \geq 0}\int_{0}^{\infty }\Phi \left( \left\| U\left( t+\xi ,\xi \right) f\left( \xi \right) \right\| \right) dt\leq \sup_{\xi \geq 0}\int_{0}^{\infty }\varphi \left( \left\| U\left( t+\xi ,\xi \right) f\left( \xi \right) \right\| \right) dt \\ &=&\sup_{\xi \geq 0}\int_{\xi }^{\infty }\varphi \left( \left\| U\left( \tau ,\xi \right) f\left( \xi \right) \right\| \right) d\tau \leq M_{\varphi }<\infty , \end{eqnarray*} where $1_{\left[ 0,h\right] }$ denotes the characteristic function of the interval $\left[ 0,h\right]$, $h>0$. Now, from \cite[Theorem 3.2.2]{Ne}, it follows that $\omega _{0}\left( \mathfrak{% T}\right) <0,$ hence $\mathcal{U}$ is uniformly exponentially stable. \begin{thebibliography}{99} \frenchspacing \bibitem{Da} R. Datko, Extending a theorem of A.M. Liapanov to Hilbert space, \textit{J. Math. Anal. Appl., }\textbf{32} (1970), 610-616. \bibitem{Pa} A. Pazy, \textit{Semigroups of Linear Operators and Applications to Partial Differential Equations, }Springer Verlag, 1983. \bibitem{R1} S. Rolewicz, On uniform $N-$equistability, \textit{J. Math. Anal. Appl., }\textbf{115 }(1986) 434-441. \bibitem{R2} S. Rolewicz, \textit{Functional Analysis and Control Theory}, D. Riedal and PWN-Polish Scientific Publishers, Dordrecht-Warszawa, 1985. \bibitem{Zh} Q. Zheng, The exponential stability and the perturbation problem of linear evolution systems in Banach spaces, \textit{J. Sichuan Univ., }\textbf{25} (1988), 401-411. \bibitem{Li} W. 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Latushkin, \textit{Evolution Semigroups in Dynamical Systems and Differential Equations, }Mathematical Surveys and Monographs, Vol. \textbf{70}, Amer. Math. Soc., Providence, RI, 1999. \bibitem{CLMR} S. Clark, Y. Latushkin, S. Montgomery-Smith and\ T. Randolph, Stability radius and internal versus external stability in Banach spaces: An evolution semigroup approach, \textit{SIAM Journal of Control and Optim., }\textbf{38}(6) (2000), 1757-1793. \bibitem{W} G. Weiss, Weakly $\ell ^{p}-$stable linear operators are power stable, \textit{Int. J. Systems Sci., }\textbf{20} (1989). \end{thebibliography} \noindent\textsc{Constantin Bu\c{s}e}\\ Department of Mathematics, West University of Timi\c{s}oara\\ Bd. V. Parvan 4\\ 1900 Timi\c{s}oara, Rom\^{a}nia \\ e-mail: buse@hilbert.math.uvt.ro \\ http://rgmia.vu.edu.au/BuseCV.html \smallskip \noindent\textsc{Sever S. Dragomir} \hfill\break School of Communications and Informatics\\ Victoria University of Technology\\ PO Box 14428\\ Melburne City MC 8001, Victoria, Australia \\ e-mail: sever@matilda.vu.edu.au \\ http://rgmia.vu.edu.au/SSDragomirWeb.html \end{document}