\input amstex \documentstyle{amsppt} \magnification=\magstephalf \hcorrection{1cm} \vcorrection{-6mm} \nologo \TagsOnRight \NoBlackBoxes \headline={\ifnum\pageno=1 \hfill\else% {\tenrm\ifodd\pageno\rightheadline \else \leftheadline\fi}\fi} \def\rightheadline{EJDE--2001/03\hfil Uniform exponential stability \hfil\folio} \def\leftheadline{\folio\hfil D. N. Cheban \hfil EJDE--2001/03} \def\pretitle{\vbox{\eightrm\noindent\baselineskip 9pt % Electronic Journal of Differential Equations, Vol. {\eightbf 2001}(2001), No. 03, pp. 1--12.\hfil\break ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \hfill\break ftp ejde.math.swt.edu (login: ftp)\bigskip} } \topmatter \title Uniform exponential stability of linear periodic systems in a Banach space \endtitle \thanks {\it 2000 Mathematics Subject Classifications:} 34C35, 34C27, 34K15, 34K20, 58F27, 34G10. \hfil\break\indent {\it Key words:} non-autonomous linear dynamical systems, global attractors, periodic systems, \hfill\break\indent exponential stability, asymptotically compact systems, equations on Banach spaces. \hfil\break\indent \copyright 2001 Southwest Texas State University. \hfil\break\indent Submitted August 28, 2000. Published January 3, 2001. \endthanks \author D. N. Cheban \endauthor \address David N. Cheban \hfill\break State University of Moldova \hfill\break Faculty of Mathematics and Informatics \hfill \break 60, A. Mateevich str. \hfill \break Chi\c sin\u au, MD-2009 Moldova \endaddress \email cheban\@usm.md \endemail \abstract This article is devoted to the study of linear periodic dynamical systems, possessing the property of uniform exponential stability. It is proved that if the Cauchy operator of these systems possesses a certain compactness property, then the asymptotic stability implies the uniform exponential stability. We also show applications to different classes of linear evolution equations, such as ordinary linear differential equations in the space of Banach, retarded and neutral functional differential equations, some classes of evolution partial differential equations. \endabstract \endtopmatter \document \head Introduction \endhead Let $A(t)$ be a $ \tau $-periodic continuous $ n\times n$ matrix-function. It is well-known that the following three conditions are equivalent: \roster \item The trivial solution of equation $$ u'=A(t)u \eqno (0.1) $$ is uniformly exponentially stable. \item The trivial solution of equation (0.1) is uniformly asymptotically stable. \item The trivial solution of equation (0.1) is asymptotically stable. \endroster For equations in infinite-dimensional spaces the statements 1)-3) are not equivalent, as shown by the examples in [15, 26]. It is clear that in general for the infinite-dimensional case condition 1) implies 2) and 2) implies 3). In this article we show that if the Cauchy operator of equation (0.1) satisfies some compactness condition, then 3) implies 1) (see Theorem 2.5 below). Applications to different classes of linear evolution equations (ordinary linear differential equations in a Banach space, retarded and neutral functional-differential equations, some classes of evolutionary partial differential equations) are given. The exponential dichotomy of asymptotically compact cocycles was studied by R. Sacker and G. Sell [29]. The general case was studied by C. Chicone and Yu, Latushkin [14] (see also their references), Yu. Latushkin and R. Schnaubelt [25], and many other authors. \head {1. Linear non-autonomous dynamical systems} \endhead Assume that $ X $ and $ Y $ are complete metric spaces, $ \Bbb R \ (\Bbb Z) $ be a group of real (integer) numbers, $ \Bbb T = \Bbb R $ or $ \Bbb Z, \Bbb T_{+} = \{ t \in \Bbb T : t \ge 0 \} , \Bbb T_{-} = \{ t \in \Bbb T | t \le 0 \}$ and $ \Bbb C$ be the set of complex numbers. For a system $(X,\Bbb T_{+},\pi)$, we defined the following concepts: (see [9,10]) \newline {\bf Point dissipative}, if there is $ K\subseteq X$ such that for all $x\in X$ $$\lim_{t\to+\infty}\rho(xt,K)=0, \eqno (1.1) $$ where $ xt=\pi ^{t}x=\pi (t,x)$; \newline {\bf Compactly dissipative}, if the equality (1.1) takes place uniformly with respect to $x$ on compacts of $X$; \newline {\bf Locally dissipative}, if for any point $p\in X$ there is $\delta_{p} > 0$ such that the equality (1.1) takes place uniformly with respect to $x\in B(p,\delta_{p}) = \{ x \in X : \rho (x,p) < \delta _{p} \}$. \smallskip Denote by $ (X, \Bbb T_{+},\pi) $\ ($ (Y,\Bbb T, \sigma) $) a semigroup (group) dynamical system on \linebreak $ \langle (X, \Bbb T_{+},\pi), (Y,\Bbb T, \sigma), h \rangle $, where $ h $ is a homomorphism of $ (X,\Bbb T_{+}, \pi) $ onto $ (Y, \Bbb T, \sigma),$ is called a non-autonomous dynamical system. A non-autonomous dynamical system $ \langle (X, \Bbb T_{+},\pi), (Y,\Bbb T, \sigma), h \rangle $ is said to be point (compactly, locally) dissipative, if the autonomous dynamical system $(X,\Bbb T_{+},\pi)$ is so. Let $ (X,h,Y) $ be a locally trivial Banach fibre bundle over $ Y $ [1]. A non-autonomous dynamical system $ \langle (X, \Bbb T_{+},\pi), (Y,\Bbb T, \sigma), h \rangle $ is said to be linear if the mapping $ \pi ^{t}: X_{y} \to X_{yt} $ is linear for every $ t\in \Bbb T_{+} $ and $ y \in Y,$ where $ X_{y}=\{ x \in X | h(x)=y \} $ and $ yt= \sigma (t,y).$ Let $\vert \cdot \vert $ be some norm on $ (X,h,Y) $ such that $\vert \cdot \vert $ is co-ordinated with the metric $ \rho $ on $ X $ (that is $ \rho (x_{1},x_{2})=\vert x_{1}-x_{2}\vert $ for any $ x_{1},x_{2} \in X $ such that $ h(x_{1})=h(x_{2}) $). Let $ E $ be a Banach space and $ \varphi :\Bbb T_{+} \times E \times Y \mapsto E$ be a continuous mapping with properties: $ \varphi (0,u,y)=u $ and $ \varphi (t+\tau,u,y) = \varphi (t,\varphi (\tau, u,y),\sigma (\tau,y)) $ for all $ u \in E $, $ y \in Y $ and $ t, \tau \in \Bbb T_{+} $. A triplet $ \langle E,\varphi , (Y,\Bbb T, \sigma) \rangle $ is called a continuous cocycle on $ (Y,\Bbb T, \sigma) $ with fibre $ E $. Let $ [E] $ be a Banach space of the all linear continuous operators acting onto $ E $ with the operator norm and $ U :\Bbb T_{+} \times Y \mapsto [E]$ be a mapping with properties: $ U(0,y)=I $, $U(t+\tau,y) = U(t,\sigma (\tau,y))U(\tau,y) $ for all $ y \in Y $ and $ t, \tau \in \Bbb T_{+} $ and the mapping $ \varphi (\cdot ,u,\cdot) :\Bbb T_{+} \times Y \to E \ (\varphi (t,u,y) = U(t,y)u $) is continuous for every $ u \in E . $ A triplet $ \langle [E], U , (Y,\Bbb T, \sigma) \rangle $ is called a $ C_{0}-$ cocycle on $ (Y,\Bbb T, \sigma) $ with fibre $ [E] $. The dynamical system $ (X, \Bbb T_{+},\pi) $ is called [17] a skew-product system if $ X=E\times Y $ and $ \pi = (\varphi ,\sigma) $ (i.e. $ \pi (t, (u,y))= (\varphi (t,u,y), \sigma (t,y)) $ for all $ u \in E $, $ y \in Y $ and $ t, \tau \in \Bbb T_{+} $). \proclaim{Theorem 1.1 [12,13]} Let $ \langle (X, \Bbb T_{+},\pi), (Y,\Bbb T, \sigma), h \rangle $ be a linear non-autonomous dynamical system and the following conditions hold: \roster \item $ Y $ is compact and minimal (i.e. $ Y=H(y) = \overline {\{ yt : t \in \Bbb T \}} $ for all $ y \in Y $); \item for any $ x \in X $ there exists $ C_{x} \ge 0 $ such that $ \vert xt \vert \le C_{x} $ for all $ t \in \Bbb T_{+} $; \item the mapping $ y \mapsto \Vert \pi ^{t}_{y} \Vert $ is continuous, where $\Vert \pi ^{t}_{y} \Vert $ is a norm of linear operator $ \pi ^{t}_{y} = \pi ^{t} |_{X_{y}} $, for every $ t \in \Bbb T_{+} $ or $ (X, \Bbb T_{+},\pi) $ is a skew-product dynamical system. \endroster Then there exists $ M \ge 0 $ such that the inequality $$ \vert \pi (t,x)\vert \le M\vert x \vert $$ holds for all $ t \in \Bbb T_{+} $ and $ x \in X.$ \endproclaim \proclaim{Lemma 1.2} Let $ \langle [E], U , (Y,\Bbb T, \sigma) \rangle $ be a $ C_{0}-$ cocycle on $ (Y,\Bbb T, \sigma) $ with fibre $ [E] $ and $ Y $ be a compact, then the following assertions hold: \roster \item For every $ \ell > 0 $ there exists a positive number $ M(\ell) $ such that $ \Vert U(t,y) \Vert \le M(\ell) $ for all $ t\in [0,\ell] $ and $y \in Y$; \item The mapping $ \varphi :\Bbb T_{+} \times E \times Y \mapsto E \quad (\varphi (t,u,y) = U(t,y)u) $ is continuous; \item There exist positive numbers $ N $ and $ \nu $ such that $ \Vert U(t,y) \Vert \le N e^{\nu t} $ for all $ t\in \Bbb T _{+} $ and $y \in Y$. \endroster \endproclaim \demo{Proof} Let $ \ell > 0$ and $ u \in E $, then there exists a positive number $ M(\ell ,u) $ such that $ \vert U(t,y)u \vert \le M(\ell ,u)$ for all $ (t,y) \in [0,\ell]\times Y $ because the mapping $ (t,y) \to U(t,y)u $ is continuous. According to principle of uniformly boundedness there exists a positive number $ M(\ell) $ such that $ \Vert U(t,y) \Vert \le M(\ell)$ for all $ (t,y) \in [0,\ell]\times Y $. Let now $ (t_{0},u_{0},y_{0}) \in \Bbb T_{+} \times E \times Y $ and $ t_{n} \to t_{0}, u_{n} \to u_{0}$ and $ y_{n} \to y_{0}$, then we have $$ \eqalign{ &\vert \varphi (t_{n},u_{n},y_{n}) - \varphi (t_{0},u_{0},y_{0}) \vert \cr &\le \vert \varphi (t_{n},u_{n},y_{n}) - \varphi (t_{n},u_{0},y_{n}) \vert + \vert \varphi (t_{n},u_{0},y_{n}) - \varphi (t_{0},u_{0},y_{0}) \vert \cr & \le \Vert U (t_{n},y_{n})(u_{n}-u_{0}) \Vert + \vert (U (t_{n},y_{n}) - U (t_{0},y_{0}))u_{0} \vert \cr} \eqno (1.2) . $$ In view of first statement of Lemma 1.2 there exists the positive number $ M $ such that $$ \Vert U(t_{n},y_{n}) \Vert \le M \eqno (1.3) $$ for all $ n \in \Bbb N $. From inequalities (1.2) and (1.3) follows the continuity of mapping $ \varphi :\Bbb T_{+} \times E \times Y \to E \quad (\varphi (t,u,y) = U(t,y)u) $. Denote by $ a = \sup \{ \Vert U(t,y) \Vert : (t,y) \in [0,1]\times Y \} $ and let $ t \in \Bbb T _{+} , t = n + \tau (n \in \Bbb N , \tau \in [0,1)) $, then we obtain $$ \Vert U(t,y) \Vert \le \Vert U(n,y\tau) \Vert \Vert U(\tau ,y) \Vert \le a^{n+1} \le N e^{\nu t } $$ for all $ t\in \Bbb T _{+} $ and $y \in Y$, where $ N=a$ and $ \nu = \ln a $. \enddemo \proclaim{Theorem 1.3 [13]} Let $ \langle (X, \Bbb T_{+},\pi), (Y,\Bbb T, \sigma), h \rangle $ be a linear non-autonomous dynamical system, $ Y $ be a compact , then the following conditions are equivalent: \roster \item The non-autonomous dynamical system $ \langle (X, \Bbb T_{+},\pi), (Y,\Bbb T, \sigma), h \rangle $ is uniformly exponentially stable, i.e. there exist two positive constants $ N $ and $ \nu $ such that $ \vert \pi (t,x) \vert \le N e^ {-\nu t}\vert x \vert $ for all $ t\in \Bbb T_{+} $ and $ x \in X $. \item $ \Vert \pi^{t} \Vert \to 0 $ as $ t \to +\infty $, where $ \Vert \pi^{t} \Vert = \sup \{ \vert \pi^{t} x \vert : x \in X, \vert x \vert \le 1 \}. $ \item The non-autonomous dynamical system $ \langle (X, \Bbb T_{+},\pi), (Y,\Bbb T, \sigma), h \rangle $ is locally dissipative. \endroster \endproclaim \head {2. Exponential stable linear periodic dynamical systems.} \endhead \proclaim{Lemma 2.1 [15, Chapter 9]} Let $ m : \Bbb T _{+} \mapsto \Bbb T _{+} $ be a positive and continuous function. If there exists a positive constant $ M $ such that $ m(t+s)\le Mm(t)$ for all $ s \in [0,1]$ and $ t \in \Bbb T _{+}$, then $ \int _{0}^{+\infty} m(t)dt < +\infty $ implies $ m(t) \to 0 $ as $ t \to + \infty $. \endproclaim \proclaim{Theorem 2.2} Let $ \langle [E], U , (Y,\Bbb T, \sigma) \rangle $ be the $ C_{0}-$ cocycle on $ (Y,\Bbb T, \sigma) $ with fibre $ [E] $ and $ (Y ,\Bbb T ,\sigma)$ be a periodical dynamical system (i.e. there are $ y_{0} \in Y $ and $ \tau \in \Bbb T \ (\tau > 0) $ such that $ Y = \{ y_{0}t : 0\le t < \tau \})$. Then the following conditions are equivalent: \roster \item"(i)" $$ \lim \limits _{t \to + \infty} \Vert U(t,y_{0}) \Vert = 0 . \eqno (2.1) $$ \item"(ii)" There exist positive constants $ N $ and $ \nu $ such that for all $ t\in \Bbb T_{+} $ and $ y \in Y $, $$ \Vert U (t,y) \Vert \le N e^ {-\nu t}\,. \eqno (2.2) $$ \item"(iii)" There exists $ p \ge 1$ such that for all $ u \in E $, $$ \int _{0}^{+ \infty} \vert U(t,y_{0}) u \vert ^{p} dt < + \infty\,. \eqno (2.3) $$ \endroster \endproclaim \demo{Proof} We remark that from equality (2.1) follows the condition $$ \lim \limits _{n \to + \infty} \sup \limits _{0\le s \le \tau} \Vert U(s+n\tau ,y_{0})\Vert =0 . \eqno (2.4) $$ In fact, by virtue of Lemma 1.2 there exists a positive constant $ M $ such that $$ \Vert U(s,y) \Vert \le M \eqno (2.5) $$ for all $ s \in [0,\tau ]$ and $ y \in Y $. Therefore, $$ \Vert U(s+n\tau ,y_{0})\Vert = \Vert U(s,y_{0}) U(n\tau ,y_{0})\Vert \le M\Vert U(n \tau ,y_{0})\Vert \eqno (2.6) $$ for all $ 0 \le s \le \tau $. Consequently, from (2.1) and (2.6) results the condition (2.4). We will show that under the condition (2.4) the equality $$ \lim \limits _{t \to +\infty}\sup \limits _{y \in Y} \Vert U(t,y)\Vert = 0 \eqno (2.7) $$ holds. In fact, let $ y \in Y $ then there exists a number $ s \in [0,\tau) $ such that $ y = y_{0}s $ and, consequently, for $ t \in \Bbb T _{+} \quad (t = n \tau + \bar{t}, \bar{t} \in [0,\tau)) $ we obtain $$ \eqalign{ \Vert U(t,y)\Vert &= \Vert U(t,y_{0}s)\Vert =\Vert U(n \tau + \bar{t},y_{0}s)\Vert \cr &= \Vert U((n-1)\tau + \bar{t} + s, y_{0}\tau) U(\tau -s,y_{0}s)\Vert \cr & \le M \max \{ \sup \limits _{0\le s \le \tau} \Vert U((n-1)\tau +s ,y_{0})\Vert ,\sup \limits _{0\le s \le \tau} \Vert U(n\tau +s ,y_{0})\Vert \}.\cr} \eqno (2.8) $$ From (2.4) and (2.8) results the equality (2.7). For finishing the proof that (i) implies (ii) is sufficient to apply Theorem 1.3 . The fact that (ii) implies (iii) is obvious. Now we prove that (iii) implies (i). Indeed, let $ u \in E $ and we consider the function $ m(t) = \vert U(t,y_{0})u \vert ^{p} \quad (t \ge 0) .$ We note that $$\aligned m(t+s)=& \vert U(t + s,y_{0})u \vert ^{p} = \vert U(s,y_{0}t)U(t,y_{0})u \vert ^{p} \\ \le & \Vert U(s,y_{0}t)\Vert ^{p} \vert U(t,y_{0})u\vert ^{p} \le M ^{p} m(t) \endaligned $$ for all $ t \in \Bbb T_{+} $ and $ s \in [0,1] $, where $ M = \sup \limits _{0\le s \le 1, y\in Y} \Vert U(s,y)\Vert $. By Lemma 2.1 $ m(t) \to 0 $ as $ t \to + \infty $ and, consequently, $$ \lim \limits _{t \to + \infty} \vert U(t,y_{0})u\vert ^{p} = 0 \eqno (2.9) $$ for all $ u \in E $. Let now $ y \in Y $, then there exists $ s \in [0,\tau) $ such that $ y=y_{0} s $ and for $ t \ge \tau - s $ we have $$ U(t,y)u= U(t,y_{0}s)u=U(t-\tau +s,y_{0})U(\tau -s,y_{0}s)u . \eqno (2.10) $$ From equalities (2.9) and (2.10), $$ \lim \limits _{t \to + \infty} \vert U(t,y)u\vert ^{p} = 0 \eqno (2.11) $$ for all $ u \in E $ and $ y \in Y .$ According to Theorem 1.1 there exists a positive number $M$ such that $ \Vert U(t,y) \Vert \le M $ for all $ t \in \Bbb T_{+} $ and $ y \in Y $. Let $ t > 0$ and $u \in E $, then we obtain $$\eqalign{ t \vert U(t,y_{0})u\vert^{p} &= \int _{0}^{t}\vert U(t,y_{0})u\vert^{p}ds \le \int _{0}^{t}\vert U(t-s,y_{0}s)\vert^{p} \vert U(s,y_{0})u\vert^{p}ds \cr & \le M^{p}\int _{0}^{t}\vert U(s,y_{0})u\vert^{p}ds \le M^{p} \int _{0}^{+\infty}\vert U(s,y_{0})u\vert^{p}ds =C_{u}\cr} $$ for all $ t \ge 0 $. By virtue of principle of uniformly boundedness there exists a positive number $ C $ such that $$ t \Vert U(t,y_{0})\Vert^{p} \le C $$ for all $ t > 0 $ and, consequently $$ \Vert U(t,y_{0})\Vert \le C ^{\frac{1}{p}} t^{-\frac{1}{p}} \to 0 $$ as $ t \to + \infty $. This completes the present proof. \enddemo \proclaim{Remark 2.3} \rm \roster \item Theorem 2.2 (the equivalence of assertions (ii) and (iii)) is a variant of the Datko-Pazy theorem (see [15-17,19]) for cocycle over periodic dynamical systems. \item Periodic, almost periodic and asymptotically almost periodic mild solutions of inhomogeneous periodic Cauchy problems considered recently by C. J. K. Batty, W.Hutter and F. R\"{a}biger [2] and W. Hutter [23]. \endroster \endproclaim The operator $U(\tau,y_{0}) $ is called operator of monodromy for $ \tau $- periodic cocycle $ U(t,y)$. The number $ 0\not= \lambda \in \Bbb C $ is called multiplicator of operator of monodromy $U(\tau,y_{0}) $ if there exists $ u_{0} \in E \ (u_{0} \not= 0) $ such that $U(\tau,y_{0})u_{0} = \lambda u_{0} $ (or, what is the same, $ U(t+\tau,y_{0})u_{0} = \lambda U(t,y_{0})u_{0}$ for all $ t \in \Bbb T _{+} $). \proclaim{Remark 2.4} \rm \roster \item"(a)" Condition (2.1) and the equality $$ \lim \limits _{n \to + \infty} \Vert U(n\tau,y_{0}) \Vert = 0 . \eqno (2.12) $$ are equivalent. We show that (2.12) implies (2.1) as follows. Let now $ t= n\tau + s , 0\le s <\tau $, then $ U(t,y_{0})= U(s+n\tau ,y_{0})= U(s,y_{0}) U(n\tau ,y_{0}) $ and, consequently, $$ \Vert U(t,y_{0})\Vert \le \max \limits _{0\le s \le \tau} \Vert U(s,y_{0})\Vert \Vert U(n\tau ,y_{0}) \Vert . \eqno (2.13) $$ From conditions (2.12) and (2.13) results (2.1). \item"(b)" Condition (2.2) and the inequality $$ \Vert U (t,y_{0}) \Vert \le N_{1} e^ {-\nu _{1}t} \quad (\forall t\in \Bbb T_{+}) \eqno (2.14) $$ are equivalent, where $ N_{1} $ and $ \nu _{1} $ are some positive constants. Indeed, from (2.14), taking into account (2.10), we obtain (2.2). \item"(c)" Condition (2.12) is satisfied if and only if $ \sigma (U(\tau ,y_{0})) \subset \Bbb D = \{ z \in \Bbb C : \vert z \vert < 1 \} $, where $\sigma (U(\tau ,y_{0})) $ is a spectrum of operator of monodromy $U(\tau ,y_{0}) $. In fact, from (2.2) results that $ r_{U(\tau ,y_{0})} = \lim \limits _{n \to + \infty} \sup (\Vert U(n\tau ,y_{0}))\Vert)^{1/n} \le e^{-\nu} <1 $, because $ U^{n}(\tau ,y_{0})= U(n\tau ,y_{0})$. If $ \gamma =r_{U(\tau ,y_{0})} < 1$, then for all $ \varepsilon > 0 $ there exists a $ n(\varepsilon) \in \Bbb N $ such that $ (\Vert U(n\tau ,y_{0})\Vert)^{1/n} \le \gamma + \varepsilon $ for all $ n \ge n(\varepsilon)$ and, consequently, $ \Vert U(n\tau ,y_{0})\Vert \le (\gamma + \varepsilon)^{n} $ for all $ n \ge n(\varepsilon)$. Thus $\Vert U(n\tau ,y_{0})\Vert \to 0 $ as $ n \to + \infty $. \endroster \endproclaim A continuous mapping $ P:E \to E$ is called [21] asymptotically compact if, for any nonempty bounded set $B\subset E$ for which $ P(B)\subseteq B,$ there is a compact set $ K \subset \overline {B} $ such that $K$ attracts $B$, i.e. $\lim \limits _{n \to + \infty} \sup \limits _{x \in B} \rho (P^{n}x,K)=0,$ where $ \rho (x,K)=\inf \limits _{y\in K} \vert x-y \vert .$ \proclaim{Theorem 2.5} Let $ \langle [E], U , (Y,\Bbb T, \sigma) \rangle $ be a $ C_{0}-$ cocycle on $ (Y,\Bbb T, \sigma) $ with fibre $ [E] $, $ (Y ,\Bbb T ,\sigma)$ be a periodic dynamical system and $ U(\tau,y_{0})$ be asymptotically compact (i.e. if $ k_{n} \to + \infty \quad (k_{n} \in \Bbb N)$, the sequences $ \{u_{n}\} \subseteq E $ and $ \{U(k_{n} \tau,y_{0})u_{n} \} $ are bounded; then the sequence $ \{U(k_{n} \tau,y_{0})u_{n} \} $ is precompact). Then the following conditions are equivalent \roster \item"(i)" Equality (2.1) holds. \item"(ii)" For all $ u \in E $, $$ \lim \limits _{t \to + \infty} \vert U(t,y_{0})u \vert = 0\,. \eqno (2.15) $$ \endroster \endproclaim \demo{Proof} It is evidently that (i) implies (ii). Now, under the conditions of Theorem 2.5 the mapping $ P=U(\tau ,y_{0}) : E \mapsto E $ is asymptotically compact because $ P^{n}=U(n\tau ,y_{0}) $. >From condition (2.15) according to uniform boundedness principle it follows that there is a positive constant $M$ such that $ \Vert P^{n}\Vert \le M $ for all $ n \in \Bbb Z _{+} $ and, consequently, the set $ B=\cup \{ P^{n}x: \vert x \vert \le 1, n \in \Bbb Z _{+} \} $ is bounded and $ P(B)\subset B $. Since the mapping $P$ is asymptotically compact in virtue of Corollary 2.2.4 from [21] the set $$ \omega (B) =\cap_{n \ge 0} \overline {\cup_{m \ge n} P^{m}(B) } $$ is nonempty, compact, and invariant and $ \omega (B) $ attracts $B$. Now we will prove that $ \lim \limits _{n \to +\infty} \Vert P^{n}\Vert =0.$ If we suppose the contrary, then there are $ \varepsilon _{0} > 0, \{x_{n}\} (\vert x_{n}\vert \le 1) $ and $ n_{k} \to +\infty (\{n_{k}\} \subset \Bbb Z _{+}) $ such that $$ \vert P^{n_{k}}x_{k}\vert \ge \varepsilon _{0} . \eqno (2.16) $$ Since $P$ is asymptotically compact without loss of generality we can suppose that the sequence $ \{ P^{n_{k}}x_{k}\} $ is convergent. Let $ \bar{x} = \lim \limits _{k \to + \infty} P^{n_{k}}x_{k},$ then $ \bar{x} \in \omega (B) $ and from (2.16) we have $ \vert \bar{x} \vert \ge \varepsilon _{0} >0.$ According to the invariance of the set $\omega (B) $ there exists a beside sequence $ \{w_{n}\}_{n \in \Bbb Z } \subset \omega (B) $ such that: $ w_{0}=\bar{x} $ and $ P(w_{n})=w_{n+1} $ for all $ n \in \Bbb Z. $ We note that $$ \inf \limits _{n \in \Bbb Z _{-}} \vert w_{n}\vert =0. \eqno (2.17) $$ Suppose that it is not true, then there is a positive number $\ell $ such that $$ \vert w_{n}\vert \ge \ell \eqno (2.18) $$ for all $ n \in \Bbb Z _{-}.$ Let $ p=\lim \limits _{k \to + \infty} w_{n_{k}} $ and $ \{z_{n}\} \subseteq \alpha _{w_{0}}, $ where $$\alpha _{w_{0}} = \bigcap_{n \le 0} \overline {\bigcup_{m \le n} w_{m} }, $$ be a beside sequence such that $z_{0}=p $ and $ P(z_{n})=z_{n+1}$ for all $ n \in \Bbb Z $. >From the inequality (2.18) results that $ \vert z_{n} \vert \ge \ell $ for all $ n \in \Bbb Z .$ On the other hand in view of (2.15) $ \lim \limits _{n \to + \infty} \vert w_{n}\vert = \lim \limits _{n \to + \infty}\vert P^{n}w_{0}\vert = 0.$ The obtained contradiction proves the equality (2.17). Let now $ n_{r} \to - \infty $ and $\vert w _{n_{r}}\vert \to 0,$ then $w_{0} = P^{-n_{r}}w_{n_{r}} $ for all $ r \in \Bbb N $ and, consequently, $ \vert w_{0} \vert =0 $ because $ \vert w_{0}\vert \le \Vert P^{-n_{r}} \Vert \vert w_{n_{r}} \vert \le M \vert w_{n_{r}} \vert .$ On the other hand $ \vert w_{0} \vert =\vert \bar{x}\vert \ge \varepsilon _{0}> 0.$ The obtained contradiction finishes the proof of our assertion. The Theorem is proved. \enddemo \proclaim{Remark 2.6} \rm C.Bu\c{s}e wrote several papers [3-5] on evolutions periodic processes that are in the spirit of the current paper. In particularly, in [5] it is proved that a trivial solution of equation $u'(t)=A(t)u(t) $ with $p -$ periodic coefficients on a separable Hilbert space $ H $ is uniformly exponentially stable if the mild solution $ u_{\mu x} $ of a well-posed inhomogeneous Cauchy problem $ u'(t)=A(t)u(t) + e^{i\mu t}x (t \ge 0), \mu \in \Bbb R , u(0)=0 $ satisfies the following condition $ \sup \limits _{\mu \in \Bbb R } \sup \limits _{t >0} \vert u_{\mu x}(t)\vert < + \infty , \forall x \in H .$ \endproclaim \head 3. Some classes of linear uniformly exponentially stable periodic differential equations. \endhead Let $ \Lambda $ be the complete metric space of linear operators that act on Banach space $ E $ and $ C(\Bbb R,\Lambda) $ be the space of all continuous operator-functions $ A: \Bbb R \to \Lambda $ equipped with open-compact topology and $ (C(\Bbb R, \Lambda),\Bbb R, \sigma) $ be the dynamical system of shifts on $ C(\Bbb R, \Lambda) .$ \subhead 3.1 Ordinary linear differential equations\endsubhead Let $ \Lambda = [E] $ and consider the linear differential equation $$ u'=\Cal A (t)u \,, \eqno (3.1) $$ where $ \Cal A \in C(\Bbb R ,\Lambda) .$ Along with equation (3.1), we shall also consider its $ H-$class, that is, the family of equations $$ v'=\Cal B (t)v \,, \eqno (3.2) $$ where $ \Cal B \in H(\Cal A) = \overline{ \{ \Cal A _{s} : s \in \Bbb R \} }, \Cal A _{s}(t)=\Cal A (t + s) \ (t \in \Bbb R) $ and the bar denotes closure in $ C(\Bbb R,\Lambda) $. Let $ \varphi (t,u,\Cal B) $ be the solution of equation (3.2) that satisfies the condition $ \varphi (0,v,\Cal B)=v.$ We put $ Y=H(\Cal A) $ and denote the dynamical system of shifts on $ H(\Cal A) $ by $ (Y, \Bbb R ,\sigma) $, then the triple $ \langle [E], U, (Y,\Bbb R, \sigma) \rangle $ is the linear cocycle on $ (Y, \Bbb R ,\sigma)$, where $ U(t,B) =\varphi (t,\cdot ,B) $ for all $t\in \Bbb R $ and $ B \in Y$. \proclaim{Lemma 3.1 [6,7]}\roster \item"(i)" The mapping $ (t,u,\Cal A) \mapsto \varphi (t,u,\Cal A)$ of $ \Bbb R \times E \times C(\Bbb R ,[E]) $ to $ E $ is continuous, and \item"(ii)" the mapping $\Cal U : \Cal A \to U(\cdot ,\Cal A) $ of $ C(\Bbb R ,[E]) $ to $ C(\Bbb R ,[E]) $ is continuous, where $ U(\cdot ,\Cal A)$ is the Cauchy operator [12] of equation (3.1). \endroster \endproclaim \proclaim{Theorem 3.2} Let $ \Cal A \in C(\Bbb R, \Lambda) $ be $ \tau -$ periodic (i.e. $ \Cal A (t+\tau)= \Cal A (t) $ for all \- $ t \in \Bbb R $), then the following conditions are equivalent: \roster \item The trivial solution of (3.1) is uniformly exponentially stable, i.e. there exist positive numbers $ N $ and $ \nu $ such that $ \Vert U(t,\Cal A)U(\tau ,\Cal A)^{-1}\Vert \le N e^{-\nu (t-\tau)} $ for all $ t \ge \tau $. \item There exist positive numbers $ N $ and $ \nu $ such that $ \Vert U(t,\Cal B)U(\tau,\Cal B)^{-1}\Vert \le N e^{-\nu (t-\tau)} $ for all $ t \ge \tau $ and $ \Cal B \in H(\Cal A)= \{ \Cal A_{s} : s \in [0,\tau)\} $. \item $ \lim \limits_{t \to + \infty} \Vert U(t,\Cal A)\Vert =0 $. \item There exists $ p \ge 1$ such that $ \int _{0}^{+ \infty} \vert U(t,\Cal A) u \vert ^{p} dt < + \infty$ for all $ u \in E $. \endroster \endproclaim \demo{Proof} Applying Theorem 2.2 to the cocycle $ \langle [E], U, (Y,\Bbb R, \sigma) \rangle $, generated by equation (3.1) we obtain the equivalence of conditions 2), 3) and 4) According to Lemma 3 [7] the conditions 1) and 2) are equivalent. The theorem is proved. \enddemo \proclaim{Theorem 3.3} Let $ \Cal A \in C(\Bbb R, \Lambda) $ be $ \tau -$ periodic and $ U(\tau, \Cal A) $ be asymptotically compact, then the following conditions are equivalent: \roster \item The trivial solution of equation (3.1) is uniformly exponentially stable. \item $ \lim \limits _{t \to + \infty}\vert U(t,\Cal A)u\vert =0 $ for every $ u \in E $. \endroster \endproclaim \demo{Proof} Applying Theorem 2.5 to non-autonomous system $ \langle (X, \Bbb R_{+},\pi), (Y,\Bbb R, \sigma), h \rangle $ generated by equation (3.1), we obtain the equivalence of conditions 1) and 2). The theorem is proved. \enddemo \subhead 3.2 Partial linear differential equations\endsubhead Let $ \Lambda $ be some complete metric space of linear closed operators acting into a Banach space $ E $ \ (for example $ \Lambda = \{ A_{0}+B | B \in [E] \} $, where $ A_{0} $ is a closed operator that acts on $ E $). We assume that the following conditions are fulfilled for equation (3.1) and its $ H-$ class (3.2): \roster \item"(a)" for any $ v \in E $ and $ \Cal B \in H(\Cal A) $ equation (3.2) has exactly one mild solution defined on $ \Bbb R_{+} $ and satisfies the condition $ \varphi (0, v, \Cal B) = v ;$ \item"(b)" the mapping $ \varphi : (t,v,\Cal B) \to \varphi (t,v,\Cal B) $ is continuous in the topology of $ \Bbb R_{+} \times E \times C(\Bbb R ; \Lambda) ;$ \endroster Under the assumptions above, (3.1) generates a linear cocycle $ \langle [E], U, (Y,\Bbb R, \sigma) \rangle $, where $U(t,B)=\varphi (t,\cdot ,B)$. Applying the results from \S\,2 to this cocycle, we will obtain the analogous assertions for different classes of partial differential equations. We will consider examples of partial differential equations which satisfy the above conditions a. and b. {\bf Example 3.1.} A closed linear operator $ \Cal A : D(\Cal A) \to E $ with dense domain of definition $ D(\Cal A) $ is said [22] to be a sectorial if one can find a $ \theta \in (0,\frac{\pi}{2}) $, an $ M \ge 1$, and a real number $ a $ such that the sector $$ S_{a,\theta } = \{ \lambda : \theta \le \vert \arg (\lambda -a) \vert \le \pi , \lambda \not= a \} $$ lies in the resolvent set $ \rho (\Cal A)$ of $ \Cal A $ and $ \Vert (\lambda I - \Cal A)^{-1}\Vert \le M\vert \lambda -a\vert^{-1} $ for all $ \lambda \in S_{a,\theta } $. If $ A $ is a sectorial operator, then there exists $a_{1} > 0 $ such that $ Re \sigma (A+a_{1}I) > 0 $ \quad $ (\sigma (A)=\Bbb C \setminus \rho (A)).$ Let $ A_{1} = A + a_{1} I . $ For $ 0 < \alpha < 1, $ one defines the operator [14] $$ A_{1}^{-\alpha} = \frac {\sin \pi \alpha }{\pi} \int _{0}^{+ \infty } \lambda ^{-\alpha}(\lambda I + A_{1})^{-1} d\lambda ,$$ which is linear, bounded, and one-to-one. Set $E^{\alpha}=D(A^{\alpha}_{1})$, and let us equip the space $ E^{\alpha}$ with the norm $ \vert u \vert _{\alpha}= \vert A_{1}^{\alpha}u\vert , E^{0}=E, X^{1}=D(A) $. Then $ E^{\alpha} $ is a Banach space with the norm $\vert \cdot \vert_{\alpha},$ and is densely continuously embedded in $E$. If the operator $A$ admits a compact resolvent, then the embedding $ E^{\alpha} \to E^{\beta } $ is compact for $ \alpha > \beta \ge 0 $ [22]. An important class of a sectorial operators is formed by elliptic operators [22,24]. Consider the differential equation $$ u'=(\Cal A_{0} + \Cal A (t))u, \eqno (3.3) $$ where $ \Cal A_{0} $ is a sectorial operator that does not depend on $ t \in \Bbb R $, and $ \Cal A \in C(\Bbb R ,[E]) $. The results of [14] imply that equation (3.3) satisfies conditions a. and b. Under the assumptions above, (3.3) generates a linear cocycle $ \langle [E], U, (Y,\Bbb R, \sigma) \rangle $, where $ Y=H(\Cal A) $ and $ U(t,B)=\varphi (t,\cdot,B)$. Applying the results from \S\,2 to this system, we will obtain the following results. \proclaim{Theorem 3.4} Let $ \Cal A_{0} $ - be the sectorial operator and $ \Cal A \in C(\Bbb R, \Lambda) $ be $ \tau $ -periodic, then the following conditions are equivalent: \roster \item The trivial solution of equation (3.3) is uniformly exponentially stable, i.e. there exist positive numbers $ N $ and $ \nu $ such that $ \Vert U(t,\Cal A_{0}+\Cal A)U(\tau ,\Cal A_{0}+\Cal A)^{-1}\Vert \le N e^{-\nu (t-\tau)} $ for all $ t \ge \tau .$ \item There exist positive numbers $ N $ and $ \nu $ such that $ \Vert U(t,\Cal A_{0}+\Cal B)U(\tau,\Cal A_{0}+\Cal B)^{-1}\Vert \le N e^{-\nu (t-\tau)} $ for all $ t \ge \tau $ and $ \Cal B \in H(\Cal A) $. \item $ \lim \limits_{t \to + \infty} \Vert U(t,\Cal A_{0}+\Cal A)\Vert =0 $. \item There exists $ p \ge 1$ such that $ \int _{0}^{+ \infty} \vert U(t,\Cal A_{0}+\Cal A) u \vert ^{p} dt < + \infty$ for all $ u \in E $. \item $ \sigma (U(\tau ,\Cal A_{0}+\Cal A)) \subset \Bbb D $. \endroster \endproclaim \proclaim{Theorem 3.5} Let $ \Cal A_{0} $ - be the sectorial operator with compact resolvent and $ \Cal A \in C(\Bbb R, \Lambda) $ be $ \tau $ - periodic, then the following conditions are equivalent: \roster \item The trivial solution of equation (3.3) is uniformly exponentially stable. \item $ \lim \limits_{t \to + \infty}\vert U(t,\Cal A_{0}+\Cal A)u\vert =0 $ for every $ u \in E $. \item $ \vert \lambda \vert <1 $ for every multiplicator $\lambda$ of operator of monodromy $ U(\tau ,\Cal A_{0}+\Cal A)$. \endroster \endproclaim \demo{Proof} Since the sectorial operator $ \Cal A_{0} $ admits a compact resolvent, then in view of Lemma 7.2.2 [14] the operator $ U(\tau,\Cal A_{0}+\Cal A) $ is compact and, consequently (see,for example [30, p.391-396]), every $ 0\not= \lambda \in \sigma (U(\tau,\Cal A_{0}+\Cal A))$ is a multiplicator for operator of monodromy $ U(\tau,\Cal A_{0}+\Cal A)$. Applying Theorem 3.4 (see also Remark 2.3) to linear cocycle $ \langle [E],U ,(Y,\Bbb R, \sigma) \rangle $ generated by equation (3.3), we obtain the equivalence of conditions 1., 2. and 3. The theorem is proved. \enddemo \subhead 3.3 Linear functional-differential equations \endsubhead Let $ r > 0, C([a,b], \Bbb R ^{n}) $ be the Banach space of all continuous functions $ \varphi : [a,b] \to \Bbb R ^{n} $ with $ \sup $-norm. If $ [a,b]=[-r,0] $, then we put $ \Cal C=C([-r,0],\Bbb R ^{n})$. Let $ \sigma \in \Bbb R , \alpha \ge 0$ and $ u \in C([\sigma - r, \sigma + \alpha],\Bbb R^{n}) $. For any $ t \in [\sigma , \sigma + \alpha ] $ we define $ u_{t} \in \Cal C $ by equality $ u_{t}(\theta)=u(t + \theta) , -r \le \theta \le 0 .$ Denote by $ \frak A = \frak A (C, \Bbb R ^{n}) $ the Banach space of all linear continuous operators acting from $ \Cal C $ into $ \Bbb R ^{n} $, equipped by operator norm. Consider the equation $$ u' = \Cal A (t)u_{t}\,, \eqno (3.4) $$ where $ \Cal A \in C(\Bbb R , \frak A) $. We put $ H(\Cal A)= \overline {\{ {\Cal A}_{\tau } : \tau \in \Bbb R \}} , {\Cal A}_{\tau }(t)=\Cal A (t + \tau) $ and the bar denotes the closure in the topology of uniform convergence on compacts of $ \Bbb R $. Along with equation (3.4) we also consider the family of equations $$ u' = \Cal B (t)u_{t} \,, \eqno (3.5) $$ where $ \Cal B \in H(\Cal A)$. Let $ \varphi _{t} (v,\Cal B) $ be a solution of equation (3.5) with condition $ \varphi _{0}(v,\Cal B)=v $ defined on $ \Bbb R_{+} $. We put $ Y=H(\Cal A) $ and denote by $ (Y,\Bbb R , \sigma) $ the dynamical system of shifts on $ H (\Cal A) $. Let $ X= C\times Y $ and $ \pi = (\varphi ,\sigma) $ the dynamical system on $ X $, defined by the equality $ \pi (\tau ,(v,\Cal B)) = (\varphi _{\tau } (v, \Cal B), \Cal B_{\tau})$. The non-autonomous dynamical system $ \langle (X, \Bbb R_{+},\pi), (Y,\Bbb R, \sigma), h \rangle $ $(h = pr_{2} : X \to Y)$ is linear. The following assertion takes place. \proclaim{Lemma 3.6 [12]} Let $ H(\Cal A) $ be compact in $ C(\Bbb R , \frak A) $, then the non-autonomous dynamical system $ \langle (X, \Bbb R_{+},\pi), (Y,\Bbb R, \sigma), h \rangle $ generated by equation (3.4) is completely continuous, i.e. for every bounded set $ A \subset X $ there exists a positive number $ \ell $ such that $ \pi ^{\ell}A $ is precompact. \endproclaim \proclaim{Theorem 3.7} Let $ \Cal A $ be $ \tau -$ periodic. Then the following assertions are equivalent: \roster \item The trivial solution of equation (3.4) is uniformly exponentially stable. \item $ \lim \limits_{t \to + \infty}\vert U(t,\Cal A)u\vert =0 $ for every $ u \in E $. \item $ \vert \lambda \vert <1 $ for every multiplicator $\lambda$ of operator of monodromy $ U(\tau ,\Cal A)$. \endroster \endproclaim \demo{Proof} Let $ \langle (X, \Bbb R_{+},\pi), (Y,\Bbb R, \sigma), h \rangle $ be the linear non-autonomous dynamical system, generated by equation (3.4). According to Lemma 3.6 this system is completely continuous and, consequently, there exists a number $ k \in \Bbb N $ such that $ U^{k}(\tau ,y_{0})= U(k\tau, y_{0})$ is precompact. By virtue of theory of Riesz-Schauder (see for example [30, p.391-395]) every $ 0 \not= \lambda \in \sigma (U(\tau ,\Cal A)) $ is a multiplicator of operator of monodromy $ U(\tau ,\Cal A)$. To finish the proof it is sufficient to refer to Theorems 2.2, 2.5 and Remark 2.3. \enddemo Consider the neutral functional differential equation $$ \frac{d}{dt}Du_{t}=\Cal A (t)u_{t} \, , \eqno (3.6) $$ where $ {\Cal A} \in C(\Bbb R , \frak A) $ and $ D \in \frak A $ is nonatomic at zero operator [20, p.67]. As well as in the case of equation (3.4), the equation (3.6) generates a linear non-autonomous dynamical system $ \langle (X, \Bbb R_{+},\pi), (Y,\Bbb R, \sigma), h \rangle $, where $ X=C\times Y, Y=H(\Cal A) $ and $ \pi = (\varphi , \sigma) $. The following statement holds. \proclaim{Lemma 3.8 [12] } Let $ H(\Cal A) $ be compact and the operator $ D $ is stable, i.e. the zero solution of homogeneous difference equation $ Dy_{t}=0 $ is uniformly asymptotically stable. Then the linear non-autonomous dynamical system $ \langle (X, \Bbb R_{+},\pi), (Y,\Bbb R, \sigma), h \rangle $, generated by equation (3.6), is asymptotically compact. \endproclaim \proclaim{Theorem 3.9} Let $ \Cal A \in C(\Bbb R , \frak A) $ be $ \tau -$ periodic and $ D $ is stable, then the following assertions are equivalent: \roster \item The trivial solution of equation (3.6) is uniformly exponentially stable; \item $ \lim \limits_{t \to + \infty}\vert U(t,\Cal A)u\vert =0 $ for every $ u \in E $; \item $ \vert \lambda \vert <1 $ for every multiplier $\lambda$ of operator of monodromy $ U(\tau ,\Cal A)$. \endroster \endproclaim \demo{Proof} Let $ \langle (X, \Bbb R_{+},\pi), (Y,\Bbb R, \sigma), h \rangle $ be the linear non-autonomous dynamical system, generated by equation (3.6). According to Lemma 3.8 this system is asymptotically compact. . According to results of [20, Chapter 12] every $ 0 \not= \lambda \in \sigma (U(\tau ,y_{0})) $ is a multiplier of operator of monodromy $ U(\tau ,y_{0})$. To finish the proof of Theorem 3.8 it is sufficient to refer to Theorems 2.2, 2.5 and Remark 2.3. The theorem is proved. \enddemo \proclaim{Remark 3.10} \rm \roster \item The equivalence of conditions 1. and 3. in Theorem 3.5 (Theorem 3.7, Theorem 3.9) was proved in [22, p.219] (resp. in [20, p.233], [20, p.365]).\newline \item All the statements from \S\, 3 hold also for difference equations and can be proved in the same way. \endroster \endproclaim \Refs \widestnumber\key{99} \ref \key 1 \by N. Bourbaki \book Vari\'et\'es diff\'erentielles et analitiques. Fascicule de r\'esultats \publ Hermann \publaddr Paris \yr 1971 \endref \ref \key 2 \by C. J. K. Batty, W. Hutter and F. 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