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\def\rightheadline{EJDE--2000/29\hfil Uniform exponential stability
\hfil\folio}
\def\leftheadline{\folio\hfil D. N. Cheban
\hfil EJDE--2000/29}
\def\pretitle{\vbox{\eightrm\noindent\baselineskip 9pt %
Electronic Journal of Differential Equations,
Vol.~{\eightbf 2000}(2000), No.~29, pp.~1--18.\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 almost periodic systems in Banach spaces
\endtitle
\thanks
{\it 2000 Mathematics Subject Classifications:} 34C35, 34C27, 34K15, 34K20, 58F27, 34G10.
\hfil\break\indent
{\it Key words and phrases:} non-autonomous linear dynamical systems,
global attractors, \hfill\break\indent
almost periodic system, exponential stability, asymptotically compact systems.
\hfil\break\indent
\copyright 2000 Southwest Texas State University and
University of North Texas.\hfil\break\indent
Submitted January 4, 2000. Published April 17, 2000.
\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 linear non-autonomous dynamical
systems possessing the property of uniform exponential stability.
We prove that if the Cauchy operator of these systems possesses
a certain compactness property, then the uniform asymptotic
stability implies the uniform exponential stability. For recurrent
(almost periodic) systems this result is precised.
We also show application for different classes of linear evolution
equations: ordinary linear differential equations in a Banach space,
retarded and neutral functional differential equations, and
some classes of evolution partial differential equations.
\endabstract
\endtopmatter
\document
\head Introduction \endhead
Let $A(t)$ be a continuous $ n\times n$ matrix-function and $ H(A)$
be the family of all matrix-functions
$ B = \lim \limits_{n \to + \infty } A_{t_n}$,
where $\{t_n\} \subset \Bbb R$, $A_{t_n}(t) = A (t_n + t) $
and the convergence
$ A_{t_n} \to B $ is uniform on every compact subset of $\Bbb R$.
The following result is well known.
\proclaim{Theorem [25,2,6]} Let $A$ be a bounded and uniformly
continuous
matrix-function on $\Bbb R$, then the following conditions are equivalent:
\item{1.} The trivial solution of equation
$$ x'=A(t)x \eqno (0.1) $$
is uniformly exponentially stable.
\item{2.} The trivial solution of equation (0.1) is uniformly asymptotically stable.
\item{3.} The trivial solution of equation (0.1) and every equation
$$ y'= B(t)y \quad ( B \in H(A) ) \eqno (0.2) $$
is asymptotically stable.
\endproclaim
\noindent For equations in infinite-dimensional spaces conditions 1, 2, and 3
are not equivalent; see examples in [10, 24, 15].
However, in the general infinite-dimensional case
condition 1 implies condition 2, and condition 2 implies condition 3.
Linear non-autonomous dynamical systems satisfying one of the
conditions 1, or 2, or 3 are studied in [10]; see also Theorem 1.1 below.
In this article we show that if the operator corresponding to the
the Cauchy problem for (0.1) satisfies some compactness condition,
then condition 3 implies condition 1 (see Theorems 2.3 and 2.4).
For recurrent (almost periodic) systems this result is made precise
in Theorems 3.2, 3.3 and 3.4.
Applications of this result to different classes of linear evolution
equations (ordinary linear differential equations
in a Banach space, retarded and neutral functional differential equations, some classes
of evolution partial differential equations) are given.
\head 1. Linear non-autonomous dynamical systems \endhead
Assume that $X$ and $Y$ are complete metric spaces, $\Bbb R$
is the set of real numbers, $\Bbb Z$ is the set of integer numbers,
$\Bbb T = \Bbb R $ or $\Bbb Z$, $\Bbb T_{+} =
\{ t \in \Bbb T : t \ge 0 \}$ and
$\Bbb T_{-} = \{ t \in \Bbb T | t \le 0 \}$.
Denote by $(X, \Bbb T_{+},\pi)$ a semigroup dynamical system on
$X$ and $ (Y,\Bbb T, \sigma ) $ a group dynamical system on $Y$.
A triple $\langle (X, \Bbb T_{+},\pi), (Y,\Bbb T, \sigma ), h \rangle$,
where $ h $ is a homomorphism of $ (X,\Bbb T_{+}, \pi ) $ onto
$ (Y, \Bbb T, \sigma ),$ is
called a non-autonomous dynamical system.
Systems $(X,\Bbb T_{+},\pi)$ have be classified as follows: (see [7,9])
\item{$\bullet$} 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)$.
\item{$\bullet$} Compactly dissipative, if (1.1) holds uniformly with
respect to $x$ on compact subsets of $X$.
\item{$\bullet$} Locally dissipative, if for any
point $p\in X$ there is $\delta_{p} > 0$ such that (1.1)
takes place uniformly with respect to $x\in B(p,\delta_{p}) = \{ x \in X :
\rho (x,p) < \delta_{p} \}$.
\noindent Let $(X,T,\pi)$ be compactly
dissipative and $K$ be a compact set that is
attractor of all compact subsets of $X$, and let
$$J=\Omega(K)=\cap_{t\geq0} \overline
{\cup_{\tau\geq t}\pi^\tau K}\,. $$
Then the set $J$ does not depend on selection of the attractor $K$,
and is characterized by the properties of the
dynamical system $(X,T,\pi)$ only; see, for example, [7,9,19,20]).
The set $J$ is called the Levinson centre of the compactly dissipative
system $(X,T,\pi)$.
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 $Y$ [3].
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 $| \cdot | $ be
some norm on $ (X,h,Y) $ such that $| \cdot | $
is co-ordinated with the metric $ \rho $ on $X$ (that is
$ \rho (x_1,x_2)=| x_1-x_2| $ for any
$ x_1,x_2 \in X $ such that $ h(x_1)=h(x_2) $).
Point, compactly, and locally dissipativity criteria for linear systems
are obtained in [10].
The entire trajectory of the semigroup dynamical system $(X, \Bbb T_{+},\pi) $
passing through the point $ x \in X $ at $ t=0$ is defined as the
continuous map $ \gamma : \Bbb T \to X $ that satisfies the conditions
$ \gamma (0)=x $ and $ \pi ^{t}\gamma (s)=\gamma (s+t) $ for all $ t \in \Bbb T_{+}$ and
$ s \in \Bbb T $. Let $\Phi_{x}$ be the set of all entire trajectories
of $(X, \Bbb T_{+},\pi) $ passing through $ x $ at $ t=0$ and
$\Phi =\cup \{ \Phi_{x} : x \in X \}$.
\proclaim{Theorem 1.1 [10]} Let
$ \langle (X, \Bbb T_{+},\pi), (Y,\Bbb T, \sigma ), h \rangle $
be a linear non-autonomous dynamical system and $Y$ be a
compact set. Then the following assertions hold:
\item {1.}$ \langle (X, \Bbb T_{+},\pi), (Y,\Bbb T, \sigma ), h \rangle $
is point dissipative if and only if
$ \lim \limits_{t \to + \infty } | xt | = 0 $ for all $ x \in X $.
\item {2.} The non-autonomous dynamical system
$ \langle (X, \Bbb T_{+},\pi), (Y,\Bbb T, \sigma ), h \rangle $
is compactly dissipative if and only if
$ \langle (X, \Bbb T_{+},\pi), (Y,\Bbb T, \sigma ), h \rangle $
is point dissipative and there exists a number $ M \ge 0 $
such that the inequality
$$ | xt | \le M| x | \eqno (1.2) $$
takes place for all $ x \in X $ and $ t \in \Bbb T_{+}$.
\item{3.} The non-autonomous dynamical system
$ \langle (X, \Bbb T_{+},\pi), (Y,\Bbb T, \sigma ), h \rangle $
is locally dissipative if and only if there exist positive numbers
$N$ and $ \nu $ such that the inequality
$ | xt | \le N e^{-\nu t}| x | $
takes place for all $ x \in X $ and $ t \in \Bbb T_{+}$.
\endproclaim
From the Banach-Steinhauss theorem it follows that point dissipativity
and compact dissipativity are equivalent for autonomous linear systems.
An example of linear autonomous dynamical system
which is compactly dissipative, but is not locally dissipative
is constructed in [10].
\proclaim{Theorem 1.2 [7,8]} Let
$ \langle (X, \Bbb T_{+},\pi), (Y,\Bbb T, \sigma ), h \rangle $
be a linear non-autonomous dynamical system, $Y$ be a compact set.
Then the following assertions take place:
\item{1.} If $ (X, \Bbb T_{+} , \pi ) $ is completely continuous (i.e. for all bounded
subset $ A \subset X $ there exists a positive number $ l= l(A) $ such that
$ \pi ^{l}A $ is precompact), then from point dissipativity of
$ \langle (X, \Bbb T_{+},\pi), (Y,\Bbb T, \sigma ), h \rangle $
follows its local dissipativity;
\item{2.} If $ (X, \Bbb T_{+} , \pi ) $ is asymptotically compact (i.e. for all bounded
sequences $\{x_n\} \subset X $ and $\{ t_n \} \to + \infty $ the sequence
$\{ x_nt_n \}$ is precompact if it is bounded), then from compact dissipativity of
$ \langle (X, \Bbb T_{+},\pi), (Y,\Bbb T, \sigma ), h \rangle $
results its local dissipativity.
\endproclaim
\noindent Recall that a measure of noncompactness [20, 27] on a complete metric
space $X$ is a function $\beta $ from the bounded sets of $X$ to the
nonnegative real numbers satisfying:
\item{(i)}$\beta (A)=0 $ for $ A \subset X $ if and only if $A$ is
precompact
\item{(ii)}$\beta (A\cup B)=\max [\beta (A),\beta (B)] $
\item{(iii)}$\beta (A + B)\le \beta (A) + \beta (B)$ for all
$A,B \subset X $ if the
space $X$ is linear.
\noindent The Kuratowski measure of non-compactness $\alpha $ is defined
by
$$\alpha (A)=\inf \{ d : A \text{ has a finite cover of diameter} < d \}.$$
The dynamical system $ (X, \Bbb T_{+}, \pi )$ is said to be
conditionally $\beta$-condensing [20]
if there exists $ t_0 > 0 $ such that $\beta (\pi ^{t_0}B) < \beta (B) $
for all bounded sets $ B $ in $X$ with $\beta (B) > 0$.
The dynamical system $ (X, \Bbb T_{+}, \pi )$ is said to be
$\beta$-condensing if it is
conditionally $\beta$-condensing and the set $ \pi ^{t_0}B $ is
bounded for all bounded sets $ B \subseteq X $.
According to Lemma 2.3.5 in [20, p.15] and Lemma 3.3 in [7] the
conditional condensing dynamical system
$ (X, \Bbb T_{+}, \pi )$ is asymptotically compact.
Let $ X=E\times Y $, $ A \subset X$, and
$ A_{y}=\{ x \in A : pr_2x=y \}$. Then
$ A= \cup \{ A_{y} : y \in Y \}$. Let $ \tilde A_{y}=
pr_1A_{y}$ and $ \tilde A= \cup \{ \tilde A_{y} : y \in Y \}$.
Note that if the space $Y$ is compact, then a set $ A \subset X $ is bounded in $X$
if and only if the set $ \tilde A $ is bounded in $ E $.
\proclaim{Lemma 1.3} The equality $\alpha (A) = \alpha (\tilde A ) $
takes place for all bounded sets $ A \subset X$, where
$\alpha (A)$ and $\alpha (\tilde A )$ are the Kuratowski measure of
non-compactness for the sets $A \subset X$ and $\tilde A \subset E$.
\endproclaim
\demo{Proof} Let $\varepsilon > 0 $ and $A$ be a bounded subset in
$X$, then there are sets $ A_1,A_2,\dots,A_n $ such that
$ A=\cup \{ A_i : i=1,2,\dots,n \}$
and $ \operatorname{diam} A_i < \alpha (A) + \varepsilon$.
Note that $ \tilde A =\cup \{ \tilde A_i : i=1,2,\dots ,n \}$ and
$\operatorname{diam} \tilde A_i \le \operatorname{diam} A_i
< \alpha (A) + \varepsilon $,
and consequently, $\alpha (\tilde A) \le \alpha (A)$.
Let $ \varepsilon$ be a positive constant, $A $ be a bounded set in $ X$,
$\tilde A = \cup \{ \tilde A_{k}: k=1,2,\dots ,m \}$ and
$\operatorname{diam} \tilde A_{k} < \alpha (\tilde A ) + \varepsilon $.
Since $Y$ is compact, there are sets
$Y_1, Y_2, \dots, Y_{\ell}$ such that
$ Y_1\cup Y_2 \cup \dots \cup Y_{\ell} =Y $ and
$\operatorname{diam}Y_j < \varepsilon$
$(j=1,2,\dots \ell )$. Let $ A_i = pr_1^{-1} (\tilde A_i)\cap A$,
and
$$ A_{ij}=pr_2^{-1} (pr_2(pr_1^{-1}(\tilde A_i)\cap A )
\cap Y_j ) \cap A_i\,. $$
Note that $ A_{ij} \subseteq \tilde A_i \times Y_j $, and
that
$$\operatorname{diam}A_{ij} \le\operatorname{diam}\tilde A_i
+\operatorname{diam}Y_j < \alpha (\tilde A ) +\varepsilon
+ \varepsilon = \alpha (A) + 2\varepsilon\,.$$
Since $ A=\cup \{ A_{ij} : i=1,2,\dots ,n , j=1,2,\dots ,\ell \}$,
it follows that $\alpha (A) \le \alpha (\tilde A) $
and $\alpha (A)=\alpha (\tilde A )$. wich concludes the present
proof.
\enddemo
Let $ E $ be a Banach space and
$ \varphi :\Bbb T_{+} \times E \times Y \mapsto E$ be a continuous mapping with
$ \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_{+}$.
The triplet
$ \langle E,\varphi , (Y,\Bbb T, \sigma ) \rangle $ is called
a continuous cocycle on $ (Y,\Bbb T, \sigma ) $ with fibre $ E $.
The dynamical system $ (X, \Bbb T_{+},\pi) $ is called
a skew-product system [25] 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_{+}$).
A cocycle $ \varphi $ is called conditionally $\alpha$-condensing
if there exists
$ t_0 > 0 $ such that for any bounded set
$ B \subseteq E $ the inequality $\alpha ( \varphi (t_0,B,Y ))
< \alpha (B) $ holds if $\alpha (B) > 0 $. The cocycle $ \varphi $ is called
$\alpha$-condensing if it is a conditional $\alpha$-condensing cocycle and
the set $ \varphi (t_0,B,Y ) = \cup \{ \varphi (t_0,u,Y )
| u \in B , y \in Y \}$ is bounded for all bounded set $ B \subseteq E $.
A cocycle $ \varphi $ is called conditional $\alpha$-contraction of order
$ k \in [0,1) $, if there exists
$ t_0 > 0 $ such that for any bounded set
$ B \subseteq E $ for which $ \varphi (t_0,B,Y ) = \cup \{ \varphi (t_0,u,Y )
| u \in B , y \in Y \}$ is bounded
the inequality $\alpha ( \varphi (t_0,B,Y ))
\le k \alpha (B) $ holds. The cocycle $ \varphi $ is called
$\alpha$-contraction if it is a
conditional $\alpha$-contraction cocycle and
the set $ \varphi (t_0,B,Y ) = \cup \{ \varphi (t_0,u,Y )
| u \in B , y \in Y \}$ is bounded for all bounded sets $ B \subseteq E $.
\proclaim{Lemma 1.4} Let $Y$ be compact and the cocycle $ \varphi $ be $\alpha$-condensing.
Then the skew-product dynamical system $ (X,\Bbb T_{+}, \pi ) $, generated by
the cocycle $ \varphi $, is $\alpha$-condensing.
\endproclaim
\demo{Proof} Let $ A \subset X $ be a bounded subset, $t_0 > 0$ and $\alpha (A) > 0, $
then
$$\eqalign{
\pi (t_0,A) =& \cup \{ \pi (t_0,A_{y} | y \in Y \} \cr
=&\cup \{ ( \varphi (t_0,A_{y},y),y t ) | y \in Y \}
\subseteq \varphi (t_0,\tilde A,Y ) \times Y .\cr} \eqno (1.3)
$$
Since $A$ is bounded, $\tilde A $ is also bounded in $ E $ and according
to the condition
of the lemma the set $ \varphi (t_0,\tilde A, Y ) $ is bounded and, consequently,
$ \pi (t_0,A) $ is bounded. According to Lemma 1.3 and (1.3) we have
$$\alpha (\pi (t_0,A))=\alpha ( \cup \{ (\varphi (t_0,A_{y},y),yt_0) |
y \in Y \} ) \le \alpha ( \varphi (t_0,\tilde A,Y )) < \alpha (\tilde A ) =\alpha (A).$$
The lemma is proved.
\enddemo
\proclaim{Theorem 1.5} Let $ E $ be a Banach space, $\varphi $ be a cocycle on
$ (Y, \Bbb T, \sigma ) $ with fibre $E$ and the following conditions be
fulfilled:
\item{1.}$ \varphi (t,u,y)=\psi (t,u,y)+\gamma (t,u,y) $ for all $ t\in
\Bbb T_{+},u \in E $ and $ y \in Y .$
\item{2.} There exists a function
$ m :\Bbb R_{+} \to \Bbb R_{+}$
satisfying the condition $ m(t) \to 0 $ as $ t \to
+ \infty $ such that
$ | \psi(t,u_1,y)- \psi(t,u_2,y)| \le m(t)
| u_1-u_2| $ for all
$ t \in \Bbb T_{+} , u_1,u_2 \in E $ and $ y \in Y $.
\item{3.}$ \gamma (t,A,Y ) $ is compact for all bounded $ A \subset X $ and $ t > 0 $.
Then the cocycle $ \varphi $ is an $\alpha$-contraction.
\endproclaim
\demo{Proof} Let $ \varepsilon>0$ and $A $ be a bounded set in $ E $,
then there are sets $ A_1, A_2,\dots ,A_n$
such that $ A=\cup \{ A_i : i=1,2,\dots ,n \}$ and
$\operatorname{diam}A_i < \alpha (A) + \varepsilon$ for $i=1,2,\dots ,n$.
Since $Y$ is compact, then there are
a sets $ Y_1, Y_2,\dots , Y_m $ such that $ Y_1 \cup
Y_2\cup \dots \cup Y_m =Y $ with condition $\operatorname{diam}Y_j <
\varepsilon $ for all $ j=1,2,\dots ,m $.
Let $ t_0 $ be a positive number such that $m(t_0) < 1 $. We note that
$$ \eqalign{\varphi (t_0,A,Y ) & \subseteq \psi (t_0,A,Y ) + \gamma (t_0,A,Y) \cr
&=
\cup \{ \psi (t_0,A_i,Y_j) | i=1,2,\dots ,n; j=1,2,..,m \} +
\gamma (t_0,A,Y ).\cr} \eqno (1.4) $$
According to the conditions of Theorem 1.5, $\alpha (\gamma (t_0,A,Y ))=0 $ and
$$\operatorname{diam}\psi (t_0,A_i,y) \le m(t_0)\operatorname{diam} A_i $$
for all $ y \in Y $. Thus we have
$$ \eqalign{| \psi(t_0,u_1,y_1)- \psi(t_0,u_2,y_2)|
&\le | \psi(t_0,u_1,y_1)- \psi(t_0,u_2,y_1)| \cr
& + | \psi(t_0,u_2,y_1)- \psi(t_0,u_2,y_2)| \cr}
\eqno (1.5) $$
and, consequently,
$$\operatorname{diam}\psi (t_0,A_i,y) \le m(t_0)\operatorname{diam} A_i +
\operatorname{diam}\psi(t_0,u_2,Y_j) \quad \text{for all} \quad y \in Y_j \,.
\eqno (1.6) $$
Since $Y$ is compact, from (1.5)-(1.6) follows the inequality
$$\operatorname{diam}\psi (t_0,A_i,Y_j) \le m(t_0)\operatorname{diam} A_i \le m(t_0)
( \alpha (A) + \varepsilon) $$
and, consequently, $\alpha (\varphi (t_0,A,Y))\le m(t_0) \alpha (A). $ The theorem
is proved.
\enddemo
\head
2. Exponential stable linear non-autonomous dynamical systems
\endhead
\proclaim{Lemma 2.1 [14]} Let $ m : \Bbb T_{+} \to \Bbb T_{+}$ satisfy the following
conditions:
\item{1.} There exists a positive constant $ M $ such that $ m(t) \le M $ for all
$ t\in \Bbb T_{+}$.
\item{2.}$ m(t) \to 0 $ as $ t \to +\infty $.
\item{3.}$ m(t+\tau)\le m(t)m(\tau)$ for all $ t,\tau \in \Bbb T_{+}$.
\noindent Then there exist two positive constants $N$ and $ \nu $ such that
$ m(t) \le N e^ {-\nu t}$ for all $ t\in \Bbb T_{+}$.
\endproclaim
\proclaim{Theorem 2.2} Let
$ \langle (X, \Bbb T_{+},\pi), (Y,\Bbb T, \sigma ), h \rangle $
be a linear non-autonomous dynamical system, $Y$ be a compact set.
Then the following conditions are equivalent:
\item{1.} 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
$ | \pi (t,x) | \le N e^ {-\nu t}| x | $ for all
$ t\in \Bbb T_{+}$ and $ x \in X $.
\item{2.}$\| \pi^{t}\| \to 0 $ as $ t \to +\infty $, where
$\| \pi^{t}\| = \sup \{ | \pi^{t} x | :
x \in X, | x | \le 1 \}. $
\item{3.} The non-autonomous dynamical system
$ \langle (X, \Bbb T_{+},\pi), (Y,\Bbb T, \sigma ), h \rangle $
is locally dissipative.
\endproclaim
\demo{Proof} According to Theorem 1.1, conditions 1 and 3 are equivalent.
Now we will prove that the conditions 1 and 2 are equivalent.
It is clear that from 1 follows 2.
According to condition 2 there exists $ L>0 $ such that
$$\| \pi^{t}\| \le 1 \eqno (1.7) $$
for all $ t \ge L $.
We claim that the family of operators
$\{ \pi ^t : t \in [0,L] \}$ is uniformly continuous, that is,
for any $ \varepsilon > 0 $
there is a $ \delta (\varepsilon ) > 0 $ such that $ | x | \le \delta $ implies
$ | xt | \le \varepsilon $ for all $ t \in [0,L]$. On the contrary, assume that
there are $ \varepsilon_0 > 0$, $\delta_n > 0$ with $\delta_n \to 0$,
$| x_n | < \delta_n $ and $ t_n \in [0,L] $ such that
$$
| x_nt_n | \ge \varepsilon_0 . \eqno (1.8) $$
Since $ (X,h,Y) $ is a locally trivial Banach fibre bundle and $Y$ is compact,
the zero section $ \Theta = \{\theta_y : y \in Y, \theta_y \in X_{y},
| \theta_{y} | = 0 \}$ of $ (X,h,Y ) $
is compact and, consequently, we can assume that the sequences $\{ x_n\}$
and $\{t_n\}$ are convergent. Put $ x_0=\lim \limits_{n \to + \infty } x_n $
and $ t_0=\lim \limits_{n \to + \infty } t_n $, then $ x_0=
\theta_{y_0} \quad ( y_0=h(x_0) )$.
Passing to the limit in (1.8) as $ n \to + \infty $, we obtain
$ 0 = | x_0t_0| \ge \varepsilon_0$. This last inequality contradicts the
choice of $ \varepsilon_0$, and hence proves the above assertion.
If $ \gamma > 0 $ is such that
$ | \pi ^tx | \le 1 $
for all $ | x | \le \gamma $ and $ t \in [0,L]$, then
$$ | xt | \le \frac{1}{\gamma} | x | \eqno (1.9)
$$
for all $ t \in [0,L] $ and $ x \in X $. We put
$ M =\max \{ \gamma ^{-1} , 1 \}$, then from (1.7) and (1.9)
follows
$$\| \pi^{t}\| \le M \eqno (1.10)
$$
for all $ t \ge 0 $ and $ x \in X $.
Consider the function $ m(t)=\| \pi^{t}\| $.
We note that
$ m(t+\tau)\le m(t)m(\tau)$ for all $ t,\tau \in \Bbb T_{+}$
and $ m(t) \le M $ for all
$ t\in \Bbb T_{+}$ and $ m(t) \to 0 $ as $ t \to +\infty $. According to Lemma 2.1
there exist positive numbers $N$ and $ \nu $ such that
$ m(t) \le N e^ {-\nu t}$ for all $ t\in \Bbb T_{+}$. Thus
$ |\pi (t,x) | \le\| \pi ^{t}\| | x | \le
N e^ {-\nu t}| x | $ for all $ t\in \Bbb T_{+}$ and $ x \in X$.
The theorem is proved.
\enddemo
Let $\Bbb B = \{ x \in X : \exists \gamma \in \Phi_{x}$
such that $ \sup \limits_{t\in \Bbb T } | \gamma (t) | < + \infty \}$.
\proclaim{Theorem 2.3} Let
$ \langle (X, \Bbb T_{+},\pi), (Y,\Bbb T, \sigma ), h \rangle $
be a linear non-autonomous dynamical system, $Y$ be compact
and $ (X, \Bbb T_{+},\pi) $ be conditionally $\alpha$-condensing. Then the following
assertions are equivalent:
\item{1.} The non-autonomous dynamical system
$ \langle (X, \Bbb T_{+},\pi), (Y,\Bbb T, \sigma ), h \rangle $
is point dissipative and this system
doesn't admit non-trivial bounded trajectories on $\Bbb T, $ i.e.
$\Bbb B \subseteq \Theta = \{\theta_y : y \in Y, \theta_y \in X_{y},
| \theta_{y} | = 0 \}.$
\item{2.} The non-autonomous dynamical system
$ \langle (X, \Bbb T_{+},\pi), (Y,\Bbb T, \sigma ), h \rangle $
is uniformly exponentially stable.
\endproclaim
\demo{Proof} Denote by $ \Theta = \{\theta_y : y \in Y, \theta_y \in X_{y},
| \theta_{y} | = 0 \}$
the zero section of the vector fibering
$ (X,h,Y) $. Since $ (X,h,Y) $ is locally trivial and $Y$ is compact, the
zero section $\Theta$ is compact and an invariant set of the dynamical system
$(X, \Bbb T_{+},\pi) $. Taking into account that the dynamical system
$ (X, \Bbb T_{+},\pi) $ is conditionally $\alpha$-condensing, according to
Theorem 2.4.8 [20]
the set $\Theta$ is orbitally stable and in particular there exists a positive
constant $N$ such that
$ | xt | \le N| x | $ for all $ t \in \Bbb T_{+}$ and
$ x \in X $.
By virtue of Theorem 1.1 the dynamical system $ (X, \Bbb T_{+},\pi) $ is compactly dissipative
and according to Theorem 1.2 $ (X, \Bbb T_{+},\pi) $ is locally dissipative.
It follows from Theorem 2.2 that $ (X, \Bbb T_{+},\pi) $ is uniformly exponentially stable.
Let now the non-autonomous dynamical system
$ \langle (X, \Bbb T_{+},\pi), (Y,\Bbb T, \sigma ), h \rangle $
be uniformly exponentially stable, then according to Theorem 2.2 it is locally dissipative.
Let $ J $ be its Levinson's centre (i.e. maximal compact invariant set
of dynamical system $ (X, \Bbb T_{+},\pi) $) . We note that according to the linearity of
non-autonomous dynamical system
$ \langle (X, \Bbb T_{+},\pi), (Y,\Bbb T, \sigma ), h \rangle $
we have $ J=\Theta $. Let $ \varphi $ be a entire bounded trajectory of dynamical
system $ (X, \Bbb T_{+},\pi) $ . Since the non-autonomous dynamical system
$ \langle (X, \Bbb T_{+},\pi), (Y,\Bbb T, \sigma ), h \rangle $
is conditionally $\alpha$-condensing, in particulary, it is asymptotically compact
and the set $ M = \varphi (\Bbb T) $ is precompact. In fact, the set $ M $ is invariant
$ \Omega (M) = \overline{M}$ and in view of Lemma 3.3 [7] the set M is precompact.
We note that $ \varphi (\Bbb T) \subseteq J= \Theta $ because $ J $ is the maximal compact invariant
set of $ (X, \Bbb T_{+},\pi) $. The theorem is proved.
\enddemo
\proclaim{Remark 2.4} Theorem A in [26] implies a version
of Theorem 2.3 under slightly stronger assumptions.
\endproclaim
\proclaim{Theorem 2.5} Let
$ \langle (X, \Bbb T_{+},\pi), (Y,\Bbb T, \sigma ), h \rangle $
be a linear non-autonomous dynamical system, $Y$ be compact
and $ (X, \Bbb T_{+},\pi) $ be completely continuous, i.e. for any bounded set
$ A \subseteq X $ there exists a positive number $ \ell $ such that
$ \pi ^{\ell}(A) $ is precompact. Then the following
assertions are equivalent:
\item{1.} The non-autonomous dynamical system
$ \langle (X, \Bbb T_{+},\pi), (Y,\Bbb T, \sigma ), h \rangle $
is uniformly exponentially stable.
\item{2.}$ \lim \limits_{t \to + \infty}| \pi ^{t}x| =0 $
for all $ x \in X $.
\endproclaim
\demo{Proof} It is clear that condition 1 implies 2. Now we will show that
condition 1 follows from 2. According to Theorem 1.1
the non-autonomous dynamical system
$ \langle (X, \Bbb T_{+},\pi), (Y,\Bbb T, \sigma ), h \rangle $
is point dissipative. Since the dynamical system $ (X, \Bbb T_{+},\pi) $
is completely continuous, by virtue of Theorem 1.2
the dynamical system $ (X, \Bbb T_{+},\pi) $ is locally dissipative. To prove the theorem
it is sufficient to refer to Theorem 2.2.
\enddemo
\head
3. Linear non-autonomous dynamical system with a minimal base
\endhead
In this section we study a linear non-autonomous dynamical system
$ \langle (X, \Bbb T_{+},\pi),$\linebreak $ (Y,\Bbb T, \sigma ), h \rangle $
with compact minimal base $ (Y,\Bbb T, \sigma ) $.
\proclaim{Theorem 3.1 [4,5]} Let
$ \langle (X, \Bbb T_{+},\pi), (Y,\Bbb T, \sigma ), h \rangle $
be a linear non-autonomous dynamical system and the following
conditions hold:
\item{1.}$Y$ is compact and minimal ( i.e. $ Y=H(y) = \overline {\{ yt : t \in \Bbb T \}}$
for all $ y \in Y $ );
\item{2.} for any $ x \in X $ there exists $ C_{x} \ge 0 $ such that
$ | xt | \le C_{x}$
for all $ t \in \Bbb T_{+}$;
\item{3.} the mapping $ y \mapsto\| \pi ^{t}_{y}\| $ is continuous, where
$\| \pi ^{t}_{y}\| $ is the norm of the 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.
\noindent Then there exists $ M \ge 0 $ such that
$$ | \pi (t,x)| \le M| x | \eqno (3.1) $$
holds for all $ t \in \Bbb T_{+}$ and $ x \in X.$
\endproclaim
\proclaim{Theorem 3.2} Let
$ \langle (X, \Bbb T_{+},\pi), (Y,\Bbb T, \sigma ), h \rangle $
be a linear non-autonomous dynamical system and the following
conditions hold:
\item{1.}$Y$ is compact and minimal.
\item{2.} The dynamical system $(X, \Bbb T_{+},\pi) $ is asymptotically compact.
\item{3.} The mapping $ y \mapsto\| \pi ^{t}_{y}\| $ is continuous, where
$\| \pi ^{t}_{y}\| $ is the norm of the 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.
\noindent Then the non-autonomous dynamical system
$ \langle (X, \Bbb T_{+},\pi), (Y,\Bbb T, \sigma ), h \rangle $
is uniformly exponentially stable if and only if
$$ \lim \limits_{t \to + \infty}| \pi ^{t}x| =0 \eqno (3.2) $$
for all $ x \in X $.
\endproclaim
\demo{Proof} It is clear that from uniform exponential stability of
the non-autonomous dynamical system
$ \langle (X, \Bbb T_{+},\pi), (Y,\Bbb T, \sigma ), h \rangle $
follows the equality (3.2).
From condition (3.2) and minimality of $ (Y,\Bbb T, \sigma ) $ by virtue of
Theorem 3.1 it follows that for the non-autonomous dynamical system
$ \langle (X, \Bbb T_{+},\pi), (Y,\Bbb T, \sigma ), h \rangle $
the inequality (3.1) holds and according to Theorem 1.1 this system is
compactly dissipative.
Since the dynamical system $ (X, \Bbb T_{+},\pi) $ is asymptotically compact, to finish
the proof of the Theorem it is sufficient to remark that according to Theorem 2.13 [8]
every compactly dissipative and asymptotically compact dynamical system is locally dissipative.
Thus a linear non-autonomous dynamical system
$ \langle (X, \Bbb T_{+},\pi), (Y,\Bbb T, \sigma ), h \rangle $
is locally dissipative and, consequently, it is uniform exponentially stable.
The theorem is proved.
\enddemo
\proclaim{Theorem 3.3 [5]} Assume that
$ \langle (X, \Bbb T_{+},\pi), (Y,\Bbb T, \sigma ), h \rangle $
is a linear non-autonomous dynamical system, generated by a cocycle $\varphi $,
$(X, \Bbb T_{+},\pi) $ is asymptotically compact,
and there is an $ M>0 $ such that
$ | \varphi (t,u,y) | \le M | u | $ for all
$ (u,y) \in X=E\times Y $ and $ t \in \Bbb T_{+}$.
Then the following assertions hold.
\item{(i)} For any $ (u,y) \in \Bbb B = \{ x \in X : \exists \gamma \in \Phi_{x}$
such that $ \sup \limits_{t\in \Bbb T } | \gamma (t) | < + \infty \}$
the set $ \Phi_{(u,y)}$ consists of a single entire recurrent trajectory.
\item{(ii)}$\Bbb B $ is closed in $X$.
\item{(iii)}$(X, \Bbb T_{+},\pi) $ induces a group dynamical system
$ (\Bbb B ,\Bbb T ,\pi ) $ on $\Bbb B $.
\item{(iv)}$ ( \Bbb B ,h,Y ) $ is a finite dimensional vector subfibering of
$ (X,h,Y) $, i.e. $ \dim \Bbb B_{y}$ does not depend on $ y \in Y $.
\endproclaim
\proclaim{Theorem 3.4} Suppose that the following conditions are satisfied:
\item{1.} A dynamical system $ (Y, \Bbb T ,\sigma ) $is compact and minimal.
\item{2.} A linear non-autonomous dynamical system
$ \langle (X, \Bbb T_{+},\pi), (Y,\Bbb T, \sigma ), h \rangle $
is generated by cocycle $ \varphi $ (i.e. $X=E\times Y $,
$ \pi = (\varphi ,\sigma) $ and $ h=pr_2:X \mapsto Y $).
\item{3.} The dynamical system $(X, \Bbb T_{+},\pi) $ is conditionally $\alpha$-condensing.
\item{4.} There exists a positive number $ M $ such that $ | \varphi (t,u,y)
| \le M | u | $ for all $ t \in Y $ and $ t \in \Bbb T_{+}$.
\noindent Then there are two vectorial positively invariant subfiberings $ (X^{0},h,Y) $
and $ (X^{s},h,Y) $ of $ (X,h,Y) $ such that:
\item{a.}$ X_{y}= X_{y}^{0}+X_{y}^{s}$ and $ X_{y}^{0}\cap X_{y}^{s}=0_{y}$ for all $ y \in Y$,
where $ 0_{y} =(0,y)\in X=E\times Y$ and $ 0 $ is the zero in the Banach space
$ E $.
\item{b.} The vector subfibering $ (X^{0},h,Y) $ is finite dimensional, invariant
(i.e. $ \pi ^{t}X^{0}=X^{0}$ for all $ t \in \Bbb T_{+}$) and every trajectory
of the dynamical system $(X, \Bbb T_{+},\pi) $ belonging to $X^{0}$ is recurrent.
\item{c.} There exist two positive numbers $N$ and $\nu$ such that
$| \varphi (t,u,y)| \le Ne^{-\nu t}| u | $ for all $ (u,y) \in
X^{s}$ and $ t \in \Bbb T_{+}$.
\endproclaim
\demo{Proof} Let $ X^{0}=\Bbb B $, then according to Theorem 3.3,
statement b holds. Denote by $ P_{y}$ the projection of $ X_{y}= h^{-1}(y) $
to $\Bbb B_{y} =\Bbb B \cap h^{-1}(y)$, then $ P_{y}(u,y)= (\Cal P (y) u, y) $
for all $ u \in E , \Cal P ^{2}(y)= \Cal P (y)$ and the mapping
$ \Cal P : Y \to [E] $ ( $ y \mapsto \Cal P (y) $)
is continuous, where by $ [E] $ denotes the set of all
linear continuous operators acting on $ E $.
Now we set $ X^{s}_{y}=\Cal Q (y) X_{y}$
and $ X^{s}=\cup \{ X^{s}_{y} : y \in Y \}$, where $ \Cal Q (y) =
Id_{E} - \Cal P (y) $. We will show that $ X^{s}$ is closed in $X$. In fact,
let $\{ x_n\}=\{(u_n,y_n)\} \subseteq X^{s}$ and $ x_0=(u_0,y_0) =
\lim \limits_{n \to \infty } x_n $. Note that $ P_{y_0}(x_0)
= ( \Cal P (y_0)u_0, y_0) = ( \lim \limits_{n \to \infty }
\Cal P (y_n)u_n,y_0) = (0, y_0)= 0_{y_0}$ and, consequently,
$ x_0 \in X^{s}_{y_0} \subseteq X^{s}$.
Let $ (X^{s},\Bbb T_{+} ,\pi ) $ be the dynamical system induced by
$ (X,\Bbb T_{+} ,\pi ) $. It is clear that under the conditions of Theorem 3.4
the dynamical system $ (X^{s},\Bbb T_{+} ,\pi ) $ is asymptotically compact and every
positive semi-trajectory is precompact and, consequently,
$ \lim \limits_{t \to \infty } | \pi (t,x) | = 0 $ for all $ x \in X^{s}$
because the dynamical system $ (X^{s},\Bbb T_{+} ,\pi ) $ doesn't have a non-trivial
entire trajectory bounded on $\Bbb T $. In fact, if we suppose that it is not true,
then there exist $ x_0 = (u_0,y_0) $ and $ t_n \to + \infty $ such that:
$ | u_0| \not= 0, \lim \limits_{n\to + \infty} \pi (t_n,x) = x_0 $ and
through point $ x_0 $ pass a non-trivial
entire trajectory bounded on $\Bbb T $. This contradiction proves the necessary assertion.
Thus we can apply Theorem 2.3 according
which there exist two positive constants $N$ and $\nu $ such that
$| \varphi (t,u,y)| \le Ne^{-\nu t}| u | $ for all $ (u,y) \in
X^{s}$ and $ t \in \Bbb T_{+}$. The theorem is proved.
\enddemo
\head
4. Some classes of linear uniformly exponentially stable
non-autonomous differential equations
\endhead
Let $ \Lambda $ be a complete metric space of linear operators that act on a Banach
space $ E $ and $ C(\Bbb R,\Lambda ) $ be the space of all continuous operator-functions
$ A: \Bbb R \to \Lambda $ equipped with the open-compact topology and
$ ( C(\Bbb R, \Lambda ),\Bbb R, \sigma ) $ be the dynamical system of shifts on
$ C( \Bbb R, \Lambda ) $.
\subhead Ordinary linear differential equations \endsubhead
Let $ \Lambda = [E] $ and consider the linear differential equation
$$ u'=\Cal A (t)u \,, \eqno (4.1) $$
where $ \Cal A \in C(\Bbb R ,\Lambda )$. Along with equation (4.1), we shall
also consider its $ H$-class, that is, the family of equations
$$ v'=\Cal B (t)v \,, \eqno (4.2) $$
where $\Cal B \in H(\Cal A )=\overline{ \{ \Cal A_{\tau} : \tau \in \Bbb R \}}$,
$\Cal A_{\tau}(t)=\Cal A (t + \tau )$ ($t \in \Bbb R$),
and the bar denotes closure in $ C(\Bbb R,\Lambda ) $. Let $ \varphi (t,u,\Cal B) $
be the solution of equation (4.2) that satisfies the condition
$ \varphi (0,u,\Cal B)=u$. 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 (X, \Bbb R_{+},\pi), (Y,\Bbb R, \sigma ), h \rangle $ is a linear
non-autonomous dynamical system, where $ X=E\times Y$,
$\pi =(\varphi , \sigma )$; i.e.,
$ \pi ((v,\Cal B),\tau ) = ( \varphi (\tau ,v, \Cal B ), \Cal B_{\tau } ) $
and $ h=pr_2 : X \to Y )$.
\proclaim{Lemma 4.1 [11,13]}
\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 A \mapsto 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 [17] of equation (4.1).
\endproclaim
\proclaim{Theorem 4.2} Let $ \Cal A \in C(\Bbb R, \Lambda ) $ be compact
(i.e. $ H(\Cal A ) $ is a compact set of
$ ( C(\Bbb R , \Lambda ), \Bbb R , \sigma )$ ),
then the following conditions are equivalent:
\item{1.} The trivial solution of equation (4.1) is uniformly exponentially stable, i.e.
there exist positive numbers $N$ and $ \nu $ such that $\|
U(t,\Cal A)U(\tau ,\Cal A)^{-1}\| \le N e^{-(t-\tau)}$
for all $ t \ge \tau .$
\item{2.} There exist positive numbers $N$ and $ \nu $ such that $\|
U(t,\Cal B)U(\tau,\Cal B)^{-1}\| \le N e^{-(t-\tau)}$
for all $ t \ge \tau $
and $ \Cal B \in H(\Cal A) $.
\item{3.}$ \lim \limits_{t \to + \infty}\sup \{\| U(t,\Cal B)\| : \Cal B \in H(\Cal A) \} =0 $.
\endproclaim
\demo{Proof} Applying Theorem 2.2 to the non-autonomous system
$ \langle (X, \Bbb R_{+},\pi), (Y,\Bbb R, \sigma ), h \rangle $,
generated by equation (4.1), we obtain the
equivalence of conditions 2 and 3. According to Lemma 3 [11] conditions
1 and 2 are equivalent. The theorem is proved.
\enddemo
\proclaim{Theorem 4.3} Let $ \Cal A \in C(\Bbb R, \Lambda ) $ be recurrent with respect to
$ t \in \Bbb T $ (i.e. $ H(\Cal A ) $ is a compact and minimal set of
$ ( C(\Bbb R , \Lambda ), \Bbb R , \sigma )$ ),
the non-autonomous system
$ \langle (X, \Bbb R_{+},\pi),$\linebreak $(Y,\Bbb R, \sigma ), h \rangle $
generated by equation (4.1) is asymptotically compact.
Then the following conditions are equivalent:
\item{1.} The trivial solution of equation (4.1) is uniformly exponentially stable, i.e.
there exist positive numbers $N$ and $ \nu $ such that $\|
U(t,\Cal A)U(\tau ,\Cal A)^{-1}\| \le N e^{-(t-\tau)} $ for all $ t \ge \tau . $
\item{2.}$ \lim \limits_{t \to + \infty}\sup | \varphi (t,u,\Cal B)| =0 $
for every $ u \in E $ and $ \Cal B \in H( \Cal A ) $.
\endproclaim
\demo{Proof} According to Lemma 4.1 the mapping
$ U(t,\cdot) : [E] \to [E] $ is continuous and, consequently, the mapping
$ \Cal B \mapsto\| U(t,\Cal B)\| $ is also continuous for every $ t \in \Bbb T$.
Now applying Theorem 3.2 to non-autonomous system
$ \langle (X, \Bbb R_{+},\pi), (Y,\Bbb R, \sigma ), h \rangle $
generated by equation (4.1), we obtain the
equivalence of conditions 1. and 2..
The theorem is proved.
\enddemo
We now formulate some sufficient conditions for the $\alpha - $ condensedness
(in particular, asymptotical compactness) of
the linear non-autonomous dynamical system generated by equation (4.1).
\proclaim{Theorem 4.4} Let $ \Cal A \in C( \Bbb R ,[E]) , \Cal A (t) = \Cal A_1(t) +
\Cal A_2(t) $ for all $ t \in \Bbb R $, and assume that
$ H(\Cal A_i)\ (i=1,2) $ are compact and the following conditions hold:
\item{(i)} The zero solution of the equation
$$ u'= \Cal A_1 (t) u \eqno (4.3) $$
is uniformly asymptotically stable, that is, there are positive numbers
$N$ and $\nu $ such that
$$\| U(t,\Cal A_1)U^{-1}(\tau,\Cal A_1)\| \le
Ne^{-\nu (t-\tau)} \eqno (4.4) $$
for all $ t \ge \tau \ (t,\tau \in \Bbb R), $ where $ U(t,\Cal A_1) $ is the
Cauchy operator of equation (4.3).
\item{(ii)} The family of operators $\{ \Cal A_2 (t) : t \in \Bbb R_{+} \}$
is uniformly completely continuous, that is, for any bounded set $ A \subset E $
the set $\{ \Cal A_2(t)A : t \in \Bbb R_{+} \}$ is precompact.
Then the linear non-autonomous dynamical system generated by equation (4.1) is
an $\alpha$-contraction.
\endproclaim
\demo{Proof} First of all we note that the set $ \varphi (t,A,Y) $ is bounded for
every $ t >0 $ and bounded set $ A \subseteq E $.
Let $ \Cal B \in H(\Cal A ) $. Then there are $\{ t_n\} \subset \Bbb T $
such that $ \Cal B (t) = \Cal B_1(t)+\Cal B_2(t) $ and $ \Cal B_i (t) =
\lim \limits_{t \to + \infty }\Cal A_i (t + t_n) $. Note that
$$ \varphi (t,v,\Cal B ) = U(t,\Cal B_1)v + \int ^{t}_0U(t,\Cal B_1)
U^{-1}(\tau,\Cal B_1)\Cal B_2(\tau)\varphi (\tau,v,\Cal B )d\tau . \eqno (4.5) $$
By Lemma 4.1,
$$ U(t,\Cal B_i)= \lim \limits_{t \to + \infty} U(t,\Cal A_{it_n}), \quad
\Cal A_{it_n} (t)= \Cal A_i (t+t_n), $$
and the equality
$$ U(t,\Cal A_{1t_n}) U^{-1}(\tau,\Cal A_{1t_n}) =
U(t+t_n,\Cal A_1) U^{-1}(\tau + t_n,\Cal A_1)$$
and inequality (4.4) imply that
$$\| U(t,\Cal B_1)U^{-1}(\tau,\Cal B_1)\| \le
Ne^{-\nu (t-\tau)} \eqno (4.6) $$
for all $ t \ge \tau $ and $ \Cal B_1 \in H(\Cal A_1) $.
By Theorem 1.5, Theorem 4.4 will be proved if we can prove that the set
$$\{ \int_0^{t}U(t,\Cal B_1)U^{-1}(\tau,\Cal B_1) \Cal B_2 (\tau )
\varphi (\tau , v, \Cal B ) d\tau : (v,\Cal B ) \in A \}
$$
is precompact for every $ t>0$ and every bounded positively invariant set
$ A \subseteq E\times Y $.
We put
$$ K^{t}_{A}= \overline{ \{ \Cal B_2 (\tau) \varphi (\tau ,v,\Cal B ) :
\tau \in [0,t] , (v,\Cal B ) \in A \} }$$
and we note that the set $ K^{t}_{A}$ is compact. Really, the set
$ \varphi ([0,t],A)= \cup \{ \varphi (\tau ,v, \Cal B ) : \tau \in [0,t],
(v,\Cal B ) \in A \}$ is bounded because
$ | \varphi (\tau ,v, \Cal B ) | \le e^{Mt}r $ for all $ \tau \in [0,t] $
and $ (v,\Cal B ) \in A $, where $ r=\sup \{ | v | : \exists \Cal B \in
H(\Cal A ) , $ such that $ (v,\Cal B ) \in A \}$ and $ M = \sup \{\| \Cal A (t)
\| : t \in \Bbb T \}$.
Then
$$ \int_0^{t}U(t,\Cal B_1)U^{-1}(\tau,\Cal B_1) \Cal B_2 (\tau )
\varphi (\tau , v, \Cal B ) d\tau $$
$$ \in t\ \overline {\operatorname{conv}} \{ U(t,\Cal B_1)U^{-1}(\tau,\Cal B_1)w :
0\le \tau \le t , \Cal B_1 \in H(\Cal A_1), w \in K^{t}_{A} \} .\eqno (4.7) $$
Since $ H(\Cal A_1), H(\Cal A ),$ and $ K^{t}_{A}$ are compact sets, then formula (4.7),
condition (ii) of Theorem 4.4 , and Lemma 4.1 imply that
$\{ U(t,\Cal B_1)U^{-1}(\tau,\Cal B_1)w :
0\le \tau \le t , \Cal B_1 \in H(\Cal A_1), w \in K^{t}_{A} \}$ is compact,
which completes the proof of the theorem.
\enddemo
\proclaim{Theorem 4.5} Let $ H(\Cal A ) $ be compact and assume that there is a
finite-dimensional projection $ P \in [E] $ such that
\item{(i)} the family of projections $\{ P(t) : t \in \Bbb R \}$, where
$ P(t)= U(t,\Cal A )PU^{-1}(t,\Cal A ),$ is precompact in $ [E]$, and
\item{(ii)} there are positive numbers $N$ and $ \nu $ such that
$$\| U(t,\Cal A )QU^{-1}(\tau ,\Cal A )\| \le Ne^{-\nu (t-\tau)}$$
for all $ t \ge \tau ,$ where $ Q = I - P $.
Then the linear non-autonomous dynamical system generated by equation (4.1)
is an $\alpha$-contraction.
\endproclaim
\demo{Proof} Since the family of projections
$ P(t)= U(t,\Cal A )PU^{-1}(t,\Cal A ) $ is precompact in $ [E]$, the family
$\Bbb H = \overline {\{ P(t) : t \in \Bbb R \}}$ is uniformly completely continuous,
where the bar denotes closure in $ [E]$. Indeed, let $A$ be a bounded subset of $ E $,
$\{x_n\} \subseteq \{ QA : Q \in \Bbb H \}$, and $ \varepsilon_n \downarrow 0$.
Then there are $ t_n \in \Bbb R $ and $ v_n \in A $ such that
$ | x_n - P(t_n)v_n |\le \varepsilon_n $. Since the sequence
$\{P(t_n)\}$ is precompact, we can assume that it converges. Let
$ L =\lim \limits_{n \to + \infty} P(t_n)$. Then $ L $ is completely continuous,
which implies that the sequence $\{x'_n\}= \{Lv_n\}$ is precompact. Note that
$$ | x_n - x'_n | \le | x_n - P(t_n)v_n|
+ | P(t_n)v_n - Lv_n| \le \varepsilon_n +\| P(t_n) - L\| | v_n | ,$$
which implies that $ | x_n - x'_n| \to 0 $ as $ n \to + \infty $. Hence,
$\{x_n\}$ is precompact.
Assume that $ \Cal B \in H(\Cal A ) $ and $\{t_n\} \subset \Bbb R $ are such that
$$ \Cal B = \lim \limits_{n \to + \infty} \Cal A_{t_n} , P(\Cal B )=
\lim \limits_{n \to + \infty} P (\Cal A_{t_n}), $$
where $ P (\Cal A_{t_n})= U(t_n,\Cal A )PU^{-1}(t_n,\Cal A )$. The assertions
proved above imply that the family $\{ P(\Cal B ) : \Cal B \in H(\Cal A ) \}$ is
uniformly completely continuous. Note that
$ Q(\Cal B )= \lim \limits_{n \to + \infty} Q (\Cal A_{t_n}), $ where
$ Q(\Cal B )= I - P(\Cal B ) $ and $ Q (\Cal A_{t_n}) = I - P (\Cal A_{t_n})$.
Moreover, condition (ii) of Theorem 4.5 implies that
$$\| U(t,\Cal A_{t_n} )QU^{-1}(\tau ,\Cal A_{t_n} )\|
\le Ne^{-\nu (t-\tau)} \eqno (4.7) $$
for all $ t \ge \tau $. Passing to the limit in (4.7) as $ n \to + \infty $ and
taking into account Lemma 4.1, we obtain that
$$\| U(t,\Cal B )QU^{-1}(\tau ,\Cal B )\|
\le Ne^{-\nu (t-\tau)} $$
for all $ t \ge \tau $ and $ \Cal B \in H(\Cal A ) $. We complete the proof of the
theorem by observing that $ U(t,\Cal B ) Q(\Cal B ) + U(t,\Cal B )P(\Cal B )=
U(t,\Cal B )$ and applying Theorem 1.5.
\enddemo
\proclaim{Theorem 4.6} Suppose that the following conditions are satisfied:
\item{1.} The operator-function $ \Cal A \in C(\Bbb R , [E]) $ is recurrent
with respect to $ t \in \Bbb R $.
\item{2.} The linear non-autonomous dynamical system
$ \langle (X, \Bbb T_{+},\pi), (Y,\Bbb T, \sigma ), h \rangle $
generated by equation (4.1) is conditionally $\alpha$-condensing.
\item{3.} the trivial solution of equation (4.1) is uniformly stable in the positive direction,
i.e. there exist a positive number $ M $ such that
$$\| U(t,\Cal A ) U^{-1}(\tau ,\Cal A )\| \le M \eqno (4.8) $$
for all $ t \ge \tau $.
\noindent Then there are two vectorial positively invariant subfibers $ (X^{0},h,Y) $
and $ (X^{s},h,Y) $ of $ (X,h,Y) $ such that:
\item{a.}$ X_{y}= X_{y}^{0}+X_{y}^{s}$ and $ X_{y}^{0}\cap X_{y}^{s}=0_{y}$ for all $ y \in Y$,
where $ 0_{y} =(0,y)\in X=E\times Y$ and $ 0 $ is the zero in the Banach space
$ E $.
\item{b.} The vectorial subfiber $ (X^{0},h,Y) $ is finite dimensional, invariant
(i.e. $ \pi ^{t}X^{0}=X^{0}$ for all $ t \in \Bbb T_{+}$) and every trajectory
of a dynamical system $(X, \Bbb T_{+},\pi) $ belonging to $X^{0}$ is recurrent.
\item{c.} There exist two positive numbers $N$ and $\nu$ such that
$| \varphi (t,u,\Cal B)| \le Ne^{-\nu t}| u | $ for all $ (u,\Cal B) \in
X^{s}$ and $ t \in \Bbb T_{+}$, where $ \varphi (t,u,\Cal B) = U(t,\Cal B )u $.
\endproclaim
\demo{Proof} Assume that $ \Cal B \in H(\Cal A ) $ and $\{t_n\} \subset \Bbb R $ are such that
$ \Cal B = \lim \limits_{n \to + \infty} \Cal A_{t_n}$ , then
condition (3) of Theorem 4.6 implies that
$$\| U(t,\Cal A_{t_n} )U^{-1}(\tau ,\Cal A_{t_n} )\|
\le N \eqno (4.8) $$
for all $ t \ge \tau $. Passing to the limit in (4.8) as $ n \to + \infty $ and
taking into account Lemma 4.1, we obtain that
$$\| U(t,\Cal B )U^{-1}(\tau ,\Cal B )\|
\le N $$
for all $ t \ge \tau $ and $ \Cal B \in H(\Cal A ) $ and, consequently,
$$\| U(t,\Cal B )\| \le N $$
for all $ t \ge 0 $ and $ \Cal B \in H(\Cal A ) $. Now to finish the proof of
Theorem 4.6 it is sufficiently to refer on Theorem 3.4.
\enddemo
\subhead Partial linear differential equations \endsubhead
Let $ \Lambda $ be some complete
metric space of linear closed operators acting into Banach space $ E $ \quad ( 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 (4.1) and its $ H-$ class (4.2):
\item{a.} for any $ v \in E $ and $ \Cal B \in H( \Cal A ) $ equation (4.2) has exactly
one mild solution $ \varphi (t,v,\Cal B)$ (i.e.
$ \varphi (\cdot ,v,\Cal B ) $ is continuous, differentiable and stisfies
of equation (4.2) ) 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 $\Bbb R_{+} \times E \times C(\Bbb R ; \Lambda ) ;$
\item{c.} for every $ t \in \Bbb R_{+}$ the mapping $ U(t, \cdot ): H( \Cal A ) \to [E] $
is continuous, where $ U(t,\cdot ) $ is the Cauchy operator of equation (4.2),
i.e. $ U(t,\Cal B )v= \varphi (t,v,\Cal B ) \quad ( t \in \Bbb R_{+} ,
v \in E $ and $ \Cal B \in H(\Cal A ) $ ).
\noindent Under the above assumptions the equation (4.1) generates a linear non-autonomous dynamical
system
$ \langle (X, \Bbb R_{+},\pi), (Y,\Bbb R, \sigma ), h \rangle $, where
$ X=E\times Y, \pi =(\varphi , \sigma ) $ and $ h=pr_2 : X \to Y.$
Applying the results from \S\,2 and \S\,3 to this system,
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-c.
{\bf Example 4.7.} A closed linear operator $ \Cal A : D(\Cal A ) \mapsto E $
with dense domain $ D(\Cal A ) $ is said [21] to be sectorial if one can find a
$ \varphi \in (0,\frac{\pi}{2} ) $, an $ M \ge 1$, and a real number $ a $
such that the sector
$$ S_{a,\varphi } = \{ \lambda : \varphi \le | arg(\lambda -a) | \le \pi ,
\lambda \not= a \}$$
lies in the resolvent set $ \rho (\Cal A)$ of $ \Cal A $ and
$\| (\lambda I - \Cal A )^{-1}\| \le M| \lambda -a|^{-1}$
for all $ \lambda \in S_{a,\varphi }$. An important class of sectorial operators is formed by
elliptic operators [21,22].
Consider the differential equation
$$ u'=(\Cal A_1 + \Cal A_2(t) )u, \eqno (4.9) $$
where $ \Cal A_1 $ is a sectorial operator that does not depend on
$ t \in \Bbb R $, and $ \Cal A_2 \in C(\Bbb R ,[E]) $.
The results of [21,23] imply that equation (4.9) satisfies conditions a.-c..
{\bf Example 4.8.} Let $ \Cal H $ be a Hilbert space with a scalar product
$ \langle \cdot , \cdot \rangle = | \cdot | ^{2} , \Cal D ( \Bbb R_{+} , \Cal H ) $ be
the set of all infinite differentiable, bounded functions on $\Bbb R_{+}$
with values into $ \Cal H $.
Denote by $ ( C(\Bbb R , [\Cal H ] ), \Bbb R , \sigma ) $ a dynamical system of
shifts on $ C(\Bbb R , [\Cal H ] ) $. Consider the equation
$$ \int \limits_{\Bbb R_{+} } [ < u(t),\varphi '(t)> + < \Cal A (t)u(t),\varphi (t) > ]dt = 0 ,
\eqno (4.10) $$
along with the family of equations
$$ \int \limits_{\Bbb R_{+} } [ < u(t),\varphi '(t)> + < \Cal B (t)u(t),\varphi (t) > ]dt = 0 ,
\eqno (4.11) $$
where $ \Cal B \in H(\Cal A ) = \overline{\{ \Cal A_{\tau} | \tau \in \Bbb R \}
}$, $\Cal A_{\tau } (t)= \Cal (t + \tau )$ and the bar denotes closure in
$ C(\Bbb R , [\Cal H ] )$.
The function $ u \in C( \Bbb R_{+} , \Cal H ) $ is called a solution of equation (4.10),
if (4.10) takes place for all $ \varphi \in \Cal D ( \Bbb R_{+} , \Cal H )$.
Assume that the operator
$ \Cal A (t) $ is self-adjoint
and negative definite, i.e., $ \Cal A(t)= - \Cal A_1(t) + i\Cal A_2(t) $ for
all $ t \in \Bbb R $, where $ \Cal A_1(t) $ and $ \Cal A_2(t) $ are self-adjoint and
$$ \langle \Cal A_1(t)u,u \rangle \ge \alpha | u | ^{2} \eqno (4.12) $$
for all $ t \in \Bbb R $ and $ u \in \Cal H $, where $\alpha > 0 $.
Let
$ ( H(\Cal A ) , \Bbb R , \sigma ) $ be a dynamical system of shifts on
$ H( \Cal A )$, $\varphi (t,v,\Cal B )$ be a solution of (4.11)
with condition
$ \varphi (0, v , \Cal B ) = v$, $\overline X = \Cal H \times H (\Cal A )$,
$X $ be a set of all the points $\langle u,\Cal B \rangle \in \overline X$
such that through point $ u \in \Cal H $ passes a solution
$ \varphi (t,u,\Cal A ) $ of equation (4.10) defined on $\Bbb R_{+}$.
According to Lemma 2.21 in [12] the set
$X$ is closed in $ \overline X $. In virtue of Lemma 2.22 in [12]
the triple $(X, \Bbb R_{+}, \pi )$
is a dynamical system on $X$ (where $ \pi = ( \varphi , \sigma )$ ) and
$ \langle (X, \Bbb R_{+},\pi), (Y,\Bbb R, \sigma ), h \rangle $ is a linear non-autonomous
dynamical system, where
$ h = pr_2 : X \to Y = H( \Cal A )$.
Applying the results from [1] it is possible to show that for every $ t $
the mapping $ \Cal B \mapsto U(t,\Cal B )$ (where
$ U(t,\Cal B )v= \varphi (t,v,\Cal B ) $)
from $ H(\Cal A ) $ into $[\Cal H ] $
is continuous and, consequently, for this system Theorem 2.2 is applicable.
Thus the following assertion takes place.
\proclaim{Theorem 4.9} Let $ \Cal A \in C(\Bbb R, [\Cal H ] ) $ be compact,
then the following assertion hold:
\item{1.} The trivial solution of equation (4.1) is uniformly exponentially stable, i.e.
there exist positive numbers $N$ and $ \nu $ such that
$\| U(t,\Cal B)\| \le N e^{-\nu t} $ for all $ t \ge 0 $ and $
\Cal B \in H(\Cal A )$.
\item{2.}$ \lim \limits_{t \to + \infty}\sup \{\| U(t,\Cal B)\| : \Cal B \in H(\Cal A) \} =0 $.
\endproclaim
\demo{Proof} According to Lemma 2.23 in [12] from (4.12) follows the
condition 1. In virtue of
Theorem 2.2 the conditions 1 and 2 are equivalent. The theorem is proved.
\enddemo
We will give the example of a boundary value problem reducing to an
equation of type (4.10). Let $ \Omega $ be a bounded domain in
$\Bbb R ^{n}$, $\Gamma $ be the boundary of $ \Omega,$
$Q = \Bbb R_{+} \times \Omega $ and $ S = \Bbb R_{+} \times \Gamma $.
Consider the first initial boundary value problem in $\Omega$ for the
equation
$$ \frac{\partial u}{\partial t}
= L (t)u \quad u|_{t=0} =\varphi ,\ u|_{S}=0\,, \eqno (4.13) $$
where
$$ L (t)u = \sum_{i,j=1}^{n}\frac{\partial}{\partial x_i}
(a_{ij}(t,x)\frac{\partial u}{\partial x_j})-a(t,x)u $$
According to the Riesz theorem, the operator $ \Cal A (t) $
is defined by
$$ \langle \Cal A (t)u, \varphi \rangle= - \int \limits_{\Omega } [ \sum_{i,j=1}^{n}
a_{ij}(t,x)\frac{\partial u}{\partial x_j}
\frac{\partial \varphi}{\partial x_i} + a(t,x)u\varphi ]dx . $$
If
$ a_{ij}(t,x)=a_{ji}(t,x)$
and the functions
$ a_{ij}(t,x) $ and $ a(t,x) $
are bounded and uniformly continuous with respect to
$ t \in \Bbb R $
uniformly with respect to $ x \in \Omega $, then for equation (4.13)
Theorem 2.2 is applicable for $\Cal H = \dot W^{1}_2(\Omega)$.
\subhead Linear functional-differential equations \endsubhead
Let $r$ be a positive number, $C([a,b], \Bbb R ^{n}) $ be the Banach
space of continuous functions
$ \varphi : [a,b] \to \Bbb R ^{n}$ with the supremum norm, and
$ \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
$ u_{t}(\theta )=u(t + \theta )$, with $-r \le \theta \le 0$.
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 with the operator norm. Consider the equation
$$ u' = \Cal A (t)u_{t} \,, \eqno (4.14) $$
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 ) $, where the bar denotes closure
in the topology of uniform convergence on every compact of
$\Bbb R$.
Along with equation (4.14) we also consider the family of equations
$$ u' = \Cal B (t)u_{t} \,, \eqno (4.15) $$
where $ \Cal B \in H(\Cal A ) .$
Let $ \varphi_{t} (v,\Cal B ) $ be a solution of equation (4.15) 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= \Cal 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 4.10[4]} 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 (4.14)
is completely continuous.
\endproclaim
\proclaim{Theorem 4.11} Let $ H(\Cal A ) $ be compact. Then the following assertions
are equivalent:
\item{1.} For any $ \Cal B \in H(\Cal A ) $ the zero solution of equation (4.15) is
asymptotically stable, i.e.
$ \lim \limits_{t \to + \infty } | \varphi_{t}(v,\Cal B ) | = 0 $
for all $ v \in \Cal C $ and $ \Cal B \in H(\Cal A ) $
\item{2.} The zero solution of equation (4.14) is uniformly exponentially stable, i.e.
there are positive numbers
$ N$ and $ \nu $ such that $ | \varphi_{t}(v,\Cal B ) | \le
Ne^{-\nu t}| v | $ for all $ t \ge 0 , v \in \Cal C $
and $ \Cal B \in H(\Cal A) $.
\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 (4.14). According
to Lemma 4.10 this system is completely continuous and to finish the proof it is
sufficient to refer to Theorem 2.5.
\enddemo
Consider the neutral functional differential equation
$$ \frac{d}{dt}Dx_{t}=\Cal A (t)x_{t} \,, \eqno (4.16) $$
where $ {\Cal A} \in C(\Bbb R , \frak A ) $ and $ D \in \frak A $
is an operator nonatomic at zero [19, p.67]. As well as in the case of equation (4.14),
the equation (4.16) generates a linear non-autonomous dynamical system
$ \langle (X, \Bbb R_{+},\pi), (Y,\Bbb R, \sigma ), h \rangle $,
where $ X=\Cal C \times Y$, $Y=H(\Cal A ) $ and $ \pi = (\varphi , \sigma ) $.
The following statement takes place.
\proclaim{Lemma 4.12} Let $ H( \Cal A ) $ be compact and the operator
$ D $ is stable; i.e., the zero solution of the 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
(4.16), is conditionally $\alpha$-condensing.
\endproclaim
\demo{Proof} According to [20, p.119, formula (5.18)] the mapping
$\varphi_{t} (\cdot,\Cal B ) : \Cal C \to \Cal C $ can be written as
$$ \varphi_{t} (\cdot,\Cal B ) = S_{t}(\cdot) + U_{t}(\cdot ,\Cal B) $$
for all $ \Cal B \in H(\Cal A ) $, where $ U_{t}(\cdot ,\Cal B) $ is
conditionally completely continuous for $t \ge r$. Also there exist
positive constants $N,\nu$
such that $ \| S_{t} \| \le N e^{-\nu t} (t \ge 0 ) $. Then the proof is
completed by referring to Theorem 1.5.
\enddemo
\proclaim{Theorem 4.13} Let $ \Cal A \in C( \Bbb R , \frak A ) $ be recurrent
(i.e. $ H(\Cal A ) $ is compact minimal set in the dynamical system of shifts
$ ( C(\Bbb R , \frak A ), \Bbb R , \sigma ) $ ) and
$ D $ is stable, then the following assertions are equivalent:
\item{1.} The zero solution of equation (4.14) and all equations
$$ \frac{d}{dt}Dx_{t}=\Cal B (t)x_{t} \,, \eqno (4.17) $$
where $ \Cal B \in H (\Cal A ) $, is asymptotically stable, i.e.
$ \lim \limits_{t \to + \infty } | \varphi_{t}(v,\Cal B ) | = 0 $
for all $ v \in \Cal C $ and $ \Cal B \in H(\Cal A ) \quad (\varphi_{t}(v,\Cal B ) $
is the solution of equation (4.17) with condition $ \varphi_0(v, \Cal B )=v $).
\item{2.} The zero solution of equation (4.16) is uniformly exponentially stable, i.e. there
are positive numbers $N$ and $ \nu $ such that
$ | \varphi_{t}(v,\Cal B) | \le N e ^{-\nu t}| v | $ for all $ t \ge 0 ,
v \in \Cal C $ and $ \Cal B \in H(\Cal A ) $.
\item{3.} All solutions of all equations (4.17) are bounded on $\Bbb R_{+}$ and they
don't have non-trivial solutions bounded on $\Bbb R$ .
\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 (4.16). According to Lemma 4.12 this system is
conditionally $\alpha $ condensing. To finish the proof of
Theorem 4.13 it is sufficient to refer to Theorems 2.3 and 3.4.
The theorem is proved.
\enddemo
\proclaim{Theorem 4.14} Let $ \Cal A \in C( \Bbb R , \frak A ) $ be recurrent
(i.e. $ H(\Cal A ) $ is compact minimal in the dynamical system of shifts
$ ( C(\Bbb R , \frak A ), \Bbb R , \sigma ) $ ),
$ D $ is stable, and
all solutions of all equations (4.17) are bounded on $\Bbb R_{+}$.
Let
$ \langle (X, \Bbb T_{+},\pi), (Y,\Bbb T, \sigma ), h \rangle $
be the linear non-autonomous dynamical system
generated by equation (4.16).
Then there are two positively invariant vector subfiberings $ (X^{0},h,Y) $
and $ (X^{s},h,Y) $ of $ (X,h,Y) $ such that:
\item{a.}$ X_{y}= X_{y}^{0}+X_{y}^{s}$ and $ X_{y}^{0}\cap X_{y}^{s}=0_{y}$ for all $ y \in Y$,
where $ 0_{y} =(0,y)\in X=E\times Y$ and $ 0 $ is the zero in the Banach space
$ E $.
\item{b.} The vector subfibering $ (X^{0},h,Y) $ is finite dimensional, invariant
(i.e$. \pi ^{t}X^{0}=X^{0}$ for all $ t \in \Bbb T_{+}$) and every trajectory
of a dynamical system $(X, \Bbb T_{+},\pi) $ belonging to $X^{0}$ is recurrent.
\item{c.} There exist two positive numbers $N$ and $\nu$ such that
$| \varphi_{t}(v,\Cal B) | \le Ne^{-\nu t}| v | $ for all $ (v,\Cal B) \in
X^{s}$, where $ \varphi_{t}(v,\Cal B) = U(t,\Cal B )v $ and $ U(t,\Cal B )$
is the Cauchy operator of equation (4.17).
\endproclaim
\demo{Proof} Let
$ \langle (X, \Bbb T_{+},\pi), (Y,\Bbb T, \sigma ), h \rangle $
be the linear non-autonomous dynamical system
generated by equation (4.16).
In virtue of Lemma 4.12 this non-autonomous dynamical system is
conditionally $\alpha$-condensing.
According to Theorem 1.6 [5] there exists a positive number $ M $ such that
$ | \varphi_{t}(v,\Cal B) | \le M | v | $ for all $ t \ge 0,
v \in \Cal C $ and $ \Cal B \in H(\Cal A ) $. To finish the proof of Theorem 4.14
it is sufficient to refer to Theorem 3.4.
\enddemo
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