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\def\rightheadline{EJDE--1999/46\hfil
Relationship between different types of stability \hfil\folio}
\def\leftheadline{\folio\hfil D. N. Cheban
 \hfil EJDE--1999/46}

\def\pretitle{\vbox{\eightrm\noindent\baselineskip 9pt %
 Electronic Journal of Differential Equations,
Vol. {\eightbf 1999}(1999), No.~46, pp.~1--9.\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
Relationship between different types of stability
for linear almost periodic systems in Banach spaces
\endtitle

\thanks
{\it 1991 Mathematics Subject Classifications:}
34C35, 34C27, 34K15, 34K20, 58F27, 34G10.\hfil\break\indent
{\it Key words and phrases:} non-autonomous linear dynamical systems,
global attractors, \hfil\break\indent
almost periodic system, stability, asymptotic stability.
\hfil\break\indent
\copyright 1999 Southwest Texas State University  and
University of North Texas.\hfil\break\indent
Submitted June 18, 1999. Published November 27, 1999.
\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
For the linear equation $ x'= A(t)x $
with recurrent (almost periodic) coefficients
in an arbitrary Banach space, we prove that the asymptotic
stability of the null solution and of all limit equations
implies the uniform stability of the null solution.
\endabstract

\endtopmatter

\document


\head Introduction \endhead

In 1962, W. Hahn [13] posed the problem of whether asymptotic stability
implies uniform stability for linear equation
$$ x'=A(t)x  \qquad ( x \in \Bbb R^{n} )   \eqno  (0.1) $$
with almost periodic coefficients.
In 1965, C. C. Conley and R. K. Miller [12] gave a negative
answer to this question, by constructing a scalar equation $x'=a(t)x$
with the property that every solution $\varphi (t,x,a) \to 0$ as
$t \to + \infty$, but the null solution is not uniformly stable (see also [4]).
From the results by R. J. Sacker and G. R. Sell [17] and I. U. Bronshteyn [2, p.141],
the uniform stability of the null solution to (0.1) holds under the following
conditions: The matrix $A(t)$ in (0.1) is recurrent
(in particular, almost periodic), and the asymptotic stability holds for
the null solution of (0.1) and for the null solutions of all systems
$$ x'= B(t)x  \,,  \eqno  (0.2) $$
where $ B \in H(A) = \overline{\{ A_{\tau} : \tau \in \Bbb R \} }$,
with $A_{\tau } $ denoting the translation of the  matrix $ A $ by
$ \tau $ and the bar denoting  the closure in the topology of the uniform
convergence on compact subsets of $\Bbb R $.
Also we want to point out that from the results of the author in [5],
the result mentioned above is valid for (0.1) with compact matrix
(i.\ e., when $ H(A) $ is compact). The goal of the
present paper is to study the relationship between the asymptotical
stability and uniform stability of the null solution of system (0.1) in
arbitrary Banach spaces.

Our main result is that for (0.1) with recurrent
coefficients in an arbitrary Banach space the following statement holds:
If the null solution of (0.1) and the null solutions of all equations (0.2)
are asymptotically stable, then the null solution of (0.1) is uniformly
stable.

\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 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 by $(Y, \Bbb T , \sigma )$ a  group 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.

The system $(X,\Bbb T_{+},\pi)$ is called:  [6-7]

\noindent{\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 ^tx=\pi (t,x)$;

\noindent{\bf compactly dissipative}, if (1.1) holds uniformly with respect to
$x$ on  compact subsets of $X$;

\noindent{\bf locally  dissipative}, if for a point $p\in X$ there is
$\delta_{p} > 0$ such that  (1.1) holds uniformly with respect to
$x\in B(p,\delta_{p})$;

A non-autonomous dynamical system
$ \langle (X, \Bbb T_{+},\pi), (Y,\Bbb T, \sigma ), h \rangle $
is said to be point (compact, local) 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
a norm on $ (X,h,Y) $, i.\ e., $| \cdot | $
is co-ordinated with the metric $ \rho $  ( 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) $).
In [8], the author obtained a point (compact, local)
dissipativity criterion for linear systems.

\proclaim{Theorem 1.1 [8]} Let
$ \langle (X, \Bbb T_{+},\pi), (Y,\Bbb T, \sigma ), h \rangle $
be a linear non-autonomous dynamical system and $ Y $ be
compact, then the following assertions hold

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 $;

2. A 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 positive number $M$
such that for all $ x \in X $ and $ t \in \Bbb T_{+} $,
$$  | xt | \le M| x |\,;    \eqno (1.2) $$

3. A 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
  $ | xt | \le N e^{-\nu t}| x | $
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.
For an example of a linear autonomous dynamical system
which is compactly dissipative but not locally dissipative, see [8].


\proclaim{Theorem 1.2 } Let
$ \langle (X, \Bbb T_{+},\pi), (Y,\Bbb T, \sigma ), h \rangle $
be a linear non-autonomous dynamical system and the following
conditions hold

1. $ Y $ is compact and minimal ( i.\ e., $ Y=H(y) =
\overline {\{ yt : t \in \Bbb T \}} $
for all $ y \in Y $ );

2. for each $ x \in X $ there exists $ C_{x} \ge 0 $ such that
for all $ t \in \Bbb T_{+} $,
$$ | xt | \le C_{x}\,;  \eqno (1.3) $$

3. the mapping $ y \mapsto \| \pi ^t_y \| $ is continuous for every
$ t \in \Bbb T_{+}$, where
$\| \pi ^t_y \| $ is the norm of the linear operator
$ \pi ^t_y = \pi ^t |_{X_y} $.

Then there exists $ M \ge 0 $ such that  (1.2) holds for all
$ t \in \Bbb T_{+} $ and $ x \in X$.
\endproclaim

\demo{Proof} From Condition 2. and the Banach-Steinhauss theorem,
it follows the uniform boundedness of the family of linear operators
$ \{ \pi ^t_y : t \in \Bbb T_{+} \} $ for every $ y \in Y$, i.\ e.,
for each $ y \in Y $ there exists $ M_y \ge 0 $ such that
$ \| \pi ^t_y \| \le M_y $ for all $ t \in \Bbb T_{+} $. We put
$$
d(y)=\sup \limits_{t \ge 0 } \| \pi ^t_y \|  \eqno (1.4)
$$
and claim that $ d: Y \to \Bbb R_{+} $ is lower semi-continuous, i.\ e.,
$ \liminf \limits_{ y_n \to y}d(y_n) \ge d(y) $
for all $ y \in Y $ and $ \{y_n\} \to y $.
Suppose that this is not true, then there exist $ y \in Y , \{y_n\} $
and $ \varepsilon > 0 $ such that
$$
\liminf \limits_{ y_n \to y}d(y_n) = d(y)- \varepsilon  \eqno (1.5) $$
From (1.4) it follows that $ d(y)= \lim \limits_{ n \to + \infty}
\| \pi ^{t_n}_y \| $ for some sequence $ \{t_n\} \subseteq \Bbb T_{+} $
and, consequently, there exists $ k $ such that
$$
| \| \pi ^{t_n}_y \| - d(y) | < \frac{\varepsilon}{4}  \eqno (1.6) $$
for all $ n \ge k $. By the the continuity of mapping
$ y \mapsto \| \pi ^t_y \| $ there exists $ n(k) $ such that
$$
| \| \pi ^{t_{k}}_{y_n} \| -\| \pi ^{t_{k}}_y \| |
< \frac{\varepsilon}{4}  \eqno (1.7) $$
for all $ n \ge n(k)$. From (1.6) and (1.7),
$$
| d(y) - \| \pi ^{t_{k}}_{y_n} \|  | < \frac{\varepsilon}{2}  \eqno (1.8) $$
for all $ n \ge n(k)$. From (1.8),
$$
| d(y)-d(y_n) | \le \frac{\varepsilon}{2}  \eqno (1.9) $$
for all $ n \ge n(k)$. Notice that (1.9) contradicts (1.5), and this
contradiction proves that $ d: Y \to \Bbb R_{+} $ is lower semicontinuous.
Hence, this function has a set
of points of continuity $ D \subset Y $ of the type $ G_{\delta}$. Let $ p \in D, $
then there exist positive numbers $ \delta_{p} $ and $ M_{p} $ such that
$ d(y)\le M_{p} $ for all $ y \in S[p,\delta_{p}]=\{ y \in Y | \rho (y,p)
\le \delta_{p} \} \subset Y$.

Since $ Y $ is minimal, there are negative numbers $ t_1,t_2,\dots, t_{m} $
such that $ Y= \bigcup_{i=1}^{m} \sigma ( S[p,\delta_{p}],t_{i}) $
\quad (see [16, p.134]). We put $ L = \max \{ t_{i} | i=1,2,\dots,m \} $.
Assume that $ m \in Y$, $y \in S[p,\delta_{p}] $ and $ t_{i} $ are such that
$ m= yt_{i}$. Then
$$ | xt | = | \pi_y^{t+t_{i}}(\pi ^{-t_{i}}_{yt_{i}}(x)) | \le
M_{p}C| x |  \eqno (1.10) $$
for all $ x \in X $ with $h(x)=m$ and $ t \ge L $, where
$$ C=\max \{ \max \{ \| \pi ^{-t_{i}}_y \| :y \in Y \}, i=1,2,\dots,m \}\,.$$
 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
$$
| x_nt_n | \ge \varepsilon_0 . \eqno (1.11) $$
Since $ (X,h,Y) $ is a locally trivial Banach fibre bundle and $ Y $ is compact,
then the zero section $ \Theta = \{\theta_y : y \in Y \} $ 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 (1.11) 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
$$ | xt | \le \frac{1}{\gamma} | x |  \eqno (1.12) $$
for all $ t \in [0,L] $ and $ x \in X $. We put
$ M =\max \{ \gamma ^{-1} , M_{p}C \}$, then from (1.10) and (1.12)
it follows (1.2) for all $ t \ge 0 $ and $ x \in X $. The theorem is proved.
\enddemo

\proclaim{Remark 1.3} a.) If the fibre bundle $ (X,h,Y) $ is finite-dimensional,
then condition~3 in Theorem 1.2 holds.

b.) Let $ X= E\times Y, $ where $ E $ is a Banach space and
$ \pi = (\varphi , \sigma ) $, i.\ e., $ \pi ^tx = (\varphi (t,u,y ), \sigma ^ty ) $
for all $ t \in \Bbb T_{+} $ and $ x = (u,y) \in X= E\times Y$. Then
condition~3 in Theorem 1.2 holds, if for every $ t \in \Bbb T_{+} $  the mapping
$ U(t,\cdot ): Y \to [E] $ is continuous, where $ U(t,y)u=\varphi (t,u,y) $
with $ (t,u,y) \in \Bbb T_{+} \times E \times Y $ and $ [E] $ the
Banach space of continuous operators acting on $ E $ equipped with
the operator norm.
\endproclaim
\proclaim{Theorem 1.4} Let
$ \langle (X, \Bbb T_{+},\pi), (Y,\Bbb T, \sigma ), h \rangle $
be a linear non-autonomous dynamical system, $ Y $ be a compact minimal set
and the mapping $ y \mapsto \| \pi ^t_y \| $ be continuous for
each $ t \in \Bbb T_{+}$. Then the point dissipativity of
$ \langle (X, \Bbb T_{+},\pi), (Y,\Bbb T, \sigma ), h \rangle $
implies its compact dissipativity.
\endproclaim

\demo{Proof} Assume that the conditions of Theorem 1.4 are fulfilled and the
non-autonomous dynamical system
$ \langle (X, \Bbb T_{+},\pi), (Y,\Bbb T, \sigma ), h \rangle $
is point dissipative, then according to Theorem 1.1 for every $ x \in X $
there exists a constant $ C_{x} \ge 0 $ such that (1.3) holds
for all $ x \in X $ and $ t \in \Bbb T_{+}$. Then by referring to Theorem 1.1,
the present proof is complete.
\enddemo

\proclaim{Theorem 1.5 [6]} Let
$ \langle (X, \Bbb T_{+},\pi), (Y,\Bbb T, \sigma ), h \rangle $
be a linear non-autonomous dynamical system, $ Y $ be  compact,
then the following assertions take place.

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 the point dissipativity of
$ \langle (X, \Bbb T_{+},\pi), (Y,\Bbb T, \sigma ), h \rangle $
follows its local dissipativity;

2. If $ (X, \Bbb T_{+} , \pi ) $ is asymptotically compact (i.\ e., for all bounded
sequence $ \{x_n\} \subset X $ and $ \{ t_n \} \to + \infty $ the sequence
$ \{ x_nt_n \} $ is precompact), then from the compact dissipativity of
$ \langle (X, \Bbb T_{+},\pi), (Y,\Bbb T, \sigma ), h \rangle $
results its local dissipativity.
\endproclaim

\head  2. Some classes of linear non-autonomous differential equations
\endhead

Let $ \Lambda $ be a complete metric space of linear operators that act on Banach
space $ E $ and $ C(\Bbb R,\Lambda ) $ be a 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 )$. \medskip

\noindent{\bf Ordinary linear differential equations.} Let $ \Lambda = [E] $ and
consider the linear differential equation
$$ u'=\Cal A (t)u\,, \eqno (2.1) $$
where $ \Cal A \in C(\Bbb R ,\Lambda ) $. Along with equation (2.1), we shall also
consider its $ H-$class, that is, the family of equations
$$ v'=\Cal B (t)v \,, \eqno (2.2) $$
where $ \Cal B \in H(\Cal A ) = \overline{ \{ \Cal A_{\tau} : \tau \in \Bbb R \} },
\Cal A_{\tau}(t)=\Cal A (t + \tau ) \quad (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 (2.2) 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 (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 )$ \quad (i.\ e., $ \pi ((v,\Cal B),
\tau ) = ( \varphi (\tau ,v, \Cal B ), \Cal B_{\tau } $) and $ h=pr_2 : X \to Y$.
Applying Theorem 1.4 to this system, we obtain the following assertion.

\proclaim{Theorem 2.1} Let $ \Cal A \in C(\Bbb R, \Lambda ) $ be recurrent
(i.\ e., $ H(\Cal A ) $ is compact minimal set of
$ ( C(\Bbb R , \Lambda ), \Bbb R , \sigma )$ )
and the zero solutions of equation (2.1) and all equations (2.2) are asymptotically
stable, i.\ e.,
$ \lim \limits_{t \to + \infty } | \varphi (t,v,\Cal B ) | =0 $ for
all $ v \in E $ and $ \Cal B \in H(\Cal A )$. Then the zero solution of equation (2.1) is
uniformly stable, i.\ e., there exists $ M \ge 0 $ such that
$ | \varphi (t,v,\Cal B ) | \le M| v | $ for all $ t \ge 0, v \in E $
and $ \Cal B \in H(\Cal A ) .$
\endproclaim

\demo{Proof} By Lemma 2 in [9], the mapping $ \Cal B \mapsto \varphi (t, \cdot ,\Cal B ) $
from $ H(\Cal A ) $ into $ [E] $ is continuous for all $ t \in \Bbb R$.
Then applying Theorem 1.3 this proof is complete.
\enddemo

\noindent{\bf Partial linear differential equations.} Let $ \Lambda $ be some complete
metric space of linear closed operators acting on a 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 (2.1) and its $ H-$ class (2.2).

a.) For every $ v \in E $ and $ \Cal B \in H( \Cal A ) $ equation (2.2) has exactly
one solution that is defined on $ \Bbb R_{+} $ and satisfies the condition
$ \varphi (0, v, \Cal B ) = v $.

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 )$.

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 (2.2),
i.\ e., $ U(t,\Cal B )v= \varphi (t,v,\Cal B ) \quad ( t \in \Bbb R_{+} ,
v \in E $ and $ \Cal \in H(\Cal A ) $ ).

Under the above assumptions,  (2.1) 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 Theorem 1.4 to this system, we will obtain the
analogue to Theorem 2.1 for different classes of partial differential equations.

We will consider an example of a partial differential equation which
satisfies conditions a)-c) above. 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
 and bounded functions on $ \Bbb R_{+} $ with values in $ \Cal H $.

Denote by $ ( C(\Bbb R , [\Cal H ] ), \Bbb R , \sigma ) $ the dynamical
system of shifts on $ C(\Bbb R , [\Cal H ] ) $. Consider the equation
$$ \int \limits_{\Bbb R_{+} }  \langle  u(t),\varphi '(t)\rangle +
\langle  \Cal A (t)u(t),\varphi (t) \rangle \,dt = 0\,,
\eqno (2.3) $$
along with the family of equations
$$ \int \limits_{\Bbb R_{+} } \langle u(t),\varphi '(t)\rangle +
\langle \Cal B (t)u(t),\varphi (t) \rangle \,dt = 0\,,
\eqno (2.4) $$
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 ] ). $

A function $ u \in C( \Bbb R_{+} , \Cal H ) $ is called a solution of (2.3),
if the equality in (2.3) is satisfied for all
$ \varphi \in \Cal D ( \Bbb R_{+} , \Cal H )$.

Assume that the operator
$ \Cal A (t) $ is self-adjoint. Let
$ ( H(\Cal A ) , \Bbb R , \sigma ) $ be the dynamical system of shifts on
$ H( \Cal A )$, $\varphi (t,v,\Cal B ) $ be a solution of (2.4)
with the 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 (2.3) defined on $ \Bbb R_{+} $. According to Lemma 2.21 in [10] the set
$ X $ is closed in $ \overline X $. By Lemma 2.22 in [10] 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, with
$ 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 we can apply Theorem 1.3.
Thus the following assertion takes place.

\proclaim{Theorem 2.2} Let $ \Cal A \in C(\Bbb R, [\Cal H ] ) $ be recurrent
and the zero solution of (2.1) and the zero solutions of (2.2)
be asymptotically stable, i.\ e.,
$ \lim \limits_{t \to + \infty } | \varphi (t,v,\Cal B ) | =0 $ for
all $ v \in E $ and $ \Cal B \in H(\Cal A )$. Then the zero solution of
(2.1) is uniformly stable, i.\ e., there exists $ M \ge 0 $ such that
$ | \varphi (t,v,\Cal B ) | \le M| v | $ for all $ t \ge 0, v \in \Cal H $
and $ \Cal B \in H(\Cal A ) .$
\endproclaim

We will give an example of a boundary-value problem reduced to an equation
 of type (2.3). Let $ \Omega $ be a bounded domain in
$ \Bbb R^n$, $\Gamma $ be boundary  of $\Omega$,
$Q = \Bbb R_{+} \times \Omega $ and $ S = \Bbb R_{+} \times \Gamma $.
In $Q$ consider the initial boundary-value problem
$$ \frac{\partial u}{\partial t}
= L (t)u  \qquad ( u|_{t=0} =\varphi , u|_{S}=0 )\,, \eqno (2.5) $$
where
$$ L (t)u = \sum_{i,j=1}^{n}\frac{\partial}{\partial x_{i}}
(a_{ij}(t,x)\frac{\partial u}{\partial x_{j}})-a(t,x)u\,.
$$
By the Riesz representation theorem,
$$ \langle \Cal A (t)u, \varphi \rangle= - \int \limits_{\Omega } [ \sum_{i,j=1}^{n}
a_{ij}(t,x)\frac{\partial u}{\partial x_{j}}
\frac{\partial \varphi}{\partial x_{i}} + a(t,x)u\varphi ]dx . $$
If
$ a_{ij}(t,x)=a_{ji}(t,x)$  and the functions
$ a_{ij}(t,x) $ and $ a(t,x) $ are recurrent (almost periodic)
with respect to $ t \in \Bbb R $
uniformly with respect to $ x \in \Omega $,
then we can apply Theorem 2.2 to equation (2.5), if
$ \Cal H = \dot W^{1}_2(\Omega) $. \medskip

\noindent{\bf Linear functional-differential equations.}
Let $ r > 0$, $C([a,b], \Bbb R^n$)  be the Banach space consisting of
continuous functions from $[a,b]$ to $\Bbb R^n$ with the supremum norm.
Then we put $ C=C([-r,0],\Bbb R^n)$.
Let $ \sigma \in \Bbb R$, $A \ge 0$ and
$ u \in C([\sigma - r, \sigma + A],\Bbb R^{n}) $.
For $ t \in [\sigma , \sigma + A ] $ we define
$ u_{t} \in C $ by $ u_{t}(\theta )=u(t + \theta )$, $-r \le \theta \le 0$.
Denote by $ \frak A = \frak A (C, \Bbb R^n) $
the Banach space consisting of linear continuous operators from
$ C $ into $ \Bbb R^n$, equipped with the operator norm.
Consider the equation
$$ u' = \Cal A (t)u_{t}  \,, \eqno (2.6) $$
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 closure in the
topology of uniform convergence on compact subsets of $ \Bbb R $.

Along with equation (2.6) we also consider the family of equations
$$ u' = \Cal B (t)u_{t}  \,, \eqno (2.7) $$
where $ \Cal B \in H(\Cal A )$.
Let $ \varphi (t,v,\Cal B ) $  be a solution of (2.7) with
$ \varphi_0(v,\Cal B )=v$ 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 ) $ be the dynamical system on
$ X $, defined by
$ \pi (t,(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, and the following assertion takes
place.

\proclaim{Lemma 2.3} 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 (2.6) is completely continuous.
\endproclaim

\demo{Proof} Let $ B$ be a bounded subset of $C[-r,0] $, and $ t \ge r $.
By the continuity of the mapping
$ \varphi : \Bbb R_{+} \times C \times H(A) \mapsto C $ and the compactness
of $ H(\Cal A) $ there exists a positive number $ M $
such that $ | \varphi_{\tau}(v,\Cal B) | \le M $ and $ | \Cal B (\tau)
\varphi_{\tau} (v,\Cal B) | \le M $ for all $ \tau \in [0,t]$,
$\Cal B \in H(\Cal A) $ and $ v \in B $. Consequently,
$ | \dot \varphi (\tau , v, \Cal B )| \le M $ for all $ \tau \in [0,t]$,
$\Cal B \in H(\Cal A) $, and $ v \in B $, i.\ e., the
family of functions
$ \{ \varphi_{t}(v, B) : \Cal B \in H(\Cal A), v \in B \} $
( for $ t \ge r $) is uniformly continuous on $ [-r,0] $.
Therefore, this family of functions is precompact, and the present proof
is complete.
\enddemo

\proclaim{Theorem 2.4} Let $ H(\Cal A ) $ be compact. Then the following
assertion are equivalent.

1. For any $ \Cal B \in H(\Cal A ) $ the zero solution of (2.7) is
asymptotically stable, i.\ e.,
$ \lim \limits_{t \to + \infty } | \varphi_{t}(v,\Cal B ) | = 0 $
for all $ v \in C $ and $ \Cal B \in H(\Cal A ) $.

2. The zero solution of (2.6) is uniformly asymptotically stable, i.\ e.,
there are the positive numbers $ N$ and $ \nu $ such that
$ | \varphi_{t}(v,\Cal B ) | \le Ne^{-\nu t}| v | $
for all $ t \ge 0$, $v \in 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 (2.6).
By Lemma 2.3 this system is completely continuous.
Then using Theorems 1.1 and 1.5, we conclude the present proof.
\enddemo

Consider the neutral functional differential equation
$$ \frac{d}{dt}Dx_{t}=\Cal A (t)x_{t} \,,     \eqno (2.8) $$
where $ {\Cal A} \in C(\Bbb R , \frak A ) $ and $ D \in \frak A $
is non-atomic at zero [14, p. 67]. As in the case of (2.6),
the equation (2.8) generates a linear 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 ) $.

\proclaim
{Theorem 2.5 [4, 11] } Let $ \langle (X, \Bbb R_{+},\pi), (Y,\Bbb R, \sigma ), h \rangle $
be a non-autonomous  dynamical system, and the mapping
$ \pi^t=\pi(\cdot,t):X \to X ( t \Bbb R_{+} ) $
be representable as a sum $ \pi (x,t)=\varphi (x,t) + \psi (x,t) $
for all $ t \Bbb R_{+} $ and  $x \in X$,  and the following conditions be
are fulfilled.

1. $| \varphi (x,t) | \le m(t,r) $ for all
$ t \Bbb R_{+}$, $r>0 $ and $ | x | \le r $,
where  $ m: \Bbb R_+ \times \Bbb R_+  \to \Bbb R_+ $
and   $ m(t,r) \to 0 $ for $ t \to + \infty $;

2. The mappings $ \psi (\cdot ,t) : X \to X ( t>0 ) $
are conditionally completely continuous, i.\ e., $ \psi (A,t)$ is relatively
compact for any   $ t>0$ and any bounded positively invariant set
$ A \subseteq X $.

Then the dynamical system $(X,\Bbb R_{+},\pi)$ is asymptotically compact.
\endproclaim

\proclaim{Lemma 2.6} Let $H( \Cal A )$ be compact and the operator
$D$ be stable, i.\ e., the zero solution of the homogeneous difference
equation $Dy_{t}=0$ is uniformly asymptotically stable.
Then a linear non-autonomous dynamical system
$ \langle (X, \Bbb R_{+},\pi), (Y,\Bbb R, \sigma ), h \rangle $,
generated by (2.8), is asymptotically compact.
\endproclaim

\demo{Proof} According to [15, p.119, formula (5.18)] the mapping
$\varphi_{t} (\cdot,\Cal B ) : C \to 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 this proof is
complete  by referring to Theorem 2.5.
\enddemo

\proclaim{Theorem 2.7} 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 ) $ ) and let
$ D $ be stable. Then the following assertions are equivalent.

1. The zero solution of (2.6) and the zero solutions of all equations
$$ \frac{d}{dt}Dx_{t}=\Cal B (t)x_{t}  \,,    \eqno (2.9) $$
where $ \Cal B \in H (\Cal A ) $, are asymptotically stable, i.\ e.,
$ \lim \limits_{t \to + \infty } | \varphi (t,v,\Cal B ) | = 0 $
for all $ v \in C $ and $ \Cal B \in H(\Cal A )$  ($\varphi (t,v,\Cal B$)
is a solution of (2.9) with $\varphi (0,v, \Cal B )=v $).

2. The zero solution of (2.8) is uniformly exponentially stable, i.\ e., there
are a positive numbers $ N $ and $ \nu $ such that
$ | \varphi (t,v, \Cal B ) | \le N e ^{-\nu t}| v | $ for all $ t \ge 0$,
$v \in C$ and $\Cal B \in H(\Cal A )$.
\endproclaim

\demo{Proof} Let $\langle (X, \Bbb R_{+},\pi), (Y,\Bbb R, \sigma ), h\rangle$
 be a linear non-autonomous dynamical system,
generated by (2.8). By Lemma 2.6  this system is asymptotically compact.
To complete this proof it is sufficient to refer to Theorems 1.1 and 1.5.
\enddemo


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