DSpace at VNU: C-semigroups and almost periodic solutions of evolution equations

journal of differential equations 152, 358 376 (1999)
Article ID jdeq.1998.3531, available online at http:ÂÂwww.idealibrary.com on

Evolution Semigroups and Spectral Criteria for Almost
Periodic Solutions of Periodic Evolution Equations
Toshiki Naito and Nguyen Van Minh
Department of Mathematics, The University of Electro-Communications, 1-5-1 Chofugaoka,
Chofu, Tokyo 182, Japan
E-mail: naitoÄe-one.uec.ac.jp; minhÄim.uec.ac.jp
Received November 17, 1997; revised June 30, 1998

We investigate spectral criteria for the existence of (almost) periodic solutions to
linear 1-periodic evolution equations of the form dxÂdt=A(t) x+f (t) with (in
general, unbounded) A(t) and (almost) periodic f. Using the evolution semigroup
associated with the evolutionary process generated by the equation under consideration we show that if the spectrum of the monodromy operator does not intersect the set e isp( f ), then the above equation has an almost periodic (mild) solution
x f which is unique if one requires sp(x f )/[*+2?k, k # Z, * # sp( f )]. We emphasize
that our method allows us to treat the equations without assumption on the existence
of Floquet representation. This improves recent results on the subject. In addition
we discuss some particular cases, in which the spectrum of monodromy operator
does not intersect the unit circle, and apply the obtained results to study the
asymptotic behavior of solutions. Finally, an application to parabolic equations is
1999 Academic Press
considered.
Key Words: periodic equation; monodromy operator; spectrum of bounded function;
(almost) periodic solution; exponential dichotomy.

1. INTRODUCTION AND PRELIMINARIES
Let us consider the following linear evolution equations
dx
=A(t) x,
dt

(1)

dx
=A(t) x+f (t),
dt

(2)

and

where x # X, X is a complex Banach space, A(t) is a (unbounded) linear
operator acting in X for every fixed t # R such that A(t)=A(t+1) for all
t # R, f : R Ä X is an almost periodic function. Under suitable conditions
Eq. (1) is well posed (see, e.g., [P]), i.e., one can associate with Eq. (1)
358
0022-0396Â99 30.00
Copyright
1999 by Academic Press
All rights of reproduction in any form reserved.


SEMIGROUPS AND ALMOST PERIODICITY

359

an evolutionary process (U(t, s)) t s which satisfies the conditions in
Definition 1.
Once the evolution equations (1) and (2) are well posed, the asymptotic
behavior of solutions at infinity is of particular interest, which has been a

central topic discussed for the past two decades. We refer the reader to the
books [Hal1, He, LZ, Na, Ne1], and the surveys [Bat, V4], and the
references therein for more complete information on the subject.
One of the interesting topics in the study of the asymptotic behavior of
solutions to well-posed evolution equations is to find (spectral) criteria for
periodicity and almost periodicity of solutions to the well posed Eqs. (1)
and (2). We refer the reader to the books [LZ, HMN, Pr2], and recent
papers [AB1, Bas, RV, V1, V2, V4, VS] for more information on this subject.
In this direction, if A(t)=A \t # R, i.e., A(t) does not depend on the time
t, it is of particular interest to find conditions on the spectra of A and f so
that Eq. (2) has an almost periodic solution which is unique in some
subspace of AP(X) (the space of all X-valued almost periodic continuous
functions in Bohr's sense). To this end, let A generate a strongly
continuous semigroup (T(t)) t 0 . Then if _(T(1)) & [e i*, * # sp( f )]=<,
Eq. (2) has an almost periodic mild solution which is unique in the subspace of AP(X) consisting of all functions whose spectra are contained in
sp( f ) (see, e.g., [Pr2, V1, V2, VS]). In particular, if T(1) is hyperbolic, i.e.,
its spectrum does not intersect the unit circle, then the unique solvability
of Eq. (2) follows. This can characterize the exponential dichotomy (or
hyperbolicity) of the semigroup (T(t)) t 0 (see [Pr1]).
Recently, several efforts have been made to extend the above mentioned
results to some classes of periodic evolution equations such as the class of
equations in the presence of a Floquet representation (see [V1]) and the
class of equations generating invertible evolution operators (see [ZM,
AMZ, Ra]). As is known, these classes of periodic evolution equations do
not include many classes of evolution equations frequently met in applications (see, e.g., [Hal2, Chap. 8; He, Chap. 7]).
The main purpose of this paper is to make an attempt to fill this gap.
The results obtained here are stated in terms of spectral properties of the
monodromy operators determined from the processes under consideration.
Our method is to employ the evolution semigroups associated with the
underlying evolutionary processes. This method has been widely used

to study the asymptotic behavior of evolution equations in recent years
(see, e.g., [LM, M1, M2, RS, S, MRS] and the references therein for
more information). In this paper we consider the evolution semigroups
associated with the underlying evolutionary processes in various subspaces
of the space of all almost periodic X-valued functions and make use of the
explicit formula for the generator of the evolution semigroup. This allows us


360

NAITO AND MINH

to state conditions for the unique solvability of Eq. (2) in terms of spectral
properties of the associated evolution semigroup. In turn, since there is a
direct relation between the spectrum of the associated evolution semigroup
and that of the monodromy operators we can find conditions for the
unique solvability of Eq. (2) in terms of spectral properties of the
monodromy operators. We then employ the unique solvability of Eq. (2) to
study the exponential dichotomy of the evolutionary process generated by
Eq. (1).
We now give an outline of the contents of the paper. Our paper consists
of three sections. Section 1 contains the introduction, preliminaries on the
theory of spectra of functions and the main notations which will be used
throughout the paper. Section 2 consists of 4 subsections corresponding to
the unique solvability of the inhomogeneous equations and the semilinear
equations in various subspaces of the space of almost periodic functions.
Our main results are stated in Theorems 2, 3 of this section. In Section 3
we consider parabolic equations with natural conditions which satisfy the
conditions of Theorems 2, 3.
Below we recall some notions and results of the spectral theory of

uniformly continuous and bounded functions on the real line. Throughout
the paper we shall denote by Z, R, C the set of integers, real and complex
numbers, respectively. We shall denote by S 1 the unit circle in the complex
plane C. L(X) will denote the space of bounded linear operators from X to
itself. _(A), \(A) will denote the spectrum and resolvent set of an operator A.
BUC(R, X) will stand for the space of all uniformly continuous and bounded
functions on the real line with sup-norm. The subspace of BUC(R, X)
consisting of all almost periodic functions in the sense of Bohr will be
denoted by AP(X) and can be defined to be the smallest closed subspace
of BUC(R, X) containing the functions of the form [e i' } x, ' # R, x # X]
(see, e.g., [LZ]). We will denote by (S(h)) h # R the group of translations in
AP(X) defined by the formula
[S(h) v](s)=v(h+s),

\v # AP(X), h, s # R.

(3)

We denote by F the Fourier transform, i.e.,
(Ff )(s) :=

|

+

e &istf (t) dt

(4)

&

(s # R, f # L 1(R)). Then the spectrum of u # BUC(R, X) is defined to be the
set
sp(u) :=[! # R : \=>0 _f # L 1(R), suppFf/(!&=, !+=), f V u{0],

(5)


SEMIGROUPS AND ALMOST PERIODICITY

361

where
f V u(s) :=

|

+

f (s&t) u(t) dt.

&

It coincides with the set _(u) consisting of ! # R such that the Fourier
Carleman transform of u

u^(*)=

{

|

e &*tu(*) dt,

(Re*>0),

0

&

(6)

|

(Re*<0),

*t

e u(&t) dt,
0

has a holomorphic extension to a neighborhood of i! (see, e.g., [Pr2,
Proposition 0.5, p. 22]). We collect some main properties of the spectrum
of a function, which we will need, in the following theorem for the reader's
convenience.
Theorem 1.
Then

Let f, g n # BUC(R, X) such that lim n Ä

(i)
(ii)

_( f ) is closed ;
_( f ( }+h))=_( f );

(iii)

If : # C"[0] _(:f )=_( f );

(iv)

If _(g n )/4 for all n # N, then _( f )/4.

&g n & f &=0.

Proof. For the proof we refer the reader to [Pr2, Proposition 0.4, p. 20
and Theorem 0.8, p. 21].

2. MAIN RESULTS
We begin this section by considering the evolutionary semigroup
associated with the given 1-periodic strongly continuous process
(U(t, s)) t s; t, s # R . We recall these concepts in the following definitions.
Definition 1. A family of bounded linear operators (U(t, s)) t s ,
(t, s # R) from a Banach space X to itself is called 1-periodic strongly continuous evolutionary process if the following conditions are satisfied
(i)

U(t, t)=I for all t # R,

(ii)

U(t, s) U(s, r)=U(t, r) for all t

(iii)

The map (t, s) [ U(t, s) x is continuous for every fixed x # X,

s

r,


362

NAITO AND MINH

(iv)
(v)

U(t+1, s+1)=U(t, s) for all t
&U(t, s)&
|(t&s)

s,

for some positive N, | independent of t

s.

If it does not cause any danger of confusion, for the sake of simplicity,
we shall often call 1-periodic strongly continuous evolutionary process
(evolutionary) process.
First we collect some results which we shall need in the paper. Recall
that for a given 1-periodic evolutionary process (U(t, s)) t s the following
operator
P(t) :=U(t, t&1), t # R

(7)

is called monodromy operator (or sometime, period map, Poincare map).
Thus we have a family of monodromy operators. Throughout the paper we
will denote P :=P(0). The nonzero eigenvalues of P(t) are called characteristic multipliers. An important property of monodromy operators is
stated in the following lemma.
Lemma 1 ([He, Lemma 7.2.2, p. 197]). P(t+1)=P(t) for all t; characteristic multipliers are independent of time, i.e., the nonzero eigenvalues of
P(t) coincide with those of P. Moreover, _(P(t))"[0]=_(P)"[0], i.e., it is
independent of t.
Definition 2. The following formal semigroup associated with a given
1-periodic strongly continuous evolutionary process (U(t, s)) t s
(T hu)(t) :=U(t, t&h) u(t&h),

\t # R,

(8)

where u is an element of some function space, is called evolutionary semigroup associated with the process (U(t, s)) t s .
Below we are mainly concerned with the following inhomogeneous equation
x(t)=U(t, s) x(s)+

|

t

U(t, !) f (!) d!,

\t

s

(9)

s

associated with a strongly continuous 1-periodic evolutionary process
(U(t, s)) t s . A continuous solution u(t) of Eq. (9) will be called mild solution to Eq. (2). The following lemma will be the key tool to study spectral
criteria for almost periodicity in this paper.
Lemma 2. Let (U(t, s)) t s be 1-periodic strongly continuous evolutionary
process. Then its associated evolutionary semigroup (T h ) h 0 is strongly continuous in AP(X). Moreover, the infinitesimal generator of (T h ) h 0 is the


SEMIGROUPS AND ALMOST PERIODICITY

363

operator L defined as follows: u # D(L) and Lu=&f if and only if u, f #
AP(X) and u is the solution to Eq. (9).
Proof. Let v # AP(X). First we can see that T h acts on AP(X). In fact
this follows from the fifth item of the definition of the 1-periodic strongly
continuous evolutionary processes and [DK, Chap. VII, Lemma 4.1]. By
definition we have to prove that

lim sup &U(t, t&h) v(t&h)&v(t)&=0.

h Ä 0+

(10)

t

Since v # AP(X) the range of v( } ) which we denote by K (consisting of x # X
such that x=v(t) for some real t) is a relatively compact subset of X.
Hence the map (t, s, x) [ U(t, s) x is uniformly continuous in the set
[1 t s &1, x # K]. Now let = be any positive real. In view of the
uniform continuity of the map (t, s, x) [ U(t, s) x in the above-mentioned
set there is a positive real $=$(=) such that
&U(t&[t], t&[t]&h) x&x&<=

(11)

for all 0n tlim sup &U(t, t&h) v(t&h)&v(t&h)&=0.

h Ä 0+

(12)

t

Now we have
lim sup &U(t, t&h) v(t&h)&v(t)&

h Ä 0+

t

lim sup &U(t, t&h) v(t&h)&v(t&h)&

h Ä 0+

t

+ lim sup &v(t&h)&v(t)&.
h Ä 0+

(13)

t

Since v is uniformly continuous this estimate and (12) imply (10), i.e., the
evolutionary semigroup (T h ) h 0 is strongly continuous in AP(X).
The remainder of the proof can be done by following the idea of [M2,
Lemma 1]. For the reader's convenience we quote it here. Let us consider
the affine semigroup (T fh ) h 0 associated with the inhomogeneous equation
(9) for f # AP(X), defined as
T fh v=T hv+

|

h

0

T h&!f d!=T hv+

|

h

0

T !f d!,

(14)


364

NAITO AND MINH

where v # AP(X), h

0. It is easily checked that for all v # AP(X) we have

[T fh v](t)=U f (t, t&h) v(t&h),

t # R, h

0,

where U f (t, s) is the evolutionary operator defined by the integral equation

(9). In other words, the assertion that g, f # AP(X) and g is a solution of
(9), is equivalent to T fh g= g (\h 0). From (15) this is equivalent to
g=T fh g=T hg+
h

T g& g=

|

h

0

!

|

h

T !f d!,

\h

0,

0

T Ag d!=&

(15)

|

h

!

T f d!,

\h

0.

0

From the general theory of linear operator semigroups [P, pp. 4 5] this is
equivalent to the assertion Ag=&f. K
In view of the strong continuity of the evolutionary semigroup (T h ) h
in C 0(R, X) (see, e.g., [LM], [Ra]) we have the following

0

Corollary 1. Let (U(t, s)) t s be a 1-periodic strongly continuous process. Then its associated evolutionary semigroup (T h ) h 0 is a C 0 -semigroup
in
AAP(X) :=AP(X) Ä C 0(R, X).
Below we shall consider the evolutionary semigroup (T h ) h
special invariant subspaces M of AP(X).

0

in some

Definition 3. The subspace M of AP(X) is said to satisfy condition H
if the following conditions are satisfied.
(i)

M is a closed subspace of AP(X).

(ii) There exists * # R such that M, contains all functions of the form
e i* } x, x # X.
(iii)

For all f # M, h>0, U( } , }&h) f ( } ) # M.

(iv) If C(t)is a norm-continuous 1-periodic operator valued function
and f # M, then C( } ) f ( } ) # M.
(v)
by (3).

M is invariant under the group of translations (S(h)) h # R defined

In what follows, technically, we need the following condition on the
evolutionary process under consideration:


SEMIGROUPS AND ALMOST PERIODICITY

365

Definition 4. The 1-periodic strongly continuous evolutionary process

(U(t, s)) t s is said to satisfy condition C if the map taking t into U(t, t&h)
is continuous in operator topology for every fixed positive h.
In the sequel we will be concerned mainly with the following concrete
examples of subspaces of AP(X) which satisfy condition H:
Example 1. Let us denote by P(1) the subspace of AP(X) consisting of
all 1-periodic functions. It is clear that P(1) satisfies condition H.
Example 2. Let (U(t, s)) t s satisfy condition C. Hereafter, for
every given f # AP(X), we shall denote by M( f ) the subspace of AP(X)
consisting of all almost periodic functions u such that sp(u)/
[*+2?n, n # Z, * # sp( f )]. Then M( f ) satisfies condition H. In fact, by
Theorem 1, it is a closed subspace of AP(X), and moreover it satisfies
conditions (ii), (v) of the definition. We now check that conditions (iii)
and (iv) are also satisfied. This can be done in the same way as in [V1,
Lemma 4.3].
The following corollary will be the key tool to study the unique
solvability of the inhomogeneous equation (9) in various subspaces M of
AP(X) satisfying condition H.
Corollary 2. Let M satisfy condition H. Then, if 1 # \(T 1 | M ), the
inhomogeneous equation (9) has a unique solution in M for every f # M.
Proof. Under the assumption, the evolutionary semigroup (T h ) h 0
leaves M invariant. The generator A of (T h | M ) h 0 can be defined as the
part of L in M. Thus, the corollary is an immediate consequence of
Lemma 2 and the spectral inclusion e _(A) /_(T 1 | M ).
Let M be a subspace of AP(X) invariant under the evolution semigroup
(T h ) h 0 associated with the given 1-periodic evolutionary process
(U(t, s)) t s in AP(X). Below we will use the following notation:
P M v(t) :=P(t) v(t),

\t # R, v # M.

If M=AP(X) we will denote P M =P.
In the sequel we need the following
Lemma 3. Let (U(t, s)) t s be a 1-periodic strongly continuous evolutionary process and M be an invariant subspace of the evolution semigroup
(T h ) h 0 associated with it in AP(X). Then the following assertions hold.


366

NAITO AND MINH

(i)

If M :=P(1), then _(P M )"[0]/_(P)"[0].

(ii) If the 1-periodic strongly
(U(t, s)) t s satisfies condition C, then

continuous evolutionary process

_(P M )"[0]=_(P)"[0]
for all invariant subspaces M satisfying condition H.
Proof. (i) Let * # \(P)"[0]. Then we show that * # \(P M )"[0]. In fact,
by the strong continuity of the underlying process and Lemma 1 it may be
noted that the map taking t into R(*, P(t)) :=(*&P(t)) &1 is strongly continuous (see, e.g., [Ra, Proposition 12]). Hence suppose that v # P(1). We
have to solve the equation (*&P(t)) x(t)=v(t) in P(1). Since
(*&P(t)) &1 v(t) is continuous and 1-periodic, it is a solution to the above
equation. Moreover, it is unique. Thus (i) is proved.
(ii) For u, v # M, consider the equation (*&P M ) u=v. It is equivalent to the equation (*&P(t)) u(t)=v(t), t # R. If * # \(P M )"[0], for
every v the first equation has a unique solution u, and &u&
&R(*, P M )& &v&. Take a function v # M of the form v(t)= ye i+t, for some

+ # R; the existence of such a + is guaranteed by the axioms of condition
H. Then the solution u satisfies &u& &R(*, P M )& &y&. Hence, for every
y # X the solution of the equation (*&P(0)) u(0)= y has a unique solution
u(0) such that
&u(0)&

sup &u(t)&
t

&R(*, P M )& sup &v(t)&

&R(*, P M )& &y&.

t

This implies that * # \(P)"[0] and &R(*, P(t))& &R(*, P M )&.
Conversely, suppose that * # \(P)"[0]. By Lemma 1 for every v the
second equation has a unique solution u(t)=R(*, P(t)) v(t). On the other
hand, it may be noted from the assumption that the map taking t into
R(*, P(t)) is norm continuous. By definition of condition H, the function
taking t into (*&P(t)) &1 v(t) belongs to M. Since R(*, P(t)) is a norm
continuous, 1-periodic function, it holds that r :=sup[&R(*, P(t))&:
t # R]< . This means that &u(t)& r&v(t)& r sup t &v(t)&, or &u& r &v&.
Hence * # \(P M ), and &R(*, P M )& r. K
We now illustrate Corollary 2 in some concrete situations.
2.1. Unique Solvability of the Inhomogeneous Equations in P(1)
In this subsection we will consider the unique solvability of Eq. (9) in
P(1).

SEMIGROUPS AND ALMOST PERIODICITY

367

Proposition 1. Let (U(t, s)) t s be 1-periodic strongly continuous. Then
the following assertions are equivalent :
(i)

1 # \(P),

(ii)

Eq. (9) is uniquely solvable in P(1) for a given f # P(1).

Proof. Suppose that (i) holds true. Then we show that (ii) holds by
applying Corollary 2. To this end, we show that _(T 1 | P(1) )"[0]/
_(P)"[0]. To see this, we note that
T 1 | P(1) =P P(1) .
In view of Lemma 3, 1 # \(T 1 | P(1) ). By Example 1 and Corollary 2 (ii)
holds true also.
The proof of the converse conclusion is suggested by [Pr1, p. 849]. In
fact, we suppose that Eq. (9) is uniquely solvable in P(1). We now show
that 1 # \(P). For every x # X put f (t)=U(t, 0) g(t) x for t # [0, 1], where
g(t) is any continuous function of t such that g(0)= g(1)=0, and

|

1

g(t) dt=1.

0

Thus f (t) can be continued to a 1-periodic function on the real line which
we denote also by f (t) for short. Put Sx=[L &1(& f )](0). Obviously, S is
a bounded operator. We have
[L &1(& f )](1)=U(1, 0)[L &1(&f )](0)+

|

1

U(1, !) U(!, 0) g(!) x d!

0

Sx=PSx+Px.
Thus
(I&P)(Sx+x)=Px+x&Px=x.
So, I&P is surjective. From the uniqueness of solvability of (9) we get
easily the injectiveness of I&P. In other words, 1 # \(P). K
Remarks. Proposition 1 in the autonomous case has been proved in
[Pr2, Example 12.1, p. 315], see also [Pr1, p. 849]. Another autonomous
form of Proposition 1 can also be found in [LM, Theorem 2.2, p. 179]
which implies Gearhart's spectral mapping theorem for C 0 -semigroups in
Hilbert spaces, and a spectral mapping theorem for C 0 -semigroups in
Banach spaces (see [LM]). See also [VS] for another proof of this
autonomous result.

368

NAITO AND MINH

2.2. Unique Solvability in AP(X ) and Exponential Dichotomy
This subsection will be devoted to the unique solvability of Eq. (9) in
AP(X) and its applications to the study of exponential dichotomy. Let us
begin with the following lemma which is a consequence of Proposition 1.
Lemma 4. Let (U(t, s)) t s be 1-periodic strongly continuous. Then the
following assertions are equivalent :
(i)

S 1 & _(P)=<,

(ii) for every given + # R, f # P(1) the following equation has a unique
solution in AP(X)
x(t)=U(t, s) x(s)+
Proof.

|

t

U(t, !) e i+!f (!) d!,

s.

\t

(16)

s

Suppose that (i) holds, i.e., S 1 & _(P)=<. Then, since
T 1 =S(&1) } P =P } S(&1)

in view of [Ru, Theorem 11.23, p. 193],
_(T 1 )/_(S(&1)) . _(P ).
It may be noted that _(S(&1))=S 1. Thus
_(T 1 )/[e i+*, + # R, * # _(P )].
Hence, in view of Lemma 3
_(T 1 ) & S 1 =<.
Let us consider the process (V(t, s)) t

s

defined by

V(t, s) x :=e &i+(t&s)U(t, s) x
for all t s, x # X. Let Q(t) denote its monodromy operator, i.e., Q(t)=
e &i+V(t, t&1) and (T h+ ) h 0 denote the evolution semigroup associated with
the evolutionary process (V(t, s)) t s . Then by the same argument as above
we can show that since _(T h+ )=e &i+_(T h ),
_(T +h ) & S 1 =<.
By Lemma 2 and Corollary 2, the equation
y(t)=V(t, s) y(s)+

|

t

V(t, !) f (!) d!,
s

\t

s


369

SEMIGROUPS AND ALMOST PERIODICITY

has a unique almost periodic solution y( } ). Let x(t) :=e i+ty(t). Then
x(t)=e i+ty(t)=U(t, s) e i+sy(s)+
=U(t, s) x(s)+

|

t

|

t

U(t, !) e i+!f (!) d!

s

U(t, !) e i+!f (!) d!,

\t

s.

s

Thus x( } ) is an almost periodic solution of Eq. (16). The uniqueness of x( } )
follows from that of the solution y( } ).
We now prove the converse. Let y(t) be the unique almost periodic
solution to the equation
y(t)=U(t, s) y(s)+

|

t

U(t, !) e i+!f (!) d!,

\t

s.

(17)

s

Then x(t) :=e &i+ty(t) must be the unique solution to the equation
x(t)=e &i+(t&s)U(t, s) x(s)+

|

t

e &i+(t&!)U(t, !) f (!) d!),

\t

s,

(18)

s

and vice versa. We show that x(t) should be periodic. In fact, it is easily
seen that x(1+} ) is also an almost periodic solution to Eq. (18). From the
uniqueness of y( } ) (and then that of x( } )) we have x(t+1)=x(t), \t. By
Proposition 1 this yields that 1 # \(Q(0)), or in other words, e i+ # \(P).
From the arbitrary nature of +, S 1 & _(P)=<. K
Remark. From Lemma 4 it follows in particular that the inhomogeneous
equation (9) is uniquely solvable in the function space AP(X) if and only if
S 1 & _(P)=<. This remark will be useful to consider the asymptotic
behavior of the solutions to the homogeneous Eq. (1). For the related
results on stability we refer the reader to [Ne1], [Ne2].
Before applying the above results to study the exponential dichotomy of
1-periodic strongly continuous processes we recall that a given 1-periodic
strongly continuous evolutionary process (U(t, s)) t s is said to have an
exponential dichotomy if there exist a family of projections Q(t), t # R and
positive constants M, : such that the following conditions are satisfied:
(i)

For every fixed x # X the map t [ Q(t) x is continuous,

(ii)

Q(t) U(t, s)=U(t, s) Q(s), \t

(iii)

\t

&U(t, s) x&

Me

&:(t&s)

s,

&x&, \t

&1 :(t&s)

s, x # ImQ(s),

(iv)

&U(t, s) y&

(v)

s.

U(t, s)| KerQ(s) is an isomorphism from KerQ(s) onto KerQ(t),

M

e

&y&, \t

s, y # KerQ(s),


370

NAITO AND MINH

Theorem 2. Let (U(t, s)) t s be given 1-periodic strongly continuous
evolutionary process. Then the following assertions are equivalent.
(i)

The process (U(t, s)) t

s

has an exponential dichotomy.

(ii) For every given bounded and continuous f the inhomogeneous
equation (9) has a unique bounded solution.
(iii) The spectrum of the monodromy operator P does not intersect the

unit circle.
(iv) For every given f # AP(X) the inhomogeneous equation (9) is
uniquely solvable in the function space AP(X).
Proof. The equivalence of (i) and (ii) has been established by Zikov
(for more general conditions, see, e.g., [LZ, Chap. 10, Theorem 1]). Now
we show the equivalence between (i), (ii), and (iii). Let the process have an
exponential dichotomy. We now show that the spectrum of the
monodromy operator P does not intersect the unit circle. In fact, from (ii)
it follows that for every 1-periodic function f on the real line there is a
unique bounded solution x( } ) to Eq. (9). This solution should be 1-periodic
by the periodicity of the process (U(t, s)) t s . According to Lemma 4,
1 # \(P). By the same argument as in the proof of Lemma 4 we can show
that e i+ # \(P), \+ # R. Conversely, suppose that the spectrum of the
monodromy operator P does not intersect the unit circle. The assertion
follows readily from [He, Theorem 7.2.3, p. 198]. The equivalence of (iv)
and (iii) is clear from Lemma 4. K
Remarks. The autonomous version of Theorem 2 can be found in
[Pr1]. The equivalence between (i) and (iii) of Theorem 2 has been proved
for the invertible periodic processes in [ZM], [AMZ] and [Ra].
2.3. Unique Solvability of the Inhomogeneous Equations in M( f )
Now let us return to the more general case where the spectrum of the
monodromy operator may intersect the unit circle.
Theorem 3. Let (U(t, s)) t s be a 1-periodic strongly continuous
evolutionary process which satisfies condition C. Moreover, let f # AP(X)
such that _(P) & [e i*, * # sp( f )]=<. Then the inhomogeneous equation (9)
has an almost periodic solution which is unique in M( f ).
Proof. From Example 2 it follows that the function space M( f )
satisfies condition H. Since (S(t)) t # R is an isometric C 0 -group, by the weak
spectral mapping theorem for isometric groups (see, e.g., [Na]) we have
_(S(1)| M( f ) )=e _(D| M( f ) ),

SEMIGROUPS AND ALMOST PERIODICITY

where D| M( f ) is the generator of (S(t)| M( f ) ) t
theory of bounded functions we have

0

371

. From the general spectral

_(D| M( f ) )=i4,
where 4=[*+2?k, * # sp( f ), k # Z] (see, e.g., the proof of this in that of
[V2, Lemma 23, p. 584 585]). Hence, since
e _(D| M( f ) ) =e i4 /e isp( f ) /e i4,
we have
_(S(1)| M( f ) )=e _(D| M( f ) ) =e isp( f ).
Thus, the condition
_(P) & e isp( f ) =<
is equivalent to the following:
1 Â _(P) . _(S(&1)| M( f ) ).
In view of the inclusion
_(T 1 | M( f ) )"[0]/_(P M( f ) ). _(S(&1)| M( f ) )"[0]
/_(P) . _(S(&1)| M( f ) )"[0],
which follows from the commutativeness of the operator P M( f ) with
S(&1)| M( f ) , the above inclusion implies that
1 Â _(T 1 | M( f ) ).
Now the assertion of the theorem follows from Corollary 2. K

2.4. Unique Solvability of Semilinear Equations
Let us consider the semilinear equation
x(t)=U(t, s) x(s)+

|

t

U(t, !) g(!, x(!)) d!.

(19)

s

We shall be interested in the unique solvability of (19) for a larger class of
the forcing term g. We shall show that the generator of evolutionary semigroup is still useful in studying the perturbation theory in the critical case
in which the spectrum of the monodromy operator P may intersect the unit
circle. We suppose that g(t, x) is Lipschitz continuous with coefficient k
and the Nemystky operator F defined by (Fv)(t)= g(t, v(t)), \t # R acts


372

NAITO AND MINH

in M. Below we can assume that M is any closed subspace of the space of
all bounded continuous functions BC(R, X). We consider the operator L in
BC(R, X). If (U(t, s)) t s is strongly continuous, then L is a single-valued
operator from D(L)/BC(R, X) to BC(R, X).
Lemma 5. Let M be any closed subspace of BC(R, X), (U(t, s)) t s be

strongly continuous and Eq. (9) be uniquely solvable in M. Then for sufficiently small k, Eq. (19) is also uniquely solvable in this space.
Proof. First, we observe that under the assumptions of the lemma we
can define a single-valued operator L acting in M as follows: u # D(L) if
and only if there is a function f # M such that Eq. (9) holds. From the
strong continuity of the evolutionary process (U(t, s)) t s one can easily see
that there is at most one function f such that Eq. (9) holds. This means L
is single-valued. Moreover, one can see that L is closed. Now we consider
the Banach space [D(L)] with graph norm, i.e., |v| =&v&+&Lv&. By
assumption it is seen that L is an isomorphism from [D(L)] onto M. In
view of the Inverse Function Theorem for Lischitz mappings (see, e.g.,
[Ma]) for sufficiently small k the operator L&F is invertible. Hence there
is a unique u # M such that Lu&Fu=0. From the definition of operator L
we see that u is a unique solution to Eq. (19). K
Corollary 3. Let M be any closed subspace of AP(X), (U(t, s)) t s be
1-periodic strongly continuous evolutionary process and for every f # M the
inhomogeneous equation (9) be uniquely solvable in M. Moreover let the
Nemytsky operator F induced by the nonlinear function g in Eq. (19) act on
M. Then for sufficiently small k, the semilinear equation (19) is uniquely
solvable in M.
Proof.

The corollary is an immediate consequence of Lemma 5. K

3. EXAMPLES
In this section we shall consider the abstract form of parabolic partial
differential equations (see, e.g., [He] for more details) and apply the results
obtained above to study the existence of almost periodic solutions to these
equations. It may be noted that a necessary condition for the existence of
Floquet representation is that the process under consideration is invertible.
It is known for the bounded case (see, e.g., [DK, Chap. V, Theorem 1.2])

that if the spectrum of the monodromy operator does not circle the origin
(of course, it should not contain the origin), then the evolution operators
admit Floquet representation. In the example below, in general, Floquet
representation does not exist. For instance, if the sectorial operator A has


SEMIGROUPS AND ALMOST PERIODICITY

373

compact resolvent, then monodromy operator is compact (see [He] for
more details). Thus, if dim X= , then monodromy operators cannot be
invertible. However, the above results can apply.
Let A be sectorial operator in a Banach space X, and the mapping
taking t into B(t) # L(X :, X) be Holder continuous and 1-periodic. Then
there is a 1-periodic evolutionary process (U(t, s)) t s associated with the
equation
du
=(&A+B(t)) u.
dt

(20)

We now check if this process satisfies condition C.
Claim 1. For any x 0 # X and { there exists a unique (strong) solution
x(t) :=x(t ; {, x 0 ) of Eq. (20) on [{, + ) such that x({)=x 0 . Moreover, if
we write x(t ; {, x 0 ) :=T(t, {) x 0 , \t {, then (T(t, {)) t { is a strongly
continuous 1-periodic evolutionary process which satisfies condition C. In
addition, if A has compact resolvent, then the monodromy operator P(t) is
compact.

Proof. This claim is an immediate consequence of [He, Theorem 7.1.3,
pp. 190 191]. In fact, it is clear that (T(t, {)) t { is strongly continuous and
1-periodic. We now check that condition C is satisfied with respect to this
process. According to [He, Theorem 7.1.3, (iii), (iv)], by choosing
;=#=0 we have
&T(t, t&h) x&T(s, s&h) x& 0
&T(t, t&h) x&T(s, t&h) x&+&T(s, t&h) x&T(s, s&h) x&
C(%)(t&s) % &x& 0
for all x # D(A 2 ), h>0, where %, C(%) are some positive constants independent of t, s, x. From this, letting t approach s, we see that this shows
that condition C is satisfied with respect to the evolutionary process
(T(t, {)) t { . The last assertion is contained in [He, Lemma 7.2.2,
p. 197]). K
Thus, in view of the above claim if dim X= , then Floquet representation does not exist for the process. However, the conclusion on the unique
existence of almost periodic solutions made in [V1, Example 2, p. 412] are
still valid in view of our Theorem 3. Namely, if the function f taking t into
f(t) # X is almost periodic and the spectrum of the monodromy operator of


374

NAITO AND MINH

the process (U(t, s)) t s is separated from the set e isp( f ), then the following
inhomogeneous equation
du
=(&A+B(t)) u+f (t)
dt
has a unique almost periodic solution u such that
sp(u)/[*+2?k, k # Z, * # sp( f )].
We now show

Claim 2. Let the conditions of Claim 1 be satisfied except for the compactness of the resolvent of A. Then
dx
=(&A+B(t)) x
dt
has an exponential dichotomy if and only if the spectrum of the monodromy
operator does not intersect the unit circle. Moreover, if A has compact resolvent, it has an exponential dichotomy if and only if all multipliers have
modulus different from one. In particular, it is asymptotically stable if and
only if all characteristic multipliers have modulus less than one.
Proof. The operator T(t, s), t>s is compact if A has compact resolvent
(see, e.g., [He, p. 196]). The claim is an immediate consequence of
Theorem 2. K

ACKNOWLEDGMENTS
The paper was written while N.V.M was the fellow of the Japan Society for the Promotion
of Science (JSPS). The support of the JSPS is gratefully acknowledged. The authors thank the
referee for the valuable remarks and suggestions to improve the presentation and to correct
inaccuracies in Theorem 3 of the paper, and for pointing out the references [Hu1, Hu2] with
closedly related results. Also, the authors thank C. J. K. Batty, F. Rabiger, and Q. P. Vu for
several remarks and discussions.

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