Exponential stability, exponential expansiveness, and exponential dichotomy of evolution equations on the half-line

Intcgr. equ. oper. theory 32 (1998) 332 - 353 0378-620X/98/030332-22 $1.50+0.20/0 �9 Birkhfiuser Verlag, Basel, 1998

i Integral Equations and Operator Theory

E X P O N E N T I A L S T A B I L I T Y , E X P O N E N T I A L E X P A N S I V E N E S S , A N D E X P O N E N T I A L D I C H O T O M Y

OF E V O L U T I O N E Q U A T I O N S O N T H E H A L F - L I N E

NGUYEN VAN MINH*, FRANK RABIGER, ROLAND SCHNAUBELT**

Let b/ = (U(t, s))t>s>0 be an evolution family on the half-line of bounded lin- ear operators on a Banach space X. We introduce operators Go, Gx and Ix on certain spaces of X-valued continuous functions connected with the integral equation u(t) = U(t, s)u(s) + f', U(t, r162162 and we characterize exponential stability, exponential expansiveness and exponential dichotomy of L /by proper- ties of Go, Gx and Ix, respectively. This extends related results known for finite dimensional spaces and for evolution families on the whole llne, respectively.

I N T R O D U C T I O N

Consider the non-autonomous linear evolution equation

-~u(t)d = A(t)u(t), t E J = •+ or R, (NCP)

on a Banach space X. If the operators A(t), t E J , are bounded (in particular, if X is

finite dimensional) there is an extensive literature initiated by the work of Perron which

connects asymptotic properties (of the solutions) of (NCP) with specific properties of the

operator L defined by

on a space of X-valued functions. For example, the existence of an exponential dichotomy

of (NCP) is related to the invertibility of L if J = ]R and to Fredholm properties of L if

J = R+ (see [BeG], [BGK], [Cop], [DaK], [MaS], [Pall and the references therein). If the

* This work was done while the first author was visiting the Department of Math- ematics of the University of Tfibingen. The support of the Alexander yon Humboldt Foundation is gratefully acknowledged. The author wishes to thank R. Nagel and the Functional Analysis group in T~bingen for their kind hospitality and constant encourage- ment.

** Support by Deutsche Forschungsgemeinschaft DFG is gratefully acknowledged.

Van Mirth, Rabigcr and Schnaubelt 333

operators A(t), t E J, are unbounded the situation becomes more delicate and usually one

has to impose rather restrictive conditions in order to obtain similar results referring to

the operator L (e.g. see [Hen], [Zha]).

A less restrictive assumption is the well-posedness of ( N C P ) in the sense that

the solutions of ( N C P ) yield an evolution family 5l = (U(t, s)),>_,, ,,seJ of bounded linear

operators on X (see [Nic], [Paz], [RRS], [RS2], [Sch] and [Tan] for sufficient conditions and

examples). Note that solution does not necessarily mean 'classical' solution. Then instead

of the operator L one investigates the integral equation

u(t) = U( t , s )u(s ) + U(t ,~)f(~)d~, t > s, t , s S J, ( IE)

and its connection with asymptotic properties of the evolution family 7,/(see [Bus], [DaK],

[Dat], [LRS], [LeZ], [Zhi]). For instance, if J = I~ then it is shown in [LRS] that U has

an exponential dichotomy if and only if for every f E Cb(]~, X) there is a unique solution

u e Cb(R, X ) of ( IZ ) .

Another approach uses the so-called evolution semigroup 7" = (T(t))~>_o on a

space of X-valued functions induced by the evolution family/X (see [AuM], [LaM], [LaR],

[LMR1], [LMR2], [Mil], [Mi2], [Nic], [RRS], [RS1], [RS2], [Ran], [Sch] and the references

therein for this concept). Roughly speaking (if J = R) the evolution semigroup T is given

by

T( t ) f (~) = U(~, ( - t ) f ( ( - t), t >_ O,

(see (1.1) for the exact definition in the present context) and in certain situations its

generator G is an extension of - L = _ d +A(-) (see [LMR1, Prop. 2.9], [Sch, Prop. 1.13]).

It turns out that in the spirit of Perron's observations asymptotic properties of U like

exponential stability or exponential dichotomy can be described by spectral properties of

the operator G (see [AuM], [LaM], [LaR], [LMR1], [LMR2], [Mil], [Mi2], [RS1], [Rau],

[Sch]). However, we have to point out that most results in this direction are restricted to

the line case J = R.

In the present paper we characterize exponential stability, exponential expan-

siveness and exponential dichotomy of an evolution family U on the half-line J -- R+. Our

approach is based on the use of (generators of) evolution semigroups and their connection

with the integral equation ( IE) (see Section 1). Exponential stability and exponential

expansiveness of U is characterized in Theorem 2.2 and Theorem 2.5, respectively. Sec-

tion 3 deals with exponential stability of individual orbits. Our main interest, however,

is directed to the exponential dichotomy of U which will be characterized in Theorem 4.3

and Theorem 4.5. Concerning the relevance of exponential dichotomy and its far-reaching

applications we refer to [SaS] and the references therein.

334 Van Minh, R~biger and Sctmaubelt

1. P R E L I M I N A R I E S

Recall that a family of operators /2 = (U(t, s))t_>,_>o of bounded linear operators on a

Banach space X is a (strongly continuous, exponentially bounded) evolution family on the

half-line if

1. U(t, t) = Id and U(t, r )U(r , , ) = U(t, ~) for t > r > ~ > O,

2. (,, s) ~+ u ( t , ~)~ is continuous for every �9 �9 X ,

3. there are constants K > 0, ~ �9 R such tha t IIv(*,~)ll < K e ~(*- ') for * > ~ > 0.

Then w(/2) := inf{cr E R : there is K > 0 such that IlU(e,s)]l _< K e ~~ e > s > 0} is

called the growth bound of/2.

Throughout the whole paper the following function spaces (endowed with the sup-norm)

play an important role.

Co := {v: [0, oo) --+ X : v is continuous, v(0) = 0 = lim v(t)}, t - -~oo

c x := { v : [0, o~) -+ x : v is continuous, li_,~ ~(t) = 0},

Cx(to) := {qI,o,oo) : ~ �9 Cx}, to >_ O,

C b := {v : [0, co) ~ X : v is continuous and bounded}.

An evolution family g on X always defines an evolution scmigroup T = (T(t))~___0 of

bounded linear operators on Co and Cx by setting

V ( s , s - t ) v ( s - t ) , s > t , (I.1) [T(t)v](s) = U( s , 0 )~ (0 ) , 0 < ~ < t,

for v in Co and Cx, respectively. It can be easily seen that 7" is strongly continuous. We

denote the infinitesimal generator of 7" on Co and Cx by Go and Gx, respectively. Note

that D(Go) = D ( G x ) N Co and Gay = Gxv for v �9 D(Go), i.e. Go is the part of Gx in

Co.

The aim of this paper is to characterize asymptotic properties of a given evolu-

tion family U by (spectral) properties of the generators G0 and Gx. The following lemma

is the key tool in our strategy. It connects the operators Go and G x with the following

inhomogeneous integral equation

f u(t) = U(t,s)u(s) + U( t , ( ) f ( ( )d ( , t > s > 0. (1.2)

1.1 L E M M A . a) Let u , f E Co. Then u E D(Go) and Gou : - f if and only if

u(t) = U( t , ( ) f ( ( )d ( , t > O. (1.3)

Van Mirth, RAbiger and Schnaubelt 335

b) Let u E Cx and f E Co. Then u e D ( G x ) and G x u = - f if and only if (1.2) holds.

P R O O F . We only prove the first assertion. The proof of the second one follows

the same lines.

Let Gou = - f . The general theory of linear semigroups (see e.g. [Paz, p.4-5]) yields

/0 /0 T( t )u - u = T(~)Goud~ = - T(~) fd~ for t > 0.

Thus,

f0 t u = T( t )u + T(~)fd~.

From the definition of T we easily obtain that u is a solution of the integral equation (1.2)

and has the form (1.3).

Conversely, if u, f �9 Co and u satisfies (1.3)0 then by reversing the above argument we

obtain

f T( t )u - u = - T(~) fd~ for t > 0.

In part icular , this implies u �9 D(Go) and Gou = - f . []

In the following remark we collect some additional propert ies of the operators

Go and G x . For sake of convenience we set U(t, a) = 0 for 0 < t < s.

1.2 R E M A R K S . a) From Lemma 1.1 we immediately obtain tha t Go is injec-

rive and that ker G x = {u 6 C x : u(t) = U(t,O)u(O), t _> 0}.

b) The range R ( G x ) of G x is always contained in Co (in part icular G x is never invertible).

In fact, if u 6 D ( G x ) then

[Cxu](0) = lira T(t)u(O) - u(0) = lira u(0) - u(0) = 0. t,o t t$o t

c) Let to > 0, x �9 X and q0 : [0, oo) ~ ~ be continuously differentiable such tha t

~l[O,to) = O. Set u(t) = T( t )U( t , to )x , t > O, and f ( t ) --- T ' ( t )U(t , to)x, t >_ O. If

u �9 C x and f �9 C0 then an immediate application of L e m m a 1.1 yields u �9 D ( G x )

and G x u = - f .

Next we define an operator I x on C x connected with the integral equation (1.2).

If u, f E C x satisfy (1.2) we set

Ixu := f,

D ( I x ) := {u C C x : there is f C C x such tha t u and f satisfy (1.2)}.

336 Van Mirth, P,/ibiger and Scknaubelt

1.3 LEMMA. ( I x ,D( I z ) ) is a well-defined closed linear operator on Cx and

an e,te.sion of ( - a x , D(ax) ) .

P R O O F . Let I xu = f and I x u = g for u , f , g E Cx. Then

f * U(t,~)(f(~) -g(~))d~ = 0 for t > s > O.

By the continuity of the integrand we obtain 0 = U(t, t )( f( t) - g(t)) = f ( t ) - g(t), t > O,

i.e. f =g . The c]osedness of I x follows immediately from the definition of I x by the integral equation

(1.2). Finally, Lemma 1.1 shows that I x is an extension o f - G x . []

1.4 R E M A R K S . a) In general, - I x is a proper extension of Gx, in particular,

I x can be surjective (cf. Theorem 4.3). Moreover, the spectra of - I x and Gx are different.

In fact, for A E C with ReA < -w(U) the function uA : t ~+ eAtU(t,O)x, t _> 0, x E X,

belongs to Cx and Ixu~ -- Aux. Thus {A E C : ReA > w(U)} is contained in the (point)

spectrum of - I x whereas the spectrum of Gx is contained in a left half-plane (see [Paz,

Remark 1.5.4]). Note, however, that we always have k e r l x = k e r G x = {u E Cx : u(t) =

v(t, o)~(o), t > o).

b) From Lemma 1.1 it easily follows that Go is the part of - I x in Co, i.e. D(Go) = {u E

D ( I x ) N Co : I x u E Co} and Gou = - I x u for u E D(Go).

2. E X P O N E N T I A L S T A B I L I T Y A N D E X P O N E N T I A L E X P A N S I V E N E S S

In this section we characterize exponentially stable and exponentially expansive evolution

families U = (U(t, s))t>s>0 by means of the operators Go and I x and the integral equation

(1.2), respectively. First we give the definition of exponential stability and exponential

expansiveness of an evolution family.

be

2.1 D E F I N I T I O N . The evolution family U on the Banach space X is said to

a) exponentially stable if there are constants N,r, > 0 such that ]]U(t,s)I I <

Ne -~(t-8) for all t > s > 0 (or, in other words, if w(U) < 0),

b) exponentially expansive if U(t, s) is invertible and there are constants N, v > 0

such tha t lIu(t ,~)zlt _> N~ ' ( t - ' ) I I z l l for x E X a~a t > . > O.

We have the following characterization of exponentially stable evolution families.

Van MirLh, Ri~biger and Sehnaubelt 337

space X .

5) 5O

5ii) 5v)

2.2 T H E O R E M . Let 11 = (U ( t, s))t_>s__O be an evolution family on the Banach

Then the following assertions are equivalent:

N is exponentially stable.

Go is invertible.

For every f ~ Co the function t ~ uf( t) = f t U(t, ~)f(~)d~ belongs to Co.

For every f E Co the function t ~-~ uz(t) = f~ U(t, ~)f(~)d~ belongs to C' .

From the proof of the theorem we separate the following l emma for later use.

2 . 3 L E M M A . Let X : [to, t1) -+ (0, oo) be a continuous function and let e > 0

and K , a k 0 be constants such that x(t) <_ Ke ~(t-t~ and f toX( t )x ( r ~ <_ c for

t E [to,Q). Then x(t) <_ max(cK, K)e~+le-�88176

P R O O F o f L e m m a 2.3. If t E [to, t~) and t _< to + 1, then x(t) <_ Ke ~ < Ke ~+ } e- ~ (t-to).

Now let t E [to,t1) and t > to + 1. Set r := ftSoX(~)-ld~ for s E [to, t l) . By our

assumpt ion r _< cr and hence r > r for t l > sl _> so > to. Thus

1 c -~(t-to-1) < cKe~+}e-}(t-to). [] x(t) = r < r + 1--) e

P R O O F o f T h e o r e m 2.2. The implications (i) ~ (iii) ~ (iv) are obvious.

The equivalence of (ii) and (iii) follows from Lemma 1.1.

(iv) :~ (i): By assumpt ion B : Co -+ C b : f ~ uf is linear and everywhere defined.

Moreover, B is closed, and hence B is bounded. Let c := IIBII.

Now fix to _> 0 and 0 # y E X. Let h := sup{t > to : U(t, to)y # 0). Then U(t, to)y # 0

for t E [t0,tl) and U(t, to)y = 0 for t > tl. Se~ x(t) := [IU(t, to)yll, t E [to,tl). For n E N

sufficiently large choose a real continuous function qo, on [0, c~) such tha t qon has compact

suppor t contained in (to,t1), 0 < qon < 1 and

Let

qOn(t) = 1 for t E [ t o + - , m i n { n , Q - - }]. n

fo(t) = { ~n(t)x(t)-l~(t, to)y

Then f,~ E Co and

x(5) IIU(t, to)yH

for i E [to,t1), for t e [o, to] u It1, ~).

--II u ( t , ~ ) f . ( ~ ) d ~ l l _< cllAll = c

338 Van Minh, R~biger and Schnaubelt

for t E [to,t1). By letting n --4 co we obtain f jox(t)x(~)- 'd~ <_ c for ~ e [to,t1). More-

over , by the exponential boundedness of U we have X(t) < IC ('- ~ on t 0 . An

application of Lamina 2.3 yields

X(t) <_ max{cK, K}e ~+} e -}(t-t~ [lyll

for t E [ t0 , t l ) . Thus

HU(t, t0)yH _< max{cK, K}e •+} e -}(t-t~

for t _ to. Since the constants K, a and c are independent of to and y the exponential

stability of H follows. []

R E M A R K . The proof of (iv) =~ (i) follows ideas used in [DaK, Proof of The-

orem IV.3.3]. By different methods the equivalence of (i) and (iv) has been shown by C.

Bu~e [Bus, Theorem 1] (see also [Dat, Theorem 8] for a related result) and the equivalence

of (i), (ii) and (iii) is proved by Y. Latushkin, S. Montgomery-Smith and T. Randolph

[LMR2, Theorem 2.2, Corollary 2.3]. Related results in the autonomous case have been

shown e.g. by J. van Neerven [Nee] and Vfi Qu6c Ph6ng [Vfi]. Note that Theorem 2.2

also holds if we replace Co and C b by LP(R+, X), 1 < p < co, and consider the evolution

semigroup T on LP(IR+, X) induced by H (see [Dat, Theorem 6] and [LMR2, Theorem 2.2,

Corollary 2.3]).

An immediate consequence of Theorem 2.2 is the spectral mapping theorem for

the evolution semigroup on Co. Recall that or(B) denotes the spectrum of a linear operator

B and p(B) := C \ or(B) is the resolvent set. Moreover, s(B) := sup{ReA : A E e(B)} is

the spectral bound and r(B) := sup{]A] : A e a(B)) is the spectral radius of B.

2.4 C O R O L L A R Y . Let gt be an evolution family on the Banach space X.

Then the evolution semigroup T on Co satisfies the spectral mapping theorem

e = \ {0}, t > 0.

Furthermore, a(G0) = {A 6 C: Re), < s(G0)} is a left half-plane and a(T(t)) = {A 6 C:

IA[ < r(T(t))}, t > O, is a disc.

P R O O F . Let A E p(Go) and # E C such that Re# >_ Re),. Note that Go - Aid

is the generator of the evolution semigroup 7~ = (e-~tT(t)),>_o on Co induced by the

evolution family U~ = (e-~(t-s)U(t, s))t>_8>o. Theorem 2.2 implies that U;~ and hence/./~,

Van Minh, Riibiger and Schnaubelt 339

is exponentially stable and, as a consequence, that Go - #Id is invertible, i.e./~ �9 p(Go).

Thus a(G0) -- {), e C : ReA _< s(G0)} is a left half-plane.

If Re,~ > s(Go), then again by Theorem 2.2 the evolution fami ly /~ is exponentially stable,

and hence the induced evolution semigroup Tx is exponentially stable as well. In particular

r(e-a'T(t)) < 1, i.e. r(T(t)) < e ae~'t for t > 0. Thus r(T(t)) < e'(G~ > O. Together

with the spectral inclusion theorem e ' ' (a~ C_ c~(T(t)), t > 0 (see [Paz, 2.2.3]), it follows

that cr(T(t)), t > 0, is a disc and that the spectral mapping theorem holds. []

R E M A R K . The same result (with a different proof) can be found in [LMR2,

Theorem 1.2, Theorem 2.2, Corollary 2.3] and [Sch, Theorem 5.3]. The observation that

on Co the spectrum of T(t), t > O, is always a disc goes back to [Ran, Proposition 2].

We now come to the characterization of exponentially expansive evolution fam-

ilies.

2.5 T H E O R E M . Let l~ = (U ( t, s) )t>s>_o be an evolution family on the Banach

space X . Then the following assertions are equivalent:

(i) Lt is exponentially expansive.

5i) For every to >_ 0 and every f e Cx(to) there is a uniqu~ u: �9 Cx(to) such that

~s t us(t) = v ( t , , ) u : ( ~ ) + u( t ,~) f (Od~, t > ~ > to. (2.1)

R E M A R K . Condition (it) implies that the operator Ix is invertible. However,

Example 4.6 shows that the exponential expansiveness of /J (and hence (it)) is not equiv-

alent to the invertibility of Ix . On the contrary, if U(t, s) is surjective for all t > ~ > 0,

then from the proof of Theorem 2.5 it follows that /4 is exponentially expansive if and only

if Ix is invertible.

As above we separate from the proof of the theorem the following lemma which

is essentially contained in [DaK, Proof of Theorem IV.3.3].

2.6 L E M M A . Let X : [0, oo) --+ (O, oo) be a continuous function and let

c > 0 and K,c~ > 0 be constants such that X(T) <_ Ke~(r-t)x(t) , r > t >_ O, and

f ~ x( t )x(T)- ldT < c, t > O. Then there exists N > 0 only dependent on g , a and c

such that X(t) > ge~(~-~)~(*) for t > ~ > O.

340 Van Minh, 1L~biger and Schnaubelt

P R O O F . Let r f t x ( r ) - l d r , t >_ O. By our assumpt ion r < - c r

Thus r < r -~( t -~) for t > s > 0. On the other hand the exponential es t imate of X

yields o o 1 o o

x(t)(~(t) ~--X(t)~ ~-~dT >_ K-If e-a(~*-t)dT oo

__ : E =:

Moreover, by ou~ assumption, X(~)r -< e, t > ~ > O. Thus

# ~7 x ( t ) > - ~ > - ~ e o " > c X ( ) for t > s > 0. []

Now we are in a position to give the proof of Theorem 2.5.

P R O O F o f T h e o r e m 2.5. (i) =~ (ii): L e t / . / b e exponential ly expansive and

to _> 0. If f E Cx(to) then the function t ~ ul(t ) := - f ~ U ( r 1 6 2 belongs

to Cx(to). It is a s traightforward computat ion to show that u/ satisfies (2.1). If u E

c x ( t 0 ) satls~es (2.1) with y = 0, then ~(t) = U(~,t0)u(t0), t > to. Thus II~(t)ll >__

Ne~(t-to)[[u(to)][ for constants N, t, > 0. This yields u(to) = 0, and hence u = 0.

(ii) ~ (i): A) The assumption implies that Ix is invertible. S i n c e I x is closed its inverse

I x 1 is bounded and c := ][Ixll[ > 0.

B) Fix 0 # x E X. The unique solvability of (2.1) for f = 0 in Cx yields u(t) := U(t, O)x 0 for t > 0. For each n E N choose a real continuous function ~ , on [0, oo) such that

0 <: ~,, < 1, Tn = 1 on [0,n] and ~o,, = 0 on [ n + 1,oo). Let f n ( t ) : = -~.(t)llu(t)ll-lu(t) for t > 0. An easy computa t ion shows tha t

f , oo ~,(~) d~u(t), t > O, ~ . ( t ) : = II~(r -

and f~ solve (1.2), i.e. Ixun = f,,. Hence, ]]u,,][ < c[]f~]l = c. Lett ing n --+ co we obtain

. 1 L c ilu(Qlfd~ <_ Hu(t)}--~, t >_ O.

The exponential boundedness of U yields II~(~-)ll -< r < ~ ( ~ - 0 II~(~)lh ~ > ~ > 0, for constants K, a > 0 independent of x. By Lemma 2.6 there is a constant N > 0 independent of x

s~ch that Ilu(t)ll _> Ne~('-~)l l~(s)[ l for t > ~ > 0. C) It remains to show that U(t, s) is surjective for t > s > 0. Fix s > 0. Let y E X and

set v(t) := U(t, s)y, t > s. We choose a real continuous function T on [0, oo) with compact

support such that I~(t)l _< 1, supp~ C (s, ~ ) and

f o~ ~(~)d~ --- 1.

Van Minh~ RANger and Schnaubelt 341

Now consider the function w defined by

/ w(t) := v(t) ~,(~)d~, t > ~.

Then w e C x ( s ) a n d w is a solution of equation (2.1) with f ( t ) := -~o(t)v(t), t > s.

Extend f continuously to [0, co) by setting f ( t ) = 0 on [0, s]. Then there is a solution

z E Cx of equation (2.1) on [0, co). By our assumpt ion zl[8,~ ) = w . Thus

y = w(s) = z(s) = U(s, 0)z(0) + U(s, ~)f(()d( = V(s, 0)z(0).

This proves the surjectivity of U(s ,0) for s > 0. Together with U(t, O) = V(t,s)U(s, O)

this yields the surjectivity of U(t, s) for t > s > 0. []

3. E X P O N E N T I A L S T A B I L I T Y O F B O U N D E D O R B I T S

In Theorem 2.2 we saw tha t exponential stability of an evolution family U = (U(t, s))t>8>0

is characterized by the invertibility of the generator Go of the evolution semigroup 7" on

Co. We now show tha t a condition on the approximate point spec t rum of Go implies

exponential stabili ty of all bounded orbits U(t, to)x, t > to > O. Recall tha t for an

opera tor B on a Banach space Y the approximate point spectrum Aa(B) of B is the set of

all complex numbers .k such tha t for every e > 0 there exists y e D(B) with Ilyll = 1 and

I I (A- B)yll <_ e.

As the whole spect rum c~(G0) (see Corollary 2.4) the approximate point spec-

t r u m Aa(Go) is invariant under translations along the imaginary axis.

3.1 L E M M A . Act(Go) = i~ + Aa(Go) for all # E R.

P R O O F . Consider the multiplication opera tor M s on Co defined by (Mvv)(t) :=

ei~tv(t), t > O, v E Co. Then M v is an isometry and

.f - - t ) , s > t > o, (MvT(t)M-vv)(8) = [ 0, 0 < s < t,

= eiVt(T(t)v)(s),

v E Co, s > 0. Thus MvT( t )M_, = el"iT(t), t > O, and hence Mv Go M- , = i# + Go.

From this the assertion immediately follows. []

We now come to the main result of this section. A special case of it has been

shown in [AuM, Theorems 7 and 7']. Our proof follows the techniques of [AuM].

342 Van Mink, Rhbiger and Sehnaubelt

3 .2 T H E O R E M . Let N = ( U ( t, s) )t>_s>_o be an evolution family on the Banach

space X such that Act(Go) f3 iN 7~ iN. Then every bounded orbit of N is exponentially

stable. Precisely, ifsupt>_to HU(t, to)xI[ < oc for fized x E X and to > O, then there exist

constants N, u > 0 independent of x and to such that

I I u ( t , to)~l l ~ 2 v e - ~ ( * - ~ ) l l U ( ~ , t o ) ~ l l , t > ~ > to.

P R O O F . Let z E X and to >_ 0 such tha t supt>to l l u ( t , t0 )x t l < o o

A ) l imt~oo t lU(t, to)zff = 0:

Choose a real continuous function ~ on [0, oo) with compact suppor t

p( to + 1) = 1 and ~[[0#oJ = O. Define

such tha t

{~,(t)u(t, to)z for t > to, v ( t ) : = 0 f o r 0 < t < t o .

Then v E Co and

IlT(t + h ) v - T ( t ) v l l

: m ~ x { s u p I lcr (~ , ~ - t - h ) v ( ~ - t - h ) - U ( ~ , ~ - t ) v ( s - t ) l l , s>_t+h

sup IlU(~,s - t )v (s - t ) l l } t+h>s>t

f o r t > O a n d h > O . From

f (s-t)v( ,to)z for -t>to, U ( s , s - t ) v ( s - t ) =

l o for O <_ s - t < tm

~(0) = 0 and the uniform boundedness of U(t, to)x, t >_ to, we obta in

l imsup sup [ [ U ( s , s - t ) v ( s - t ) l l = 0 hJ~0 t > 0 t+h>s>t

On the other hand

sup s>tWh

I [ u ( s , s - t - h ) v ( s - t - h ) - U ( s , s - t ) v ( s - t)l l

= sup ]l~(s - t - h)U(~, to)x - ~(s - t )U(s, to)Xi[ s>_t+h

< sup [r -- t -- h) - q~(s - t)l sup I lCr (~ , to )z l l . s>tq-h s>to

By the uniform cont inui ty of ha we have

l imsup sup I [ U ( s , s - t - h ) v ( s - t - h ) - U ( s , s - t ) v ( s - t ) ] [ = O . h,~0 t > 0 s>t+h

Van Mirth, IL~biger and Schnaubelt 343

Thus t ~-~ T(t)v is a uniformly continuous Co-valued function on [0, co). Our assumpt ion

on Go and L e m m a 3.1 imply Aa(Go) M iN = O. An application of a result of Bat ty and

Vfi [BaV, Theorem 1] yields limt-~oo T(t)v = 0. In part icular,

0 = t-+~lim II(T(t)v)(t + to + 1)It = 11% Ilu(t + to + 1,to + 1)v(to + 1)ll

= lira IIV(t + to + 1,to)xlt. t - 4 - ~

B) IIU(t, to)xll <_ N,-~( ' - '~ t _> to: Without loss of generali ty we may assume Ilzll = 1. Since o r Aa(ao) there is a constant

u > 0 such tha t

I l a o v l l _> ~,l lvl l for v E D ( a o ) . (3.1)

Let u(t) := U(t, to)x, t > to, and h := sup{t > to : U(t, to)x # 0). The exponential

boundedness of U implies tha t there are constants K, a > 0 (independent of x and to) such

tha t Iiu(t)ll < Ke o(t-to), t > to. For n E N sufficiently large we choose a real continuous

function r on [0, oo) such that r has compact support contained in ( t 0 ,h ) , 0 _< r _< 1

and 1 1

en( t ) = 1 for t E [to + - , m i n { n , tl - - ~ } ] . n

Let 0 for t E [O, to],

~ . ( t ) := f/oen(~)llu(~)ll-:d~ for t E (tO,tl),

f/o ~ r for t ~ h , or

un(t) := ~n(t)U(t, to)x, t > 0, and f~(t) := ~( t )U( t , t0)x, t > 0, where we set U(t, to) = 0

for 0 _< t < to. From A) and the definition of ~ it follows tha t un and fn satisfy the

assumptions of Remark 1.2 c). Hence un E D(Go) and Goun = - f n . Thus, by (3.1),

i = sup IiA(t)ll = IIAII -> .llu,~ll = v s u p Ibn(t)l[. t_>to t_>to

In part icular ,

~-: > Ilu=(t)ll = I~=(t)llb(t)ll = r Ib(t)ll for t E [t0,tl).

Lett ing n --+ c~ we obtain

l tb(t)lllb(~:)ll-~d~ < . -~ .

By L e m m a 2.3 there is a constant N only dependent on v, K and a such tha t

tlU(t, to),ll = Ib(t)ll _< N~ - " " - ' ~ t __ to.

344 Van Mirth, R,5,biger and Sehnaubelt

C) Fix s _> to and set y := U(s, to)x. Then supt>, IIU(t,~Dyll < ~ and from A) and B)

we obtain

ilu(t, to)xlt = Ilu(t,s)yll ~ Ne-~(t-*)llyl] = Ne-~(*-~)llU(s,to)xll, t ~ s. []

Denote by Xo(to) := {x e X : limt~ccU(t, to)x = 0} the stable subspace cor-

responding to /4 and to :> 0. From Theorem 3.2 we obtain the following properties of

Xo(to).

3.3 C O R O L L A R Y . Under the conditions of Theorem 3.2 we have

Xo(to) = {z e x : sup Ilu(t, to)xll < ~ } t>to

= {x e x : IIu(t, to).ll < x~-~(t- '~ So > 0,

for fixed constants N, u > O, and Xo(to) is a closed linear subspace of X .

R E M A R K S . a) Lemma 3.1 and Lemma 1.1 imply that the following assertions

are equivalent:

(i) An(Go) N ilR r JR.

(ii) 0 r A~(Go).

(iii) A~r(ao) N iR = ~.

(iv) There is a constant u > 0 such that for any pair u , f C Co with u(t) =

f~ U(t,C)f(~)d~, t >_ O, one has t[fH >- ul[ulI-

b) An evolution family with an exponential dichotomy (see Definition 4.1) always satisfies

A~(a0) n iR = ~ (see the Remark after Theorem 4.3).

c) If B is strongly continuous and uniformly bounded function from [0, oo) into the space of

bounded linear operator on X, then there is a unique evolution family b/B = (UB(t, s))t>s>o satisfying the variation of constants formula

uB(t ,s)x = U(t,~)~ + U(t,r162162162 t > ~ > 0 , x e X

(see [RS2], [Sch] and the references therein). The generator of the evolution semigroup

TB on Co induced by UB is given by Go + B(.). Suppose that Go is invertible resp.

0 (~ Aa(Go). Then Go + B(.) has the same property for every perturbation B(.) such that

tIB(.)II = suptk0 IIB(t)l I is sufficiently small (ef. [Kat, IV.3.1]). In other words exponential

stability of 5/resp. of all bounded orbits of/4 is robust with respect to small perturbations

B(.) of/,/ (see also [LMR2, Section 3.1], [Sch, Section 5.4]).

Van Mirth, R~biger and Schnaubelt 345

4. E X P O N E N T I A L D I C H O T O M Y

In this section we characterize the exponential dichotomy of an evolution family/ / / by

properties of the operators Go, Gx and Ix , and of the operators Gz and I z to be defined

below. At first we recall the following definition.

4.1 D E F I N I T I O N . An evolution family/ , / = (U(t, s))t>,>o on the Banach

space X is said to have an exponential dichotomy if there exist bounded linear projections

P( t ) , t > 0, on X and constants N, v > 0 such that

a) v ( t , s )P(s ) = p ( t ) v ( t , s), t > s > o,

b) the restriction U(t , s ) l : kerP(s ) -+ ke rP( t ) , t > s > 0, is an isomorphism (and

we denote its inverse by U(s, t)l : ker P(t) --+ ker P(s)) ,

c) [Iu(t,s)~ll _< Ne-""-~) l lxl l for x �9 P ( s ) X , t > s > O, d) IIV(s,t)t~l[ < Ne-V(t-s)[[xH for x �9 kerP( t ) , t > s > O.

In the following lemma we collect some properties of the family P( t ) , t _> 0. By

s X) we denote the space of bounded linear operators between the Banach spaces Y

and X.

4.2 L E M M A . Let bl be an evolution family having an exponential dichotomy

with corresponding family of projections P(t), t >_ O, and constants N, v > O. Then the

following holds:

a) M := supt_> o IIP(t)[[ < ee,

b) [0,t] ~ s ~ U(s,t)l S s P ( t ) ,X ) is strongly continuous for t > O.

c) t ~-+ P(t) is strongly continuous,

d) llU(t,~)P(s)rl <_ M N e -~(~-s) for t > s > O,

~) I I u ( s , t ) t ( I - P(t))ll _< M N e -v(t-s) for t > s > O.

P R O O F . a) can be shown as in [DaK, Lemma IV.I.1 and Lemma IV.3.2]. For

sake of completeness we present the details. Fix to >_ 0. Let Po := P(to), P1 := I d - P ( t o )

and Xk := PkX, k = 0, 1. Set ")'to := inf{lIxo + xll[ : Xk e Xk, ]lxoll = Plxltl = 1}. If

x E X a n d P k x # 0 , k = 0 , 1 , then

Pox PlX I - 1 [Pozll n ,, % -< IPoxll + Nix IIPoxH IlP0x + ~ / ~ l X

1 IIPoxll- IlPlxll 211xlt - [ipoxl----- ~ [Ix + HP-~-x-H PlXll < - - " -IIPoxll

a s a consequence [[Pol[ -< 2")'t~ 1. It remains to show that there is a constant c > 0

(independent of to) such that 7to >- c. For this fix xk E Xk, k = 0, 1, with Ilxo II = HXlll = 1.

346 Van Mirth, I~biger and Schnaubek

By the exponential boundedness of U w e have IIU(t, to)(xo + ~1)11 ~ Ke~'(t-t~ + =1 II for t >_ to and constants K, a >__ O. Thus

11=o + ~111 -> K-~e-~(*-*~ to)=o + U(t, to)=lll > K - 1 e-a( t - to) (N-1 eg(t-to) _ Ne-~( t - to) )

= : ct-to~ t ~ to~

and hence 7to -> ct-to. Obviously cm > 0 for m sufficiently large. Thus 0 < C m <~ "[to-

b) Fix t > 0, 0 _< so _< t and z e ke rP( t ) , and let (s,,) be a sequence in [0,t] converging

to so. There is y E ker P(0) such that U(t, O)y = x. By the strong continuity of/,( we have

1 ~ l l u ( ~ . , t ) f z - U(~o,t)lxll = l i p IIu(~., 0)ty - Y(~o, 0)tyll = 0.

c) Note tha t

liP(t) x - P(s)zl i -< liP(t) x - P( t )U(t , s )x l l + l lU(t ,~)P(s)x - P(s)xBt

<_ (sup I[P(r)li)li~ - u ( t , s)~ll + llY(t, ~)P(~)~ - P(Ox[I r>_O

for x E X and t > s > 0. By the strong continuity of U we obtain tha t P( .) is strongly con-

tinuous f rom the right. In order to show strong continuity from the left set Q(.) := I d - P ( . )

and fix t > 0 and x E X . For 0 < s < t we have O(~)~ = U ( ~ , t ) i U ( t , s ) O ( ~ ) ~ =

U(s,t)lQ(t)U(t , s)x. By b) the family (U(s,t)l),e[o.t ] C s P ( t ) , Z ) is strongly continu-

ous and uniformly bounded. The strong continuity of N yields lim~1-t Q(t)U(t, s)x = Q(t)x.

Thus

lira O(s)x = lira U(s, t)l e ( t )U( t , s)x = O(t)=, sit s~t

i.e. Q( . )~ and h e . c e p ( . ) ~ is continuous from the Ie~t.

Assertions d) and e) are immediate consequences of a). []

We come to our first main result. It characterizes evolution families with an

exponential dichotomy by conditions on the operators Go, Gx and Zx, respectively.

4.3 T H E O R E M . Let/d = (U(t, s))t>~>o be an evolution family on the Banach

space X . Then the following assertions are equivalent:

(i) ld has an exponential dichotomy.

(ii) The range R(Gx) of Gx coincides with Co and X0(0) is complemented in X .

(iii) I x is surjective and Xo(0) is complemented in X .

Van Minh, P~biger and Schnaubelt 347

P R O O F . (i) :# (ii): Let P(t), t > O, be the family of projections given by

the exponential dichotomy. Then X0(0) = P(O)X, and hence Xo(0) is complemented. If

f E Co define v : [0, oo) --> X by

/0' v(t) = U(t,~)P(~)f(~)d~ - U(t , ( ) l ( Id - P(())f(~)d~. (4.1)

An easy computa t ion shows tha t v E Cx and v is a solution of equation (1.2). By Lemma

1.1 we have G z v = - f . Together with Remark 1.2 b) this implies R ( G x ) = Co.

(i) ~ (iii): We can use the same arguments as in the proof of (i) =r (ii). Note tha t for

f E Cx the function v defined by (4.1) is also in Cx and satisfies (1.2). Hence by the

definition of I x we have v E D ( I x ) aad I x v = f , i.e. I x is surjective.

(ii) ~ (i): A) Let Z C X be a complement of X0(0) in X , i.e. Z = X0(0) @ Z. Set

X l ( t ) := U(t,O)Z, t > O. Clearly,

u ( t , , ) X o ( , ) c_ Xo(t) , t 2> , , (4.2)

U(t,S)Xl(S) = Xl ( t ) , t 2> s.

B) There are constants N, u > 0 such tha t

llU(t,O)xt[ 2> Ne~('-8)llU(*,0)xll for x E XI(0) and t > s > O. (4.3)

In fact, let Y := {v E D ( G x ) : v(0) E XI(0)} endowed with the graph norm I[vflGx :=

IMI + I laxvll . Then Y is a closed subspace of the Banach space (D(Gx) ,

[l" [lax), and hence Y is complete. By Remark 1.2 a) we have k e r G x = {v E Cx :

v(t) = u ( t , o > fo~ so~e ~ ~ Xo(O)}. Since X = Xo(O) | X~(O) and n ( a x ) = Co we

obtain tha t Gx : Y --+ Co is bijective and hence an isomorphism. Thus there is a constant

> 0 such tha t

]IGxvll 2> ullvllax 2> vllvll for v E Y. (4.4)

Let now 0 5d x E XI(0) and set u(t) := U(t,O)x, t 2> O. By Remark 1.2 a) we have

u(t) ~ 0 for all t 2> 0. For each n E N choose a real continuous function ~o,~ on [0, oo)

such tha t 0 <_ ~o~ _< 1, ~n = 1 o n [1,n] and~o~ = 0 on { 0 } U [ n + l , oo). Set f~(t) :=

-~ , ( t )nu( t ) I i - lu ( t ) , t 2> O, and

u. ( t ) :-- i t - ~ l d ~ ( t ) , t 2> 0.

Then fn E Co and u , E Cx. An easy computat ion shows that fn and u,~ satisfy (1.2). By

L e m m a 1.1 we have u,, E D ( G x ) and Gxu,, =- -f,~. From (4.4) we obtain [Ifnll 2> ~ll~dl.

Lett ing n -+ co this yields

HvJ,t"~'r162 < . - ' II~(t)ll -~, t 2> 0.

348 Van MinlL Ra'biger and Selmaubelt

Now the exponential boundedness of U and Lemma 2.6 imply tha t there is a constant

N > 0 independent o f x such that Ilu(t)]] _> Nd'(t-S)ll~(s)]l, t > s > O.

C) X = Xo(• ) @Xl ( t ) , t > 0:

Let Y C_ Cx be as in B). Then D(Go) = D(Gx) M Co C y and (4.4) yields IiG0vil _> vlivll

for v E n(Go). Thus 0 ~ Aa(Go) n iN and CoroUary 3.3 implies tha t Xo(t) is closed for

4_>0. From (4.2), (4.3) and the closedness of Xl(0),we derive that Xl(t) is closed and Xo(t) n X1(4) = {0} for t _> 0.

Finally, fix g0 > 0 and x E X. Choose a real continuous function T on [0, oo) such tha t ~ has oo d eo compact support contained in [t0, ec) and fro W(r r = 1. Set v(t) := fi ~(~)d~ U(t, to)x

and f( t) := -T(t)U(t , to)x, t >_ to. Then v is a solution of equation (2.1), and v E Cx(to).

Extend f continuously to [0, ~ ) by setting fif0,t0) -- 0. Then f E Co and by assumpt ion

there exists w E Cx such that Gxw = - f . In view of Lemma 1.1 w is a solution of

equation (1.2). In particular, Wl[to,oo ) satisfies (2.1). Thus

v ( t ) - w ( t ) = U( t , to ) (v ( to ) - w( to ) ) = U(t , t o ) ( x - w( to ) ) , t > to.

Since v - wl[to,~ ) E Cx(to) this implies x - w(to) E X0(t0). On the other hand w(0) =

w0 + wl with wk E Xk(0), k = 0, 1. Then w(to) = U(to,O)wo + U(to,O)wl and by (4.2)

we have U(to, O)wk E Xk(to), k = 0, 1. Hence x = x - w(to) + w(to) E Xo(to) + Xl(to).

This proves C).

D) Let P(t) be the projection from X onto Xo(t) with kernel Xl ( t ) , t __ 0. Then (4.2)

implies P(t)U(t,s) = U(t,s)P(s), t >_ s >_ O. From (4.2) and (4.3) we obtain that

U(t,s)l : k e r P ( s ) --+ ker P( t ) , t > s > 0 is an isomorphism. Finally, by (4.3), Theorem 3.2

and our assumpt ion Acy(Go) M iR # iN there exist constmats N, v > 0 such tha t

tlcr(t,,)~H _< Ne-~(*-~)ll~ll for �9 E P ( , ) X , t > ~ > O,

tlU(s,t)lxll <_ Ne-V(t-8)ltxll for x E ke rP ( t ) , t > s > 0.

Thus /2 has an exponential dichotomy.

(iii) ~ (i): We use exactly the same arguments as in (ii) =~ (i). We only have to replace

the opera tor Gx by Ix . []

R E M A R K . Par t C) of the proof shows that for an evolution family/2 with an

exponential dichotomy the generator Go of the evolution semigroup on Co always satisfies

0 r Aa(Go), and hence Az(Go) n iN = 0 (see Section 3, Remark a)).

If X is a Hilbert space we only have to assume the closedness of the stable

subspace.

Van Minh, R/ibiger and Schnaubelt 349

4.4 C O R O L L A R Y . Let l,l = (U(t,s))t>8>o be an evolution family on the

Hilbert space H. Then the following assertions are equivalent:

(i) l~ has an exponential dichotomy.

(ii) R ( G H ) = Co and H0(0) is closed.

5ii) Iu is surjective and Ho(0) is closed.

In the proof of Theorem 4.3, (ii) =~ (i), we saw that the invertibility of Gx

restricted to a certain subspace of Cx plays a crucial role in order to prove the existence of

an exponential dichotomy for a given evolution family/A. In our next result we show that

the existence of an exponential dichotomy can be even characterized by such an invertibility

condition. Let us introduce the following notion. For a closed linear subspace Z of X let

Cz := { f e c x : f(o) e z } .

Denote by Gz the part of Gx in Cz, i.e. D(Gz) = D(Gx) N Cz and Gzv = Gxv for

v E D(Gz) . Then Go is the part of Gz in Co. In the same way let I z be the restriction

of I x to C2, i.e. D(Iz ) = D( I x ) N Cz and I zv = I x v for v E D(Iz) . Notice that the

evolution semigroup 7" l~ves Cz invariant and that Gz is the generator of the restriction

of T to Cz.

With this notat ion we obtain the following characterization of evolution families

with exponential dichotomy. A similar result is shown in [BGK, Theorem 1.1] for the finite

dimensional case (see also [Pal]). We point out that in contrast to Theorem 4.3 we do not

have to assume that X0(0) is complemented.

4.5 T H E O R E M . Let l.f = (U(t, s) ),>8>_o be an evolution family on the Banach

space X and let Z be a closed linear subspace of X . Then the following assertions are

equivalent:

(i) bl has an exponential dichotomy with ker P(0) = Z.

Oi) Gz : D(Gz) C Cz --+ Co is invertibIe.

6ii) I z : D( I z ) :_ Cz --+ Cx is invertible.

P R O O F . (i) ~ (ii) : Let P(t), t > O, be a family of projections given by the

exponential dichotomy such that ker P(0) = Z. Then P ( 0 ) X - X0 (0) and X = X o ( 0 ) � 9 Z.

Fix f E Co. By Theorem 4.3 there is v E D(Gx) such that Gxv = f . On the other hand

u : [0, oo) --+ X : t ~+ U(t, 0)P(0)v(0) belongs to Cx and Cxu = 0 (cf. Remark 1.2 a)).

Then v - u E D(Gz) and Cz(v - u) = Gxv = f . Hence Gz : D(Cz) -+ Co is surjective.

If w E k e r G z then w(t) = U(t,O)w(O), t > 0 (cf. Remark 1.2 a)). Since w 6 Cx we have

w(0) E Z N Xo(0) = {0} and hence w = 0, i.e. Gz is injective.

350 Van Mirth, IEa'biger and Schnaubelt

(i) ~ (iii) can be shown by the same arguments. One only has to use Remark 1.4 a)

instead of Remark 1.2 a).

(ii) :* (i) : Let a z : D ( a z ) -~ Co be invertible. By Remark 1.2 b) we have Co =

R(Gz) = R(Gx) . The closedness of Gz implies that G z 1 is bounded, and hence there

exists v > 0 such that Ilazvll >_ ullvll for all v E D(Gz) . Since Go is the part of Gz

in Co we obtain 0 ~ A~r(Go). Finally we show that X = Xo(0) @ Z. Corollary 3.3

implies that Xo(0) is closed. Now let x E X. If U(t, 0)x = 0 for some t > 0, then

x E Xo(0). Otherwise U(t,O)z 7 ~ 0 for all t >_ 0. Let ~o : [0, oo) --+ IR be continuously

differentiable such that ~O][od I = 1 and TI[2,~) = 0. Set u(t) := T(t)U(t,O)x, t >_ O, and

f ( t ) := T'(t)U(t, O)z, t >_ O. By Remark 1.2 c) we have u E D ( G x ) and Gx u = - f . On

the other hand there exists v E D(Gz) such that Gzv = - f . Thus u - v E ker Gx and

hence

- v)(0 = - = - t > o .

Since u - v E CN this implies x - v(0) E Xo(0). Thus x = x - v(0) + v(0) E Xo(0) + Z.

If y E Z N X0(0), then w defined by w(t) := U(t,O)y, t >_ O, belongs to Cz n k e r G x (see

Remark 1.2 a)). Hence Gzw = 0 and by the invertibility of Gz we have w = 0. Thus

0 = w(0) = y, i.e. Xo(0) M Z = {0}. This shows X = Xo(0) | Z. The assertion now follows

from Theorem 4.3.

(iii) ~ (i) is shown by the same arguments as (ii) ~ (i). []

R E M A R K . As a special case of Theorem 4.5 the invertibility of Gx resp.

I x characterizes evolution families with an exponential dichotomy such that Xo(0) =

P ( 0 ) X = {0}.

We conclude with an example of an evolution family U with non-trivial expo-

nential dichotomy such that I x is invertible. In particular this shows that invertibility of

I x is not equivalent to the exponential expansiveness of/d (el. Theorem 2.5).

4.6 E X A M P L E . Let X = L 1[0,0~). For 0 < s < t < 1 set

and f o r l < s < t s e t

0, 0 < ~ < t - s , Ul( t , s ) f ( ( ) := / ( ~ - t + . s ) , t - s < ~ ,

U2(t,s)f(~) > 1,

Van Mirth, RRbiger and Schnaubclt 351

f E X. Then L/= (U(t, s))t>s>o defined by

{ U~(t,s), 0 < s < t < 1, U(t,s):= U2(t, 1)Ul(1,s), s < l <t,

u 2 ( t , s ) , 1 < s < t,

is an evolution family on X with an exponential dichotomy. The corresponding family of

projections P(t), t >_ 0, is given by

P(t) f = X[o,min{1,t}]f, f E X,

where Xc denotes the characteristic function of a set C. In particular, {0} = P(O)X = X0(0). Theorem 4.5 implies that Ix is invertible. Since ]lU(t, 1)P(1)]] < e -(t-l), t >__ 1,

the evolution family H cannot be exponentially expansive.

R E F E R E N C E S

[AuM]

[BaV]

[Bee]

[BGK]

[Bus]

[Cop]

[DaK]

[Dat]

[Hen]

[Kat]

[L~M]

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Nguyen Van Minh, Department of Mathematics, University of Hanoi, 90 Nguyen Trai, Hanoi, Vietnam. Frank Rs and Roland Schnaubelt, Mathematisches Institut, Unlversitgt Tiibingen, Auf der Morgenstelle 10, D-72076 Tiibingen, Germany email: [email protected] and

[email protected]

1991 Mathematics Subject Classification: Primary: 34G10, 47D06, Secondary: 47H20

Submitted: August 15, 1997