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ASYMPTOTIC ALMOST PERIODICITY OF C -SEMIGROUPS
LINGHONG XIE, MIAO LI, and FALUN HUANG Received 6 August 2001 and in revised form 2 March 2002
Let{T (t)}t≥0be aC-semigroup on a Banach spaceXwith generatorA. We will investigate the asymptotic almost periodicity of{T (t)}via the Hille-Yosida space of its generator.
2000 Mathematics Subject Classification: 47D60, 47D62, 47D06.
1. Introduction. Motivated by the abstract Cauchy problem d
dtu(t)=Au(t) (t≥0), u(0)=x, (1.1) a generalization of strongly continuous semigroups, C-semigroups, has re- cently received much attention (see [6, 7, 17]). The operator Agenerating a C-semigroup leads to (1.1) having a unique solution, wheneverx =Cy for someyin the domain ofA. It is well known that the class of operators that generateC-semigroups is much larger than the class of operators that generate strongly continuous semigroups.
On the other hand, the asymptotic almost periodicity ofC0-semigroup has been studied systematically in [2,4,8,11,12,14,15, 16]. It was shown that whenAgenerates an asymptotically almost periodicC0-semigroup on a Banach spaceX, thenXcan be decomposed into the direct sum of two subspacesXa
and Xs, and the mild solutions with initial values taken fromXa are almost periodic and thus can be extended to the whole line while the mild solutions fromXsare vanishing at infinity.
In this paper, we will discuss the asymptotic almost periodicity ofC-semi- groups. We show that if A generates an asymptotically almost periodic C- semigroup, then the range ofC has an analogous decomposition. The tech- nique we use here is the Hille-Yosida space forA, which is a maximal imbed- ded subspace ofXsuch that the part ofAon this subspace generates aC0- semigroup. The crucial facts are that a mild solution of the abstract Cauchy problem is asymptotically almost periodic in the Hille-Yosida space if and only if it is asymptotically almost periodic inX, and the mild solution is vanishing at infinity in the Hille-Yosida space if and only if it is vanishing inX.
Under suitable spectral conditions, we obtain a theorem for asymptotic al- most periodicity ofC-semigroups that is more easily testified (Theorem 3.7).
At last, we give a theorem for asymptotically almost periodic integrated semi- groups (Theorem 3.9).
Although there are papers devoted to asymptotic properties of individual solution [1,2,3,4], our main concern is some kind of global property.
Throughout the paper, all operators are linear. We writeD(A)for the domain of an operatorA, R(A)for the range, andρ(A)for the resolvent set;X will always be a Banach space, and the space of all bounded linear operators onX will be denoted byB(X)whileCwill always be a bounded, injective operator onX. Finally,R+ will be the half-line[0,+∞), and C+ will be the half-plane {z|z=τ+iw,τ, w∈Randτ >0}.
2. Preliminaries. We start with the definitions and properties ofC-semi- groups.
Definition2.1. A family{T (t)}t≥0⊂B(X)is aC-semigroupif (a) T (0)=C,
(b) the maptT (t)x, from[0,+∞)intoX, is continuous for allx∈X, (c) CT (t+s)=T (t)T (s).
Thegenerator of{T (t)}t≥0,Ais defined by
Ax=C−1
limt↓0
1 t
T (t)x−Cx
(2.1)
with
D(A)=
x∈X|lim
t↓0
1 t
T (t)x−Cx
exists and is inR(C)
. (2.2)
The complex numberλis inρC(A), theC-resolvent setofA, if(λ−A)is injective andR(C)⊆R(λ−A). And we denote byσC(A)the set of all points in complex plane which are not in theC-resolvent ofA.
For the basic properties ofC-semigroups and their generators, we refer to [7]. Next, we introduce the Hille-Yosida space for an operator.
Definition2.2. Suppose thatAhas no eigenvalues in(0,∞)and is a closed linear operator. TheHille-Yosida space for A, Z0is defined byZ0= {x∈X| The Cauchy problem (1.1) has a bounded uniformly continuous mild solution u(·, x)}with
xZ0=supu(t, x): t≥0 forx∈Z0. (2.3) Lemma2.3(see [7]). Suppose thatAgenerates a strongly uniformly contin- uous bounded C-semigroup {T (t)}t≥0 on X. Then, the Hille-Yosida space for
A,Z0is a Banach space and
Z0=
x|t →C−1T (t)xis uniformly continuous and bounded (2.4)
with
xZ0=supC−1T (t)x: t≥0 . (2.5)
Moreover,Z0X, that is,Z0can be continuously imbedded inX,A|Z0generates a contractionC0-semigroupS(t)=C−1T (t)onZ0, andT (t)x=S(t)Cxfor all x∈X.
LetJ=R+ or R. The spaces of all bounded continuous functions fromJ intoXwill be denoted byCb(J, X), C0(R+, X)will designate the set of those ϕ∈Cb(R+, X)that vanish at infinity onR+, and we will hereafter assume that each of these spaces is equipped with the supremum norm. Moreover, forf∈ Cb(J, X),w∈J, we putfw(t)=f (t+w) (t∈J)and letH(f )= {fw:w∈J} denote the set of all translates off.
Definition2.4. (a) A functionf∈C(J, X)is said to bealmost periodic if for everyε >0, there existsl >0 such that every subinternal ofJof lengthl contains, at least, oneτsatisfyingf (t+τ)−f (t) ≤εfort∈J. The space of all almost periodic functions will be denoted by AP(J, X).
(b) A functionf∈Cb(R+, X)is said to beasymptotically almost periodic if for everyε >0, there existl >0 andM >0 such that every subinterval ofR+of lengthlcontains, at least, oneτ satisfyingf (t+τ)−f (t) ≤εfor allt≥M.
The space of all asymptotically almost periodic functions will be denoted by AAP(R+, X).
Lemma 2.5(see [10, 15, 16]). For a function f ∈C(R+, X), the following statements are equivalent:
(a)f∈AAP(R+, X);
(b)there exist uniquely determined functionsg∈AP(R, X)andϕ∈C0(R+, X) such thatf=g|R++ϕ;
(c)H(f )is relatively compact inCb(R+, X).
Definition2.6. Let{F (t)}t∈J ⊆B(X) be a strongly continuous operator family.
(a)F (t) (t∈J)isalmost periodicifF (·)xis almost periodic for everyx∈X.
(b)F (t) (t∈R+)isasymptotically almost periodicifF (·)xis asymptotically almost periodic for everyx∈X.
3. Main results. In order to characterize the asymptotic almost periodicity ofC-semigroups, we need the following result.
Lemma3.1. Assume that{T (t)}t≥0is aC-semigroup onXgenerated byA andAhas no eigenvalues in(0,∞). Suppose thatT (·)x:R+Xis asymptoti- cally almost periodic for somex∈X. Then,
(a) there existy∈Xandϕ∈C0(R+, X)such that, for all t≥0, T (t)y∈ R(C),C−1T (·)y∈AP(R+, X), and
T (t)x=C−1T (t)y+ϕ(t); (3.1) (b) if {T (t)}is strongly uniformly continuous and bounded, then there exist
y, z∈Z0such thatS(·)y∈AP(R+, Z0),S(·)z∈C0(R+, Z0), and T (t)x=S(t)y+S(t)z ∀t≥0, (3.2) whereZ0is the Hille-Yosida space forAand{S(t)}is theC0-semigroup generated byA|Z0.
Proof. (a) ByLemma 2.5, there exist uniquely determined functionsh∈ AP(R, X)andϕ∈C0(R+, X)such thatT (·)x=h|R++ϕ, and, from the proof ofLemma 2.5(see [14]), we know that there exists 0< tn→ ∞such that
h(t)=lim
n→∞T t+tn
x ∀t≥0. (3.3)
Lety=h(0), then we have limn→∞T (tn)x=y. Therefore, for eacht≥0, Ch(t)=lim
n→∞CT t+tn
x=T (t)lim
n→∞T tn
x=T (t)y, (3.4) which impliesT (t)y∈R(C)for all t≥0 andC−1T (·)y=h|R+ ∈AP(R+, X), so that
T (t)x=C−1T (t)y+ϕ(t) ∀t≥0. (3.5) (b) Letyandϕ(t)be the same as in (a). SinceC−1T (·)y=h|R+∈AP(R+, X), we have thatC−1T (t)yis uniformly continuous and bounded in[0,+∞), and so it follows thaty∈Z0byLemma 2.3. Settingz=Cx−y, then obviously, z∈Z0. Hence,S(t)y=C−1T (t)yand
S(t)z=C−1T (t)z=C−1T (t)Cx−C−1T (t)y=T (t)x−C−1T (t)y=ϕ(t) (3.6) so that
T (t)x=S(t)y+S(t)z ∀t≥0. (3.7) Next, we show thatS(·)y∈AP(R+, Z0)andS(·)z∈C0(R+, Z0). By the defi- nition of almost periodicity, we know, for everyε >0, there existsl >0 such that every subinterval ofR+of lengthlcontains, at least, oneτsatisfying
sup
t≥0
S(t+τ)y−S(t)y< ε, (3.8)
hence, sup
t≥0
S(t+τ)y−S(t)yZ
0
= sup
t≥0, s≥0
C−1T (s)S(t+τ)y−C−1T (s)S(t)y
= sup
t≥0, s≥0
C−1T (s)C−1T (t+τ)y−C−1T (s)C−1T (t)y
= sup
t≥0, s≥0
C−1T (t+s+τ)y−C−1T (t+s)y
= sup
t≥0, s≥0
S(t+s+τ)y−S(t+s)y
=sup
t≥0
S(t+τ)y−S(t)y< ε,
(3.9)
which yieldsS(t)y∈AP(R+, Z0). Also, S(t)zZ
0=sup
s≥0
C−1T (s)S(t)z=sup
s≥0
S(t+s)z=sup
s≥t
S(s)z (3.10) converges to 0 as t→ +∞since S(t)z∈C0(R+, X), and this yieldsS(t)z ∈ C0(R+, Z0).
Now, we can prove the following theorem which characterizes the asymp- totic almost periodicity ofC-semigroups.
Theorem 3.2. Let{T (t)}be a C-semigroup generated byA on X. Then, {T (t)}is asymptotically almost periodic if and only ifR(C)⊆X0a+X0s, where X0a = {x | x ∈ Z0, S(t)x ∈ AP(R+, Z0)} and X0s = {x |x ∈Z0, S(t)x ∈ C0(R+, Z0)}, whereZ0is the Hille-Yosida space forA,{S(t)}is theC0-semigroup generated byA|Z0.
Proof. Necessity. First, it follows from [17, Lemma 1.6.(a)] and the uni- form boundedness theorem that{T (t)}is bounded and strongly uniformly continuous. By Lemma 3.1(b), for every x ∈ X, there exist y, z∈ Z0 such that S(·)y∈AP(R+, Z0),S(·)z∈C0(R+, Z0), andT (t)x=S(t)y+S(t)z for allt≥0. Choosingt=0, we obtainCx=y+z, that is,R(C)⊆X0a+X0s.
Sufficiency. SinceR(C)⊆X0a+X0s, for anyx∈X, there existy∈X0a and z∈X0ssuch thatCx=y+z. Hence, byLemma 2.3, we haveT (t)x=S(t)Cx= S(t)y+S(t)z whileS(t)y ∈AP(R+, Z0)and S(t)z∈C0(R+, Z0). Since Z0 is continuously imbedded inX, it follows thatT (t)x is asymptotically almost periodic byLemma 2.5, that is,{T (t)}is an asymptotically almost periodic C-semigroup.
From the proof of Lemma 3.1(b), we know that for y, z ∈ Z0, S(t)y ∈ AP(R+, X)andS(t)z∈C0(R+, X)if and only ifS(t)y∈AP(R+, Z0)andS(t)z∈ C0(R+, Z0), respectively. Hence, we have the following corollary.
Corollary3.3. Let{T (t)}t≥0be aC-semigroup generated byAonX. Then, {T (t)}is asymptotically almost periodic if and only if for any x ∈X, there
existy, z∈Z0such thatCx=y+z,C−1T (t)y∈AP(R+, X), andC−1T (t)z∈ C0(R+, X), whereZ0is the Hille-Yosida spaces forA.
In the case ofR(C)=X, we have the following theorem.
Theorem3.4. Assume that{T (t)}t≥0is aC-semigroup onXandR(C)=X.
Then,{T (t)}is asymptotically almost periodic if and only if R(C)⊆Xa+Xs, whereXa = {x |x∈X,T (t)x ∈AP(R+, X)}andXs = {x |x∈X, T (t)x ∈ C0(R+, X)}.
Proof. Necessity. ByLemma 3.1(a), for everyx∈X, there existy∈Xand ϕ∈C0(R+, X)such that for allt≥0,T (t)y∈R(C),C−1T (·)y∈AP(R+, X), and
T (t)x=C−1T (t)y+ϕ(t). (3.11) It follows thatT (t)Cx=CT (t)x=T (t)y+Cϕ(t). Settingz=Cx−y, then T (t)z=T (t)Cx−T (t)y=Cϕ(t)∈C0(R+, X); on the other hand,T (·)y ∈ AP(R+, X)sinceCis bounded. So, we haveR(C)⊆Xa+Xs.
Sufficiency follows from the fact thatR(C)is dense inXand AAP(R+, X)is closed in the space of all bounded uniformaly continuous functions fromR+ toX.
The following result clarifies the relations between the generator of an asymptotically almost periodicC-semigroup and of an asymptotically almost periodicC0-semigroup.
Theorem3.5. Suppose thatAis closed and has no eigenvalues in(0,∞), and assume thatC−1AC=A. Then,Agenerates an asymptotically almost periodicC- semigroup if and only ifR(C)⊆Zaap≡(Z0)a+(Z0)s, where(Z0)a= {x|x∈X, the Cauchy problem (1.1) has an almost periodic mild solution u(·, x)}and (Z0)s = {x |x ∈X, the Cauchy problem (1.1) has a mild solution u(·, x)∈ C0(R+, X)}. AndZaapis the maximal continuously imbedded subspace on which Agenerates an asymptotically almost periodicC0-semigroup.
Proof. Necessity holds byLemma 3.1and the relations betweenC-semi- group and solutions of the corresponding Cauchy problem [7, Theorem 3.13].
Sufficiency. ByDefinition 2.2, we know that both(Z0)a and(Z0)s are con- tained inZ0; soR(C)⊆Z0. Thus, by [7, Theorem 5.17] and [9, Corollary 3.14], Agenerates a boundedC-semigroup; since all mild solutions with initial data taken fromR(C)are asymptotically almost periodic, so is theC-semigroup.
Now, suppose thatY Xand A|Y generates a contraction asymptotically almost periodicC0-semigroup, thenY Z0sinceZ0is maximal (cf. [7, Theorem 5.5]). So,YZaapfollows from the fact that the asymptotic almost periodicity of the mild solution of the abstract Cauchy problem inZ0is equivalent to the same property inX, and the mild solutionu(t)converges to 0 ast→ ∞inX is equivalent tou(t)→0 inZ0.
Remark3.6. (a)(Z0)a is the maximal subspace on whichAgenerates an almost periodicC0-semigroup.
(b) From the proof ofLemma 3.1and Theorems3.2and3.5, it is clear that (Z0)a=X0a,(Z0)s=X0s.
WhenσC(A)∩iRis countable, we obtain a result which is more easily testi- fied for asymptotic almost periodicity ofC-semigroups.
Letf:R+→Xbe strongly measurable, and letfbe the Laplace transform off,
f (z) = +∞
0 e−tzf (t) dt. (3.12) We assume thatf (z) exists for allzinC+, sofis holomorphic inC+(usually,f will be bounded). A pointλ=iηiniRis said to be a regular point forfif there is an open neighborhoodUofλinCand a holomorphic function g:U→X such thatg(z)=f (z) wheneverz∈U∩C+. The singular set E offis the set of all points ofiRwhich are not regular points.
Theorem3.7. Let{T (t)}t≥0be aC-semigroup generated byAinX, and let σC(A)∩iRbe countable. Then, the following assertions are equivalent:
(a) {T (t)}is asymptotically almost periodic;
(b) {T (t)}is bounded, strongly uniformly continuous and, for every r ∈ σC(A)∩iR,x∈X,limλ→0λ+∞
0 e−(λ+ir )tT (t+s)x dtexists uniformly for s≥0.
Proof. (a)⇒(b). It follows from the properties of asymptotically almost pe- riodic functions (cf. [4]).
(b)⇒(a). Givenx∈X, letf (t)=T (t)x, and then we have thatf (t)is bounded, uniformly continuous andf (λ) =(λ−A)−1Cx(Reλ >0). LetEbe the singular set offiniR, thenE⊆σC(A)∩iR, and then it follows thatEis countable by the assumption. Moreover, for eachir∈σC(A)∩iR,
limλ→0λfs(λ+ir )=lim
λ→0λ +∞
0
e−(λ+ir )tT (t+s)x dt (3.13)
exists uniformly fors≥0, wherefs(t)=f (s+t). Therefore,f (t)=T (t)x is asymptotically almost periodic by [5, Theorem 4.1]; so,{T (t)}is asymptotically almost periodic.
Remark3.8. The result ofTheorem 3.7can be deduced directly from [8, Theorem 4] with the assumption onσC(A) replaced by that onσ (A); while with the aid of [5, Theorem 4.1], the result can be improved.
We end this paper with a theorem for integrated semigroups, see [13] for the definitions and basic properties of integrated semigroups.
Theorem3.9. Suppose thatAgenerates a boundedn-times integrated semi- group{T (t)}t≥0andσ (A)∩iRis at most countable. Then, the following asser- tions are equivalent:
(a) {T (t)}is asymptotically almost periodic;
(b) for everyr∈σ (A)∩iR,x∈X, the limit limλ→0λ
+∞
0 e−(λ+ir )tT (t+s)x dt (3.14)
exists uniformly fors≥0.
Proof. We only need to show (b)⇒(a).
We first recall that for bounded integrated semigroup{T (t)}, we have (λ−A)−1x=λn
+∞
0 e−λtT (t)x dt (3.15) for Reλ >0, that is,
+∞
0 e−λtT (t)x dt= 1
λn(λ−A)−1x (3.16) for Reλ >0,so thatT (λ) :=+∞
0 e−λtT (t)x dtcan be extended holomorphically to a connected open neighborhood V of (iR\σ (A))\{0}, hence the singular set ofT (λ) iniRis contained in(σ (A)∩iR)
{0}. By our assumption and [5, Theorem 4.1], we derive (a) from (b).
Acknowledgments. We are indebted to Professor Ralph deLaubenfels for many important and valuable suggestions;Theorem 3.5was, in fact, suggested by him. We also thank the referees for their helpful suggestions and for the simplification of the proof ofTheorem 3.7.
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Linghong Xie: Department of Applied Mathematics, Southwest Jiaotong University, Chengdu 610031, China
Miao Li: Department of Mathematics, Sichuan University, Chengdu 610064, China E-mail address:[email protected]
Falun Huang: Department of Mathematics, Sichuan University, Chengdu 610064, China
