Math. J. Okayama Univ. 48 (2006), 145–157

ALMOST PERIODIC SOLUTIONS OF C-WELL-POSEDEVOLUTION EQUATIONS
Nguyen Van MINH
Abstract. This paper is concerned with the existence and uniquenessof almost periodic mild solutions of evolution equations of the formu̇(t) = Au(t) + f (t) where A is the generator of a holomorphic Csemigroup on a Banach space and f is an almost periodic function. A
suﬃcient condition in terms of spectral properties of A and f is obtainedthat extends a well known result in this subject.

1. Introduction
In this paper we deal with conditions for the existence and uniqueness ofalmost periodic mild solutions with speciﬁc spectra to evolution equationsof the form
du(1.1)
= Au + f (t),dtwhere A is a (unbounded) linear operator which generates a holomorphic
C-semigroup of linear operators on a Banach space X and f is an almost
periodic function taking values in X.
This kind of problems has been of great interest to many mathematicians
for decades. Actually, it goes back to the far-reaching result of O. Perron on
the characterization of exponential dichotomy of linear ordinary diﬀerential
equations. We refer the reader to [3, 4, 5, 9, 14, 17, 19, 20, 21, 23, 24, 28, 29]
and the references therein for more information on recent and related results.
Notice that most results (except for those in [4]) were obtained for evolution
equations of the form (1.1) that are well-posed. As shown in many recent
works (see, for instance, [6, 7, 10, 11, 13, 15, 16, 18, 25, 26, 27, 30] and thereferences therein) an important class of ill posed evolution equations canbe treated in the framework of C-semigroups that was introduced by Da
Prato [7], Davies and Pang [6].
As is well-known (see, e.g.,[9, 14, 19, 23, 28]), if A generates a holomorphic
C0 -semigroup and σ(A) ∩ iσb (f ) = ®, then Eq. (1.1) has a unique almost
periodic mild solution u such that σb (u) ⊂ σb (f ) (here σb (f ) denotes the
Mathematics Subject Classiﬁcation. Primary: 47D60, 34G10; Secondary: 34C27.
Key words and phrases. Holomorphic C-semigroup, almost periodic solution, sums ofcommuting operators.
The author thanks the referee for carefully reading the manuscript and for his suggestions to improve the presentation.
145

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N. V. MINH

Bohr spectrum of the given almost periodic function f whose deﬁnitionis given in the next section). In the simplest case, when A generates a C0 semigroup and the spectrum of A does not cut the imaginary axis, the above
problem is directly concerned with the exponential dichotomy or stabilityof the semigroup generated by A (see e.g. [22]). Hence, for arbitrary closed
operator A the above condition is not suﬃcient for the existence of an almost
periodic mild solution.
In this paper, we aim at extending this result to the case where A is thegenerator of a holomorphic C-semigroup. Our main result (Theorem 3.8)
becomes the above when C = I. To this end, we will use the method ofsums of commuting operators (see [8, 12]). A modiﬁcation of a result in [1]on the spectra of sums of commuting operators will be made in the next
section that is the key tool to derive the main result of this paper (Theorem
3.8).
2. Preliminaries
2.1. Notation. Throughout the paper, R, C, X stand for the sets of real,complex numbers and a complex Banach space, respectively; L(X),
BC(R, X), BU C(R, X), AP (X) denote the spaces of all linear bounded operators on X, all X-valued bounded and continuous functions, all X-valued
bounded uniformly continuous and all almost periodic functions in Bohr’s
sense (see the deﬁnition below) with sup-norm, respectively. For a linear
operator A, we denote by D(A), σ(A) the domain and the spectrum of A.
2.2. Almost periodic functions. In this paper by almost periodic functions we mean the almost periodic functions in the sense of Bohr (we refer
the reader to [17] for the deﬁnition and basic properties of such almost periodic functions). If f is an almost periodic function, the following limit
Z T
1f (t)e−iλt dt,
for all λ ∈ R(2.1)
a(λ, f ) := lim
T →∞ 2T −T
exists and is called the Bohr transform of f . As is known, there is an at mostcountable set of reals λ such that the above limit diﬀers from zero. This set
will be denoted by σb (f ) and called Bohr spectrum of f . The Approximation
Theorem (see [17]) says that for every almost periodic function f there exists
PNn
λk,n t , wherea sequence of trigonometric polynomials Pn (t) =k=1 ak,n e
λk,n ∈ σb (f ) and ak,n ∈ X for all k, n, that converges uniformly in t ∈ R to
f as n → ∞.
2.3. Spectral theory of functions. In the present paper, for a given u ∈BC(R, X), sp(u) stands for the Carleman spectrum of u, which consists of

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147

all ξ ∈ R such that the Carleman-Fourier transform of u, deﬁned by
 R∞
−λt

(Reλ > 0) 0 e u(t)dtû(λ) :=

 R ∞ λt
− 0 e u(−t)dt (Reλ < 0),
has no holomorphic extension to any neighborhood of iξ (see [2, 23]).
Proposition 2.1. Let u, un , v ∈ BC(R, X) such that limn→∞ kun − uk = 0.
Then(i) sp(u(· + h)) = sp(u) for any h ∈ R;(ii) sp(Bu(·)) ⊂ sp(u) for any B ∈ L(X);(iii) sp(αu(·) ⊂ sp(u) for any α ∈ C;(iv) If sp(un ) ⊂ Λ, for all n, then sp(u) ⊂ Λ, (here Λ is a subset of R);(v) If for u ∈ BU C(R, X), sp(u) is countable, and X does not contain any subspace which is isomorphic to the space of numericalsequences c0 , then u is almost periodic;(vi) If u is uniformly continuous and sp(u) is discrete, then u is almost
periodic;(vii) If u is almost periodic, then sp(u) = σb (u).
Proof. We refer the reader to [2], [23, Prop. 0.4, Prop. 0.6, Theorem 0.8, p.
20-25] and [17, Chap. 6] for the proofs.
¤
If Λ is a closed subset of R, then Λ(X) stands for the subspace ofBU C(R, X) consisting of all g ∈ BU C(R, X) such that sp(g) ⊂ Λ. As a consequence of the above properties, Λ(X) is a closed subspace of BU C(R, X)
that is invariant under translations. In particular, we have (see e.g. [19])Lemma 2.2. Let Λ be a closed subset of the real line. Then
(2.2)

σ(DΛ(X) ) = iΛ.

Let us denote by APΛ (X) the subspace of AP (X) consisting of all almost
periodic functions g such that sp(g) ⊂ Λ. Then we can see that if f is almost
periodic, then Λ(X) is exactly APΛ (X).
2.4. C-semigroups.
Deﬁnition 2.3. Let X be a Banach space and let C be an injective operatorin L(X). A family {T (t); t ≥ 0} in L(X) is called an exponentially bounded
C-semigroup if the following conditions are satisﬁed:(i) T (0) = C,(ii) T (t + s)C = T (t)T (s) for t, s ≥ 0,(iii) T (·)x : [0, ∞) → X is continuous for any x ∈ X,(iv) There are M ≥ 0 and a ∈ R such that kT (t)k ≤ M eat for t ≥ 0.

148

N. V. MINH

We deﬁne an operator A as follows:
D(A) = {x ∈ X : lim (T (h)x − Cx)/h ∈ R(C)}h→0+Ax = C−1lim (T (h)x − Cx)/h,h→0+

for all x ∈ D(A).

This operator is called the generator of (T (t))t≥0 . It is known that A isclosed but not necessarily densely deﬁned. Next we deﬁne the operator
Gx=

lim C −1

t→0

T (t)x − Cxt

T (t)x − Cx}.
t
The complete inﬁnitesimal generator of (T (t))t≥0 is deﬁned to be the operator G if R(C) is dense in X.
As in this paper we consider only exponentially bounded C-semigroups
which, for the sake of brevity, will be referred to as C-semigroups unlessotherwise stated.
D(G) := {x ∈ R(C) : ∃ lim C −1
t→0+

Lemma 2.4. Let C be an injective linear operator and let (T (t))t≥0 be a
C-semigroup with generator A. Then, the following assertions hold true:(i) T (t)T (s) = T (s)T (t),for all t, s ≥ 0,(ii) If x ∈ D(A), then T (t)x ∈ D(A), AT (t)x = T (t)Ax andZ t
T (ξ)Axdξ = T (t)x − Cx,
for all t ≥ 0,
(iii)

Rt

0

Rt
0 T (ξ)xdξ ∈ D(A) and A 0 T (ξ)xdξ = T (t)x−Cx for every x ∈ X

and t ≥ 0,(iv) A is closed and satisﬁes C −1 AC = A,(v) R(C) ⊂ D(A),(vi) If R(C) is dense in X, then D(G) = X and G ⊂ A.

For more information about C-semigroups and the relations between themand the so-called integrated semigroups we refer the reader to [10, 15, 18, 16]
and the references therein.
2.5. Holomorphic C-semigroups. Let C be an injective operator in L(X)with range R(C) dense in X.
Deﬁnition 2.5. A holomorphic C-semigroup is a family of operators{T (t), t ∈ C, arg t < δ} in L(X) (where 0 < δ ≤ π2 ) satisfying(i) T (t)T (s) = T (t + s)C, for | arg t| < δ; | arg s| < δ and T (0) = C;(ii) T (t) is holomorphic on | arg t| < δ;(iii) limt→0,| arg t|≤δ−² T (t)x = Cx for all x ∈ X and ² ∈ (0, δ);ALMOST PERIODIC SOLUTIONS

149(iv) For each ² ∈ (0, δ) there exist a real number a and a positive constant M² such that kT (t)k ≤ M² eaRet for | arg t| ≤ δ − ².
The following result is due to Tanaka [25].
Theorem 2.6. A closed linear operator A is the complete inﬁnitesimal generator of a holomorphic C-semigroup {T (t), | arg t| < δ} (where 0 < δ ≤ π2 )if and only if the following conditions hold(A1 ) There is a real number a such that for any | arg(λ − a)| < π2 + δ,
λ−A is injective, R(λ−A) ⊃ R(C) and (λ−A)−1 C is holomorphic,(A2 ) For ² ∈ (0, π) there is a positive constant M² such thatk(λ − A)−1 Ck ≤ M² |λ − a|−1
for | arg(λ − a)| ≤ π2 + δ − ².
(A3 ) (λ − A)−1 Cx = C(λ − A)−1 x for all x ∈ D((λ − A)−1 ) and | arg(λ −a)| < π2 + δ,(A4 ) D(A) is dense in X,(A5 ) C(D(A)) is a core for A.
Proof. For the proof see [25, Theorem 1].

¤

In the sequel, we need the following
Theorem 2.7. Let C be an injective bounded linear operator with dense
range in X, and let A be a closed operator that satisﬁes the conditions (A1 )−(A4 ) in Theorem 2.6. Then the family of operators T (t) deﬁned as
(
R λt
1
−1
for | arg t| ≤ δ − 2²,2πi
γ e (λ − A) Cdλ,T (t) :=
C, for t = 0,
where ² ∈ (0, δ/2), δ is the constant deﬁned in Theorem 2.6, and γ is a
curve running in the sector {| arg λ| < π/2 + δ} from ∞e−iφ to ∞eiφ with
φ = π/2 + δ − ², forms a C-semigroup (with t ≥ 0) (whose generator may
be diﬀerent from A).
Proof. For the proof and related remarks see [7] and [25].

¤2.6. Sums of commuting operators and holomorphic C-semigroups.
In this section we will extend a result by Arendt-Rabiger-Sourour [1, Theorem 7.3] to the case of holomorphic C-semigroups.
Deﬁnition 2.8. Let A and B be operators on a Banach space G with nonempty resolvent set. We say that A and B commute if one of the following
equivalent conditions hold:(i) R(λ, A)R(µ, B) = R(µ, B)R(λ, A) for some (all) λ ∈ ρ(A), µ ∈ρ(B) ,

150

N. V. MINH(ii) x ∈ D(A) implies R(µ, B)x ∈ D(A) and AR(µ, B)x = R(µ, B)Ax
for some (all) µ ∈ ρ(B).
For any θ ∈ (0, π) and R > 0 we let Σ(θ, R) := {z ∈ C : |z| ≥ R, | arg z| ≤θ}.
Theorem 2.9. Let A and B be commuting operators on a Banach space X
that both commute with the given injective bounded linear operator C, andlet D(A) be dense in X. Assume further that there exist R > 0 and θ ∈ (0, π)
such that(i) The operator A is the generator of a holomorphic C-semigroup;(ii) There exist R > 0 and θ1 ∈ (0, θ) such thatπΣ( − θ1 , R) ⊂ ρ(B)
2
and
sup
kλR(λ, B)k < ∞.
λ∈Σ( π2 −θ1 ,R)(iii) σ(A) ∩ σ(−B) = ®.
Then, A + B is closable and there exists an injective bounded linear operator
Q commuting with A, B, C such that
QA + B ⊂ A + BQ = C.

(2.3)

Proof. The proof is suggested by [1, Theorem 7.3]. Choose a rectiﬁable path
Γ0 lying in {z ∈ ρ(A) ∩ ρ(B) : |z| ≤ R} starting from the point Re−i(π/2+θ)and ending at Rei(π/2+θ) . Then, consider the oriented contour Γ0 consistingof{re−i(π/2+θ):r≥R }, Γ0 , {rei(π/2+θ) : r ≥ R}
Then there exists a partition C = Ω− ∪ Γ0 ∪ Ω+ , where Ω− , Ω+ are open
such thatint(Σ(θ + π/2, R)) ⊂ Ω+ ,{reiα : r ≥ R, α ∈ (θ + π/2, 3π/2 − θ)} ⊂ Ω− .

So, we have{λ ∈ σ(A) : |λ| ≥ R} ⊂ Ω− ,{λ ∈ σ(−B) : |λ| ≥ R} ⊂ Ω+ .

There exist compact subsets K+ , K− with oriented (piecewise C 1 ) boundary
Γ+ and Γ− , respectively such thatΩ− ∩ σ(−B) ⊂ intK− ⊂ K− ⊂ Ω− \σ(A)andΩ+ ∩ σ(A) ⊂ intK+ ⊂ K+ ⊂ Ω+ \σ(−B)

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151

Let Γ = Γ0 ∪ (−Γ− ) ∪ Γ+ . Since supλ∈Γ λ|2 |kR(λ, A)CR(λ, −B)k < ∞,Z
1
Q :=R(λ, A)CR(λ, −B)dλ
2πi Γ
Z
1
Qt :=
eλt R(λ, A)CR(λ, −B)dλ, (t > 0)2πi Γdeﬁne bounded operators on X that satisfy limt→0+ Qt x = Qx, for all x ∈ X.
By Theorem 2.7, the following family of operators(
R λt
1
for | arg t| ≤ δ − 2²,2πiΓ e R(λ, A)Cdλ,T (t) :=
C, for t = 0,
forms a C-semigroup (with t ≥ 0). In particular, we have limt→0 T (t)x = Cx
for all x ∈ X. Since A, B, C commute with each other,(A + B)R(λ, A)CR(λ, −B) = (R(λ, A) − R(λ, −B))C.
This yields in particular that Qt x ∈ D(A + B), for all x ∈ X. Next, we
have
Z
Z
1
1
λt
(A + B)Qt x =e R(λ, A)Cxdλ −eλt R(λ, −B)Cxdλ.
2πi Γ
2πi Γ
R
Since Γ eλt R(λ, −B)dλ = 0 by Cauchy’s theorem, we have
(2.4)

(A + B)Qt x = T (t)x,

for all x ∈ X.

Moreover, for x ∈ D(A + B), since A is closed and A, B, C are commuting,Z1eλt R(λ, A)CR(λ, −B)Axdλ
Qt Ax =
2πi ΓZ
1eλt R(λ, A)CR(λ, −B)xdλ = AQt x
= A
2πi Γ
Similarly, we can show that Qt Bx = BQt x for all x ∈ D(A + B). This yields
that for x ∈ D(A + B) we have Qt (A + B)x = (A + B)Qt x. By the same
argument, we can show that for all x ∈ D(A + B), Q(A + B)x = (A + B)Qx.
Next, for x ∈ D(A + B) by (2.4) we have
Qt (A + B)x = (A + B)Qt x = T (t)x.
Thus, letting t ↓ 0, since T (t)x → Cx, we obtain
Q(A + B)x = Cx,

for all x ∈ D(A + B)

Therefore, for all x ∈ D(A + B), we have (A + B)Qx = Q(A + B)x = Cx.
From the commutativeness of A, B, C for any µ ∈ ρ(A) ∩ ρ(B) it follows
that
R(µ, A)R(µ, B)Cx = R(µ, A)R(µ, B)(A + B)Qx
= (R(µ, A) − R(µ, B))Qx.

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N. V. MINH

Hence, if Qx = 0, then R(µ, A)R(µ, B)Cx = 0. The injectiveness ofC, R(µ, A), R(µ, B) yields that x = 0, that is the injectiveness of Q. Next,
to show the closability of A + B we suppose that xn ∈ D(A + B) such that
xn → 0 and (A + B)xn → y ∈ X as n → ∞. Then
Qy = lim Q(A + B)xn = lim Cxn = 0.
n→∞

n→∞

Since Q is injectiveness, y = 0, that is, the closability of the operator A + B.
Next, let x ∈ X. Then, limt↓0 Qt x = Qx andlim(A + B)Qt x = lim T (t)x = Cx.
t↓0

t↓0

Therefore, Qx ∈ D(A + B) and(2.5)( for all x ∈ X)

A + BQx = Cx,

Conversely, let x ∈ D(A + B) and let xn ∈ D(A + B) such that xn → x and(A + B)xn → A + Bx. ThenQA + Bx ==lim Q(A + B)xnn→∞lim Cxn = Cx.

n→∞¤This and (2.5) prove the theorem.
3. Almost periodic mild solutions

In this section we consider the equationdx(3.1)= Ax + f (t), x ∈ X, t ∈ R,
dt
where A generates an exponentially bounded C-semigroup of bounded linear
operators on X and f is a bounded and uniformly continuous function.
We always assume in this section that C is an injection with R(C) dense
in X unless otherwise stated. The main purpose of this section is to ﬁnd
conditions for the existence and uniqueness of almost periodic mild solutions
to Eq. (3.1) when f is almost periodic. To this end, we ﬁrst recall thefollowing deﬁnitions:Deﬁnition 3.1.
(i) An X-valued function u on R is said to be a solution on R to Eq.(3.1) for given linear operator A and f ∈ BC(R, X)(or sometimes, classical solution) if u ∈ BC 1 (R, X), u(t) ∈ D(A),for all t and u satisﬁes Eq.(3.1) for all t ∈ R.
(ii) An X-valued continuous function u on R is said to be a mild solution
on R to Eq.(3.1) for a given f ∈ BC(R, X) if u satisﬁes
Z t
(3.2)
Cu(t) = T (t − s)u(s) +T (t − ξ)f (ξ)dξ, for all t ≥ s.
s

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153

Lemma 3.2. Let (T (t))t≥0 be a C-semigroup with generator A and let f bea continuous function on the real line. Then, u is a solution of Eq. (3.2) if
and only if both of the following assertions hold:
Z t
(3.3)
Cu(ξ)dξ ∈ D(A),for all t ∈ R
0
Z t
Z t
(3.4)
Cu(t) − Cu(0) = A
Cu(s)ds +Cf (s)ds,
for all t ∈ R.
0

0¤

Proof. For the proof see [4, Lemma 2.9].

From this lemma, by the injectivity of the operator C, it follows in particular that every classical solution is a mild one.
Deﬁnition 3.3. Let Λ be a closed subset of R and let (T (t))t≥0 be a
C-semigroup on X. Then the following family of bounded linear operators (T h )h≥0 on APΛ (X) is called the evolution semigroup associated with(T (t))t≥0 on APΛ (X):
(3.5)

(T h v)(t) := T (h)v(t − h),v ∈ APΛ (X), h ≥ 0.

Let us denote by C̃ the operator of multiplication by C on APΛ (X).
Lemma 3.4. Under the above notation the evolution semigroup (T h )h≥0 is
a C̃-semigroup with generator −LM , where M := AP (X).
Proof. For the proof see [4, Lemma 2.6].

¤

We introduce the following operator LM .
Deﬁnition 3.5. Let M be a closed subspace of BU C(R, X). We deﬁne theoperator LM on M as follows: u ∈ D(LM ) if and only if u ∈ M and there
is f ∈ M such that
Z t
T (t − r)f (r)dr, for all t ≥ s(3.6)
Cu(t) = T (t − s)u(s) +sand in this case LM u := f .
Let us denote the diﬀerential operator d/dt in APΛ (X) by D and thefunction space APΛ (X) by M. It is obvious that D, Ã and C̃ commute witheach other. Moreover, we have
Lemma 3.6. Let A be the generator of a C-semigroup. Then, under theabove notations, the operator LM is well-deﬁned single valued operator.
Moreover,(3.7)

LM = D − Ã154

N. V. MINH

Proof. The fact that LM is a single valued operator was established in [4].
By Lemma 2.6 of [4], −LM is the generator of the evolution C̃-semigroup(T h )h≥0 . On the other hand, by [16, Theorem 2.1], the generator of thisevolution semigroup is nothing but −D + Ã. So, LM = D − Ã.
¤
We are now in a position to apply the method of sums of commuting
operators to derive conditions for the existence of almost periodic mild solutions to Eq. (1.1). To this end, assume that the operator A in Eq. (1.1)
generates a holomorphic C-semigroup T (t) on the Banach space X. Let us
denote by Λ the closure of the Bohr spectrum of the almost periodic function
f . Consider the operators T̃ (t) of multiplication by T (t) and the operatorÃ of multiplication by A on M := APΛ (X). Recall that, by deﬁnition,D(Ã) consists of all g ∈ APΛ (X) such that g(t) ∈ D(A), for all t ∈ R andAg(·) ∈ APΛ (X).
Lemma 3.7. Under the above assumptions and notations, the family of operators of multiplication T̃ (t) is a holomorphic C̃-semigroup with generatorÃ on APΛ (X);
Proof. First, we show that the range of C̃ is dense in APΛ (X). In fact, by
the assumption, every trigonometric monomial eiλt a, λ ∈ Λ, a ∈ X canbe approximated by a sequence of trigonometric monomials eiλt ak withak ∈ R(C), k = 1, 2, · · · . So, every trigonometric polynomial Pn (t) :=Pniλk,n t , with λa∈ X, can be approximated by ak,n ∈ Λ and
k=1 ak,n ePnk,ntrigonometric polynomial Qn (t) := k=1 bk,n eiλk,n t with bk,n ∈ R(C). Notethat Qn ∈ R(C̃). On the other hand, by the Approximation Theorem of Almost Periodic Functions, every function in APΛ (X) can be approximated bya sequence of trigonometric polynomials in X with exponents in Λ. Therefore, it can be approximated by a sequence of trigonometric polynomials
with coeﬃcients in R(C) and exponents in Λ (that are in R(C̃)). So, therange of C̃ is dense in APΛ (X).
The precompactness of the range of each function in APΛ (X) yields theconvergence in APΛ (X) of the limit limt→0,| arg t|≤δ−² T̃ (t)g = C̃g for allg ∈ APΛ (X) and ² ∈ (0, δ). Therefore, T̃ (t) is a holomorphic C̃-semigroup.
It remains to show that its generator is exactly Ã. First of all, suppose thatg ∈ D(Ã). Then, for each t ∈ R there exists limh↓0 (T (h)g(t) − Cg(t))/h ∈R(C). From the precompactness of the range of g and the strong continuity of (T (h))h≥0 it follows that this limit is uniform in t ∈ R. Hence,limh↓0 (T̃ (h)g −Cg)/h exists as an element w of APΛ (X) with range in R(C).
Since C −1 w(·) = Ag(·) = Ãg ∈ APΛ (X), we have that g = Cw ∈ R(C̃).
Hence, g ∈ D(A), where A denotes the generator of the C-semigroup T̃ (t).
Obviously, Ãg = Ag. This shows that Ã ⊂ A. Conversely, let g ∈ D(A).

ALMOST PERIODIC SOLUTIONS

155

Then, by deﬁnition, we can easily see that for each t ∈ R, g(t) ∈ D(A) andAg(·) ∈ APΛ (X), that is, A ⊂ Ã. Finally, we obtain that Ã is exactly thegenerator of the C-semigroup T̃ (t).
¤We now state the main result of this paper.
Theorem 3.8. Let A be the generator of a holomorphic C-semigroup with
R(C) dense in X, and let the following conditions be satisﬁed:(i) There exists an almost periodic function g such that f (t) = Cg(t),for all t ∈ R;(ii) σ(A) ∩ isp(g) = ®.
Then, the following assertions hold:(i) There exists an almost periodic mild solution u to Eq. (3.1) suchthat sp(u) ⊂ sp(g);(ii) If there exists another bounded mild solution v to Eq. (3.1) suchthat sp(v) ⊂ sp(g), then u(t) = v(t), for all t ∈ R.
Proof. (i) Let us denote Λ := sp(g). We can easily show σ(Ã) ⊂ σ(A).
And by Lemma 2.2 we obtain that σ(D) = iΛ = iσ(g). By Theorem 2.9,
there exists an injective bounded linear operator Q on APΛ (X) such that
D − ÃQ = C̃. Setting u = Qg, we have D − ÃQg = C̃g = f . Therefore,u = Qg is an almost periodic mild solution of Eq. (3.1) such that sp(u) ⊂Λ = sp(g).
(ii) The uniqueness of the solution u is an immediate consequence of thefollowing estimate: For any bounded mild solution w of Eq. (3.1) we have
(3.8)

sp(Cw) ⊂ σi (A) ∪ sp(g) ,where σi (A) := {ξ ∈ R : iξ ∈ σ(A)}. Assume that there is ξ0 ∈ R such thatiξ0 ∈ ρ(A). Taking the Laplace transforms of both sides of (3.4) we caneasily show that (for Reλ 6= 0 and λ close to iξ0 ),(3.9)

c (λ).
dCw(λ)= R(λ, A)Cw(0) + R(λ, A)Cf

Therefore, if ξ0 6∈ sp(g), (so ξ0 6∈ sp(f ) by Proposition 2.1 (ii)), the functiondCw(λ)
has an analytic extension around iξ. Thus, sp(Cw(·)) ⊂ σi (A) ∪sp(Cf (·)). And hence we have (3.8).
Obviously [Cu(·) − Cv(·)] is an almost periodic mild solution of Eq. (3.1)
with f replaced by 0. In view of (3.8), isp(Cu(·) − Cv(·)) ⊂ σi (A). On theother hand, since both u and v are in Λ(X) we have isp(Cu(·) − Cv(·)) ⊂σi (A) ∩ iΛ = σi (A) ∩ isp(g) = ®. This yields that [Cu(·) − Cv(·)] = 0.
¤

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N. V. MINH

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Nguyen Van Minh
Department of Mathematics
University of West Georgia
Carrollton, GA 30118
e-mail address: vnguyen@westga.edu(Received August 5, 2005 )