For the differential equation u′(t)=Au
(t)+f(t), t≥0 on a
Hilbert space H, we find the necessary and sufficient conditions that
the above-mentioned equation has a unique almost periodic solution. Some applications are also
given.

1. Introduction

In this paper we are concerned with the almost periodicity of solutions of the differential equation
u′(t)=Au(t)+f(t)t∈ℝ,
where A is a linear, closed operator on a Hilbert space H and f is a function from ℝ to H. The asymptotic behavior and, in particular, the almost periodicity of solutions of (1.1) has been a subject of intensive study for recent decades; see, for example, [1–5] and references therein. A particular condition for almost periodicity is the countability of the spectrum of the solution. In this paper we investigate the almost periodicity of mild solutions of (1.1), when A is a linear, unbounded operator on a Hilbert space H. We use the Hilbert space AP(ℝ,H) introduced in [4], defined by what follows. Let (·,·) be the inner product of H, and let APb(ℝ,E) be the space of all almost periodic functions from ℝ to H. The completion of APb(ℝ,E) is then a Hilbert space with the inner product defined by
〈f,g〉:=limT→∞12T∫-TT(f(s),g(s))ds.
First, we establish the relationship between the Bohr transforms of the almost periodic solutions of (1.1) and those of the inhomogeneity f. We then give a necessary and sufficient condition so that (1.1) admits a unique almost periodic solution for each almost periodic inhomogeneity f. As applications, in Section 4 we show a short proof of the Gearhart's theorem. If A is generator of a strongly continuous semigroup T(t), then 1∈ϱ(T(1)) if and only if 2kπi∈ϱ(A) and supk∈ℤ∥(2kπi-A)-1∥<∞.

2. The Hilbert Space of Almost Periodic Functions

Let us fix some notations. Recall that a bounded, uniformly continuous function f from ℝ to a Banach space H is almost periodic, if the set {S(t)f:t∈ℝ} is relatively compact in BUC(ℝ,H), the space of bounded uniformly continuous functions with sup-norm topology. Let H be now a Hilbert space with (·,·), and let ∥·∥ be the inner product and the norm in H, respectively. Let APb(ℝ,H) be the space of all almost periodic functions from ℝ to H. In APb(ℝ,H) the following expression
〈f,g〉:=limT→∞12T∫-TT(f(s),g(s))ds
exists and defines an inner product. Hence, APb(ℝ,H) is a pre-Hilbert space and its completion, denoted by AP(ℝ,H), is a Hilbert space. The inner product and the norm in AP(ℝ,H) are denoted by 〈f,g〉 and ∥·∥AP, respectively.

For each function f∈AP(ℝ,H), the Bohr transform is defined by
a(λ,f):=limT→∞12T∫-TTf(s)e-iλsds.
The set
σ(f):={λ∈ℝ:a(λ,f)≠0}
is called the Bohr spectrum of f. It is well known that σ(f) is countable for each function f∈AP(ℝ,H), and the Fourier-Bohr series of f is
∑λ∈σ(f)a(λ,f)eiλt,
and it converges to f in the norm topology of AP(ℝ,E). The following Parseval's equality also holds
∥f∥AP(ℝ,H)2=∑λ∈σ(f)∥a(λ,f)∥.
For more information about the almost periodic functions and properties of the Hilbert space AP(ℝ,H), we refer readers to [2, 4].

Let W2(AP) be the space consisting of all almost periodic functions f, such that f′∈AP(ℝ,H). W2(AP) is then a Hilbert space with the norm
∥f∥W2(AP)2:=∥f∥AP(ℝ,H)2+∥f′∥AP(ℝ,H)2.
Note that the W2(AP)-topology is stronger than the sup-norm topology (see [6]). We will use the following lemma.

Lemma 2.1.

If F is a function in W2(AP) and f=F′, then we have
a(λ,f)=λi·a(λ,F).

Proof.

If λ≠0, using the integration by part we have
12T∫-TTe-iλsf(s)ds=12TF(t)e-iλt|-TT+iλ2T∫-TTF(s)e-iλsds=F(T)e-iλT-F(-T)eiλT2T+iλ12T∫-TTF(s)e-iλsds.
Let T→∞, and note that F(t) is bounded, we have (2.7).

If λ=0, then
a(0,f)=limT→∞12T∫-TTf(s)ds=limT→∞F(T)-F(-T)2T=0,
which also satisfies (2.7).

Finally, for a linear, closed operator A in a Hilbert space H, we denote the domain, the range, the spectrum, and the resolvent set of A by D(A), Range(A), σ(A), and ϱ(A), respectively.

3. Almost Periodic Mild Solutions of Differential Equations

We now turn to the differential equation
u′(t)=Au(t)+f(t),t∈ℝ.
First we define two types of solutions to (3.1).

Definition 3.1.

(1) A continuous function u is called a mild solution of (3.1) if
u(t)=u(0)+A∫0tu(s)ds+∫0tf(s)ds,
for all t∈ℝ.

(2) A function u is a classical solution of (3.1), if u(t)∈D(A), u is continuously differentiable, and (3.1) holds for t∈ℝ.

Remark 3.2.

The mild solution to (3.1) defined by (3.2) is really an extension of classical solution in the sense that every classical solution is a mild solution and conversely, if a mild solution is continuously differentiable, then it is a classical solution.

If A is the generator of a C0 semigroup T(t), then a continuous function u:ℝ→E is a mild solution of (1.1) if and only if it has the form (see [7])
u(t)=T(t-s)u(s)+∫stT(t-r)f(r)dr,fors<t.
We now consider the almost periodic mild solutions of (3.1). The following proposition describes the connection between the Bohr transforms of such solutions and those of f(t).

Proposition 3.3.

Suppose f∈AP(ℝ,H) and u is an almost periodic mild solution of (3.1). Then
(λi-A)a(λ,u)=a(λ,f),
for every λ∈ℝ.

Proof.

Suppose λ is a nonzero real number. Multiplying each side of (3.2) with e-iλt and taking definite integral from -T to T on both sides, we have
∫-TTe-iλtu(t)dt=∫-TTe-iλtu(0)dt+A∫-TTe-iλt∫0tu(s)dsdt+∫-TTe-iλt∫0tf(s)dsdt.
Here we used the fact that ∫abAu(t)dt=A∫abu(t)dt for a closed operator A. It is easy to see that
∫-TTe-iλtu(0)dt=-e-iλTu(0)-eiλTu(0)iλ
and, applying integration by part for any integrable function g(t), we have
∫-TTe-iλt∫0tg(s)dsdt=e-iλt∫0tg(s)ds|-TT+1iλ∫-TTe-iλtg(t)dt=e-iλT∫0Tg(t)dt-eiλT∫0-Tg(t)dt+1iλ∫-TTe-iλtg(t)dt.
Using (3.7) for g(t)=u(t) and g(t)=f(t) in (3.5), respectively, we have
12T∫-TTe-iλtu(t)dt=-e-iλTu(0)-eiλTu(0)iλ2T+e-iλT2T(A∫0Tu(t)dt+∫0Tf(t)dt)-eiλT2T(A∫0-Tu(t)dt+∫0-Tf(t)dt)+1iλ2T(A∫-TTe-iλtu(t)dt+∫-TTe-iλtf(t)dt)=I1+I2+I3,
where
I1=-e-iλTu(0)-eiλTu(0)iλ2T→0
as T→∞;
I2=e-iλT2T(A∫0Tu(t)dt+∫0Tf(t)dt)-eiλT2T(A∫0-Tu(t)dt+∫0-Tf(t)dt)=e-iλT2T(u(T)-u(0))-eiλT2T(u(-T)-u(0))→0
as T→∞, and
I3=1iλ(12TA∫-TTe-iλtu(t)dt+12T∫-TTe-iλtf(t)dt).
Let uT:=(1/2T)∫-TTe-iλtu(t)dt. It is clear that
limT→∞uT=a(λ,u),
and from (3.11), we have
AuT=12TA∫-TTe-iλtu(t)dt=iλI3-12T∫-TTe-iλtf(t)dt=iλ(uT-I1-I2)-12T∫-TTe-iλtf(t)dt→iλa(λ,u)-a(λ,f)as T→∞.
Since A is a closed operator, from (3.12) and (3.13) we obtain a(λ,u)∈D(A) and Au(λ,u)=iλa(λ,u)-a(λ,f), from which (3.4) is followed.

Finally, if λ=0, let uT=(1/2T)∫-TTu(s)ds. Then, limt→∞uT=a(0,u) and, using the definition of u in (3.2),
AuT=12TA∫-TTu(s)ds=u(T)-u(-T)2T-12T∫-TTf(s)ds→-a(0,f)as T→∞.
Again, since A is a closed operator, it implies a(a,u)∈D(A) and Au(0,u)=-a(0,f), from which (3.4) is followed, and this completes the proof.

Note that Proposition 3.3 also holds in a Banach space. We are now going to look for conditions that (3.1) has an almost periodic mild solution.

Theorem 3.4.

Suppose f is an almost periodic function, which is in W2(AP). Then the following statements are equivalent.

(i) Equation (3.1) has an almost periodic mild solution, which is in W2(AP).

(ii) For every λ∈σ(f), a(λ,f)∈Range(A) and there exists a series {xλ}λ∈σ(f) in H satisfying (iλ-A)xλ=a(λ,f), for which the following holds
∑λ∈σ(f)∥xλ∥2<∞,∑λ∈σ(f)|λ|2∥xλ∥2<∞.

Proof.

(i)⇒(ii) Let u(t) be an almost periodic solution to (3.1), which is in W2(AP). By Proposition 3.3, (iλ-A)a(λ,u)=a(λ,f). Hence a(λ,f)∈Range(A) for all λ∈σ(f).

Put now xλ:=a(λ,u) for λ∈σ(f). Then it satisfies (iλ-A)xλ=a(λ,f). Moreover, iλxλ=a(λ,u′); hence,
∑λ∈σ(f)∥xλ∥2=∥u∥AP2,∑λ∈σ(f)|λ|2∥xλ∥2=∥u′∥AP2,
which imply (3.15).

(ii)⇒(i) Let {xλ}λ∈σ(f) be a series in H satisfying (iλ-A)xλ=a(λ,f), for which (3.15) holds. Put
fN(t):=∑λ∈σ(f),|λ|<Neiλta(λ,f),uN(t):=∑λ∈σ(f),|λ|<Neiλtxλ.
It is then easy to find their norms:
∥uN∥2=∑λ∈σ(f),|λ|<N∥xλ∥2,∥uN′∥2=∑λ∈σ(f),|λ|<N|λ|2∥xλ∥2.
From (3.15) it implies that uN→u and uN′→v as N→∞ for some function u and v in the topology of AP(ℝ,H). Since the differential operator is closed, we obtain u∈W2, u′=v and limN→∞uN=u in the topology of W2(AP). Hence, u is almost periodic. It remains to show that u is a mild solution of (1.1). In order to do that, note uN is a classical solution of (3.1), and hence, a mild one, that is,
uN(t)=uN(0)+A∫0tuN(s)ds+∫0tfN(s)ds.
For each t∈ℝ, we have
limN→∞∫0tfN(s)ds=∫0tf(s)ds,limN→∞∫0tuN(s)ds=∫0tu(s)ds,
and, using (3.19),
limT→∞A∫0tuN(s)ds=limT→∞(uN(t)-uN(0)-∫0tfN(s)ds)=u(t)-u(0)-∫0tf(s)ds.
Since A is a closed operator, we obtain ∫0tu(s)ds∈D(A) and
A∫0tu(s)ds=u(t)-u(0)-∫0tf(s)ds,
which shows that u is a mild solution of (1.1) and the proof is complete.

Note that if condition (ii) in Theorem 3.4 holds, (3.1) may have two or more almost periodic mild solutions. We are going to find conditions such that for each almost periodic function f, (3.1) has a unique almost periodic mild solution. We are now in the position to state the main result.

Theorem 3.5.

Suppose A is a closed operator on a Hilbert space H and M is a closed subset of ℝ. The following are equivalent.

For each function f∈W2(AP) with σ(f)⊆M, (3.1) has a unique almost periodic mild solution u in W2(AP) with σ(u)⊆M.

For each λ∈M, iλ∈ϱ(A) and
supλ∈M∥(iλ-A)-1∥<∞.

Proof.

(i)⇒(ii) Let W2(AP)|M be the subspace of all functions f in W2(AP) with σ(f)∈M. Then W2(AP)|M is a Hilbert space by nature. Let x be any vector in H, let λ be a number in M, and let f(t)=eiλtx. Then f∈W2(AP)|M and hence, (3.1) has a unique almost periodic solution u. By Theorem 3.4, x=a(λ,f)∈Range(iλ-A), hence (iλ-A) is surjective for all λ∈M. On the other hand, (iλ-A) is injective; otherwise, u2(t)=u(t)+eiλtx, where x is a nonzero vector in H satisfying (iλ-A)x=0, would be another almost periodic mild solution to (3.1) with σ(u2)=σ(u)⊆M. Hence (iλ-A) is bijective and iλ∈ϱ(A) for all λ∈M.

In W2(AP)|M we define the operator L by what follows. For each f∈W2(AP)|M, L(f) is the unique almost periodic mild solution to (1.1) corresponding to f. By the assumption, L is everywhere defined. We will prove that L is a bounded operator by showing L is closed in W2(AP)|M. Let fn→f and Lfn→u in W2(AP)|M, where
(Lfn)(t)=(Lfn)(0)+A∫0t(Lfn)(s)ds+∫0tfn(s)ds.
For each t∈ℝ, we have limn→∞Lfn(t)=u(t), limN→∞∫0tfn(s)ds=∫0tf(s)ds, and limn→∞∫0tLfn(s)ds=∫0tu(s)ds. Moreover, from (3.24) we have
A∫0t(Lfn)(s)ds=(Lfn)(t)-(Lfn)(0)-∫0tfn(s)ds→n→∞u(t)-u(0)-∫0tf(s)ds,
for each t∈ℝ. Since A is a closed operator, ∫0tu(s)ds∈D(A) and
A∫0tu(s)ds=u(t)-u(0)-∫0tf(s)ds,
which means u is a mild solution to (3.1) corresponding to f. Thus, f∈D(L), Lf=u and hence, L is closed.

Next, for any x∈H and λ∈M, put f(t)=eiλtx, then u(t)=eiλt(2kπi-A)-1x is the unique almost periodic solution to (3.1), that is, u=Lf. Using the boundedness of operator L, we obtain
(|λ|+1)∥(iλ-A)-1x∥=∥u∥W2(AP)≼∥L∥∥u∥W2(AP)=∥L∥(|λ|+1)∥x∥,
which implies
∥(iλ-A)-1x∥≼∥L∥·∥x∥.
for any x∈E and any λ∈M. Thus, (3.33) holds.

(ii)⇒(i) Suppose f is a function in W2(AP)|M. Put xλ:=(iλ-A)-1a(λ,f). Then
∑λ∈σ(f)∥xλ∥2≼supλ∈σ(f)∥(iλ-A)-1∥2∑λ∈σ(f)∥a(λ,f)∥2≼supλ∈M∥(iλ-A)-1∥2∥f∥2<∞,∑λ∈σ(f)λ2∥xλ∥2≼supλ∈σ(f)∥(iλ-A)-1∥2∑λ∈σ(f)λ2∥a(λ,f)∥2=supλ∈M∥(iλ-A)-1∥2∥f′∥2<∞.
By Proposition 3.3, (3.1) has an almost periodic mild solution in W2(AP)|M. That solution is unique, since its Bohr transforms are uniquely determined by a(λ,u)=(iλ-A)-1a(λ,f) for all λ∈M.

We can apply Theorem 3.5 to some particular sets for M. First, if M=ℝ, we have the following.

Corollary 3.6.

Suppose A is a closed operator on a Hilbert space H. The following are equivalent.

(i) For each function f∈W2(AP), (3.1) has a unique 1-periodic mild solution in W2(AP).

(ii) iℝ⊆ϱ(A) and
supλ∈ℝ∥(iλ-A)-1∥≤∞.

Let now L2(0,1) be the Hilbert space of integrable functions f from (0,1) to H with the norm
∥f∥L2(0,1)2=∫01∥f(t)∥2dt<∞.
If M={2kπ:k∈ℤ}, then the space W2(AP)|M becomes W21(1), the space of all periodic functions f of period 1 with f′∈L2(0,1). W21(1) is then a Hilbert space with the norm
∥f∥W21(1)2=∥f∥L2(0,1)2+∥f′∥L2(0,1)2.

Corollary 3.7.

Suppose A is a closed operator on a Hilbert space H. The following are equivalent.

(i) For each function f∈W21(1), (3.1) has a unique 1-periodic mild solution in W21(1).

(ii) For each k∈ℤ, 2kiπ∈ϱ(A) and
supk∈ℤ∥(2kiπ-A)-1∥<∞.

4. Application: A <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M289"><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant="bold-italic">C</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant="bold">0</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>-Semigroup Case

If A generates a C0-semigroup (T(t))t≥0, then (see [7, Theorem 2.5]), mild solutions of (3.1) can be expressed by
u(t)=T(t-s)u(s)+∫stT(t-τ)f(τ)dτ,
for t≥s. If f is a 1-periodic function, then it is easy to see that the above solution u is 1-periodic if and only if u(1)=u(0). Hence, to consider 1-periodic solution, it suffices to consider u in [0,1] and in this interval we have
u(t)=T(t)u(0)+∫0tT(t-s)f(s)ds.
We obtain the following results, in which we show the Gearhart's theorem (the equivalence (iv)⇔(v) with a short proof.

Theorem 4.1.

Let A generate a C0-semigroup (T(t)) on a Hilbert H, then the following are equivalent.

(i) For each function f∈L2(0,1), (3.1) has a unique 1-periodic mild solution.

(ii) For each function f∈W21(1), (3.1) has a unique 1-periodic classical solution.

(iii) For each function f∈W21(1), (3.1) has a unique 1-periodic solution contained in W21(1).

(iv) For each k∈ℤ, 2kπi∈ϱ(A) and
supk∈ℤ∥(2kπi-A)-1∥<∞.

(v) 1∈ρ(T(1)).

Proof.

The equivalence (iii)⇔(iv) is shown in Corollary 3.7, (i)⇔(ii) can be easily proved by using standard arguments, (i)⇔(v) has been shown in [8], and (ii)⇒(iii) is obvious. So, it remains to show the inclusion (iii)⇒(ii).

Let f be any function in W21(1) and let u(t) be the unique mild solution of (3.1), which is in W21(1). Since for each f∈W21(1), the function g(t):=∫0tT(t-s)f(s)ds is continuously differentiable and g(t)∈D(A) for all t∈[0,1] (see [9]), to show u is a classical solution, it suffices to show u(0)∈D(A).

From the above observation and from formula (4.2), the function t↦T(t)u(0)=u(t)-∫0tT(t-s)f(s)ds is differentiable almost everywhere on [0,1]. It follows that T(t)u(0)∈D(A) for almost everywhere t (since t↦T(t)x is differentiable at t0 if and only if T(t0)x∈D(A)). Hence, T(1)u(0)∈D(A). By formula (4.2), u(1), and thus, u(0)=u(1), belongs to D(A). The uniqueness of this 1-periodic classical solution is obvious and the proof is complete.

Acknowledgment

The author would like to express his gratitude to the anonymous referee for his/her helpful suggestions.

ArendtW.BattyC. J. K.Almost periodic solutions of first- and second-order Cauchy problemsLevitanB. M.ZhikovV. V.MurakamiS.NaitoT.Van MinhN.Evolution semigroups and sums of commuting operators: a new approach to the admissibility theory of function spacesPhongV. Q.A new proof and generalizations of Gearhart's theoremRuessW. M.PhongV. Q.Asymptotically almost periodic solutions of evolution equations in Banach spacesTriebelH.ArendtW.BattyC. J. K.HieberM.NeubranderF.PrüssJ.On the spectrum of C0-semigroupsNagelR.SinestrariE.BierstedtK. D.PietschA.RuessW. M.VogtD.Inhomogeneous Volterra integrodifferential equations for Hille-Yosida operators