On the Almost Periodic Solutions of Differential Equations on Hilbert Spaces

Abstract

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, (1.1) 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 𝐴𝑃_𝑏(ℝ,𝐸) be the space of all almost periodic functions from ℝ to 𝐻. The completion of 𝐴𝑃_𝑏(ℝ,𝐸) is then a Hilbert space with the inner product defined by ⟨𝑓,𝑔⟩∶=lim_𝑇→∞12𝑇^𝑇_−𝑇(𝑓(𝑠),𝑔(𝑠))𝑑𝑠. (1.2) 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 sup_𝑘∈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 𝐴𝑃_𝑏(ℝ,𝐻) be the space of all almost periodic functions from ℝ to 𝐻. In 𝐴𝑃_𝑏(ℝ,𝐻) the following expression ⟨𝑓,𝑔⟩∶=lim_𝑇→∞12𝑇^𝑇_−𝑇(𝑓(𝑠),𝑔(𝑠))𝑑𝑠 (2.1) exists and defines an inner product. Hence, 𝐴𝑃_𝑏(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)∶=lim_𝑇→∞12𝑇^𝑇_−𝑇𝑓(𝑠)𝑒^{−𝑖𝜆𝑠}𝑑𝑠. (2.2) The set 𝜎(𝑓)∶={𝜆∈ℝ∶𝑎(𝜆,𝑓)≠0} (2.3) 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)𝑒^𝑖𝜆𝑡, (2.4) and it converges to 𝑓 in the norm topology of 𝐴𝑃(ℝ,𝐸). The following Parseval's equality also holds ‖𝑓‖²_{𝐴𝑃(ℝ,𝐻)}=_{𝜆∈𝜎(𝑓)}‖𝑎(𝜆,𝑓)‖. (2.5) 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‖²_{𝑊2(𝐴𝑃)}∶=‖𝑓‖²_{𝐴𝑃(ℝ,𝐻)}+‖𝑓^′‖²_{𝐴𝑃(ℝ,𝐻)}. (2.6) 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). (2.7)

Proof. If 𝜆≠0, using the integration by part we have 12𝑇^𝑇_−𝑇𝑒^{−𝑖𝜆𝑠}1𝑓(𝑠)𝑑𝑠=2𝑇𝐹(𝑡)𝑒^{−𝑖𝜆𝑡}|^𝑇_−𝑇+𝑖𝜆2𝑇^𝑇_−𝑇𝐹(𝑠)𝑒^{−𝑖𝜆𝑠}=𝑑𝑠𝐹(𝑇)𝑒^{−𝑖𝜆𝑇}−𝐹(−𝑇)𝑒^𝑖𝜆𝑇12𝑇+𝑖𝜆2𝑇^𝑇_−𝑇𝐹(𝑠)𝑒^{−𝑖𝜆𝑠}𝑑𝑠. (2.8) Let 𝑇→∞, and note that 𝐹(𝑡) is bounded, we have (2.7).
If 𝜆=0, then 𝑎(0,𝑓)=lim_𝑇→∞12𝑇^𝑇_−𝑇𝑓(𝑠)𝑑𝑠=lim_𝑇→∞𝐹(𝑇)−𝐹(−𝑇)2𝑇=0, (2.9) 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. (3.1) 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^𝑡₀𝑢(𝑠)𝑑𝑠+^𝑡₀𝑓(𝑠)𝑑𝑠, (3.2) 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)+^𝑡_𝑠𝑇(𝑡−𝑟)𝑓(𝑟)𝑑𝑟,for𝑠<𝑡. (3.3) 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), (3.4) 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 ^𝑇_−𝑇𝑒^{−𝑖𝜆𝑡}𝑢(𝑡)𝑑𝑡=^𝑇_−𝑇𝑒^{−𝑖𝜆𝑡}𝑢(0)𝑑𝑡+𝐴^𝑇_−𝑇𝑒^{−𝑖𝜆𝑡𝑡}₀+𝑢(𝑠)𝑑𝑠𝑑𝑡_𝑇−𝑇𝑒^{−𝑖𝜆𝑡𝑡}₀𝑓(𝑠)𝑑𝑠𝑑𝑡. (3.5) Here we used the fact that ∫^𝑏_𝑎𝐴𝑢(𝑡)𝑑𝑡=𝐴∫^𝑏_𝑎𝑢(𝑡)𝑑𝑡 for a closed operator 𝐴. It is easy to see that ^𝑇_−𝑇𝑒^{−𝑖𝜆𝑡}𝑒𝑢(0)𝑑𝑡=−^{−𝑖𝜆𝑇}𝑢(0)−𝑒^𝑖𝜆𝑇𝑢(0)𝑖𝜆 (3.6) and, applying integration by part for any integrable function 𝑔(𝑡), we have ^𝑇_−𝑇𝑒^{−𝑖𝜆𝑡𝑡}₀𝑔(𝑠)𝑑𝑠𝑑𝑡=𝑒^{−𝑖𝜆𝑡𝑡}₀𝑔(𝑠)𝑑𝑠|^𝑇_−𝑇+1𝑖𝜆^𝑇_−𝑇𝑒^{−𝑖𝜆𝑡}𝑔(𝑡)𝑑𝑡=𝑒^{−𝑖𝜆𝑇𝑇}₀𝑔(𝑡)𝑑𝑡−𝑒^𝑖𝜆𝑇₀^−𝑇1𝑔(𝑡)𝑑𝑡+𝑖𝜆^𝑇_−𝑇𝑒^{−𝑖𝜆𝑡}𝑔(𝑡)𝑑𝑡. (3.7) Using (3.7) for 𝑔(𝑡)=𝑢(𝑡) and 𝑔(𝑡)=𝑓(𝑡) in (3.5), respectively, we have 12𝑇^𝑇_−𝑇𝑒^{−𝑖𝜆𝑡}𝑒𝑢(𝑡)𝑑𝑡=−^{−𝑖𝜆𝑇}𝑢(0)−𝑒^𝑖𝜆𝑇𝑢(0)+𝑒𝑖𝜆2𝑇_{−𝑖𝜆𝑇}2𝑇(𝐴^𝑇₀𝑢(𝑡)𝑑𝑡+^𝑇₀−𝑒𝑓(𝑡)𝑑𝑡)_𝑖𝜆𝑇2𝑇(𝐴₀^−𝑇𝑢(𝑡)𝑑𝑡+₀^−𝑇+1𝑓(𝑡)𝑑𝑡)𝑖𝜆2𝑇(𝐴^𝑇_−𝑇𝑒^{−𝑖𝜆𝑡}𝑢(𝑡)𝑑𝑡+^𝑇_−𝑇𝑒^{−𝑖𝜆𝑡}𝑓(𝑡)𝑑𝑡)=𝐼₁+𝐼₂+𝐼₃, (3.8) where 𝐼₁𝑒=−^{−𝑖𝜆𝑇}𝑢(0)−𝑒^𝑖𝜆𝑇𝑢(0)𝑖𝜆2𝑇→0 (3.9) as 𝑇→∞; 𝐼₂=𝑒^{−𝑖𝜆𝑇}2𝑇(𝐴^𝑇₀𝑢(𝑡)𝑑𝑡+^𝑇₀𝑒𝑓(𝑡)𝑑𝑡)−^𝑖𝜆𝑇2𝑇(𝐴₀^−𝑇𝑢(𝑡)𝑑𝑡+₀^−𝑇=𝑒𝑓(𝑡)𝑑𝑡)_{−𝑖𝜆𝑇}𝑒2𝑇(𝑢(𝑇)−𝑢(0))−^𝑖𝜆𝑇2𝑇(𝑢(−𝑇)−𝑢(0))→0 (3.10) as 𝑇→∞, and 𝐼₃=1(1𝑖𝜆2𝑇𝐴^𝑇_−𝑇𝑒^{−𝑖𝜆𝑡}1𝑢(𝑡)𝑑𝑡+2𝑇^𝑇_−𝑇𝑒^{−𝑖𝜆𝑡}𝑓(𝑡)𝑑𝑡). (3.11) Let 𝑢_𝑇∶=(1/2𝑇)∫^𝑇_−𝑇𝑒^{−𝑖𝜆𝑡}𝑢(𝑡)𝑑𝑡. It is clear that lim_𝑇→∞𝑢_𝑇=𝑎(𝜆,𝑢), (3.12) and from (3.11), we have 𝐴𝑢_𝑇=12𝑇𝐴^𝑇_−𝑇𝑒^{−𝑖𝜆𝑡}𝑢(𝑡)𝑑𝑡=𝑖𝜆𝐼₃−12𝑇^𝑇_−𝑇𝑒^{−𝑖𝜆𝑡}𝑓(𝑡)𝑑𝑡=𝑖𝜆(𝑢_𝑇−𝐼₁−𝐼₂1)−2𝑇^𝑇_−𝑇𝑒^{−𝑖𝜆𝑡}𝑓(𝑡)𝑑𝑡→𝑖𝜆𝑎(𝜆,𝑢)−𝑎(𝜆,𝑓)as𝑇→∞. (3.13) 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 𝑢_𝑇=(1/2𝑇)∫^𝑇_−𝑇𝑢(𝑠)𝑑𝑠. Then, lim_𝑡→∞𝑢_𝑇=𝑎(0,𝑢) and, using the definition of 𝑢 in (3.2), 𝐴𝑢_𝑇=12𝑇𝐴^𝑇_−𝑇𝑢(𝑠)𝑑𝑠=𝑢(𝑇)−𝑢(−𝑇)−12𝑇2𝑇^𝑇_−𝑇𝑓(𝑠)𝑑𝑠→−𝑎(0,𝑓)as𝑇→∞. (3.14) 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 {𝑥_𝜆}_{𝜆∈𝜎(𝑓)} in 𝐻 satisfying (𝑖𝜆−𝐴)𝑥_𝜆=𝑎(𝜆,𝑓), for which the following holds _{𝜆∈𝜎(𝑓)}‖𝑥_𝜆‖²<∞,_{𝜆∈𝜎(𝑓)}|𝜆|²‖𝑥_𝜆‖²<∞. (3.15)

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 𝑥_𝜆∶=𝑎(𝜆,𝑢) for 𝜆∈𝜎(𝑓). Then it satisfies (𝑖𝜆−𝐴)𝑥_𝜆=𝑎(𝜆,𝑓). Moreover, 𝑖𝜆𝑥_𝜆=𝑎(𝜆,𝑢^′); hence, _{𝜆∈𝜎(𝑓)}‖𝑥_𝜆‖²=‖𝑢‖²_𝐴𝑃,_{𝜆∈𝜎(𝑓)}|𝜆|²‖𝑥_𝜆‖²=‖𝑢^′‖²_𝐴𝑃, (3.16) which imply (3.15).
(ii)⇒(i) Let {𝑥_𝜆}_{𝜆∈𝜎(𝑓)} be a series in 𝐻 satisfying (𝑖𝜆−𝐴)𝑥_𝜆=𝑎(𝜆,𝑓), for which (3.15) holds. Put 𝑓_𝑁(𝑡)∶=_{𝜆∈𝜎(𝑓),|𝜆|<𝑁}𝑒^𝑖𝜆𝑡𝑢𝑎(𝜆,𝑓),_𝑁(𝑡)∶=_{𝜆∈𝜎(𝑓),|𝜆|<𝑁}𝑒^𝑖𝜆𝑡𝑥_𝜆. (3.17) It is then easy to find their norms: ‖𝑢_𝑁‖²=_{𝜆∈𝜎(𝑓),|𝜆|<𝑁}‖𝑥_𝜆‖²,‖𝑢^′_𝑁‖²=_{𝜆∈𝜎(𝑓),|𝜆|<𝑁}|𝜆|²‖𝑥_𝜆‖². (3.18) From (3.15) it implies that 𝑢_𝑁→𝑢 and 𝑢^′_𝑁→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 lim_𝑁→∞𝑢_𝑁=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 𝑢_𝑁 is a classical solution of (3.1), and hence, a mild one, that is, 𝑢_𝑁(𝑡)=𝑢_𝑁(0)+𝐴^𝑡₀𝑢_𝑁(𝑠)𝑑𝑠+^𝑡₀𝑓_𝑁(𝑠)𝑑𝑠. (3.19) For each 𝑡∈ℝ, we have lim_𝑁→∞^𝑡₀𝑓_𝑁(𝑠)𝑑𝑠=^𝑡₀𝑓(𝑠)𝑑𝑠,lim_𝑁→∞^𝑡₀𝑢_𝑁(𝑠)𝑑𝑠=^𝑡₀𝑢(𝑠)𝑑𝑠, (3.20) and, using (3.19), lim_𝑇→∞𝐴^𝑡₀𝑢_𝑁(𝑠)𝑑𝑠=lim_𝑇→∞(𝑢_𝑁(𝑡)−𝑢_𝑁(0)−^𝑡₀𝑓_𝑁(𝑠)𝑑𝑠)=𝑢(𝑡)−𝑢(0)−^𝑡₀𝑓(𝑠)𝑑𝑠. (3.21) Since 𝐴 is a closed operator, we obtain ∫^𝑡₀𝑢(𝑠)𝑑𝑠∈𝐷(𝐴) and 𝐴^𝑡₀𝑢(𝑠)𝑑𝑠=𝑢(𝑡)−𝑢(0)−^𝑡₀𝑓(𝑠)𝑑𝑠, (3.22) 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.(i)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.(ii)For each 𝜆∈𝑀, 𝑖𝜆∈𝜚(𝐴) and sup_𝜆∈𝑀‖(𝑖𝜆−𝐴)⁻¹‖<∞. (3.23)

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)=𝑒^𝑖𝜆𝑡𝑥. 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)+𝑒^𝑖𝜆𝑡𝑥, 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 𝑓_𝑛→𝑓 and 𝐿𝑓_𝑛→𝑢 in 𝑊₂(𝐴𝑃)_|𝑀, where (𝐿𝑓_𝑛)(𝑡)=(𝐿𝑓_𝑛)(0)+𝐴^𝑡₀(𝐿𝑓_𝑛)(𝑠)𝑑𝑠+^𝑡₀𝑓_𝑛(𝑠)𝑑𝑠. (3.24) For each 𝑡∈ℝ, we have lim_𝑛→∞𝐿𝑓_𝑛(𝑡)=𝑢(𝑡), lim_𝑁→∞∫^𝑡₀𝑓_𝑛(𝑠)𝑑𝑠=∫^𝑡₀𝑓(𝑠)𝑑𝑠, and lim_𝑛→∞∫^𝑡₀𝐿𝑓_𝑛(𝑠)𝑑𝑠=∫^𝑡₀𝑢(𝑠)𝑑𝑠. Moreover, from (3.24) we have 𝐴^𝑡₀(𝐿𝑓_𝑛)(𝑠)𝑑𝑠=(𝐿𝑓_𝑛)(𝑡)−(𝐿𝑓_𝑛)(0)−^𝑡₀𝑓_𝑛(𝑠)𝑑𝑠_𝑛→∞−−−−−→𝑢(𝑡)−𝑢(0)−^𝑡₀𝑓(𝑠)𝑑𝑠, (3.25) for each 𝑡∈ℝ. Since 𝐴 is a closed operator, ∫^𝑡₀𝑢(𝑠)𝑑𝑠∈𝐷(𝐴) and 𝐴^𝑡₀𝑢(𝑠)𝑑𝑠=𝑢(𝑡)−𝑢(0)−^𝑡₀𝑓(𝑠)𝑑𝑠, (3.26) 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)=𝑒^𝑖𝜆𝑡𝑥, then 𝑢(𝑡)=𝑒^𝑖𝜆𝑡(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‖, (3.27) which implies ‖(𝑖𝜆−𝐴)⁻¹𝑥‖≼‖𝐿‖⋅‖𝑥‖. (3.28) for any 𝑥∈𝐸 and any 𝜆∈𝑀. Thus, (3.33) holds.
(ii)⇒(i) Suppose 𝑓 is a function in 𝑊₂(𝐴𝑃)_|𝑀. Put 𝑥_𝜆∶=(𝑖𝜆−𝐴)⁻¹𝑎(𝜆,𝑓). Then _{𝜆∈𝜎(𝑓)}‖𝑥_𝜆‖²≼sup_{𝜆∈𝜎(𝑓)}‖(𝑖𝜆−𝐴)⁻¹‖²_{𝜆∈𝜎(𝑓)}‖𝑎(𝜆,𝑓)‖²≼sup_𝜆∈𝑀‖(𝑖𝜆−𝐴)⁻¹‖²‖𝑓‖²<∞,_{𝜆∈𝜎(𝑓)}𝜆²‖𝑥_𝜆‖²≼sup_{𝜆∈𝜎(𝑓)}‖(𝑖𝜆−𝐴)⁻¹‖²_{𝜆∈𝜎(𝑓)}𝜆²‖𝑎(𝜆,𝑓)‖²=sup_𝜆∈𝑀‖(𝑖𝜆−𝐴)⁻¹‖²‖𝑓^′‖²<∞. (3.29) 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 sup_𝜆∈ℝ‖(𝑖𝜆−𝐴)⁻¹‖≤∞. (3.30)
Let now 𝐿₂(0,1) be the Hilbert space of integrable functions 𝑓 from (0,1) to 𝐻 with the norm ‖𝑓‖²_𝐿2(0,1)=¹₀‖𝑓(𝑡)‖²𝑑𝑡<∞. (3.31) If 𝑀={2𝑘𝜋∶𝑘∈ℤ}, then the space 𝑊₂(𝐴𝑃)_|𝑀 becomes 𝑊¹₂(1), the space of all periodic functions 𝑓 of period 1 with 𝑓^′∈𝐿₂(0,1). 𝑊¹₂(1) is then a Hilbert space with the norm ‖𝑓‖²_𝑊12(1)=‖𝑓‖²_𝐿2(0,1)+‖𝑓^′‖²_𝐿2(0,1). (3.32)

Corollary 3.7. Suppose 𝐴 is a closed operator on a Hilbert space 𝐻. The following are equivalent.
(i) For each function 𝑓∈𝑊¹₂(1), (3.1) has a unique 1-periodic mild solution in 𝑊¹₂(1).
(ii) For each 𝑘∈ℤ, 2𝑘𝑖𝜋∈𝜚(𝐴) and sup_𝑘∈ℤ‖(2𝑘𝑖𝜋−𝐴)⁻¹‖<∞. (3.33)

4. Application: A 𝐂_𝟎-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)+^𝑡_𝑠𝑇(𝑡−𝜏)𝑓(𝜏)𝑑𝜏, (4.1) 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)+^𝑡₀𝑇(𝑡−𝑠)𝑓(𝑠)𝑑𝑠. (4.2) 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∈𝑊¹₂(1), (3.1) has a unique 1-periodic classical solution.
(iii) For each function 𝑓∈𝑊¹₂(1), (3.1) has a unique 1-periodic solution contained in 𝑊¹₂(1).
(iv) For each 𝑘∈ℤ, 2𝑘𝜋𝑖∈𝜚(𝐴) and sup_𝑘∈ℤ‖(2𝑘𝜋𝑖−𝐴)⁻¹‖<∞. (4.3)
(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 𝑊¹₂(1) and let 𝑢(𝑡) be the unique mild solution of (3.1), which is in 𝑊¹₂(1). Since for each 𝑓∈𝑊¹₂(1), the function 𝑔(𝑡)∶=∫^𝑡₀𝑇(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)−∫^𝑡₀𝑇(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.