#### Abstract

For the differential equation , on a Hilbert space , 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 where is a linear, closed operator on a Hilbert space and is a function from to . 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 is a linear, unbounded operator on a Hilbert space . We use the Hilbert space introduced in [4], defined by what follows. Let be the inner product of and let be the space of all almost periodic functions from to . The completion of is then a Hilbert space with the inner product defined by First, we establish the relationship between the Bohr transforms of the almost periodic solutions of (1.1) and those of the inhomogeneity . We then give a necessary and sufficient condition so that (1.1) admits a unique almost periodic solution for each almost periodic inhomogeneity . As applications, in Section 4 we show a short proof of the Gearhart's theorem. If is generator of a strongly continuous semigroup , then if and only if and .

#### 2. The Hilbert Space of Almost Periodic Functions

Let us fix some notations. Recall that a bounded, uniformly continuous function from to a Banach space is almost periodic, if the set is relatively compact in , the space of bounded uniformly continuous functions with sup-norm topology. Let be now a Hilbert space with and let be the inner product and the norm in , respectively. Let be the space of all almost periodic functions from to . In the following expression exists and defines an inner product. Hence, is a pre-Hilbert space and its completion, denoted by , is a Hilbert space. The inner product and the norm in are denoted by and , respectively.

For each function , the Bohr transform is defined by The set is called the Bohr spectrum of . It is well known that is countable for each function and the Fourier-Bohr series of is and it converges to in the norm topology of . The following Parseval's equality also holds For more information about the almost periodic functions and properties of the Hilbert space , we refer readers to [2, 4].

Let be the space consisting of all almost periodic functions , such that . is then a Hilbert space with the norm Note that the -topology is stronger than the sup-norm topology (see [6]). We will use the following lemma.

Lemma 2.1. *If is a function in and , then we have
*

*Proof. *If , using the integration by part we have
Let , and note that is bounded, we have (2.7).

If , then
which also satisfies (2.7).

Finally, for a linear, closed operator in a Hilbert space , we denote the domain, the range, the spectrum, and the resolvent set of by , , and , respectively.

#### 3. Almost Periodic Mild Solutions of Differential Equations

We now turn to the differential equation First we define two types of solutions to (3.1).

*Definition 3.1. *(1) A continuous function is called a mild solution of (3.1) if
for all .

(2) A function is a classical solution of (3.1), if , is continuously differentiable, and (3.1) holds for .

*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 is the generator of a semigroup , then a continuous function is a mild solution of (1.1) if and only if it has the form (see [7])
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 .

Proposition 3.3. *Suppose and is an almost periodic mild solution of (3.1). Then
**
for every .*

*Proof. *Suppose is a nonzero real number. Multiplying each side of (3.2) with and taking definite integral from to on both sides, we have
Here we used the fact that for a closed operator . It is easy to see that
and, applying integration by part for any integrable function , we have
Using (3.7) for and in (3.5), respectively, we have
where
as ;
as , and
Let . It is clear that
and from (3.11), we have
Since is a closed operator, from (3.12) and (3.13) we obtain and , from which (3.4) is followed.

Finally, if , let . Then, and, using the definition of in (3.2),
Again, since is a closed operator, it implies and , 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 is an almost periodic function, which is in . Then the following statements are equivalent.**(i) Equation (3.1) has an almost periodic mild solution, which is in **(ii) For every , and there exists a series in satisfying , for which the following holds
*

*Proof. *(i)(ii) Let be an almost periodic solution to (3.1), which is in . By Proposition 3.3, . Hence for all .

Put now for . Then it satisfies . Moreover, ; hence,
which imply (3.15).

(ii)(i) Let be a series in satisfying , for which (3.15) holds. Put
It is then easy to find their norms:
From (3.15) it implies that and as for some function and in the topology of . Since the differential operator is closed, we obtain , and in the topology of . Hence, is almost periodic. It remains to show that is a mild solution of (1.1). In order to do that, note is a classical solution of (3.1), and hence, a mild one, that is,
For each , we have
and, using (3.19),
Since is a closed operator, we obtain and
which shows that 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 , (3.1) has a unique almost periodic mild solution. We are now in the position to state the main result.

Theorem 3.5. *Suppose is a closed operator on a Hilbert space and is a closed subset of . The following are equivalent.*(i)*For each function with , (3.1) has a unique almost periodic mild solution in with .*(ii)*For each , and
*

*Proof. *(i)(ii) Let be the subspace of all functions in with . Then is a Hilbert space by nature. Let be any vector in , let be a number in and let . Then and hence, (3.1) has a unique almost periodic solution . By Theorem 3.4, , hence is surjective for all . On the other hand, is injective; otherwise, , where is a nonzero vector in satisfying , would be another almost periodic mild solution to (3.1) with . Hence is bijective and for all .

In we define the operator by what follows. For each , is the unique almost periodic mild solution to (1.1) corresponding to . By the assumption, is everywhere defined. We will prove that is a bounded operator by showing is closed in . Let and in , where
For each , we have , and . Moreover, from (3.24) we have
for each . Since is a closed operator, and
which means is a mild solution to (3.1) corresponding to . Thus, , and hence, is closed.

Next, for any and , put , then is the unique almost periodic solution to (3.1), that is, . Using the boundedness of operator , we obtain
which implies
for any and any . Thus, (3.33) holds.

(ii)(i) Suppose is a function in . Put . Then
By Proposition 3.3, (3.1) has an almost periodic mild solution in . That solution is unique, since its Bohr transforms are uniquely determined by for all .

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

Corollary 3.6. *Suppose is a closed operator on a Hilbert space . The following are equivalent.**(i) For each function , (3.1) has a unique 1-periodic mild solution in .**(ii) and
**Let now be the Hilbert space of integrable functions from to with the norm
**
If , then the space becomes , the space of all periodic functions of period with . is then a Hilbert space with the norm
*

Corollary 3.7. *Suppose is a closed operator on a Hilbert space . The following are equivalent.**(i) For each function , (3.1) has a unique 1-periodic mild solution in .**(ii) For each , and
*

#### 4. Application: A -Semigroup Case

If generates a -semigroup , then (see [7, Theorem 2.5]), mild solutions of (3.1) can be expressed by for . If is a 1-periodic function, then it is easy to see that the above solution is 1-periodic if and only if . Hence, to consider 1-periodic solution, it suffices to consider in and in this interval we have 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 generate a -semigroup on a Hilbert , then the following are equivalent.**(i) For each function , (3.1) has a unique 1-periodic mild solution.**(ii) For each function , (3.1) has a unique 1-periodic classical solution.**(iii) For each function , (3.1) has a unique 1-periodic solution contained in .**(iv) For each , and
**(v) .*

*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 be any function in and let be the unique mild solution of (3.1), which is in . Since for each , the function is continuously differentiable and for all (see [9]), to show is a classical solution, it suffices to show .

From the above observation and from formula (4.2), the function is differentiable almost everywhere on . It follows that for almost everywhere (since is differentiable at if and only if ). Hence, . By formula (4.2), , and thus, belongs to . 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.