The aim of this work is to investigate the existence and uniqueness of a weakly almost periodic solution in the sense of Eberlein for the following linear equation :

for, where X is a complex Banach space, is (unbounded) linear operator acting on X for every fixed such that for all, for some, and the input function is weakly almost periodic in the sense of Eberlein (Eberlein-weakly almost periodic). In the sequel, we essentially assume that:

is a family of infinitesimal generators for which the corresponding homogeneous equation of (1) is well posed and stable in the following sense: there exists a T-periodic strongly continuous evolutionary process, which is uniformly bounded and strongly continuous such in particular that:

for all

and

Further, if is given and, then and

We also assume

The corresponding homogeneous equation of (1) has an exponential dichotomy, i.e., there exist a family of projections and positive constants such that the following conditions are satisfied :

1) For every fixed the map is continuous and T-periodic

2)

3) where for all and

4)

5) is an isomorphism from onto,

The problem of the existence of almost periodic solutions has been extensively studies in the literature [1-6]. Eberlein-weak almost periodic functions are more general than almost periodic functions and they were introduced by Eberlein [7], for more details about this topics we refer to [8-11] where the authors gave an important overview about the theory of Eberlein weak almost periodic functions and their applications to differential equations. In the literature, many works are devoted to the existence of almost periodic and pseudo almost periodic solutions for differential equations (a pseudo almost periodic function is the sum of an almost periodic function and of an ergodic perturbation), but results about Eberlein weak almost periodic solutions are rare [7,12-16].

In ([17], Chap. 3) the authors investigate the existence and uniqueness of an almost periodic solution for equation (1) when the corresponding homogeneous equation of (1) has an exponential dichotomy and the function f is almost periodic. In ([17], Chap. 3) the authors showed that, if the corresponding homogeneous equation of (1) has an exponential dichotomy and the function f is almost periodic, the equation (1) has a unique bounded integral solution on which is also almost periodic. Here we propose to extend the result in [17] to the Eberleinweakly almost periodic case.

In the sequel, we give some properties about weak almost periodic functions in the sense of Eberlein (Eberlein-weak almost periodic functions).

Let X and Y be two Banach spaces. Denote by the space of all continuous functions from X to Y. Let be the space of all bounded and continuous functions from to X, equipped with the norm of uniform topology.

Definition 2.1 A bounded continuous function is said to be almost periodic, if the orbit of x, the set of translates of x:

is a relatively compact set in with respect to the supremum norm.

We denote these functions by

Definition 2.2 A function, for is said to be weakly almost periodic in the sense of Eberlein (Eberlein-weakly almost periodic) if the orbit of x with respect to J:

is relatively compact with respect to the weak topology of the sup-normed Banach space.

For the sequel, will denote the set of Eberlein-weakly almost periodic X valued functions.

Theorem 2.3 Equipped with the norm

the vector space is a Banach space.

In [18,19] Deleeuw and Glicksberg proved that if we consider the subspace of those Eberlein weakly almost periodic functions, which contain zero in the weak closure of the orbit (weak topology of), i.e.;

the following decomposition

holds. Moreover, if, , and with then

uniformly in.

For a more detailed information about the decomposition and the ergodic result we refer to the book of Krengel [20,21].

In order to prove the weak compactness of the translates, Ruess and Summers extended the double limits criterion of Grothendieck [22] to the following:

Proposition 2.4 A subset is relatively weakly compact if and only if

1) H is bounded in, and

2) for all and the following double limits condition holds:

whenever the iterated limits exist.

This result will be the main tool in verifying weak almost periodicity. For the other task we will use.

Proposition 2.5 For every Eberlein weakly almost periodic function f there exists a sequence such that if g is the almost periodic part

In this section, we state a result of the existence and uniqueness of an Eberlein-weakly almost periodic solution of the Periodic Inhomogeneous Linear Equation (1). The existence and uniqueness of an almost periodic and bounded solution has been studied by M. N’Guérékata ([17]). More precisely, the author proved the following result.

Theorem 3.1 ([17]) Assume that and hold. If the function f is continuous and bounded on then Equation (1) has a unique bounded mild solution on Moreover, if f is almost periodic, then is almost periodic.

We propose to extend the above theorem to the case where f is Eberlein-weakly almost periodic.

Theorem 3.2 Assume that and hold. If the function f is Eberlein-weakly almost periodic with a relatively compact range, then Equation (1) has a unique bounded mild solution on which is Eberleinweakly almost periodic.

For the proof of theorem (3.2), we use the following lemmas.

Lemma 3.3 Let be a bounded uniformly continuous function with relatively compact range, , and If , or is bounded, then one has

whenever the iterated limits exist.

Proof. Noting that only the equality of the iterated limits has to be proved, we may pass to subsequences. Therefore we assume that the following limits exists

1)

2)

3)

4)

here 1) and 2) can be obtained by a diagonalization argument. Since is separable, we may assume that 3) holds.

Let then by the uniform continuity of f, we find

and

Again by uniform continuity of f, and by the choice of subsequences we find for the interchanged limits.

Lemma 3.4  Let such that for a subcompact set  

and

Then

Proof. We first prove that the set is weakly relatively compact in Thus, for given sequences we have to verify the following identity :

whenever the iterated limits exist. Since for all as a consequence of the metric weak compactness of K, we may pass to subsequences of and such that the iterated limits of exist in X, without loss of generality the sequences are chosen in this way. The characterization of weak compactness gives,

Since (the convergence holds in norm), hence one will obtain that is weakly relatively compact in Using the fact that :

a standard trick of topology gives

Lemma 3.5 Let, for a Banach space (the space of all bounded linear operators acting on X) and periodic. Then, for any given Eberlein-weakly almost periodic with a relatively compact range,

Proof. In Order to prove that is Eberlein-weakly almost periodic, by W. M. Ruess and W. H. Summers’s criterion (2.4), we have to verify that for given sequences and

whenever the iterated limits exist. Assuming that the iterated limits exist and by the fact that we only have to prove the equality of them, we may pass to subsequences for the verification. Since g is Eberlein weakly almost periodic with a relatively compact range, by a use of a diagonalization routine, we may assume that

for a suitable choice of subsequences and. We define

By hypothesis, we have is periodic, thus satisfies the double limits condition. Let be the double limit.

Now,

From the convergence of andwe derive that for every there exists an such that for there exist an such that

Using the double limits condition of the sequence, for given there exists such that for all there is an such that

for all Applying the continuity of the map for we find a and according to the previous observation, there exists an such that for all we find an with

This yields, by a standard estimate, that and hence

The following example shows that the compactness assumption on the range of g is essential and that the periodicity of is not sufficient even if additional algebraic structure is given.

Example 3.6 We let and choose

and

Further, if denotes the indicator function for the set A, we choose

Using Lemma 2. 16 in ([13]), we obtain that g is Eberlein weakly almost periodic. Now, for the sequences

some calculations lead to the identity :

hence is not uniformly continuous, hence not Eberlein weakly almost periodic.

Proof. (of Theorem 3.2) Since f is Eberlein-weakly almost periodic, then f is continuous and bounded on. The existence and uniqueness of the bounded mild solution on result of theorem (3.1).

We claim that

In fact, for any we have

On the other hand, we have

which ends the claim.

Now, to complete the proof, it remains for us to prove that is Eberlein-weakly almost periodic. By Ruess and Summers’s double limits criterion, we have to verify that for given sequences

and

whenever the iterated limits exist. Assuming that the iterated limits exist and by the fact that we only have to prove the equality of them, we may pass to subsequences.

Since is uniformly continuous, by Lemma (3.3), we may assume that and Furthermore without loss of generality otherwise we have and by going over to subsequence the uniform continuity gives us that the double limits for these both sequences coincide. Bringing the equality (2) into play we obtain:

Since we obtain:

thus,

where

and

Since by Lemma (3.5)

and

are Eberlein weakly almost periodic, we may assume that

and

Bringing the last estimate into play we obtain

Thus,

The uniform boundedness of the sequences of linear functional

and

and the fact that Lemma (3.4) applies to

By going to appropriate subsequences, we can assume that the iterated double limits for (resp. for) exist. Since they have to coincide, they have to be zero. By the triangle inequality we find,

Starting with, then, and at last, we obtain

which concludes the proof.

The authors would like to thank the I. R. D. (Institut de Recherche pour le Développement , UMI 209) for its hospitality and support.