Exponential Dichotomy and Eberlein-Weak Almost Periodic Solutions ()
in a Banach space X, where 
is a family of infinitesimal generators such that for all
, 
for some 
for which the homogeneuous linear equation 
is well posed, stable and has an exponential dichotomy, and 
is Eberlein-weakly amost periodic.
1. Introduction
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 :
(1)
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.
2. Eberlein-Weak Almost Periodic Functions
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 
3. Statement of the Main Result
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 
and
we 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
(2)
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.
4. Acknowledgements
The authors would like to thank the I. R. D. (Institut de Recherche pour le Développement , UMI 209) for its hospitality and support.
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