Exponential Dichotomy and Eberlein-Weak Almost Periodic Solutions *

We give sufficient conditions ensuring the existence and uniqueness of an Eberlein-weakly almost periodic solution to the following linear equation         d , d x t A t x t f t t   in a Banach space X, where is a family of infinitesimal generators such that for all ,    t A t  t     A t T A t   for some for which the homogeneuous linear equation 0, T        d d x t A t x t t  is well posed, stable and has an exponential dichotomy, and : f X   is Eberlein-weakly amost periodic.

in a Banach space X, where is a family of infinitesimal generators such that for all , for some for which the homogeneuous linear equation 0, A t x t t  is well posed, stable and has an exponential dichotomy, and : f X   is Eberlein-weakly amost periodic.

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 : for x X  , where X is a complex Banach space,   A t is (unbounded) linear operator acting on X for every fixed such that for all t , t    

    A t T   :
A t 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 genera-    t A t  tors 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 , ,

U t s x A t U t s x t U t s x U t s A s x s
, .
The corresponding homogeneous equation of (1) has an exponential dichotomy, i.e., there exist a family of projections  , Q t t   and positive constants , M  such that the following conditions are satisfied : 1) For every fixed x X  the map The problem of the existence of almost periodic solutions has been extensively studies in the literature [1][2][3][4][5][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][9][10][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][13][14][15][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.

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.
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 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 .
  will denote the set of Eberlein-weakly almost periodic X valued functions.
Theorem 2.3 Equipped with the norm 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 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: 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.

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 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) 3) lim and here 1) and 2) can be obtained by a diagonalization argument.Since

 
f J is separable, we may assume that 3) holds.
Let then by the uniform continuity of f, we find lim ; Again by uniform continuity of f, and by the choice of subsequences we find for the interchanged limits.
By hypothesis, we have is periodic, thus satisfies the double limits condition.Let be the double limit.
. that for all there is an such that

 
, and according to the previous observation, there exists an such that for all we find an with 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 : 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 In fact, for any we have On the other hand, we have 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.
which ends the claim.Now, to complete the proof, it remains for us to prove that f x is Eberlein-weakly almost periodic.By Ruess and Summers's double limits criterion, we have to verify that for given sequences Since f x 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: Bringing the last estimate into play we obtain are Eberlein weakly almost periodic, we may assume that

Proposition 2 . 5
For every Eberlein weakly almost periodic function f there exists a sequence   n n t  .such that if g is the almost periodic part n t f g 

From
We propose to extend the above theorem to the case where f is Eberlein-weakly almost periodic.  We first prove that the set , :  is not sufficient even if additional algebraic structure is given. d 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, The authors would like to thank the I. R. D. (Institut d