(ω, c)- Pseudo almost periodic distributions

The theory of uniformly almost periodic functions was introduced and studied by H. Bohr in the beginning of the last century [6], since then, many authors contributed to the development of this theory in di erent directions, see [5], [8] and [9]. The concept of (w, c)−almost periodicity introduced in [10] is a generalization of (w, c)−periodicity whichmotivated by someknown results regarding the qualitative properties of solutions to theMathieu linear second-order di erential equation


Introduction
The theory of uniformly almost periodic functions was introduced and studied by H. Bohr in the beginning of the last century [6], since then, many authors contributed to the development of this theory in di erent directions, see [5], [8] and [9].
The concept of (w, c) −almost periodicity introduced in [10] is a generalization of (w, c) −periodicity which motivated by some known results regarding the qualitative properties of solutions to the Mathieu linear second-order di erential equation y ′′ (t) + [a − q cos t] y (t) = , arising in seasonally forced population dynamics, see [1].
In [14], C. Zhang introduced an extension of the almost periodic functions, the so-called pseudo almost periodic functions. These pseudo almost periodic functions are related to many applications in the theory of di erential equations.
The classes of (w, c) −pseudo almost periodic functions were introduced and studied in [11], by taking into consideration the concept of pseudo ergodicity introduced by C. Zhang [14]. These classes generalizes the concept of (w, c) −pseudo periodicity introduced by Alvarez, Gömez and Pinto in [2].
The theory of Sobolev-Schwartz distributions is a very powerful tool in mathematics and their applications. Almost periodic distributions extending the classical Bohr and Stepano almost periodic functions are due to L. Schwartz, see [13]. The paper [12] introduce and investigate the (w, c) −almost periodicity in the setting of Sobolev-Schwartz distributions.
As mentioned in the abstract, the main aim of this paper is to introduce a new space of (w, c)−pseudo almost periodic distributions containing (w, c)−pseudo almost periodic functions as well as (w, c)−almost periodic Schwartz distributions. The paper is brie y described as follows: In section , we recall the basic de nitions and results of the concept of (w, c)−pseudo almost periodic functions. The main results of this work are given in the section 3 and 4. In the rst one, we introduce and study the (w, c)−pseudo almost periodicity in the setting of Sobolev-Schwartz distributions, by recalling some new basic spaces of functions and distributions in which we can study this concept of (w, c)−pseudo almost periodicity. The second one is devoted to some interesting applications to linear di erential equations, we propose results about the existence of distributional (w, c)−pseudo almost periodic solutions of linear di erential systems.

(w, c)−Pseudo almost periodic functions
Unless speci ed otherwise, throughout the paper, we will assume that c ∈ C\{ }, w > and we will use the following notations: and where the equality is taken in the usual (resp. Lebesgue, distributional) sense. First, let us recall C b , · L ∞ the Banach algebra of bounded and continuous complex valued functions on R endowed with the norm · L ∞ of uniform convergence on R. Denote by AP the well-known space of Bohr almost periodic functions on R; is the closed subalgebra of C b , · L ∞ that contains all the continuous functions f : R −→ C, satisfying: the following structural property: For any ε > , the set of ε-almost periods of f , de ned by We recall also the space APw,c of (w, c)− almost periodic functions which has been introduced in [ When c = and w > arbitrary, we obtain APw,c := AP. The space APw,c is a vector space together with the usual operations of addition and pointwise multiplication with scalars. Some properties of (w, c) −almost periodic functions are summarized in the following proposition.
Proof. See [10]. Now, we recall the space of pseudo almost periodic functions, introduced by C. Zhang, see [14] and [15]. Set De nition 2. A function f ∈ C b is called pseudo almost periodic if it can be expressed as where g ∈ AP and h ∈ PAP .
Denote by PAP the set of all such functions. The decomposition ( . ) is unique, so, the functions g and h are respectively called the almost periodic component and the ergodic perturbation of the pseudo almost periodic function f . We have The (w, c) −pseudo almost periodicity of continuous functions and their Stepanov generalizations is recently introduced by M. T. Khalladi, M. Kostic, A. Rahmani and D. Velinov, see [11]. We recall the de nition and some basic properties of (w, c) −pseudo almost periodic functions. For the proofs and more details see [10] and [11].
Consider the (w, c)−mean of a function h : R −→ C, given by whenever the limit exists. Let us de ne the space A function h is said to be c−ergodic if and only if belongs to this space, i.e., if and only if c − (·) w h (·) ∈ PAP . Therefore, the ergodic space of C. Zhang [12] can be recovered by plugging c = in the above de nition.

De nition 3.
A function f ∈ C is said to be (w, c) −pseudo almost periodic if it can be expressed as where g ∈ APw,c and h ∈ PAP ;w,c . The space of all such functions will be denoted by PAPw,c .
We can easily show that the decomposition ( . ) is unique.

(w, c)−Pseudo almost periodic distributions
In this section, we introduce and study a new generalization of Schwartz (w, c)−almost periodic distributions introduced in [12]. To this end, we rst recall that the L p −distributions denoted by D ′ L p are introduced by L. Schwartz in [13], further developed by J. Barros-Neto in [3]. The space D ′ L p , < p ≤ +∞, is de ned to be the topological dual of the di erential Fréchet space endowed with the topology de ned by the countable family of norms The topological dual of D L , denoted by D ′ L ∞ , is called the space of bounded distributions. The space is endowed with the topology de ned by the countable family of norms The space is the closure of the space of test functions D in the topology de ned by |·| k,∞ .
The topological dual of · D L ∞ , denoted by D ′ L , is called the space of integrable distributions. We recall the following characterizations of L p −distributions, ≤ p ≤ +∞.
L. Schwartz has also introduced the space B ′ ap of almost periodic distributions. This space is based on the topological de nition of Bochner's almost periodic functions. Let h ∈ R and T ∈ D ′ , the translated of T by h, denoted by τ h T, is de ned as: The de nition and characterizations of Schwartz almost periodic distributions are given in the following result.

Theorem 3. For any bounded distribution T ∈ D
′ L ∞ , the following statements are equivalent: Proof. See [13].
The authors in [12] have introduced and studied a new weighted spaces of distributions D ′ L p w,c and their test To recall the concept of (w, c) −almost periodicity in the setting of Sobolev-Schwartz distributions, we need to present the space D L w,c and their dual space. Let q = , ∞ We denote by L q w,c , the set of (w, c) −Lebesgue functions with exponent q, i.e.
When c = , L q w,c = L q the classical Lebesgue space over R.
endowed with the topology de ned by the following countable family of norms Proof. Since D is dense in the space Cc of continuous functions with compact support it su ces to show that Cc is dense in L w,c . Let S be the set of all simple measurable functions s, with complex values, de ned on R and such that mes {t : First it is clear that S is dense in L w,c . Indeed, as c − t w s ∈ L , then S ⊂ L w,c . Suppose f ∈ L w,c is positive and de ne the sequence (sn) n such that ≤ s ≤ s ≤ ... ≤ f and for all t ∈ R, sn (t) −→ f (t) when n −→ +∞.
On the other hand, by Lusin's theorem, for s ∈ S and ε > , there exists g ∈ Cc such that g (t) = s (t) , except on a set of measure less than ε, and |g| ≤ s ∞ , and since s takes only a nite number of values, there exists a constant C > which depends on c and w such that The density of S in L w,c completes the proof.
Taking into account the notation ( . ) , we have the following consequence of Theorem 4. Proof. From the equivalence (i) ⇐⇒ (ii) of Theorem 4 and Theorem 2, we have The concept of (w, c) −almost periodicity of Schwartz distributions is given by the following Characterizations of (w, c) −almost periodic distributions are given in the following result.
Returning to the notation ( . ), we have the following proposition.
We recall also the following space of smooth (w, c) −almost periodic functions is the space of smooth almost periodic functions introduced by L. Schwartz. We endow B APw,c with the topology induced by D L ∞ w,c . The main properties of B APw,c and B ′ APw,c are summarized in the following Proof. See [12].
We recall that a sequence T j j converges to zero in D ′ L ∞ if τ −h T j j converges to in D ′ uniformly in h ∈ R. We have the following properties of B ′ APw,c .

APw,c if and only if for every sequence c j j ⊂ R there exists a subsequence b j j of c j j such that
Proof. (i) It follows from the relation and the equivalence (i) ⇐⇒ (ii) of Theorem 5.
(ii) Suppose that S = . Then Therefore, ∀ε > , there exists jε ∈ Z+ such that i.e. Tw,c , φ = , ∀φ ∈ D, thus by Lemma 1, we obtain T = , which is in contradiction with the hypothesis. The last part of (ii) , follows straightforward from inequality and sup Now, we recall the space B ′ PAP of pseudo almost periodic distributions. The space of bounded distributions with mean value vanishing at in nity is denoted and de ned by

De nition 6. A distribution T ∈ D
′ L ∞ is said to be a pseudo almost periodic, if there are R ∈ B ′ ap and S ∈ B ′ such that T = R + S.
The decomposition of a pseudo almost periodic distribution is unique. In addition, we have the following characterization of pseudo almost periodic distributions. Theorem 7. Let T ∈ D ′ L ∞ , the following statements are equivalent : Proof. It follows by using the same arguments as in [7], Theorem 1 and its proof by replacing the space B ′ of bounded distributions with mean value vanishing at in nity instead of the space B ′+ of bounded distributions tending to zero at in nity.
It was shown in [12], Theorem 3.4 that, if T ∈ B ′ APw,c ,Tw,c * φ ∈ AP,∀φ ∈ D, thus Some properties, easy to prove, of B ′ ;w,c distributions are given in the following. Proof. It follows from the de nition of (w, c) −almost periodic distributions and Proposition 6-(i) .
The following result gives a characterization of (w, c) −pseudo almost periodic distributions.

Theorem 8. Let T ∈ B
′ w,c , the following statements are equivalent : Proof. According to Proposition 8, a distribution T ∈ B Another characterization of (w, c) −pseudo almost periodic distributions is given by the following result.
and the fact that τ −b j φψ, b j ∈ R, j ∈ Z+ is a bounded set in D.
(ii) It is clear that which gives lim

Application to linear di erential systems
In this section we will apply the above theoritical results to study the distributional (w, c) −pseudo almost periodic solutions of some linear di erential equations. Let us rst consider the linear system of di erential equations with constant coe cients where A = a ij ≤i,j≤k , k ∈ N, is a given square matrix of complex numbers, W = (W i ) ≤i≤k ∈ D ′ k is a vector distribution and X = (X i ) ≤i≤k is the unknown vector distribution.
Let us consider the system ( . ) with W ∈ B ′ APw,c k and recall the following main result obtained in [12]. Proof. See [12].
The following result gives the (w, c) −pseudo almost periodicity of the (w, c) −bounded solution of system ( . ).  Proof. Since the su cient condition is trivial, it su es to prove that the given condition is necessary. Let c j j ⊂ R such that lim By the Proposition , using the limit in D ′ when j −→ +∞ in the equation

Theorem 11. Assume the matrix A is such that
we obtain U ′ = gU + V , and by Property (ii) of Theorem 6, we have P ′ = fP + Y .
Thanks to the linearity of equation ( . ), it follows that Q ′ = fQ + Z.