James Larrouy
email: james.larrouy@univ-antilles.fr

Gaston M N'guérékata
email: nguerekata@aol.com

Measure (ω, c)-pseudo almost periodic functions and Lasota-Wazewska model with ergodic and unbounded oscillating oxygen demand

Keywords: Periodic oscillations, (ω, c)-periodic, Unboundedness, Lasota-Wazewska model, Composition Theorem AMS 2020 Subject Classification: 34C27, 34K14, 44A35, 47D06, 92-10

The primary aim of this work is to introduce a new class of functions called µ-(ω, c)-pseudo almost periodic functions. Using the measure theory, we generalize in a natural way some recent works and study some properties of those µ-(ω, c)-pseudo almost periodic functions including two new composition results which play a crucial role for the existence of some µ-(ω, c)-pseudo almost periodic solutions of certain semilinear differential equations and partial differential equations. We also investigate the existence and uniqueness of the µ-(ω, c)pseudo almost periodic solutions for some models of Lasota-Wazewska equation with measure (ω, c)-pseudo almost periodic coefficient and mixed delays.

Introduction

Most of the natural phenomena we consider as periodic are in fact almost periodic, in other words, they are periodic up to epsilon. The concept of almost periodic functions was introduced in the literature in the mid-1920s by the Danish mathematician Harald Bohr [START_REF] Bohr | Almost Periodic Functions[END_REF]. It was later generalized in various directions by many may researchers [1-3, 7, 8, 10, 12, 14, 18, 24, 25]. As we all know, many phenomena in nature have oscillatory character and their mathematical models have led to the introduction of certain classes of functions to describe them. Such a class form pseudo almost periodic functions which is a natural generalization of the concept of almost periodicity (in Bohr's sense). In this work we introduce the notion of measure (ω, c)pseudo almost periodic functions (or µ-(ω, c)-pseudo almost periodic functions) with values in a complex Banach space and enlighten their applications throughout the study of a biological model. This work generalizes the concept of µ-Pseudo almost periodic functions introduced by Blot et al. [START_REF] Blot | New approach for weighted pseudo-almost periodic functions under the light of measure theory, basic results and applications[END_REF] which already generalizes the class of weighted pseudo almost periodic functions of Diagana [START_REF] Diagana | Weighted pseudo-almost periodic functions and applications[END_REF][START_REF] Diagana | Weighted pseudo-almost periodic solutions to some differential equations[END_REF]. Here, we investigate many interesting properties of this new class of functions and present new and more general results based on measure theory that extend the existing ones.

The concept of (ω, c)-periodicity was introduced by Alvarez et al. [START_REF] Alvarez | ω, c)-periodic functions and mild solutions to abstract fractional integro-differential equations[END_REF] motivated by the qualitative properties of solutions to the Mathieu linear second-order differential equation y ′′ (t) + [a -2q cos(2t)] y(t) = 0, arising in seasonally forced population dynamics. Further on, Alvarez et al. proposed a new concept of (ω, c)-pseudo periodicity and proved the existence of positive (ω, c)-pseudo periodic solutions to the Lasota-Wazewska equation with (ω, c)-pseudo periodic coefficients y ′ (t) = -δy(t) + h(t)e -a(t)y(t-τ ) , t ≥ 0. This equation describes the survival of red blood cells in the blood of an animal. The works of Khalladi et al. [START_REF] Khalladi | Pseudo almost periodic functions, (ω,c)-pseudo almost automorphic functions and applications[END_REF] have shown that (ω, c)-pseudo periodic functions can be also solutions time varying impulsive differential equations and linear delayed equations.

First, pseudo-almost periodicity was introduced in the literature in the early nineties by Zhang [START_REF] Zhang | Integration of vector-valued pseudo-almost periodic functions[END_REF][START_REF] Zhang | Pseudo almost periodic solutions of some differential equations[END_REF][START_REF] Zhang | Pseudo almost periodic solutions of some differential equations II[END_REF], as a natural generalization of the classical almost periodicity in the sense of Bohr. Then, Diagana [START_REF] Diagana | Weighted pseudo-almost periodic functions and applications[END_REF][START_REF] Diagana | Weighted pseudo-almost periodic solutions to some differential equations[END_REF], introduced the concept of weighted pseudo almost periodicity which generalizes the latter and the author gave some properties of the space of weighted pseudo almost periodic functions such as the completeness and a composition theorem. The concept of weighted pseudo almost periodic functions became an interesting field of dynamical systems that attracted many authors. A few years later, Blot et al. [START_REF] Blot | New approach for weighted pseudo-almost periodic functions under the light of measure theory, basic results and applications[END_REF] came up with a new concept of weighted pseudo almost periodic functions under the light of measure theory. Giving a positive measure µ on R, they defined the concept of µ-pseudo almost periodic functions as follows : it is said that a function f is µ pseudo almost periodic if

f = g + φ
where g is almost periodic and φ is µ-ergodic in the sense that lim r→∞ 1 µ([-r, r]) [-r,r] ∥φ(t)∥dµ(t) = 0 where µ([-r, r]) is the measure of the set [-r, r]. Here, the classical theory of weighted pseudo almost periodicity became a particular case of Blot et al. approach. Indeed, one can observe that a weighted pseudo almost periodic function of weight ρ is µ-pseudo almost periodic where the measure µ is absolutely continuous with respect to the Lebesgue measure, and its Radon-Nikodym derivative is ρ : dµ(t) = ρ(t)dt.

In their work, Blot et al. have investigated many important results on the theory of µ-pseudo almost periodicity ; they studied the completeness and provided a composition theorem on the functional space of µ-pseudo almost periodic functions. They also gave some applications for evolution equations which inclued reaction-diffusion systems and partial differential equations.

In this work, we introduce a new class of µ-(ω, c)-ergodic components and we investigate many important results on the new theory of µ-(ω, c)-pseudo almost periodic functions. We study the completeness and the composition theorem on the functional space of µ-(ω, c)-pseudo almost periodic functions.

The organization of this work is as follows : in the next section, we recall the basic definitions and properties of µ-pseudo almost periodic functions. In Section 3, we give the new concept of µ-(ω, c)-pseudo almost periodicity and study the convolution product on the spaces of c-bounded functions, µ-(ω, c)-ergodic functions and µ-(ω, c)-pseudo almost periodic functions. In Section 4, we introduce the concept of (ω, c)-type compactness and then we study a composition theorem which plays a crucial role to study the existence of µ-(ω, c)-pseudo almost periodic solution for a perturbed semilinear system. In Section 5, we propose a more realistic Lasota-Wazewska model than the existing ones due to (ω, c)-periodicity and then we study the existence and uniqueness of µ-(ω, c)-pseudo almost periodic solutions for the model, using the completeness and composition results.

Terminology and definitions

In this section we review a few notations, definitions and lemmas which will be utilized throughout this paper.

Let (X, ∥ • ∥), (Y, ∥ • ∥) be complex Banach spaces. Throughout this work, C(R, X) and BC(R, X) (respectively C(R × Y, X) and BC(R × Y, X)) denotes the Banach spaces consisting of all continuous functions and all bounded continuous functions from R to X (respectively from R × Y to X) equipped with the supremum norm

∥f ∥ ∞ = sup t∈R ∥f ∥
Let's first recall the notion of (ω, c)-periodicity.

Definition 2.1 ( [1]). Let ω > 0 and c ∈ C \ {0}. A function f ∈ C(R, X) is said to be (ω, c)-periodic if f (t + ω) = cf (t), for each t ∈ R.
In this case, ω is called a c-period of the function f .

We denote by P (ω,c) (R, X) the vector space of all (ω, c)-periodic functions from R to X. One can note that the space P (ω,c) (R, X) contains the spaces of periodic, antiperiodic and Bloch periodic functions among others (respectively taking c = 1, c = -1 and c = e ikt ) . (see [START_REF] Larrouy | ω,c)-periodic and asymptotically (ω,c)-periodic mild solutions to fractional Cauchy problems[END_REF] for more details)

Proposition 2.2 ( [1]). Let f ∈ C(R, X). Then, f ∈ P (ω,c) (R, X) if and only if f (t) = c t ω u(t), u(t) ∈ P ω,1 (R, X).
Using the principal branch of the complex Logarithm, c t ω is defined as

c t ω := exp( t ω Log(c)) = c ∧ (t)
and we will use the notation |c| ∧ (t

) := |c ∧ (t)| = |c| t ω .
Now we recall some properties of almost periodic and (ω, c)-pseudo almost periodic functions. (Bohr) almost periodic if for each ε > 0 there exists l > 0, such that for all α ∈ R, there exists τ ∈ [α, α + l] with

Definition 2.3. A function f ∈ C(R, X) is called
sup t∈R ∥f (t + τ ) -f (t)∥ < ε. (1)
The vector space consisting of all (Bohr) almost periodic functions is denoted by AP(X).

It is well known that a continuous function f : R → X is almost periodic if and only if the set

{f τ : τ ∈ R} is relatively compact in BC(R, X)
, where the function f τ is defined by

f τ (t) = f (t + τ ), t ∈ R.
Such number τ in ( 1) is called ε-translation number of f (t) and we denote by T ε (f ) the set of all ε-translation numbers of f . This set has the following property : Given any f ∈ AP(X),

1. if τ ∈ T ε (f ), then -τ ∈ T ε (f ),
This concept has been extended by Khalladi et al. [START_REF] Khalladi | Almost periodic type functions and applications[END_REF] as follows :

Definition 2.4 ( [13]). A function f ∈ C(R, X) is called (ω, c)-almost periodic if and only if the function f (ω,c) (t) := c -t w f (t), t ∈ R belongs to AP(X).
The vector space consisting of all (ω, c)-almost periodic functions is denoted by AP ω,c (X).

Unless specified otherwise, in the remainder of the paper, we will always assume that c ∈ C \ {0} and ω ∈ R ⋆ + ∩ T ε (f ). Furthermore, the principal branches are always used for taking powers of complex numbers.

In the following, we will keep the notation :

f (ω,c) (t) := c -t w f (t). Remark 2.5. When c = 1, AP ω,c (X) = AP(X).
Remark 2.6. One can note that in our paper, contrary to the paper [START_REF] Khalladi | Almost periodic type functions and applications[END_REF], ω is not only positive but it belongs also to the set of all ε-translation number of f . This condition yields AP ω,1 (X) := AP ω (X) = AP(X).

In order to conserve the periodic structure of (ω, c)-periodic type functions, we need to use an (ω, c)-norm which can be defined as

∥f ∥ (ω,c) := sup t∈R ∥c ∧ (-t)f (t)∥
(ω, c)-norms was introduced in the literature by Alvarez et al. taking the supremum norm not on the whole R but on the principal c-period interval [0, ω] of the (ω, c)-periodic considered function in order to handle the (ω, c)-periodicity properties of f (see in [START_REF] Alvarez | ω, c)-periodic functions and mild solutions to abstract fractional integro-differential equations[END_REF][START_REF] Abadias | Periodic Mild Solutions to Non-Autonomous Abstract Differential Equations[END_REF][START_REF] Larrouy | ω,c)-periodic and asymptotically (ω,c)-periodic mild solutions to fractional Cauchy problems[END_REF] for more details). We have the following completeness result.

Remark 2.7. We say that f is c-bounded when ∥f ∥ (ω,c) < ∞.

Proposition 2.8 ( [15]). (AP ω,c (X), ∥ • ∥ (ω,c) ) is a Banach space.
Proposition 2.9 ( [START_REF] Kostić | Mutli-dimensional (ω, c)-almost periodic type functions and applications[END_REF]). AP ω,c (X) is translation invariant and closed under the multiplication with complex scalars. Now we recall the concept of µ-pseudo almost periodic functions introduced by Blot et al. [START_REF] Blot | New approach for weighted pseudo-almost periodic functions under the light of measure theory, basic results and applications[END_REF].

We denote by B the Lebesgue σ-field of R and by M teh set of all positive measures µ on B satisfying µ

(R) = +∞ and µ([a, b]) < +∞, for all a, b ∈ R (a ≤ b). Definition 2.10 ( [3]). Let µ ∈ M. A function f ∈ BC(R, X) is said to be µ-ergodic if lim r→∞ 1 µ([-r, r]) [-r,r] ∥f (t)∥dµ(t) = 0
We denote the space of all such functions by E(R, X, µ).

Definition 2.11 ( [3]). Let µ ∈ M. A function f ∈ C(R, X) is said to be µ-pseudo almost periodic if f is written in the form f = g + φ
where g ∈ AP(X) and φ ∈ E(R, X, µ).

We denote the space of all such functions by PAP(R, X, µ).

Proposition 2.12 ( [3]). Let µ ∈ M. Then (E(R, X, µ), ∥ • ∥ ∞ ) is a Banach space.
We end this section with the following result.

Lemma 2.13. If f, g ∈ PAP(R, C, µ), then f g ∈ PAP(R, C, µ).

Proof. Since f, g ∈ PAP(R, C, µ) then they have following decompositions f = f 1 + f 2 and g = g 1 + g 2 where f 1 , g 1 ∈ AP(C) and f 2 , g 2 ∈ E(R, C, µ). Then we have

f g = f 1 g 1 + f 1 g 2 + g 1 f 2 + f 2 g 2
First we show that the product f 1 g 1 ∈ AP(C). If we take f 1 = g 1 we have

∥(f 1 ) 2 (t + τ ) -(f 1 ) 2 (t)∥ = ∥(f 1 )(t + τ ) + (f 1 )(t)∥ × ∥(f 1 )(t + τ ) -(f 1 )(t)∥
It can be easily seen that since f 1 is bounded, then there exists M ∈ R + such that

∥f 1 ∥ ≤ M
Thus, it comes the following

∥(f 1 ) 2 (t + τ ) -(f 1 ) 2 (t)∥ ≤ 2M ε ≤ ε ′ Then (f 1 ) 2 ∈ AP(C). Now, one can note that f 1 g 1 = 1 4 (f 1 + g 1 ) 2 -(f 1 -g 1 ) 2 . Since (f 1 + g 1 ) 2 ∈ AP(C) and (f 1 -g 1 ) 2 ∈ AP(C), then f 1 g 1 ∈ AP(C). Now, for (f 1 g 2 + g 1 f 2 + f 2 g 2 ) one has that : 1 µ([-r, r]) r -r ((f 1 )(g 2 ) + (g 1 )(f 2 ) + (f 2 )(g 2 ))(t) dµ(t) ≤ 1 µ([-r, r]) r -r ∥f 1 ∥ ∞ |(g 2 )(t)| + ∥g 1 ∥ ∞ |(f 2 )(t)| + ∥f 2 ∥ ∞ |(g 2 )(t)| dµ(t) And consequently, since f 2 , g 2 ∈ E(R, C, µ) we have lim r→+∞ 1 µ([-r, r]) r -r ((f 1 )(g 2 ) + (g 1 )(f 2 ) + (f 2 )(g 2 ))(t) dµ(t) = 0
The proof is complete.

Measure (ω, c)-pseudo almost periodic functions

In this section, we introduce the new concepts of µ-(ω, c)-ergodic functions and the µ-(ω, c)pseudo almost periodic functions. The notion of µ-(ω, c)-pseudo almost periodic functions is a generalization of µ-pseudo almost periodic functions introduced by Blot et al. [START_REF] Blot | New approach for weighted pseudo-almost periodic functions under the light of measure theory, basic results and applications[END_REF] which now becomes the particular case c = 1 of our work. It is also a generalization of the concept of weighted pseudo almost periodicity given by Diagana [START_REF] Diagana | Weighted pseudo-almost periodic functions and applications[END_REF][START_REF] Diagana | Weighted pseudo-almost periodic solutions to some differential equations[END_REF] and consequently this work generalizes that of Zhang [START_REF] Zhang | Integration of vector-valued pseudo-almost periodic functions[END_REF][START_REF] Zhang | Pseudo almost periodic solutions of some differential equations[END_REF][START_REF] Zhang | Pseudo almost periodic solutions of some differential equations II[END_REF] on the classical pseudo almost periodicity.

Here we introduce the space BC(R, X, c) (resp. BC(R × Y, X, c)), where BC(R, X, c) (resp. BC(R×Y, X, c)) denotes the Banach space consisting of all c-bounded continuous functions from R to X (resp. from R × Y to X) equipped with the (ω, c)-norm ∥ • ∥ (ω,c) defined in Section 2.

Remark 3.1. One can note that in the case c = 1,

(BC(R, X, c), ∥ • ∥ (ω,c) ) = (BC(R, X), ∥ • ∥ ∞ ), (BC(R × Y, X, c), ∥ • ∥ (ω,c) ) = (BC(R × Y, X), ∥ • ∥ ∞ ).
Moreover we have the following results.

Theorem 3.2. Let f ∈ BC(R, X, c). Then f ∈ BC(R, X) is and only if c = 1.
We begin this part with the following helpful convolution theorem for c-bounded functions.

Let L(X) be the space of bounded linear maps from the complex Banach space X into itself. We denote L 1 (R, L(X)) the Lebesgue space with respect to the Lebesgue measure on R.

Remark 3.3. One can note that if h ∈ L 1 (R, L(X)), then ϕ(•) := c ∧ (•)h(•) ̸ ∈ L 1 (R, L(X)) but φ := (c ∧ (-•)ϕ(•)) ∈ L 1 (R, L(X)). Theorem 3.4. Let f ∈ BC(R, X, c) and φ := (c ∧ (-•)ϕ(•)) ∈ L 1 (R, L(X)), then the convolution product of f ⋆ ϕ defined by (f ⋆ ϕ)(t) = +∞ -∞ ϕ(s)f (t -s)ds, for t ∈ R. = +∞ -∞ c ∧ (s) φ(s)f (t -s)ds, for t ∈ R. is c-bounded.
Proof. Let f ∈ BC(R, X, c). In order to state that (f ⋆ ϕ) ∈ BC(R, X, c), we consider the function ϱ n : R → X (n ∈ N, n ≥ 1) defined by

ϱ n (t) = n -n ϕ(s)f (t -s)ds. Observing that c ∧ (-t)ϱ n (t) = n -n c ∧ (-s)ϕ(s)c ∧ (-t + s)f (t -s)ds.
it is clear that ϱ n is c-bounded on R. We deduce that ϱ n (t) is continuous by using the uniform continuity of f on all compact subsets of R. Consequently, ϱ n (t) ∈ BC(R, X, c), which means that c ∧ (-t)ϱ n (t) ∈ BC(R, X), and from the following inequality :

c ∧ (-t) (f ⋆ ϕ)(t) -ϱ n (t) ≤ ∥f ∥ (ω,c) -n -∞ ∥c ∧ (-s)ϕ(s)∥ds + +∞ n ∥c ∧ (-s)ϕ(s)∥ds , we deduce that lim n→+∞ c ∧ (-t)ϱ n (t) = c ∧ (-t)(f ⋆ ϕ)(t) uniformly on R. Therefore f ⋆ ϕ ∈ BC(R, X, c).
3.1 On µ-(ω, c)-ergodicity.

First, we introduce the new concept of µ-(ω, c)-ergodic functions.

Definition 3.5. Let µ ∈ M. A function f ∈ BC(R, X, c) is said to be µ-(ω, c)-ergodic if lim r→∞ 1 µ([-r, r]) [-r,r] ∥f (ω,c) (t)∥dµ(t) = 0
We denote the space of all such functions by E (ω,c) (R, X, µ).

We give a completeness result and a characterization of µ-(ω, c)-ergodic functions.

Proposition 3.6. Let µ ∈ M. Then (E (ω,c) (R, X, µ), ∥ • ∥ (ω,c) ) is a Banach space. Proof. It is clear that E (ω,c) (R, X, µ) is a vector subspace of BC(R, X, c). We show that E (ω,c) (R, X, µ) is closed in BC(R, X, c). Let (f n ) ⊂ E (ω,c) (R, X, µ
) be a Cauchy sequence converging to f uniformly in R. From µ(R) = +∞, it follows that µ([-r, r]) > 0 for r sufficiently large. We have that

1 µ([-r, r]) [-r,r] ∥f (ω,c) (t)∥dµ(t) ≤ ∥f -f n ∥ (ω,c) + 1 µ([-r, r]) [-r,r] ∥(f n ) (ω,c) (t)∥dµ(t) then lim sup r→∞ 1 µ([-r, r]) [-r,r] ∥f (ω,c) (t)∥dµ(t) ≤ ∥f -f n ∥ (ω,c) , for all n ∈ N. Since lim n→+∞ ∥f -f n ∥ (ω,c) = 0, we deduce that lim r→∞ 1 µ([-r, r]) [-r,r] ∥f (ω,c) (t)∥dµ(t) = 0 Theorem 3.7.
Let µ ∈ M and I be a bounded interval (eventually I = ∅).

Let f ∈ BC(R, X, c), then following assertions are equivalent :

1. f ∈ E (ω,c) (R, X, µ), 2. lim r→∞ 1 µ([-r, r] \ I) [-r,r]\I ∥f (ω,c) (t)∥dµ(t) = 0, 3. For any ε > 0, µ( t ∈ [-r, r] \ I : ∥f (ω,c) (t)∥ > ε ) µ([-r, r] \ I) = 0. Proof. (1 ⇔ 2) Let Γ := µ(I) and Λ := I ∥f (ω,c) ∥dµ(t) then Γ, Λ ∈ R since I is bounded and
f is c-bounded and continuous. For r > 0 such that I ⊂ [-r, r] and µ([-r, r] \ I) > 0, we have

1 µ([-r, r] \ I) [-r,r]\I ∥f (ω,c) (t)∥dµ(t) = 1 µ([-r, r]) -Γ [-r,r] ∥f (ω,c) (t)∥dµ(t) -Λ = µ([-r, r]) µ([-r, r]) -Γ 1 µ([-r, r]) µ([-r,r]) ∥f (ω,c) (t)∥dµ(t) - Λ µ([-r, r])
Assume that 2 holds, then using the first equality and since µ(R) = +∞, we have that

0 = lim r→∞ 1 µ([-r, r]) -Γ [-r,r] ∥f (ω,c) (t)∥dµ(t) = lim r→∞ 1 µ([-r, r]) [-r,r] ∥f (ω,c) (t)∥dµ(t)
then 1 is true.

Conversely, if assumption 1 holds, using the second equality and since lim

r→∞ µ([-r, r]) µ([-r, r]) -Γ = 1
then assumption 2 is obviously holds.

(2 ⇔ 3) We set :

Γ ε r := t ∈ [-r, r] \ I : ∥f (ω,c) (t)∥ > ε and Λ ε r := t ∈ [-r, r] \ I : ∥f (ω,c) (t)∥ ≤ ε Using that [-r,r] ∥f (ω,c) (t)∥dµ(t) = Γ ε r ∥f (ω,c) (t)∥dµ(t) + Λ ε r ∥f (ω,c) (t)∥dµ(t)
we have for sufficiently large r > 0

1 µ([-r, r] \ I) [-r,r]\I ∥f (ω,c) (t)∥dµ(t) ≤ ∥f ∥ (ω,c) µ(Γ ε r ) µ([-r, r] \ I) + ε
Now assume that assumption 3 holds. Then from previous inequality, we have that for all ε > 0,

lim r→∞ 1 µ([-r, r] \ I) [-r,r]\I ∥f (ω,c) (t)∥dµ(t) ≤ ε
and consequently, assumption 2 holds. In order to prove the last implication, we use the following inequality

1 µ([-r, r] \ I) [-r,r]\I ∥f (ω,c) (t)∥dµ(t) ≥ 1 µ([-r, r] \ I) Γ ε r ∥f (ω,c) (t)∥dµ(t) ≥ ε µ(Γ ε r ) µ([-r, r] \ I)
assume that assumption 2 holds, we obtain assumption 3 when making r → +∞. The proof is complete.

We recall here the notion of equivalent measures.

Definition 3.8 ( [3]). Let µ 1 , µ 2 ∈ M. µ 1 is equivalent to µ 2 (or ρ 1 ∼ ρ 2 ) if there exists constants α, β > 0 and a bounded interval I (eventually I = ∅) such that αµ 1 (A) ≤ µ 2 (A) ≤ βµ(A), when A ∈ B satisfies A ∩ I = ∅
We have following result:

Theorem 3.9. Let µ 1 , µ 2 ∈ M. If µ 1 ∼ µ 2 then E (ω,c) (R, X, µ 1 ) = E (ω,c) (R, X, µ 2 ).
Proof. Assume that µ 1 ∼ µ 2 and B is the Lebesgue σ-field. Then we obtain for r sufficiently large

α β µ 1 ( t ∈ [-r, r] \ I : ∥f (ω,c) (t)∥ > ε ) µ 1 ([-r, r] \ I) ≤ µ 2 ( t ∈ [-r, r] \ I : ∥f (ω,c) (t)∥ > ε ) µ 2 ([-r, r] \ I) ≤ β α µ 1 ( t ∈ [-r, r] \ I : ∥f (ω,c) (t)∥ > ε ) µ 1 ([-r, r] \ I) We deduce that E (ω,c) (R, X, µ 1 ) = E (ω,c) (R, X, µ 2 ).
From ( 2), ( 3), ( 5) we obtain

M τ (r) = 1 µ([-r + τ, r + τ ]) [-r+τ,r+τ ] ∥f (ω,c) (t -τ )∥dµ(t).
If we denote by τ + := max(τ, 0) and τ -:= max(-τ, 0), we have that

|τ | + τ = 2τ + and |τ | -τ = 2τ -and then [-r + τ -|τ |, r + τ = |τ |] = [-r -2τ -, r + 2τ + ].
Therefore we obtain

M τ (r + |τ |) = 1 µ([-r -2τ -, r + 2τ + ]) [-r-2τ -,r+2τ + ] ∥f (ω,c) (t -τ )∥dµ(t). (6)
From ( 6) and the following inequality :

1 µ([-r, r]) [-r,r] ∥f (ω,c) (t -τ )∥dµ(t) ≤ 1 µ([-r, r]) [-r-2τ -,r+2τ + ] ∥f (ω,c) (t -τ )∥dµ(t) it comes that 1 µ([-r, r]) [-r,r] ∥f (ω,c) (t -τ )∥dµ(t) ≤ µ([-r -2τ -, r + 2τ + ]) µ([-r, r]) M τ (r + |τ |), that implies 1 µ([-r, r]) [-r,r] ∥f (ω,c) (t -τ )∥dµ(t) ≤ µ([-r -2|τ |, r + 2|τ |]) µ([-r, r]) M τ (r + |τ |). (7)
From ( 4), [START_REF] Corduneanu | Almost Periodic Functions[END_REF] and using Lemma 3.12, we deduce that

lim r→+∞ 1 µ([-r, r]) [-r,r] ∥f (ω,c) (t -τ )∥dµ(t) = 0.
The proof is complete.

We end this section by giving a convolution theorem for µ-(ω, c)-ergodic functions.

Theorem 3.14. Let µ ∈ M satisfying (H2). If f ∈ E (ω,c) (R, X, µ) and φ(•) := (c ∧ (-•)ϕ(•)) ∈ L 1 (R, L(X)), then the convolution product of f ⋆ ϕ defined by (f ⋆ ϕ)(t) = +∞ -∞ ϕ(s)f (t -s)ds, for t ∈ R, = +∞ -∞ φ(s)c ∧ (s)f (t -s)ds, for t ∈ R. is µ-(ω, c)-ergodic. Proof. Let f ∈ E (ω,c) (R, X, µ). According to Theorem 3.4, (f ⋆ ϕ) ∈ BC(R, X, c).
From µ(R) = +∞, we deduce the existence of r 0 ≥ 0 such that µ([-r, r]) > 0 for all r ≥ r 0 . Using the inequality :

1 µ([-r, r]) [-r,r] ∥(f ⋆ϕ) (ω,c) (t)∥dµ(t) ≤ 1 µ([-r, r]) [-r,r] +∞ -∞ ∥ φ(s)∥∥c ∧ (-t+s)f (t-s)∥dsdµ(t)
where φ := ϕ (ω,c) ∈ L 1 (R, L(X)), In view of the Fubini's Theorem, we deduce

1 µ([-r, r]) [-r,r] ∥(f ⋆ ϕ) (ω,c) (t)∥dµ(t) ≤ +∞ -∞ ∥ φ(s)∥ µ([-r, r]) [-r,r] ∥f (ω,c) (t -s)∥dµ(t)ds
Invoking Theorem 3.13, we have that

lim r→+∞ 1 µ([-r, r]) [-r,r] ∥f (ω,c) (t -s)∥dµ(t) = 0, for all s ∈ R.
And since

0 ≤ ∥ φ(s)∥ µ([-r, r]) [-r,r] ∥f (ω,c) (t -s)∥dµ(t) ≤ ∥ φ(s)∥∥f ∥ (ω,c)
in view of the Lebesgue dominated convergence Theorem, we obtain

lim r→+∞ +∞ -∞ ∥ φ(s)∥ µ([-r, r]) [-r,r] ∥f (ω,c) (t -s)∥dµ(t)ds = 0 It comes that lim r→+∞ 1 µ([-r, r]) [-r,r] ∥(f ⋆ ϕ) (ω,c) (t)∥dµ(t) = 0 and consequently, (f ⋆ ϕ) ∈ E (ω,c) (R, X, µ).
Example 3.2. The unique solution of the heat equation

u t (x, t) = u xx (x, t), x ∈ R, t ≥ 0,
with the initial condition u(x, 0) = f (x) is given by

u(x, t) = 1 2 √ πt +∞ -∞ e -(x-s) 2 4t f (s)ds, x ∈ R, t ≥ 0, If c ∧ (-•)e - (•) 2 4t 0 ∈ L 1 (R) and f ∈ E (ω,c) (R, R, µ
), then by Theorem 3.14, the solution

[x → u(x, t 0 ), x ∈ R] ∈ E (ω,c) (R, R, µ).
Now we are ready to define measure (ω, c)-pseudo almost periodic functions.

Measure (ω, c)-pseudo almost periodic function

In this subsection, we introduce the new class of measure (ω, c)-pseudo almost periodic function and we study some properties of such functions. Let us define this new notion.

Definition 3.15. Let µ ∈ M. A function f ∈ C(R, X) is said to be measure (ω, c)-pseudo almost periodic (or µ-(ω, c)-pseudo almost periodic) if f can be written in the form f = g + φ where g ∈ AP ω,c (X) and φ ∈ E (ω,c) (R, X, µ).
We denote the space of all such functions by PAP (ω,c) (R, X, µ).

We will say that g is the (ω, c)-almost periodic part of f and φ the µ-(ω, c)-pseudo ergodic perturbation of f .

We have the following spaces inclusions

AP ω,c (X) ⊂ PAP (ω,c) (R, X, µ) ⊂ BC(R, X, c) Remark 3.16. Observe that AP ω,c (X) is a proper subspace of PAP (ω,c) (R, X, µ) since the function ϕ(t) = (0.2) t (sin 2 (πt) + sin 2 ( √ 5t) + e -t t cos 2 (t) ) ∈ PAP (1,0.2) (R, X, µ) but ϕ / ∈ AP (1,0.2) (X) since [t → sin 2 (πt) + sin 2 ( √ 5t) + e -t t cos 2 (t) ] / ∈ AP(X).
The following theorem gives a characterization of the measure (ω, c)-pseudo almost periodic functions.

Theorem 3.17. Let f ∈ C(R, X). Then, f ∈ PAP (ω,c) (R, X, µ) if and only if f (t) ≡ c ∧ (t)u(t), with c ∧ (t) := c t ω and u ∈ PAP(R, X, µ). Proof. Obviously, if f (t) = c ∧ (t)u(t) with u ∈ PAP(R, X, µ) then f ∈ PAP (ω,c) (R, X, µ). Conversely, let f ∈ PAP (ω,c) (R, X, µ). Then ∃(g, φ) ∈ AP ω,c (X) × E (ω,c) (R, X, µ) such that f = g + φ. Therefore, taking u(t) := c ∧ (-t)f (t) it comes that u ∈ PAP(R, X, µ).
In view of Definition (3.15), for any f ∈ PAP (ω,c) (R, X, µ) we say that c ∧ (t)u(t) is the cfactorization of f . We give the first basic result Proposition 3.18. Let µ ∈ M. Then PAP (ω,c) (R, X, µ) is a vector space.

Proof. Obvious.

Now we intend to show that PAP

(ω,c) (R, X, µ) = AP ω,c (X) E (ω,c) (R, X, µ).
In order to prove Proposition 3.20, we will need following lemma.

Lemma 3.19. Assume f ∈ AP ω,c (X), write B ε := {τ ∈ R : ∥f (ω,c) (t 0 + τ ) -f (ω,c) (t 0 )∥ < ε} where ε > 0 and t 0 ∈ R is fixed. Then there exists s 1 , s 2 , . . . , s m ∈ R such that m i=1 (s i + B ε ) = R.
We have following result. Proposition 3.20. Let µ ∈ M satisfying (H1) and f ∈ PAP (ω,c) (R, X, µ) be such that

f = g + φ
where g is its (ω, c)-almost periodic component, then we have

g(R) ⊂ f (R) (8) Therefore, ∥f ∥ (ω,c) ≥ ∥g∥ (ω,c) ≥ inf t∈R |g (ω,c) (t)| ≥ inf t∈R |f (ω,c) (t)|.
Proof. Suppose that ( 8) is not true. Then there exists t 0 ∈ R, ε > 0 such that

∥g (ω,c) (t 0 ) -f (ω,c) (t)∥ ≥ 2ε, t ∈ R. (9)
Let s 1 , s 2 , . . . , s m be as in Lemma 3.19 and write

τ i = s i -t 0 , i = 1, 2, . . . , m, η = max 1≤i≤m |τ i |.
For r ∈ R with |r| > η, we let

B (i) ε,r := [-r + η -τ i , r -η -τ i ] ∩ (t 0 + B ε ), i = 1, 2, . . . , m.
where

B ε is as in Lemma 3.19. It is clear that m i=1 (τ i + B (i) ε,r ) = [-r + η, r -η].
Thus, we obtain

2(r -η) = µ([-r + η, r -η]) ≤ m i=1 µ(τ i + B (i) ε,r ) = m i=1 µ(B (i) ε,r ) ≤ m • max 1≤i≤m {µ(B (i) ε,r )} ≤ m • µ([-r, r] ∩ (t 0 + B ε )) (10)
since for each i = 1, 2, . . . , m

B (i) ε,r ⊂ [-r, r] ∩ (t 0 + B ε )
Using inequality (9), we have

∥ϕ (ω,c) (t)∥ = ∥f (ω,c) (t) -g (ω,c) (t)∥ ≥ ∥g (ω,c) (t 0 ) -f (ω,c) (t)∥ -∥g (ω,c) (t) -g (ω,c) (t 0 )∥ > ε for any t ∈ t 0 + B ε .
This and inequality (10) together give

1 µ([-r, r]) [-r,r] ∥ϕ (ω,c) (t)∥dµ(t) ≥ r -η mr ε -→ ε m , as r -→ ∞.
This is a contradiction since h ∈ E (ω,c) (R, X, µ) and establishes our claim [START_REF] Diagana | Weighted pseudo-almost periodic functions and applications[END_REF].

We can now establish the uniqueness of the decomposition in Definition 3.15.

Theorem 3.21. Let µ ∈ M satisfying (H1). Then the decomposition of a µ-(ω, c)-pseudo almost periodic function in the form f = g + φ, where g ∈ AP ω,c (X) and

φ ∈ E (ω,c) (R, X, µ) is unique. Proof. Assume that f = g 1 + φ 1 and f = g 2 + φ 2 . Then, 0 = (g 1 -g 2 ) + (φ 1 -φ 2 ). Since g 1 -g 2 ∈ AP ω,c (X) and φ 1 -φ 2 ∈ PAP (ω,c) (R, X, µ)
, in view of proposition 3.20, we deduce that g 1 -g 2 = 0 and consequently, φ 1 = φ 2 which proves the uniqueness of the decomposition.

From above it is clear that AP ω,c (X) ∩ E (ω,c) (R, X, µ) = {0}
Furthermore, we have following results

Theorem 3.22. The space PAP (ω,c) (R, X, µ) is a translation invariant.

Proof. This is a direct consequence of Proposition 2.9, Theorem 3.13 and Theorem 3.21.

Theorem 3.23. The space

PAP (ω,c) (R, X, µ) is a translation invariant C * -subalgebra of BC(R, X, c)
without the constant functions. Furthermore,

PAP (ω,c) (R, X, µ)/E (ω,c) (R, X, µ) ∼ = AP ω,c (X)
Proof. We show that PAP (ω,c) (R, X, µ) is a closed subspace of BC(R, X, c).

Let (f n ) ⊂ PAP (ω,c) (R, X, µ) be Cauchy. By proposition 3.20, the sequence (g n ) ⊂ AP ω,c (X) is Cauchy too and so is (φ n ) ⊂ E (ω,c) (R, X, µ). Since AP ω,c (X) and E (ω,c) (R, X, µ) are closed in BC(R, X, c), there are g ∈ AP ω,c (X) and

φ ∈ E (ω,c) (R, X, µ) such that ∥(g n ) (ω,c) -g (ω,c) ∥ → 0 and ∥(φ n ) (ω,c) -φ (ω,c) ∥ → 0 as n → ∞. Set f = g + φ, then f ∈ PAP (ω,c) (R, X, µ) and ∥(f n ) (ω,c) -f (ω,c) ∥ → 0 as n → 0.
The rest of the proof is clear.

Now we show the completeness of PAP (ω,c) (R, X, µ) with the following result.

Theorem 3.24. Let µ ∈ M satisfying (H1). Then (PAP (ω,c) (R, X, µ), ∥ • ∥ (ω,c) ) is a Banach space.

Proof. Let (f n ) be a Cauchy sequence in PAP (ω,c) (R, X, µ). Then, given ε > 0 ∃n 0 ∈ N such that for all p, q ≥ n 0 ,

∥(f p )(t) -(f q )(t)∥ (ω,c) < ε.
Invoking Theorem 3.17, ∃(u p , u q ) ∈ (PAP(R, X, µ)) 2 such that f p (t) = c ∧ (t)u p (t) and f q (t) = c ∧ (t)u q (t) for all t ∈ R and since p, q ≥ n 0 we have

∥u p (t) -u q (t)∥ ≤ ∥f p (t) -f q (t)∥ (ω,c) < ε.
Consequently, (u n ) be a Cauchy sequence in PAP(R, X, µ). Using the completeness of PAP(R, X, µ), we know that ∃u ∈ PAP(R, X, µ) such that ∥u n -u∥ → 0 as n → ∞.

We take f (t) := c ∧ (t)u(t). We claim that ∥u n -u∥ → 0 as n → ∞. And it can be easily seen that :

∥f n -f ∥ (ω,c) = sup t∈R ∥u n -u∥ → 0 (n → ∞)
Which completes the proof.

We end this subsection giving a general convolution theorem for our new class of functions. In their paper, Blot et al. obtained the following convolution result.

Theorem 3.25 ( [3]). Let µ ∈ M satisfying (H2). If f ∈ PAP(R, X, µ) and ϕ(•) ∈ L 1 (R, L(X)),
then the convolution product f ⋆ ϕ defined by

(f ⋆ ϕ)(t) = +∞ -∞ ϕ(s)f (t -s)ds, for t ∈ R. is µ-pseudo almost periodic.
We generalize this result as follows.

Theorem 3.26. Let µ ∈ M satisfying (H2). If f ∈ PAP (ω,c) (R, X, µ) and φ := (c ∧ (-•)ϕ(•)) ∈ L 1 (R, L(X)), then the convolution product of f ⋆ ϕ defined by (f ⋆ ϕ)(t) = +∞ -∞ ϕ(s)f (t -s)ds, for t ∈ R. (11)
= +∞ -∞ φ(s)c ∧ (s)f (t -s)ds, for t ∈ R. is µ-(ω, c)-pseudo almost periodic. Proof. Let f ∈ PAP (ω,c) (R, X, µ) and φ := (c ∧ (-•)ϕ(•)) ∈ L 1 (R, L(X)).
First, note that using Theorem 3.4, (f ⋆ ϕ) ∈ BC(R, X, c). Furthermore, according to Theorem 3.17, there exists a u ∈ PAP(R, X, µ) such that f (t) := c ∧ (t)u(t), for any t ∈ R. It comes that

(f ⋆ ϕ)(t) = +∞ -∞ ϕ(s)f (t -s)ds = c ∧ (t) +∞ -∞ c ∧ (-s)ϕ(s)c ∧ (-t + s)f (t -s)ds = c ∧ (t) +∞ -∞ φ(s)u(t -s)ds (12)
Invoking succesively Theorems 3.25 and 3.17, we have that ( 12) is µ-(ω, c)-pseudo almost periodic. The proof is complete.

Example 3.4. The unique solution of the heat equation

u t (x, t) = u xx (x, t), x ∈ R, t ≥ 0,
with the initial condition u(x, 0) = f (x) is given by

u(x, t) = 1 2 √ πt +∞ -∞ e -(x-s) 2 4t f (s)ds, x ∈ R, t ≥ 0, If c ∧ (-•)e - (•) 2 4t 0 ∈ L 1 (R) and f ∈ PAP (ω,c) (R, R, µ
), then by Theorem 3.14, the solution

[x → u(x, t 0 ), x ∈ R] ∈ PAP (ω,c) (R, R, µ).
4 Jointly continuous case.

This section is devoted to the study of a composition result well-suited for the introduced (ω, c)periodicity concept. The main results of this section are Theorems 4.7 and 4.16. But first, let's define some new notions.

Basic definitions and properties.

First of all, reader should be aware that the already known concept of compactness for subsets seems to be irrelevant when it comes to deal with (ω, c)-periodicity where c ̸ = 1 since (ω, c) periodic type functions are not bounded on R (i.e. 1-bounded on R) but c-bounded on R. With the following definition, we propose a new concept of compactness for subset well-suited for (ω, c)-periodic calculus.

Definition 4.1. Let K be a non-empty set. We say that K is an (ω, c)-type compact subset of Y if and only if following assumptions are satisfied :

1. K is compact, 2. Every k ∈ K admits following decomposition k := c ∧ (-•)y ∈ K where y ∈ Y.
One can note that a compact subset of AP(X) is in fact a (ω, c)-type subset of AP ω,c (X) since if K is a compact subset of AP(X) we have the following equality

Σ : = {u(t) : t ∈ R, u ∈ K} = {(c ∧ (t)u(t)) (ω,c) : t ∈ R, (c ∧ (t)u(t)) ∈ AP ω,c (X), u ∈ K} for any c ∈ C -{0}. Definition 4.2 ( [22]). A function F ∈ C(R × Y, X) is called (Yoshizawa) almost periodic in t ∈ R uniformly in y ∈ Y if for each ε > 0 and any compact K ⊂ Y there exists L > 0, such that for all β ∈ R, there exists τ ∈ [β, β + L] with sup t∈R sup y∈K ∥F (t + τ, y) -F (t, y)∥ < ε. ( 13
)
for all t ∈ R and all y ∈ K.

The collection of such functions will be denoted by AP(Y, X).

Such number τ in ( 13) is called ε-translation number of F (t, y) and we denote by T ε (F, K) the set of all ε-translation numbers of F for y ∈ K. This set has the following properties : For a fixed compact set K,

1. an ε-translation number is also an ε ′ -translation number if ε ′ > ε, and hence we have the inclusion

T ε (F, K) ⊂ T ε ′ (F, K), 2. if τ ∈ T ε (F, K), then -τ ∈ T ε (F, K), 3. if (τ 1 , τ 2 ) ∈ T ε 1 (F, K) × T ε 2 (F, K) then τ 1 + τ 2 ∈ T ε 1 +ε 2 (F, K).
In what follows, we assume that ω ∈ R ⋆ + ∩ T ε (F, K).

In [START_REF] Khalladi | Almost periodic type functions and applications[END_REF] authors have introduced two concepts of (ω, c)-almost periodic functions in the case of jointly continuous functions, but in this paper use a novel approach.

Definition 4.3. A function F ∈ C(R × Y, X) is called (ω, c)-almost periodic in t ∈ R uniformly in y ∈ Y if for each ε > 0 and any (ω, c)-type compact subset K of Y there exists L > 0, such that for all β ∈ R, there exists τ ∈ [β, β + L] with sup t∈R sup y∈K ∥c ∧ (-t -τ )F (t + τ, y) -c ∧ (-t)F (t, y)∥ < ε. ⇐⇒ sup t∈R sup y∈K ∥F (ω,c) (t + τ, y) -F (ω,c) (t, y)∥ < ε
for all t ∈ R and all y ∈ K, where F (ω,c) (t, •) := c -t ω F (t, •). The space of all such functions will be denoted by AP ω,c (Y, X).

In the following, we use the notation :

F (ω,c) (t, •) := c -t w F (t, •). Remark 4.4. When c = 1, AP ω,c (Y, X) = AP(Y, X). Proposition 4.5. (AP ω,c (Y, X), ∥ • ∥ (ω,c) ) is a Banach space.
We need to develop some tools in order to propose a composition theorem for measure (ω, c)pseudo almost periodic functions.

We give the following results

Lemma 4.6. If K is an (ω, c)-type compact subset of AP ω,c (Y), then

Σ := {u (ω,c) (t) : t ∈ R, u ∈ AP ω,c (Y), u (ω,c) ∈ K} is a relatively compact subset of Y.
Proof. Let ε > 0. Since K is compact, it is also precompact, thus there exists {(u 1 ) (ω,c) , . . . , (u m ) (ω,c) } a finite (ω, c)-type subset of AP ω,c (Y) (i.e. a finite subset of AP(Y)) such that

K ⊂ 1≤i≤m z ∈ AP ω,c (Y) : ∥z -u i ∥ (ω,c) ≤ ε 2 . Since, (u i ) (ω,c) (R) is relatively compact in X for all i = 1, . . . , m, then 1≤i≤m (u i ) (ω,c) (R) is also
relatively compact and consequently, there exists a finite subset of R {t 1 , . . . , t k } such that

1≤i≤m (u i ) (ω,c) (R) ⊂ 1≤i≤m 1≤j≤k y ∈ Y : ∥y -(u i ) (ω,c) (t j )∥ ≤ ε 2 .
If y ∈ Y there exists z ∈ K and t ∈ R such that y = z(t), and there exists i ∈ {1, . . . , m} such that ∥z -(u i ) (ω,c) ∥ ≤ ε 2 and consequently ∥y -(u i ) (ω,c) (t)∥ ≤ ε 2 . Now, using the previous inclusion, there exists p ∈ {1, . . . , m} and j ∈ {1, . . . , k} such that

∥(u i ) (ω,c) (t) -(u p ) (ω,c) (t j )∥ ≤ ε 2 It comes that ∥y -(u i ) (ω,c) (t j )∥ ≤ ∥y -(u i ) (ω,c) (t)∥ + ∥(u i ) (ω,c) (t) -(u p ) (ω,c) (t j )∥ ≤ 2ε 2 = ε This proves that Σ ⊂ 1≤i≤m 1≤j≤k y ∈ Y : ∥y -(u i ) (ω,c) (t j )∥ ≤ ε 2
or in other words, Σ is precompact, and since Y is complete, we obtain that Σ is relatively compact.

Now, for a given function F ∈ AP ω,c (Y, X), we define the Nemytskii's superposition operator

N F : AP ω,c (Y) -→ AP ω,c (X) such that [t -→ u(t)] -→ N F (u) := [t -→ F (t, u(t))].
The first main result of this section is the following theorem.

Theorem 4.7. Let F ∈ AP ω,c (Y, X). Then the Nemytskii superposition operator N F is continuous from AP ω,c (Y) into AP ω,c (X).

Proof. Let K be an (ω, c)-type compact subset of AP ω,c (Y) let ũ ∈ K and ε > 0.

We set Σ := {u (ω,c) (t) : t ∈ R, u ∈ AP ω,c (Y), u (ω,c) ∈ K}. According to Lemma 4.6, the closure Σ is compact. Since F ∈ AP ω,c (Y, X), there exists l > 0 such that for α ∈ R, there exists -τ ∈ [α, α + l] satisfying :

∥F (ω,c) (t + τ, y) -F (ω,c) (t, y)∥ ≤ ε 3 , for all (y, t) ∈ Σ × R. (14)
Since Σ×[0, l] is compact then F (ω,c) is uniformly continuous on it and consequently there exists δ > 0 such that, for all t 1 , t 2 ∈ [0, l] and for all y 1 , y 2 ∈ Σ

∥y 1 -y 2 ∥ ≤ δ, |t 1 -t 2 | ≤ δ ⇒ ∥F (ω,c) (t 1 , y 1 ) -F (ω,c) (t 2 , y 2 )∥ ≤ ε 3 .
And this implies that :

∥y 1 -y 2 ∥ ≤ δ ⇒ ∥F (ω,c) (t, y 1 ) -F (ω,c) (t, y 2 )∥ ≤ ε 3 , for all t ∈ [0, l]. (15)
If ṽ ∈ K satisfies ∥ṽ -ũ∥ ∞ ≤ δ ⇐⇒ ∥c ∧ (t)ṽ(t) -c ∧ (t)ũ(t)∥ (ω,c) ≤ δ.

We set u(t) := c ∧ (t)ũ(t) and v(t) := c ∧ (t)ṽ(t) for all t ∈ R.

Then we have u, v ∈ AP ω,c (Y), and using ( 14) and ( 15), we obtain, for all t ∈ R

∥F (ω,c) (t, u(t)) -F (ω,c) (t, v(t))∥ ≤ ∥F (ω,c) (t, u(t)) -F (ω,c) (t -τ, u(t))∥ + ∥F (ω,c) (t -τ, u(t)) -F (ω,c) (t -τ, v(t))∥ + ∥F (ω,c) (t -τ, v(t)) -F (ω,c) (t, v(t))∥ ≤ 3 ε 3 = ε.
And so, by taking the supremum on the t ∈ R, we obtain

∥N F (u) -N F (v)∥ (ω,c) ≤ ε.
This proves that the restriction of J f to K is continuous for all (ω, c)-type compact subset K of AP ω,c (Y). And since (AP ω,c (Y), ∥ • ∥ (ω,c) ) and (AP ω,c (X), ∥ • ∥ (ω,c) ) are Banach Spaces, this proves the continuity of J F on AP ω,c (Y).

The following proposition is a generalization of Cieutat, Fatajou and N'Guérékata's Theorem in [START_REF] Cieutat | Composition of pseudo-almost periodic and pseudo-almost automorphic functions and applications to evolution equations[END_REF] which becomes the particular case c = 1 of our result. 1. for all y ∈ Y, F (ω,c) (•, y) ∈ AP(X),

F (ω,c

) is uniformly continuous on each (ω, c)-type compact set K in Y with respect to the second variable, namely, for each (ω, c)-type compact set K in Y, for all ε > 0, there exists δ > 0 such that for all y 1 , y 2 ∈ K, one has

∥y 1 -y 2 ∥ ≤ δ =⇒ sup t∈R ∥F (ω,c) (t, y 1 ) -F (ω,c) (t, y 2 )∥ ≤ ε.
Here, we propose a concept of µ-(ω, c)-ergodicity for the jointly continuous functions case. c) is said to be µ-(ω, c)-ergodic in t uniformly with respect to y ∈ Y if the two following conditions are true :

Definition 4.9. Let µ ∈ M. A function F ∈ BC(R × Y, X,
1. for all y ∈ Y, F (ω,c) (•, y) ∈ E(R, X, µ), 2. F (ω,c
) is uniformly continuous on each (ω, c)-type compact set K in Y with respect to the second variable, namely, for each (ω, c)-type compact set K in Y, for all ε > 0, there exists δ > 0 such that for all y 1 , y 2 ∈ K, one has

∥y 1 -y 2 ∥ ≤ δ =⇒ sup t∈R ∥F (ω,c) (t, y 1 ) -F (ω,c) (t, y 2 )∥ ≤ ε.
We denote the space of all such functions by E ω,c,3 (R × Y, X, µ).

Remark 4.10. When c = 1, we write

E(R × Y, X, µ) instead of E ω,1,3 (R × Y, X, µ). Definition 4.11. Let µ ∈ M. A function F ∈ C(R × Y, X
) is said to be µ-(ω, c)-pseudo almost periodic in t uniformly with respect to y ∈ Y if F is written in the form

F = G + Φ where G ∈ AP ω,c (Y, X) and Φ ∈ E ω,c,3 (R × Y, X, µ).
PAP ω,c (R × Y, X, µ) denotes the set of such that functions.

The following inclusion hold

AP ω,c (Y, X) ⊂ PAP ω,c (R × Y, X, µ) ⊂ BC(R × Y, X, c) Remark 4.12. When c = 1, we write PAP(R × Y, X, µ) instead of PAP ω,1 (R × Y, X, µ).
As in the previous section, we propose a characterization result which hold for (ω, c)-almost periodic, µ-(ω, c)-ergodic and µ-(ω, c)-pseudo almost periodic functions in t uniformly with respect to y ∈ Y.

Theorem 4.13. Let F ∈ C(R × Y, X). Then, F ∈ PAP ω,c (R × Y, X, µ) (resp. AP ω,c (Y, X) or E ω,c,3 (R × Y, X, µ)) if and only if F (t, y) ≡ c ∧ (t)u(t, y), with c ∧ (t) := c t ω and u ∈ PAP(R × Y, X, µ)(resp. AP(Y, X) or E(R × Y, X, µ)).
Proof. The proof is similar to the one of Theorem 3.17.

We end this section with this result which is a consequence of Proposition 4.8 and definition 4.9.

Theorem 4.14. Let µ ∈ M and F : R × Y → X be µ-(ω, c)-ergodic (of second kind) in t uniformly with respect to y ∈ Y. Then ,c) is uniformly continuous on each (ω, c)-type compact set K in Y with respect to the second variable.

1. for all y ∈ Y, F (ω,c) (•, y) ∈ PAP(R, X, µ), 2. F (ω
We are now in a position to give a composition theorem for measure (ω, c)-pseudo almost periodic functions.

Composition of measure (ω, c)-pseudo almost periodic functions.

The proof of our result of composition of µ-(ω, c)-pseudo almost periodic functions is based on the following lemma due to Schwartz [START_REF] Schwartz | Topologie Générale et Analyse Fonctionnelle[END_REF].

Lemma 4.15. If Ψ ∈ C(X, Y), then for each compact set K in X and all ε > 0, there exists δ > 0 such that for any x 1 , x 2 ∈ X, one has

x 1 ∈ K and ∥x 1 -x 2 ∥ ≤ δ ⇒ ∥Ψ(x 1 ) -Ψ(x 2 )∥ ≤ ε.
We now can state and prove our composition result.

Theorem 4.16. Let µ ∈ M, F ∈ PAP ω,c (R × Y, X, µ) and y ∈ PAP (ω,c) (R, Y, µ). Assume that the following hypothesis holds 1. For all bounded subset Ω of Y, F is c-bounded on R × Ω (i.e. F (ω,c) is bounded on R × Ω). Then [t -→ F (t, y (ω,c) (t))] ∈ PAP ω,c (R × Y, X, µ). Proof. First note that the function [t -→ F (t, y (ω,c) (t))] is continuous and by Hypothesis (1) it is c-bounded. Since y ∈ PAP (ω,c) (R, Y, µ) there exists y 1 ∈ AP ω,c (Y) and y 2 ∈ E (ω,c) (R, Y, µ) such that y = y 1 + y 2 .
Moreover, since F ∈ PAP ω,c (R × Y, X, µ), there exists

F 1 ∈ AP ω,c (Y, X) and F 2 ∈ E ω,c,3 (R × Y, X, µ) such that F = F 1 + F 2 .
and there exists

F ∈ PAP(R × Y, X, µ) such that F (t, y(t)) := c ∧ (t) F (t, y(t)), ∀t ∈ R. F1 ∈ AP(Y, X) such that F 1 (t, y(t)) := c ∧ (t) F1 (t, y(t)), ∀t ∈ R. F2 ∈ E(R × Y, X, µ) such that F 2 (t, y(t)) := c ∧ (t) F2 (t, y(t)), ∀t ∈ R.
So we have the following decomposition for F (t, y(t))

F (t, y(t)) = c ∧ (t) F1 (t, y 1 (t)) + [ F (t, y(t)) -F (t, y 1 (t))] + [ F (t, y 1 (t)) -F1 (t, y 1 (t))] = c ∧ (t) F1 (t, y 1 (t)) + [ F (t, y(t)) -F (t, y 1 (t))] + F2 (t, y 1 (t))
Invoking Theorem 4.7 and keeping in mind that

(y 1 ) (ω,c) ∈ AP(Y) then F1 (t, (y 1 ) (ω,c) (t)) ∈ AP(X) (16)
and consequently we have that [t -→ F 1 (t, (y 1 ) (ω,c) (t))] ∈ AP ω,c (Y, X). Now denote K the closure of the range of (y 1 ) (ω,c) : K = {(y 1 ) (ω,c) (t) : t ∈ R}. One can easily note that since y 1 ∈ AP ω,c (Y), using Theorem 3.17, we have (y 1 ) (ω,c) ∈ AP(Y) and consequently, K is a compact subset of Y [START_REF] Corduneanu | Almost Periodic Functions[END_REF].

In what follows, we denote by Ψ the function defined by

Ψ : Y -→ PAP (ω,c) (R, X, µ) y -→ Ψ(y) = c ∧ (•) F (•, y) Since F (•, y) := c ∧ (•) F (•, y) ∈ PAP ω,c (R × Y, X, µ)
, by using Theorem 4.14 we deduce that the restriction of Ψ on all (ω, c)-type compact subset K of Y is uniformly continuous, which is equivalent to say that the function Ψ is continuous on Y. From Lemma 4.15 applied to Ψ, we deduce that for given ε > 0, there exists δ > 0 such that, for all t ∈ R, u 1 , u 2 ∈ Y, one has

u 1 ∈ K and ∥u 1 -u 2 ∥ ≤ δ ⇒ ∥ F (t, u 1 ) -F (t, u 2 )∥ ≤ ε.
Since, y(t) = y 1 (t) + y 2 (t) and (y 1

) (ω,c) (t) ∈ K, it comes that t ∈ R and ∥(y 2 ) (ω,c) (t)∥ ≤ δ ⇒ ∥ F (t, y (ω,c) (t)) -F (t, (y 1 ) (ω,c) (t))∥ ≤ ε.
therefore, the following inequality holds

µ( t ∈ [-r, r] : ∥ F (t, y (ω,c) (t)) -F (t, (y 1 ) (ω,c) (t))∥ > ε ) µ([-r, r]) ≤ µ( t ∈ [-r, r] : ∥(y 2 ) (ω,c) (t)∥ > δ ) µ([-r, r])
Since y 2 ∈ E (ω,c) (R, Y, µ) and according to Theorem 3.7 with c = 1 we have for the abovementioned δ

lim r→+∞ µ( t ∈ [-r, r] : ∥(y 2 ) (ω,c) (t)∥ > δ ) µ([-r, r]) = 0,
and consequently

lim r→+∞ µ( t ∈ [-r, r] : ∥ F (t, y (ω,c) (t)) -F (t, (y 1 ) (ω,c) (t))∥ > ε ) µ([-r, r]) = 0,
Invoking Theorem 3.7 with c = 1 and previous inequality it is proved that

[t -→ F (t, y (ω,c) (t)) -F (t, (y 1 ) (ω,c) (t))] ∈ E ω,1 (R, Y, µ) = E(R, Y, µ).
which is equivalent to say that (according to Theorem 4.13)

[t -→ F (t, y (ω,c) (t)) -F (t, (y 1 ) (ω,c) (t))] ∈ E (ω,c) (R, Y, µ).
In order to complete the proof, we prove that [t -→ F 2 (t, (y 1

) (ω,c) (t))] ∈ E (ω,c) (R, Y, µ). Since F 2 := c ∧ (•) F2 is uniformly continuous on the (ω, c)-type compact set K = {(y 1 ) (ω,c) (t) : t ∈ R}
with respect to the second variable, we deduce that for given ε > 0, there exists η > 0 such that, for all t ∈ R, ξ 1 , ξ 2 ∈ K, one has

∥ξ 1 -ξ 2 ∥ ≤ η =⇒ ∥ F2 (t, ξ 1 ) -F2 (t, ξ 2 )∥ ≤ ε.
Therefore, there exists ϑ ε and {(y

i ) (ω,c) } ϑε i=1 ⊂ K, such that K ⊂ ϑε i=1
B((y i ) (ω,c) , η) and then

∥ F2 (t, (y 1 ) (ω,c) (t))∥ ≤ ε + ϑε i=1 ∥ F2 (t, (y i ) (ω,c) (t))∥. Since, ∀i ∈ {1, . . . , ϑ ε }, lim r→+∞ 1 µ([-r, r]) [-r,r] ∥ F2 (t, (y i ) (ω,c) (t))∥dµ(t) = 0.
we deduce that ∀ε > 0, lim

r→+∞ 1 µ([-r, r]) [-r,r] ∥ F2 (t, (y 1 ) (ω,c) (t))∥dµ(t) ≤ ε, which implies that ∀ε > 0, lim r→+∞ 1 µ([-r, r]) [-r,r] ∥ F2 (t, (y 1 ) (ω,c) (t))∥dµ(t) = 0.
Therefore, using Theorem 3.7 with c = 1, we have that [t -→ F2 (t, (y 1

) (ω,c) (t))] ∈ E(R, Y, µ).
In conclusion, invoking Theorem 4.13, we showed that

[t -→ F 2 (t, (y 1 ) (ω,c) (t))] ∈ E (ω,c) (R, Y, µ).
Finally, it comes that

[t -→ F (t, y (ω,c) (t))] ∈ PAP ω,c (R × Y, X, µ).
The proof is complete.

We end this section with the following result.

Corollary 4.17.

Let µ ∈ M, Ψ ∈ C(X, Y), ω > 0 and c ∈ C -{0}.
Assume that for all bounded subset B of X, Ψ is c-bounded on B, (i.e. Ψ(

•) := c ∧ (•) Ψ(•) where Ψ is bounded on B). Then if x ∈ PAP (ω,c) (R, X, µ), [t -→ Ψ(x (ω,c) (t)) := c ∧ (t) Ψ(x (ω,c) (t))] ∈ PAP (ω,c) (R, X, µ).
Proof. It is a consequence of Theorem 4.16 with

F (t, x (ω,c) (t)) = Ψ(x (ω,c) (t)).
5 Application : Measure (ω, c)-Pseudo Almost Periodic solutions to a Lasota-Wazewska model.

First, Wazewska-Czyzewska and Lasota [START_REF] Wazewska-Czyzewska | Mathematical problems of the dynamics of the red blood cells systems[END_REF] proposed in 1976 the delay logistic equations with one constant concentrated delay

N ′ (t) = -µN (t) + pe -rN (t-τ )
in order to describe the survival of red blood cells in an animal. Here N (t) denotes the number of red blood cells at time t, µ is the probability of death of a red blood cell, p and r are positive constants related to the production of red blood cells per unit time and τ is the time required to produce a red blood cell. Few years later, Gopalsamy and Trofimchuk [START_REF] Gopalsamy | Almost periodic solutions of Lasota-Wazewska-type delay differential equation[END_REF] obtained that the Lasota-Wazewska model with one discrete delay

x ′ (t) = -α(t)x(t) + β(t)e -νx(t-τ )
has a globally attractive almost periodic solution under some additional assumptions.

The aim here is to study the existence and uniqueness of a generalized Lasota-Wazewska model with µ-(ω, c)-pseudo almost periodic coefficients and with mixed delay which is in the form :

y ′ (t) = -α(t)y(t) + m j=1 a j (t)e -c ∧ (-t)ω j (t) t -∞ K j (t-s)y(s)ds + n i=1 b i (t)e -c ∧ (-t)β i (t) p j=1 y(t-τ ij ) , t ∈ R (17)
where y(t) stands for the number of red blood cells at time t and α(t) is the average part of red blood cells population being destroyed in time t. For all 1 ≤ j ≤ m and 1 ≤ i ≤ n, a j (t) and b i (t) are the connected with demand for oxygen at time t, ω j (t) and β i (t) characterize excitability of haematopoietic system at time t, K j is the probability kernel of the distributed delays and τ ij is the time required to produce a red blood cell. One can note that we consider in our new approach the µ-(ω, c)-pseudo almost periodic for the connected with demand for oxygen at time t and the µ-pseudo almost periodic for the excitability of haematopoietic system at time t since it is more realistic for the description of the physical and biological phenomena.

The method consists to reduce the existence of the unique solution for the Lasota-Wazewska model [START_REF] Larrouy | ω,c)-periodic and asymptotically (ω,c)-periodic mild solutions to fractional Cauchy problems[END_REF] to the search for the existence of the unique fixed point of an approriate operator on the Banach space PAP (ω,c) (R, R + , µ). Notice that we restrict our selves to R + -valued functions since only non-negative solutions are biologically meaningful. In what follows, given a c-bounded continuous function f defined on R, f (ω,c) and f (ω,c) are defined by

f (ω,c) = sup t∈R f (ω,c) (t) = sup t∈R c ∧ (-t)f (t), and f (ω,c) = inf t∈R f (ω,c) (t) = inf t∈R c ∧ (-t)f (t)
Remark 5.1. If c = 1 we use the notations

f := f (ω,1) = sup t∈R f (t) and f := f (ω,1) = inf t∈R f (t)
First, we give sufficient conditions which ensures existence and uniqueness of µ-(ω, c)-pseudo almost periodic solution of [START_REF] Larrouy | ω,c)-periodic and asymptotically (ω,c)-periodic mild solutions to fractional Cauchy problems[END_REF].

(C1) 0 < c ≤ 1, (C2) α ∈ AP(R + ), (C3) β i , ω j ∈ PAP(R, R + , µ), for all (i, j) ∈ [[1, n]] × [[1, m]], (C4) a j , b i ∈ PAP (ω,c) (R, R + , µ), for all (i, j) ∈ [[1, n]] × [[1, m]], (C5) m j=1 (a j ) (ω,c) ω j + pξ n i=1 (b i ) (ω,c) β i α < 1, (C6) For all 1 ≤ j ≤ m, (K j ) (ω,c) : [0, +∞) → R + are continuous, integrable and ∞ 0 (K j ) (ω,c) (u)du = 1, and ∞ 0 (K j ) (ω,c) (u)e λu du < +∞,
where λ is a sufficiently non negative small constant.

Lemma 5.2. Let f ∈ PAP (ω,c) (R, R + , µ) and g ∈ PAP(R, R + , µ). If c > 0, then f g ∈ PAP (ω,c) (R, R + , µ).
Proof. According to Theorem 3.17 there exists a unique u ∈ PAP(R, R + , µ) such that

f (t) := c ∧ (t)u(t)
for all t ∈ R. Using Lemma 2.13 it is clear that u × g ∈ PAP(R, R + , µ). Then,

(f g)(t) := c ∧ (t)(u(t)g(t)
), where ug ∈ PAP(R, R + , µ).

Invoking Theorem 3.17, we complete the proof.

Lemma 5.3. Let µ ∈ M satisfying (H2). For all x(•) ∈ PAP (ω,c) (R, R + , µ), the function

ψ i : t → ψ i (x (ω,c) (t)) = a(t)e -c ∧ (-t)ω i (t) p j=1 x(t-τ ij ) = c ∧ (t)a (ω,c) (t)e -ω i (t) p j=1 c ∧ (-τ ij )x (ω,c) (t-τ ij )
belongs to PAP (ω,c) (R, R + , µ) for all 1 ≤ i ≤ n.

Proof. First, by Lemma 3.22, the function

t → x(t -τ ij ) ∈ PAP (ω,c) (R, R + , µ)
for all 1 ≤ i ≤ n and 1 ≤ j ≤ p. Then according to Proposition 3.18

t → p j=1 x(t -τ ij ) ∈ PAP (ω,c) (R, R + , µ)
Furthermore, by Lemma 5.2

t → ω i (t) p j=1 x(t -τ ij ) ∈ PAP (ω,c) (R, R + , µ).
for all 1 ≤ i ≤ n. Now, using the fact that the function (x → e -x ) is Lipschitzian and bounded, and a (ω,c) ∈ PAP(R, R + , µ) is also bounded then invoking the Corollary 4.17 it is clear that

ψ i : t → c ∧ (t)a (ω,c) (t)e -c ∧ (-t)ω i (t) p j=1 x(t-τ ij ) ∈ PAP (ω,c) (R, R + , µ)
for all 1 ≤ i ≤ n.

By using condition (C6) and Theorem 3.26, we can deduce the following Lemma.

Lemma 5.4. Suppose that (H2) and (C6) hold. If x ∈ PAP (ω,c) (R, R + , µ), then the function defined by

t → t -∞ K j (t -s)x(s)ds ∈ PAP (ω,c) (R, R + , µ)
for all 1 ≤ j ≤ m.

Theorem 5.5. Suppose that (H2) and (C1) -(C4) are satisfied. Then the nonlinear operator Γ defined for each x ∈ PAP (ω,c) (R, R + , µ) by

(Γx)(t) = t -∞ e -t s α(ξ)dξ m j=1 a j (s)e -c ∧ (-s)ω j (s) s -∞ K j (s-σ)x(σ)dσ + n i=1 b i (s)e -c ∧ (-s)β i (s) p j=1 x(s-τ ij ) ds maps PAP (ω,c) (R, R + , µ) into itself.
Proof. Using Lemmas 2.13, 5.2, 5.3, 5.4 and Corrolary 4.17, then the function χ defined by

χ(s) = m j=1 a j (s)e -c ∧ (-s)ω j (s) s -∞ K j (s-σ)x(σ)dσ + n i=1 b i (s)e -c ∧ (-s)β i (s) p j=1 x(s-τ ij ) = c ∧ (s) m j=1 (a j ) (ω,c) (s)e -c ∧ (-s)ω j (s) s -∞ K j (s-σ)x(σ)dσ + n i=1 (b i ) (ω,c) (s)e -c ∧ (-s)β i (s) p j=1 x(s-τ ij )
is measure (ω, c)-pseudo almost periodic. Consequently, we can write χ = χ 1 + χ 2 where χ 1 ∈ AP ω,c (R + ) and

χ 2 ∈ E (ω,c) (R, R + , µ). It follows that (Γχ)(t) := t -∞ e -t s α(ξ)dξ χ(s)ds = (Γχ 1 )(t) + (Γχ 2 )(t)
Let us show that (Γχ 1 ) ∈ AP ω,c (R + ). We recall that by applying condition (C1) to the model ( 17

|(χ 1 ) (ω,c) (ξ + η) -(χ 1 ) (ω,c) (ξ)| < ε It comes that (Γχ 1 )(t + η) -(Γχ 1 )(t) = t+η -∞ e -t+η s α(ξ)dξ χ 1 (s)ds - t -∞ e -t s α(ξ)dξ χ 1 (s)ds = t+η -∞ e -t s-η α(ξ+η)dξ χ 1 (s)ds - t -∞ e -t s α(ξ)dξ χ 1 (s)ds = t -∞ e -t s α(ξ+η)dξ χ 1 (s + η)ds - t -∞ e -t s α(ξ)dξ χ 1 (s)ds = t -∞ e -t s α(ξ+η)dξ χ 1 (s + η)ds - t -∞ e -t s α(ξ)dξ χ 1 (s + η)ds + t -∞ e -t s α(ξ)dξ χ 1 (s + η)ds - t -∞ e -t s α(ξ)dξ χ 1 (s)ds
So there exists δ ∈ (0, 1) such that

|(Γχ 1 ) (ω,c) (t + η) -(Γχ 1 ) (ω,c) (t)| ≤ |χ 1 | (ω,c) t -∞ e -t s α(ξ+η)dξ -e -t s α(ξ)dξ ds + t -∞ e -t s α(ξ)dξ (χ 1 ) (ω,c) (s + η) -(χ 1 ) (ω,c) (s) ds ≤ |χ 1 | (ω,c) t -∞ e -t s α(ξ+η)dξ -e -t s α(ξ)dξ ds + ε t -∞ e -t s α(ξ)dξ ds ≤ |χ 1 | (ω,c) t -∞ e -t s α(ξ+η)dξ -e -t s α(ξ)dξ ds + ε t -∞ e -(t-s)α ds ≤ |χ 1 | (ω,c) t -∞ e - t s α(ξ+η)dξ+δ t s α(ξ)dξ-t s α(ξ+η)dξ × t s α(ξ + η) -α(ξ)dξ ds + ε t -∞ e -(t-s)α ds ≤ |χ 1 | (ω,c) t -∞ e -t s α(ξ+η)dξ e -δ( t s α(ξ)dξ-t s α(ξ+η)dξ) t s α(ξ + η) -α(ξ) dξds + ε t -∞ e -(t-s)α ds ≤ ε|χ 1 | (ω,c) t -∞ [e -(t-s)α e -δ( t s α(ξ)dξ-t s α(ξ+η)dξ) (t -s)]ds + ε t -∞ e -(t-s)α ds ≤ ε|χ 1 | (ω,c) t -∞ [e -(t-s)α (t -s)]ds + ε t -∞ e -(t-s)α ds ≤ ε|χ 1 | (ω,c) α 2 + ε t -∞ e -(t-s)α ds ≤ ε|χ 1 | (ω,c) α 2 + ε α = |χ 1 | (ω,c) α 2 + 1 α ε This proves that (Γχ 1 ) ∈ AP ω,c (R + ). Now, let us show that (Γχ 2 ) ∈ E (ω,c) (R, R + , µ). We have that lim r→+∞ 1 µ([-r, r]) r -r t -∞ e -t s α(ξ)dξ χ 2 (s)ds (ω,c) dµ(t) ≤ lim r→+∞ 1 µ([-r, r]) r -r t -∞ e -(t-s)α χ 2 (s) (ω,c) dsdµ(t) ≤ lim r→+∞ 1 µ([-r, r]) r -r ∞ 0 e -αξ χ 2 (t -ξ) (ω,c) dξdµ(t) = lim r→+∞ ∞ 0 e -αξ 1 µ([-r, r]) r -r χ 2 (t -ξ) (ω,c) dµ(t) dξ.
By the Lebesgue dominated convergence Theorem and (H2), we obtain that

lim r→+∞ 1 µ([-r, r]) r -r t -∞ e -t s α(ξ)dξ χ 2 (s)ds (ω,c) dµ(t) = 0.
Then, (Γχ 2 ) ∈ E (ω,c) (R, R + , µ) and consequently, Γ ∈ PAP (ω,c) (R, R + , µ). Proof. First, we proves that the operator Γ is a mapping from R to R. We set

γ j (s) = a j (s)e -c ∧ (-s)ω j (s) s -∞ K j (s-σ)x(σ)dσ θ i (s) = b i (s)e -c ∧ (-s)β i (s) p j=1 x(s-τ ij )
In fact, we have

|Γx(t)| (ω,c) ≤ t -∞ e -t s α(ξ)dξ m j=1 γ j (s) + n i=1 θ i (s) (ω,c) ds = t -∞ e -t s α(ξ)dξ m j=1 γ j (s) + n i=1 θ i (s) (ω,c) ds ≤ t -∞ e -t s α(ξ)dξ m j=1 (a j ) (ω,c) + n i=1 (b i ) (ω,c) ds ≤ t -∞ e -α(t-s) m j=1 (a j ) (ω,c) + n i=1 (b i ) (ω,c) ds = m j=1 a j (ω,c) + n i=1 b i (ω,c) α
In the other hand, if we set

ξ = max c ∧ (τ i,j ) for (i, j) ∈ [[1, n]] × [[1, p]], δ j (s) = c ∧ (-s)ω j (s) s -∞ K j (s -σ)x(σ)dσ, for j ∈ [[1, m]]. ϕ i (s) = c ∧ (-s)β i (s) p j=1 x(s -τ ij ) for i ∈ [[1, n]]. then, we have for x ∈ R |Γx(t)| (ω,c) = t -∞ e -t s α(ξ)dξ m j=1 a j (s)e -δ j (s) + n i=1 b i (s)e -ϕ i (s) ds (ω,c) ≥ c ∧ (-t) t -∞ e -t s α(ξ)dξ m j=1 a j (s)e -ω j s -∞ (K j ) (ω,c) (s-σ)x (ω,c) (σ)dσ + n i=1 b i (s)e -β i p j=1 c ∧ (-τ ij )x (ω,c) (s-τ ij ) ds ≥ c ∧ (-t) t -∞ e -t s α(ξ)dξ m j=1 a j (s)e -ω j U B s -∞ (K j ) (ω,c) (s-σ)dσ + n i=1 b i (s)e -β i U B p j=1 c ∧ (-τ ij ) ds ≥ c ∧ (-t) t -∞ e -t s α(ξ)dξ m j=1 a j (s)e -ω j U B + n i=1 b i (s)e -pξβ i U B ds ≥ t -∞ c ∧ (-s)e -t s α(ξ)dξ m j=1 a j (s)e -ω j U B + n i=1 b i (s)e -pξβ i U B ds = t -∞ e -t s α(ξ)dξ m j=1 (a j ) (ω,c) (s)e -ω j U B + n i=1 (b i ) (ω,c) (s)e -pξβ i U B ds ≥ t -∞ e -α(t-s) m j=1 (a j ) (ω,c) e -ω j U B + n i=1 (b i ) (ω,c) e -pξβ i U B ds = m j=1 (a j ) (ω,c) e -ω j U B + n i=1 (b i ) (ω,c) e -pξβ i U B α
which implies that the operator Γ is a mapping from R to itself. To end the proof, it suffices to prove that Γ is a contraction mapping. Let x, y ∈ R. Then (a j ) (ω,c) (s) e -c ∧ (-s)ω j (s) s -∞ K j (s-σ)x(σ)dσ -e -c ∧ (-s)ω j (s) s -∞ K j (s-σ)y(σ)dσ (a j ) (ω,c) e -c ∧ (-s)ω j (s) s -∞ K j (s-σ)x(σ)dσ -e -c ∧ (-s)ω j (s) s -∞ K j (s-σ)y(σ)dσ (a j ) (ω,c) e -c ∧ (-s)ω j (s) s -∞ K j (s-σ)x(σ)dσ -e -c ∧ (-s)ω j (s) s -∞ K j (s-σ)y(σ)dσ

Example

In order to illustrate some features of our theoretical study, we will apply our main results to a special system and demonstrate the efficiencies of our criteria. We consider the following Lasota-Wazewska model with mixed delays y ′ (t) = -α(t)y(t) + 3 j=1 a j (t)e -c ∧ (-t)ω j (t) t -∞ K j (t-s)y(s)ds + b i (t)e -c ∧ (-t)β i (t)y(t-τ i )

where α(t) = 8 + cos 2 + e -0.9 1 2 3

2 + e -0.9 1 2 3

2 + e -0.9

Proposition 4 . 8 .

 48 Let F : R × Y → X be a continuous function. Then F ∈ AP ω,c (Y, X) if and only if the following conditions hold :

5. 1

 1 Existence and uniqueness of µ-(ω, c)-pseudo almost periodic solution to the model.

  ), α is almost periodic (i.e. (ω, c)almost periodic with constant c = 1). Now, in the view of the almost periodicity of the function α and the (ω, c)-almost periodicity of the function χ 1 , there exists a number l ε such that in any interval [δ, δ + l ε ] one finds a number η, such that sup ξ∈R |α(ξ + η) -α(ξ)| < ε and sup ξ∈R

Theorem 5 . 6 .

 56 Assume that (H2) and (C1) -(C6) hold, then the Lasota-Wazewska model with mixed delays (17) possess a unique measure (ω, c)-pseudo almost periodic solution y and we have y (ω,c) in the regionR = {ψ ∈ PAP (ω,c) (R, R + , µ), LB ≤ |ψ (ω,c) ) (ω,c) e -ω j U B + n i=1 (b i ) (ω,c) e -pξβ i U B α

  |Γx(t) -Γy(t)| (ω,c) s) e -c ∧ (-s)ω j (s) s -∞ K j (s-σ)x(σ)dσ -e -c ∧ (-s)ω j (s) s -∞ K j (s-σ)y(σ)dσ + n i=1 b i (s) e -c ∧ (-s)β i (s) p j=1 x(s-τ ij ) -e -c ∧ (-s)β i (s) p j=1 y(s-τ ij ) s) e -c ∧ (-s)ω j (s) s -∞ K j (s-σ)x(σ)dσ -e -c ∧ (-s)ω j (s) s -∞ K j (s-σ)y(σ)dσ + n i=1 b i (s) e -c ∧ (-s)β i (s) p j=1 x(s-τ ij ) -e -c ∧ (-s)β i (s) p j=1 y(s-τ ij )

  ) (ω,c) (s) e -c ∧ (-s)β i (s) p j=1 x(s-τ ij ) -e -c ∧ (-s)β i (s) p j=1 y(s-τ ij )

  ) (ω,c) e -c ∧ (-s)β i (s) p j=1 x(s-τ ij ) -e -c ∧ (-s)β i (s) p j=1 y(s-τ ij )

  ) (ω,c) e -c ∧ (-s)β i (s) p j=1 x(s-τ ij ) -e -c ∧ (-s)β i (s) p j=1 y(s-τ ij ) dsObviously, for u, v ∈ [0, +∞) |e -u -e -v | ≤ |u -v| then it comes that |Γx(t) -Γy(t)| (ω,c) ) (ω,c) c ∧ (-s)ω j (s) s -∞ K j (s -σ)(x(σ) -y(σ))dσ + n i=1 (b i ) (ω,c) c ∧ (-s)β i (s) p j=1 (x(s -τ ij ) -y(s -τ ij )) ds ≤ ) (ω,c) ω j s -∞ (K j ) (ω,c) (s -σ)dσ|x -y| (ω,c) + pξ|x -y| (ω,c) n i=1 (b i ) (ω,c) β i ds = ) (ω,c) ω j + pξ n i=1 (b i ) (ω,c) β i ds|x -y| (ω,c) ≤ m j=1 (a j ) (ω,c) ω j + pξ n i=1 (b i ) (ω,c) β i α |x -y| (ω,c)which implies (invoking (C5)) that the mapping Γ is a contraction mapping of R. Consequently, Γ possess a unique fixed point x ⋆ ∈ R. Hence, x ⋆ is the unique measure (ω, c)-pseudo almost periodic solution of Equation[START_REF] Larrouy | ω,c)-periodic and asymptotically (ω,c)-periodic mild solutions to fractional Cauchy problems[END_REF] in R.

3 i=1

 3

τ 1 = 1

 11 τ 2 = τ 3 = 1, K j = (0.9) ∧ (t)e -t . Thenm j=1 (a j ) (ω,c) ω j + pξ n i=1 (b i ) (ω,c) β i If the Radon-Nikodym derivative ρ of the measure µ is ρ(t) = esin t with respect to the Lebesgue measure on R (i.e. dµ = ρ(t)dt), then µ ∈ M and satisfies (H1), since µ([-r, r]) = r -r e sin(t) → +∞, if r → +∞, and µ(τ + a) ≤ e 2 µ(A), ∀τ ∈ R and a ∈ A. Hence, conditions (C1) -(C6) and (H2) are satisfied then according to the Theorem 5.6, the Lasota-Wazewska model with a mixed delays (18) has a unique µ-(ω, c)-pseudo almost periodic solution in the region R = {y ∈ PAP (ω,c) (R, R + , µ), LB ≤ |y (ω,c) | ≤ U B} where ) (ω,c) e -ω j U B + n i=1 (b i ) (ω,c) e -pξβ i U B

1

 1

  ∧ (t) 1 + 0.25 cos 2 ( √ 5t) + 0.25 cos 2 (πt) + 0.5e -t 2 cos 2 (t) (0.9) ∧ (t) 1 + 0.25 cos 2 ( √ 5t) + 0.25 cos 2 (πt) + 0.5e -t 2 cos 2 (t) (0.9) ∧ (t) 1 + 0.25 cos 2 ( √ 5t) + 0.25 cos 2 (πt) + 0.5e -t 2 cos 2 (t) (πt) + 0.25e -t 2 cos 2 (t)

					 	ω 1 (t) ω 2 (t) ω 3 (t)	  =	  	0.125 cos 2 ( √ 0.125 cos 2 ( √ 0.250 cos 2 ( 2t) + 0.125 cos 2 (πt) + 0.25 1+t 2 2t) + 0.125 cos 2 (πt) + 0.25 1+t 2 √ 2t) + 0.25e -t 2 cos 2 (t)	  
						 (0.9) 
	 	b 1 (t) b 2 (t) b 3 (t)   =	      	      
			 	β 1 (t) β 2 (t) β 3 (t)	  =	   0.125 cos 2 ( 0.125 cos 2 ( √ 0.125 cos 2 ( √ 2t) + 0.125 cos 2 (πt) + 0.25 1+t 2 2t) + 0.125 cos 2 (πt) + 0.25 1+t 2 √ 2t) + 0.125 cos 2
						2 ( √	5t) + cos 2 (t), c = 0.9, ω = 1,
	 	a 1 (t) a 2 (t) a 3 (t)	  =	        (0.9) ∧ (t) 0.5 + 0.25 cos 2 ( (0.9) ∧ (t) 1 + 0.25 cos 2 ( (0.9) ∧ (t) 0.5 + 0.25 cos 2 ( √ 2t) + 0.25 cos 2 (πt) + 0.5 1+t 2 √ 3t) + 0.25 cos 2 (πt) + 1 1+t 2 √ 5t) + 0.25 cos 2 ( √ 2t) + e -t 2 cos 2 (t)	       

Now, we intend to prove that E (ω,c) (R, X, µ) is translation invariant.

For µ ∈ M and τ ∈ R, we denote µ τ the positive measure on (R, B) defined by

(

We need to formulate the following hypotheses for µ ∈ M and to recall some lemmas.

(H1) For all a, b, c ∈ R such that 0 ≤ a < b ≤ c, there exists τ 0 ≥ 0 and α 0 such that

(H2) For all ∈ R, there exists β > 0 and a bounded interval I such that

). (H2) implies (H1).

Lemma 3.11 ( [3]

). Let m ∈ M satisfying (H2). Then the measures µ and µ τ are equivalent for all τ ∈ R.

We can prove the following result.

Theorem 3.13. Let µ ∈ M satisfying (H2). Then E (ω,c) (R, X, µ) is translation invariant.

Proof. Let f ∈ E (ω,c) (R, X, µ) and τ ∈ R. Since µ(R) = +∞, there exists r 0 > 0 such that µ [-r -|τ |, r + |τ |] > 0 for all r ≥ r 0 . In this proof, we always assume that r ≥ r 0 . Let us denote by

∥f (ω,c) (t)∥dµ(t), for r > 0 and τ ∈ R

where µ τ is the positive measure defined by [START_REF] Abadias | Periodic Mild Solutions to Non-Autonomous Abstract Differential Equations[END_REF]. By using Lemma 3.11, it follows that µ τ and µ are equivalent, then by using Theorem 3.9 we have

For all A ∈ B, we denote by χ A the characteristic functions of A; By using definition of µ τ we have that

for all A ∈ B and since t → ∥f (ω,c) (t)∥ is the pointwise limit of an increasing sequence of linear combination of characteristic functions (see [START_REF] Rudin | Real and Complex Analysis[END_REF] Theorem 1.17), we deduce that ∥f (ω,c) (t -τ )∥dµ(t) [START_REF] Cherif | New results for a Lasota-Wazewska model[END_REF]


L Manceau

Keywords: . 25.75.-q Relativistic heavy-ion collisions -12.38.Mh Quark-gluon plasma -25.75.Nq Quark deconfinement, quark-gluon plasma production, and phase transitions

ALICE (A Large Ion Collider Experiment) is the LHC detector designed to measure nucleusnucleus (AA) collisions where the formation of the Quark Gluon Plasma is expected. The experiment will also study proton-proton (pp) collisions at 14 TeV. Amongst the relevant observables to be investigated in pp collisions, the B-hadron cross-section is particularly interesting since it provides benchmarks for theoretical models and it is mandatory for understanding heavy flavour production in AA collisions. The performances of the ALICE muon spectrometer to measure B-hadron cross-section in pp collisions at 14 TeV via single muons are presented.

B-hadron cross-section measurement in pp collisions at √ s = 14 TeV with the ALICE muon spectrometer

Introduction

The prime goal of relativistic heavy-ion collisions is to study QCD matter in extreme conditions of temperature. With a nucleon-nucleon center-of-mass energy of √ s N N = 5. 5 TeV which exceeds the one reached at RHIC by about a factor 30, the LHC (Large Hadron Collider) will bring new insights in heavy-ion physics. The energy density should reach 15 -60 GeV/fm 3 . At such energy densities, QGP is expected to be created. One of the most important aspects of this energy range will be the abundant production of hard probes which could be used, for the first time, as high statistics probes of the medium. The charm and beauty production crosssection (see Tab. 1) should be 10 and 100 times, respectively, larger than at RHIC [1].

Table 1. NLO pQCD predictions in pp collisions at √ s = 14 TeV for the number of charm and beauty quark-antiquark pairs per event and corresponding production cross-section [2]. The main motivation to measure the B-hadron crosssection in pp collisions is to test Next-to-Leading-Order perturbative QCD (NLO pQCD) calculations which still predict cross-sections with large uncertainties [3]. Measuring the B-hadron cross section in pp collisions is also mandatory for understanding:

N/evt. σ Q Q (mb) cc 0.
-B-hadron cross-section in pA collisions, where the study of shadowing and anti-shadowing should give information on gluon Parton Distribution Funtions (PDF) in the nucleus;

-B-hadron cross-section in AA collisions where the study of heavy flavour energy loss through hot QCD matter should give information on the initial density of gluons and on the dissipative properties of the medium.

One can mention that one expects that light quarks exhibit a weak sensitivity to these effects. Indeed, they lose more energy than heavy quarks;

production cross-sections of bottomonia in pp, pA and AA collisions;

production cross-section of secondary J/ψ in pp collisions which is a benchmark for understanding its production in pA and AA collisions1 .

2 The ALICE detector ALICE (A Large Ion Collider Experiment) is the LHC experiment dedicated to the study of nucleus-nucleus collisions [4]. More than 1000 collaborators from about 30 countries are presently participating to ALICE. The AL-ICE experiment is designed to perform simultaneous and high precision measurements of numerous observables based on hadrons, leptons and photons, in a broad acceptance.

The detector consists of a central barrel, a forward muon spectrometer and forward/backward small acceptance detectors. The central barrel of ALICE is made of four layers of detectors placed in the solenoidal field (B < 0.5 T) provided by the L3 magnet. From the inner side to the outer side, these detectors are i) the Inner Tracker System (ITS) consisting of six layers of silicon detectors, ii) the large-volume Time Projection Chamber (TPC), iii) the high-granularity Transition Radiation Detector (TRD) and iv) the high-resolution Time of Flight system (TOF) based on multi-gap resistive plate chambers. They provide charged particle reconstruction and identification in the pseudo-rapidity range |η| < 0.9, with full azimuthal coverage and a broad transverse momentum (p t ) acceptance. The ALICE central barrel will later also be equipped with a large acceptance (|η| < 1.4, ∆Φ = 110 • ) Electro-Magnetic CALorimeter (EMCAL). These large area devices are complemented by two smaller acceptance detectors: the High Momentum Particle IDentification (HMPID) and the PHOton Spectrometer (PHOS). In the forward/backward region, additional detectors allow fast characterization and selection of the events as well as charged particle measurement in the pseudo-rapidity ranges -3.4 < η < -1.7 and 1.7 < η < 5.0. The Photon Multiplicity Detector (PMD) will measure photon multiplicity in the pseudo-rapidity range 1.8 < η < 2.6. At large rapidities, spectator nucleons in heavy-ion collisions will be measured by the Zero-Degree Calorimeters. Finally a forward muon spectrometer [5] covering the pseudorapidity range -4 < η < -2.5 complements the central part. It consists of a front absorber, a dipole magnet, ten high-granularity tracking chambers, a muon filter and four large area trigger chambers.

Simulation inputs

We use the simulated data produced within the Physics Data Challenge 2006 (PDC06) via the computing grid. Events have been simulated and reconstructed with the AliRoot framework [6]. The simulation is done by means of the PYTHIA 6.2 Monte Carlo event generator with CTEQ5L PDFs (parameters are described in [1]). In order to optimize the statistics and to reduce the computing time, events were reconstructed only if at least one muon with p t larger than 0.5 GeV/c is produced in the acceptance of the muon spectrometer. This cut is applied at the generation of the events, when π and K are not yet desintegrated in muons. Thus, the yield of muons from light hadron decays is strongly underestimated. In the analysis presented here, we focus on p t > 2 GeV/c range and background from π, K decay is perfectly substracted. The reconstructed muon distributions then consist exclusively of muons from charm and beauty decay. These are corrected for detection efficiency by means of the method described in [START_REF] Del Valle | Conesa del Valle "Performance of the ALICE muon spectrometer. Weak boson production and measurement in heavy-ion collisions at LHC[END_REF]8]. The mean value of the efficiency is about 96%.

Whereas the Pythia setting in the PDC06 simulations allows to consistently reproduce the beauty production cross-section as predicted by NLO pQCD calculations (Tab. 1), it underestimates the charm production cross-section by a factor ∼ 2. This is taken into account by scaling up the muon distributions accordingly. Furthermore, the statistics available in the PDC06 production allows to exploit the muon p t distribution up to ∼ 10 GeV/c only. In order to compensate for this lack of statistics, the muon distributions have been fitted and then extrapolated up to p t = 20 GeV/c. A successful description of the muon p t distributions is obtained for both charm and beauty with the following function [1]:

f = a × 1 (1 + ( pt b ) 2 ) c (1)
where a, b and c are free parameters. Note that beyond p t = 20 GeV/c muons from W ± decay dominate [9]. Figure 1 shows the p t distributions of muons from D and Bhadron decays scaled and extrapolated as described previously. The charm/beauty crossing point is located at p t ∼ 6 GeV/c. Beauty dominates at high transverse momentum and charm dominates at low transverse momentum. It is worth noticing that the value of the crossing point depends on the quark masses, the QCD factorization and renormalization scales, the fragmentation functions and the PDFs. For example, using the inputs provided in the report by the joined HERA-LHC groups [3], a shift of the crossing point by approximately ±2 GeV/c is expected.

Method

The method to extract the differential inclusive B-hadron cross-section is widely used and well documented. It was developed by the UA1 collaboration [10] and is used by the CDF [11] and the D0 [12] collaborations. It has also been used by the ALICE collaboration to investigate the feasibility of measuring the B-hadron cross-section in pp collisions at √ s = 14 TeV with single electrons [START_REF] Antinori | ALICE collaboration internal note[END_REF], in Pb-Pb collisions at √ s N N = 5.5 TeV with single electrons [START_REF] Antinori | ALICE collaboration internal note[END_REF], single muons and dimuons [START_REF] Guernane | ALICE collaboration internal note[END_REF]. This method consists in three steps:

1. Extraction of the number of muons from b semileptonic decays by performing a combined fit of the muon p t distribution. The fitting function is:

(T -B)(f c + R × f b ). (2)
In this formula, T is the total number of muons, B is the number of muons from b semileptonic decays (it is a free parameter), f c and f b are model dependent shapes of the p t distribution of muons from c and b semileptonic decays, respectively; R is the number of muons from b semileptonic decays over the number of muons from c semileptonic decays (it is a free parameter fixed within 30% as we assume that models [3] predict R with uncertainties smaller than 30%). 2. Correction of this number of muons for integrated luminosity ( Ldt), detection efficiency ( ), acceptance and decay kinematics (F (µ ← B)). This correction is achieved according to Eq.( 3). The F (µ ← B) factor is obtained from a Monte Carlo simulation.

σ B = dσ B dη (p B t > p min t )| -4≤η≤-2.5 = Nµ←B(Φ µ ) Ldt × 1 × dσ B dη (p B t >p min t )| -4≤η≤-2.5 σ B (Φ µ ) M C = Nµ←B(Φ µ ) Ldt × 1 × F (µ ← B.) (3) In Eq. 3 i) dσ B dη (p B t > p min t )| -4≤η≤-2.
5 is the integral of the p t spectrum of B-hadrons with rapidity -4 ≤ η B ≤ -2.5 and p B t > p min t , ii) σ B (Φ µ ) is the integral of the p t distribution of B-hadrons decaying into muons within a given phase space Φ µ (-4 ≤ η µ ≤ -2.5, p µ ≥ 4 GeV/c and a given p µ t range), and iii) p min t is defined such that 90% of accepted B-hadrons have a p t larger than that value (see Fig. 2). We choose this definition to minimize the acceptance and decay kinematics correction dependence on shapes used in Pythia for the Monte Carlo computation of F (µ ← B). 3. Repetition of step one and two for different Φ µ (i.e. different p µ t ranges) to extract differential inclusive Bhadron cross-section (Fig. 5).

Expected yields

We performed statistics estimates for three different scenarios of data taking: scenario 1: one month of data taking at low luminosity, L = 10 30 cm -2 s -1 , t = 10 6 s;

scenario 2: one month of data taking at nominal luminosity, L = 3 • 10 30 cm -2 s -1 , t = 10 6 s;

scenario 3 (nominal scenario): one year of data taking at nominal luminosity, L = 3•10 30 cm -2 s -1 , t = 10 7 s.

Figure 3 shows that the muon yield is large over a broad p t range even in scenario 1. This yield leads to low statistical uncertainties. Even in scenario 1 at a p t of 20 GeV/c, the relative statistical uncertainty is as low as 2%. The p t range shown in Fig. 3 goes from 2 GeV/c to 20 GeV/c. Indeed, below 2 GeV/c and beyond 20 GeV/c muons from π, K and W decay dominate, respectively.

Systematic uncertainties

In order to determine the systematic uncertainties from the method, we have used NLO pQCD predictions with different assumptions on the quark masses, the QCD factorization and renormalization scales and the PDFs [3]. As these predictions are available at the hadron level only, we have first performed Pythia simulations in order to reproduce them. Then, we have extracted from the corresponding decay muon p t distributions biased f c and f b shapes. The latter are used as inputs for the combined fit from which we have extracted the systematic uncertainties. Figure 4 shows two examples of fits obtained with these biased f c and f b shapes. The systematic uncertainties run from 17% to 22%. It is important to note from Fig. 4 that the χ 2 gives a good indication of the fit quality and should therefore allow to constrain the systematics. reconstructed (squares). Statistical errors are negligible (section 5). The main source of uncertainties is the systematics from the fit (section 6) which is about 20%. As the choice of p min t (section 4) minimize the F (µ ← B) dependence on shapes used in Pythia the systematic errors from F (µ ← B) are negligible. Statistical errors from F (µ ← B) run from 0.4% to 15%. Note that this error can be further reduced since it only depends on the Monte-Carlo simulation used for computing the F (µ ← B) factor. A 5% normalisation uncertainty (uncertainties on luminosity, pp cross-section) is not included in figure 5.

Conclusion

The B-hadron cross-section in pp collisions is an essential measurement for a number of physics items in pp, pA as well as in AA collisions at the LHC. We have shown that this measurement can be successfully performed from single muons in pp collisions at √ s = 14 TeV using a method first developed by the UA1 collaboration and further used by the CDF and the D0 collaborations. The inclusive differential B-hadron cross-section can be reconstructed from low p t up to p t ∼ 20 GeV/c. The statistical error is negligible and the systematic one is of the order of 20%. In the present work the contribution from muons from π, K decays were assumed to be negligible. In the next steps, a realistic background subtraction should be performed. We note that the same method can be applied in the electron channel in the central barrel of the AL-ICE experiment with different statistical and systematical uncertainties. Thus, performing the two analysis will provide useful cross-checks. We finally note that the Dhadron cross-section can be simultaneously extracted with the exposed method.

a

  Loic.Manceau@clermont.in2p3.fr

Fig. 1 .

 1 Fig. 1. Transverse momentum distributions of muons from c and b quark decays (statistics from scenario 1, see section 5).

Fig. 2 .

 2 Fig. 2. Illustration of F (µ ← B) computation. Horizontal line area: inclusive B-hadron production in the acceptance. Veritcal line area: B hadrons which yield muons of 2 GeV/c ≤ pt ≤ 3 GeV/c in the acceptance of the spectrometer.

1 Fig. 3 .

 13 Fig. 3. Muon yield versus pt for the three scenarios of data taking described in the text.

Fig. 4 .

 4 Fig.4. Examples of combined fits for two combinations of biased shapes as described in the text. The dotted lines schow the result of the fit and the histograms are input distributions. The χ 2 of the fits are reported on the plots.

Figure 5 Fig. 5 .

 55 Figure5shows the inclusive differential B-hadron crosssection. As it can be seen, the input (histogram) is well

The ratio of secondary J/ψ from B-hadron decay to direct J/ψ is expected to reach

20% at the LHC (assuming no hot medium effects).