Convolutions in μ-pseudo almost periodic and μ-pseudo almost automorphic function spaces and applications to solve Integral equations

Abstract:The aimof thiswork is to give su cient conditions ensuring that the space PAP(R, X, μ)of μ-pseudo almost periodic functions and the space PAA(R, X, μ) of μ-pseudo almost automorphic functions are invariant by the convolution product ζf = k * f , k ∈ L1(R). These results establish su cient assumptions on k and the measure μ. As a consequence, we investigate the existence and uniqueness of μ-pseudo almost periodic solutions and μ-pseudo almost automorphic solutions for some abstract integral equations, evolution equations and partial functional di erential equations.


Introduction
µ-pseudo almost periodic functions and µ-pseudo almost automorphic functions have been studied by several authors in the last decade. They have produced extensive literature on the theory of almost periodic, pseudo-almost periodic, almost automorphic, pseudo-almost automorphic functions and their applications to di erential equations. Details can be found in [4-6, 9-11, 14, 20-23] and the references therein. Recently, Ezzinbi, Blot and Cieutat [1,2] introduced the concepts of µ-pseudo-almost periodic functions and µ-pseudo-almost automorphic functions, which are natural generalizations of the classical weighted pseudoalmost periodic functions introduced by Diagana [32] and weighted pseudo-almost automorphic functions in the sense of Blot et al. [8] respectively. Let us recall the meaning of these notions as introduced by Ezzinbi et al. [1,2]. Let µ be a positive measure on R and X a Banach space. A continuous function f : R → X is µ-pseudo almost periodic (respectively µ-pseudo almost automorphic) if where g is almost periodic (respectively almost automorphic) and ϕ is ergodic with respect to the measure in the sense that ϕ : R → X is a bounded continuous function such that When the measure µ is absolutely continuous with respect to the Lebesgue measure dt and its Radon-Nikodym derivative is ρ, a µ-pseudo almost periodic (respectively µ-pseudo almost automorphic) function is simply said to be a ρpseudo almost periodic (respectively ρ-pseudo almost automorphic) function. Motivated by the recent work of Ezzinbi et al. [1,2] who gave one su cient condition for the convolution invariance of PAP(R, X, µ) and PAA(R, X, µ), and the work of Coronel et al. [3] who gave several di erent conditions for convolution invariance of PAP(X, ρ) where ρ is a weight, the goal of this work is to give new su cient assumptions for convolution invariance of PAP(R, X, µ) and PAA(R, X, µ). Indeed, since k * g is almost periodic (respectively almost automorphic) [1,2], then the following two assertions are equivalent: (i) PAP(R, X, µ) or PAA(R, X, µ) is convolution invariant.
(ii) E(R, X, µ) is convolution invariant. The results of this study are more general than [1][2][3] and give new concepts of weighted pseudo almost periodic functions and weighted pseudo almost automorphic functions in the context of measures. Furthermore, we investigate the existence and uniqueness of µ-pseudo almost periodic (respectively µpseudo almost automorphic) mild solutions to the following equations: The rest of this work is organized as follows. In section we introduce the concepts of µ-pseudo almost periodic and µ-pseudo almost automorphic functions and we recall composition results that will be used. In section , we state and prove our main results about convolution on µ-pseudo almost periodic and µ-pseudo almost automorphic function spaces; we also recall results of translation invariance of the spaces above. In section we illustrate our results with some applications to integral equations, evolution equations, partial functional di erential equations in Banach spaces.

Preliminaries . µ-pseudo almost periodic and µ-pseudo almost automorphic notions
In this section we recall the concept of µ-ergodic, µ-pseudo almost periodic, µ-pseudo almost automorphic under the light of measure theory developed in [1,2] and we also recall useful results. Let (X, · ), (Y , · ) two Banach spaces and let BC(R, X) (respectively, BC(R × Y , X)) be the space of bounded continuous functions f : R → X (respectively, jointly bounded continuous functions f : We denote by B the Lebesgue σ-eld of R and by M the space of all positive measures µ on B satisfying µ(R) = +∞ and µ([a, b]) < ∞ for all a, b ∈ R (a ≤ b).
We denote the space of all such functions by E(R, X, µ). Then E(R, X, µ), . ∞ is a Banach space.

De nition 2.2. [2] Let µ ∈ M.
A continuous function f : R × Y → X is said to be µ-ergodic in t uniformly with respect to x ∈ Y, if the following two conditions are true The space of such functions is denoted by EU(R × Y , X, µ).

De nition 2.3. [4]
A continuous function f : R → X is said to be almost periodic if for every ε > there exists a positive number lε such that every interval of length lε contains a number τ such that where g ∈ AP(R, X) and ϕ ∈ E(R, X, µ).
We denote the space of all such functions by PAP(R, X, µ). We have AP(R, X) ⊂ PAP(R, X, µ) ⊂ BC(R, X).
Equivalently, a continuous function f : R → X is said to be almost automorphic if and only if We denote such functions by AA(R, X). We recall that (AA (R, X) , . ∞) is a Banach space.
Then k is almost automorphic, but not almost periodic since it is not uniformly continuous on R. Then AP (R, X) AA (R, X).

Convolution on E(R, X, µ)
De nition 3.1.  We shall give new general assumptions on µ ∈ M, ρ ∈ L (µ) and k ∈ L (R) such that maps E(R, X, µ) into itself. We consider that the µ-mean, exists. In this section, by using the Lebesgue-Radon-Nikodym Theorem above, we consider that µ ∈ M, µ = µ + µ where µ is the µ-measure component which is absolutely continuous with respect to the Lebesgue measure dt and its Radon-Nikodym derivative is ρ, that is dµ (t) = ρ(t)dt and µ is the µ-measure component such that µ is singular to the Lebesgue measure dt.
Theorem 3.5. Let k ∈ L (R) and µ ∈ M, with Radon-Nikodym derivative ρ and ζ be de ned by ( ). Assume that ρ, µ and k satisfy the following assumptions: Proof. We adapt Coronel et al. in [3]. By the properties of convolution we have that f ∈ BC(R, X) implies that ζf ∈ BC(R, X), ∀k ∈ L (R). Then, in order to get that ζf ∈ E(R, X, µ) we must to prove that M( ζf X ) = .
We consider µ ∈ M and ρ its Radon-Nikodym derivative. In the rst stage we assume that k(t) = on R * − . From µ(R) = +∞, we deduce the existence of r ≥ such that µ([−r, r]) > , ∀r ≥ r . Then by applying the Fubini Theorem, for r ≥ r , we deduce that for f ∈ BC(R, X) Using assumptions ( . ), ( . ) and the fact that f ∈ E(R, X, µ), this concludes this stage. Now, in the second stage, proceeding similarly like in the rst stage, we assume that k(t) = on R * + we obtain: We conclude using the fact that f ∈ E(R, X, µ) and assumptions ( . ), ( . ). The case of a general k we deduce the result similarly using the fact that k(t) = kχ t≥ (t) + kχ t< (t).
In the rst stage we assume that k(t) = on R * − . From µ(R) = +∞, we deduce the existence of r ≥ such that µ([−r, r]) > , ∀r ≥ r . Then by applying the Fubini-Tonelli Theorem, we deduce that By assumption ( . ) we have that We have, for all r ≥ r So in view of the Lebesgue dominated convergence Theorem, we obtain This concludes this stage of ( . ). Now, in the second stage, proceeding like in the rst stage, we assume that k(t) = on R * + and by applying the Fubini's Theorem, we deduce that Like in the rst stage, we use assumptions ( . ), and the Lebesgue dominated convergence Theorem. This concludes this second stage of ( . ).
Corollary 3.7. Let µ ∈ M be such that the nonnegative B-measurable function ρ be its Radon-Nikodym derivative. Assume that for all k ∈ L (R) the following requirements are satis ed: Then E(R, X, µ) is convolution invariant. Corollary 3.8. Let µ ∈ M be such that the nonnegative B-measurable function ρ be its Radon-Nikodym derivative. Assume that the following requirements are satis ed: Then E(R, X, µ) is convolution invariant.
which implies that ( . ) is satis ed. We also have that Now we give another assumption about the convolution invariance of E(R, X, µ). This condition was focused by Ezzinbi et al. in [1,2]. For µ ∈ M and τ ∈ R, we denote by µτ the positive measure on (R, B) de ned by From µ ∈ M, we formulate the following hypothesis (H ): For all τ ∈ R, there exists β > and a bounded interval I such that Theorem 3.10. [2] Let µ ∈ M satisfy (H ). Then E(R, X, µ) is convolution invariant.
We also give some consequences of the Theorem . . Let us start by recalling that Ezzinbi et al. [1,2] proved that if f ∈ AP(R, X) (or f ∈ AA(R, X)), then f is invariant by ζ i.e. ζf ∈ AP(R, X) (or ζf ∈ AA(R, X)). Moreover, they prove the following two results.

. Integral equations
Consider the following integral equation where f and R satisfy the following hypothesis  Proof. We use the proof of Coronel et al. in [3]. Let us consider the operator

. Evolution equations
In this section, we study the existence of µ-pseudo almost automorphic (respectively µ-pseudo almost periodic) solutions of the following evolution equation in a Banach space X where A : D(A) ⊂ X → X is the in nitesimal generator of a c -semigroup (T(t)) t≥ of bounded linear operators on X and f : R × X → X a continuous function. Moreover, we assume that (H ):

It follows from (H )
Then, we have Since α < ω M , then ψ is a strict contraction. Using the Banach xed point Theorem, there exists a unique x ∈ PAA(R, X, µ) (respectively x ∈ PAP(R, X, µ)) such that ψ(x)(t) = x, that is x satis es ( ), so x is a mild solution on R of equation ( ), which is µ-pseudo almost automorphic (µ-pseudo almost periodic). This complete the proof.

. Partial functional di erential equations
Here we propose another application to partial functional di erential equations. We consider the following equation where A : D(A) → X is a linear operator (not necessarily densely de ned) on a Banach space X and e : R → X is a continuous function. Let us denote by C := C([−r, ]; X); r > the Banach space of continuous functions from [−r, ] to X endowed with the uniform norm topology. The operator L is a bounded linear operator from C to X and for x ∈ C(R; X) and t ∈ R, the history function x t ∈ C is de ned by We assume that A satis es the Hille-Yosida condition, which means that there exist M ≥ and ω ∈ R such that (ω, +∞) ⊂ ρ(A) and || (λI − A) −n ||≤ M (λ − ω) n , for n ∈ N * and λ > ω, where ρ(A) is the resolvent set of A. To equation ( ), we associate the following initial value problem x ′ (t) = Ax(t) + L(x t ) + e(t), ∀t ≥ , For more details about this topic, we refer the reader to the book [29] which provides several examples to illustrate the fact that partial functional di erential equations arise from various physical systems.

Lemma 4.4. [9] A generates a C -semigroup T (t) t≥ on D(A).
For the existence of integral solutions, we have the following result.

Theorem 4.5. [9, 28] For all φ( ) ∈ D(A), equation ( ) has a unique integral solution x on [r, +∞).
Moreover, The phase space C of equation ( ) is de ned by For all t ≥ de ne the linear operator V(t) on C by φ) is the integral solution of the following linear equation

Then, A V is the in nitesimal generator of the semigroup V(t) t≥ on C .
In order to give the variation of constants formula, we need to recall some notations and results which are taken from [27]. Let ≺ X be the space de ned by ≺ X = {X c : c ∈ X}, where the function X c is de ned by The space C ⊕ ≺ X endowed with the norm is a Banach space. We denote by A V the continuous extension of the operator A V de ned on C ⊕ ≺ X by In the following, we assume: (E1) the operator T (t) is compact on D(A) for every t > .
As a consequence from the compactness property of the operator V(t), we deduce that spectrum σ(A T ) is the point spectrum. Moreover, we have the following lemma. De nition 4.12. [1] We say that the semigroup (V(t)) t≥ is hyperbolic if Theorem 4.13. [9] Assume that (E ) holds and the semigroup (V(t)) t≥ is hyperbolic. Then the space C is decomposed as the direct sum of the stable and unstable subspaces C = S ⊕ U of two V(t)-invariant closed subspaces S and U and there exist K > and c > such that Theorem 4.14. [9] Assume that (E ) holds and the semigroup (V(t)) t≥ is hyperbolic. If e is bounded continuous on R, then equation ( ) has a unique bounded integral solution on R which is given by the following formula where P s (respectively P u ) is the projection of C onto S the stable (respectively U the unstable) subspace.
We give the main result of this part, which shows the existence and uniqueness of the µ-pseudo almost automorphic (respectively µ-pseudo almost periodic) solution if the input function e is µ-pseudo almost automorphic (respectively µ-pseudo almost periodic).
With those notations, the unique bounded solution x t can be written in the following form We have to show that both functions Q S e and Q U e are µ-pseudo almost automorphic (respectively µ-pseudo almost periodic). In fact, e is µ-pseudo almost automorphic (respectively µ-pseudo almost periodic), then e = g + ϕ where g ∈ AA(R, X) (respectively g ∈ AP(R, X)) and ϕ ∈ E(R, X, µ). Consequently and we have to show that Q S g and Q U g are almost automorphic(respectively almost periodic); and Q S ϕ and Q U ϕ are µ-ergodic.
Firstly if e is µ-pseudo almost automorphic, let (sm) m≥ be a real sequence. Since g is almost automorphic, then there exists a sub-sequence of (sm) m≥ denoted by (σm) m≥ and a function g such that Now for t ∈ R, we have which give for t ∈ R We will show that lim m→+∞ Q S g (t + σm) = Q S g (t) and lim m→+∞ Q S g (t − σm) = Q S g t) and by using the Hille-Yosida condition on A T one can nd a positive constant K such that Using the Lebesgue dominated convergence Theorem, we obtain that Proceeding as previously, one can show that lim m→+∞ Q S g (t − σm) = Q S g (t), for t ∈ R.
Consequently,Q S g is almost automorphic. Similar, for t ∈ R, one can show that Q U g is almost automorphic. secondly if e is µ-pseudo almost periodic, then e = g + ϕ where g is almost periodic and ϕ ∈ E(R, C, µ). Therefore We have (Q S gτ) = (Q S g)τ, for τ ∈ R. By using the continuity of the operator Q S , we deduce that Q S ({gτ : τ ∈ R}) is relatively compact in BC(R, C), this implies that Q S g ∈ AP(R, C). Using same argument as above, we can prove that Q U g ∈ AP(R, C). It remains to prove that Q S ϕ and Q U ϕ are µ-ergodic. Let e, a µ-pseudo almost automorphic (respectively µpseudo almost periodic) function. By using the Hille-Yosida condition on A T , one can nd a positive constant K such that Since t → ϕ(t) ∈ E(R, R, µ), we deduce from Corollary . , Corollary . that and from ( ) we obtain that Q S ϕ ∈ E(R, C, µ). Using the argument as above, we can prove that Q U ϕ ∈ E(R, C, µ).

Examples
Example 5.1. Let X = L ([ , π]) be equipped with its natural norm . . In order to illustrate Theorem . , we consider the heat equation with Dirichlet condition given by the system where λ is a positive parameter and G is a function de ned as follows G(t) = cos(t) + cos( √ t) + ϕ(t) for each t ∈ R and with ϕ bounded. Let A be de ned by It is well-known that A is the in nitesimal generator of a c -semigroup (T(t)) t≥ on L ([ , π]), de ned by for φ ∈ L ([ , π]) and ψn(t) = π sin(nt) with n ∈ N * . Set f (t, u(t)) = λG(t) sin(t, u(t)), for t ∈ R.
We check that g ∈ µ − PAA(R × X, X), with the function (t, x) −→ u(t)(x) sin + cos t + cos √ t as it almost automorphic component and e −t sin(u(t, x)) as it µ-ergodic component. Moreover, So g is µ-pseudo almost automorphic in t ∈ R and Lipschitzian uniformly in the second variable with α < .
Using example above and the Theorem . , the evolution equation ( ) has a unique µ-pseudo almost automorphic solution.
The partial functional problem ( ) can be reformulated as the following initial value problem u ′ (t) = Au(t) + L(u(t − r)) + e(t), t ∈ R where D(A) = {u ∈ C(Ω, R); ∆u ∈ C(Ω, R) and u = on ∂Ω} Au = ∆u, with the bounded linear operator L : C X → X, is de ned by (Lφ)(x) = γφ(−r)(x), and e : R → X, is a continuous function de ned by e(t) = sin + cos t + cos √ t + e −t , for t ∈ R, φ ∈ C X and x ∈ Ω. This function e is µ-pseudo almost automorphic(PAA(R, X, µ)). The operator A generates a semigroup (T(t)) t on X, which is exponentially stable, namely, Assume that semigroup (V(t)) t is hyperbolic. It follows from Theorem . that equation ( ) has a unique mild solution in PAA(R, X, µ), then system ( ) has a unique solution which is µ-pseudo almost automorphic.