J Vanterler

Da C Sousa

G M N'guerekata

STEPANOV TYPE µ-PSEUDO ALMOST AUTOMORPHIC MILD SOLUTIONS OF SEMILINEAR FRACTIONAL INTEGRODIFFERENTIAL EQUATIONS

Keywords: 2010 Mathematics Subject Classification. 35R11, 34K14, 35B15, 45N05, 58D25. almost automorphic functions, Fractional integro-differential equations, bounded solutions, fixed point theorems

In this paper, using necessary and sufficient conditions, the new concept of Stepanov µ-pseudo almost automorphic functions and ergodicity results, we investigate the existence of mild bounded solutions for a class of fractional integro-differential equations in the sense of the Weyl fractional derivative in a Banach space.

Introduction

The concept of almost automorphic functions was introduced by S. Bochner [START_REF] Bochner | Continuous mappings of almost automorphic and almost periodic functions[END_REF] in the early sixties while working on some problems in differential geometry. It turns out that it generalizes the concept of almost periodic functions in the sense of H. Bohr. Then came the concept of pseudo-almost automorphic function, which is a natural generalization of that of almost automorphic function. In this sense, N'Guérékata and Pankov [START_REF] N'guérékata | Stepanov-like almost automorphic functions and monotone evolution equations[END_REF] introduced the concept of Stepanov-like almost automorphic. Many important concepts and new generalizations over the years have been discussed and consequently, numerous discussions on problems of existence of mild bounded solutions [START_REF] Chang | Some new results on bounded solutions to a semilinear integrodifferential equation in Banach spaces[END_REF][START_REF] Chang | Stepanov-like weighted pseudo almost automorphic functions via measure theory[END_REF][START_REF] Diagana | Existence of pseudo-almost automorphic solutions to some abstract differential equations with S p -pseudo-almost automorphic coefficients[END_REF][START_REF] Diagana | Almost automorphic type and almost periodic type functions in abstract spaces[END_REF][START_REF] Diagana | Existence of weighted pseudo almost periodic solutions to some classes of differential equations with S p -weighted pseudo almost periodic coefficients[END_REF][START_REF] Henríquez | Compact almost automorphic solutions to integral equations with infinite delay[END_REF][START_REF] Ding | Almost automorphic solutions to nonautonomous semilinear evolution equations in Banach spaces[END_REF][START_REF] Granas | Elementary fixed point theorems[END_REF][START_REF] Chang | Weighted pseudo almost automorphic solutions to a semilinear fractional differential equation with Stepanov-like weighted pseudo almost automorphic nonlinear term[END_REF].

Blot et al. [START_REF] Blot | Measure theory and pseudo almost automorphic functions: New developments and applications[END_REF] established a new concept of weighted pseudo almost automorphic functions using the measure theory and investigated many interesting properties of such functions. Weighted pseudo almost automorphic functions have been studied recently and have become an interesting field. In addition, this paper by Blot el al. [START_REF] Blot | Measure theory and pseudo almost automorphic functions: New developments and applications[END_REF], made clear the new and general concept about automorphic function, that is, the pseudo almost automorphic function is a µ-pseudo almost automorphic function in the particular case where the measure µ is the Lebesgue measure.

In 2011 Lizama and Ponce [START_REF] Lizama | Bounded solutions to a class of semilinear integro-differential equations in Banach spaces[END_REF], investigated the existence, uniqueness and regularity of solutions for the following integro-differential equations given by u (t) = Au (t) + α t -∞ e -β(t-s) Au (s) ds + f (t, u (t))

where α, β ∈ R, A : D(A) ⊂ X → X is a closed linear operator defined on a Banach space X, and f belongs to a closed subspace of the space of continuous and bounded functions. The results were discussed when f (•, •) and u are almost periodic (resp. almost automorphic), asymptotically periodic (resp. almost periodic), pseudo-almost periodic (resp. almost automorphic).

Em 2015 Chang et al. [START_REF] Chang | Some new results on bounded solutions to a semilinear integrodifferential equation in Banach spaces[END_REF], discussed the new existence results of mild solutions via concept of Stepanov type µ-pseudo almost automorphic functions to a semilinear integro-differential equations given by (1.1)

u (t) = Au (t) + α t -∞ e -β(t-s) Au (s) ds + f (t, u (t))
where α, β ∈ R with α > 0, α = 0 and α + β > 0, A : D(A) ⊂ X → X is the generator of an immediately norm continuous C 0 -semigroup defined on the Banach space X, and f : R × X → X belongs to a closed subspace of the space of continuous and bounded functions satisfying some Lipschitz type conditions. In 2018 Ezzinbi et al. [START_REF] Ezzinbi | Eberlein weak almost periodic solutions for a class of integro-differential equations with infinite delay[END_REF], investigated weak almost periodic solutions for class of integro-differential equations of the form Eq.(1.1) with conditions on A, f , u and α, β.

The theory of fractional differential and integro-differential equations has been used to describe physical and biological phenomena [START_REF] Chaudhary | Design of multi innovation fractional LMS algorithm for parameter estimation of input nonlinear control autoregressive systems[END_REF][START_REF] Luo | Iterative learning control for fractional-order multi-agent systems[END_REF][START_REF] Aghayan | Robust stability analysis of uncertain fractional order neutral-type delay nonlinear systems with actuator saturation[END_REF][START_REF] Sweilam | Optimal control of variable-order fractional model for delay cancer treatments[END_REF] and references therein. In addition, investigating the properties of existence, uniqueness, stability and attractivity of solutions (classic, mild and strong), has been gaining increasing prominence in the scientific community [START_REF] Abbas | Fractional differential inclusions of Hilfer type under weak topologies in Banach spaces[END_REF][START_REF] Sousa | Attractivity for differential equations of fractional order and ψ-Hilfer type[END_REF][START_REF] Sousa | Existence of mild solutions to Hilfer fractional evolution equations in Banach space[END_REF][START_REF] You | Relative controllability of fractional delay differential equations via delayed perturbation of Mittag-Leffler functions[END_REF][START_REF] Abbas | Darboux problem for fractional-order discontinuous hyperbolic partial differential equations in Banach algebras[END_REF][START_REF] Kilbas | Theory and applications of fractional differential equations[END_REF] and references therein.

In 2013 Ponce [START_REF] Ponce | Bounded mild solutions to fractional integro-differential equations in Banach spaces[END_REF] investigated the existence and uniqueness of bounded solutions for the semilinear fractional differential equation

D α u (t) = Au (t) + t -∞ a(t -s)Au (s) ds + f (t, u (t)) , t ∈ R
where A is a closed linear operator defined on a Banach space X, α > 0, a ∈ L 1 (R + ) is a scalar-valued kernel and f : R × X → X satisfies some Lipschitz type conditions. In this work, Ponce established sufficient conditions for the existence and uniqueness of an almost periodic, almost automorphic and asymptotically almost periodic solution, among others. Other interesting results can be obtained in the following references [START_REF] Cao | Asymptotically Almost Periodicity for a Class of Weyl-Liouville fractional Evolution Equations[END_REF][START_REF] Chen | On fractional resolvent operator functions[END_REF][START_REF] Lizama | Regularized solutions for abstract Volterra equations[END_REF][START_REF] Lizama | An operator theoretical approach to a class of fractional order differential equations[END_REF][START_REF] Araya | Almost automorphic mild solutions to fractional differential equations[END_REF][START_REF] Cuevas | Almost automorphic solutions to a class of semilinear fractional differential equations[END_REF][START_REF] Ding | Almost automorphic solutions to abstract fractional differential equations[END_REF][START_REF] Agarwal | Weighted pseudo-almost periodic solutions of a class of semilinear fractional differential equations[END_REF][START_REF] Cuevas | Existence of S-asymptotically ω-periodic solutions for fractional order functional integrodifferential equations with infinite delay[END_REF][START_REF] Agarwal | Weighted pseudo-almost periodic solutions of a class of semilinear fractional differential equations[END_REF][START_REF] Cuevas | S-asymptotically ω-periodic solutions of semilinear fractional integro-differential equations[END_REF][START_REF] Cuevas | S-asymptotically ω-periodic solutions of semilinear fractional integro-differential equations[END_REF][START_REF] De Andrade | S-asymptotically ω-periodic and asymptotically ω-periodic solutions to semi-linear Cauchy problems with non-dense domain[END_REF].

On the other hand, Xia [START_REF] Xia | Pseudo almost periodicity of fractional integro-differential equations with impulsive effects in Banach spaces[END_REF] investigated the existence and uniqueness of a pseudo almost periodic P C-mild solution for the impulsive fractional integro-differential equations involving Caputo fractional derivative in a Banach space given by

c D α u (t) + Au (t) = f (t, u (t)) + (Ku)(t) + ∞ k=-∞ G k (u(t))δ(t -τ k ) where (Ku)(t) = t -∞ k(t -s)g (s, u(s)) ds. 0 < α ≤ 1, -A : D(A) ⊂ X → X is a linear infinitesimal operator of an analytic semigroup S(t), f, g are pseudo almost periodic in t ∈ R uniformly in the second variable, G k : D(G k ) ⊂ X → X are continuous impulsive operators, δ(•) is the Dirac's delta-function, τ k ∈ T ,
where T will be defined later. Here the fractional derivative is understood in Caputo's sense. Other results on about almost periodic, almost automorphic, asymptotically almost periodic involving fractional differential and integrodifferential equations, for example, can be obtained [START_REF] Chang | Properties on measure pseudo almost automorphic functions and applications to fractional differential equations in Banach spaces[END_REF][START_REF] Kavitha | Existence of weighted pseudo almost automorphic mild solutions to fractional integro-differential equations[END_REF][START_REF] Chang | Weighted pseudo almost automorphic mild solutions to semilinear fractional differential equations[END_REF][START_REF] Mophou | Weighted pseudo almost automorphic mild solutions to semilinear fractional differential equations[END_REF][START_REF] Alvarez-Pardo | Weighted pseudo almost automorphic mild solutions for two-term fractional order differential equations[END_REF][START_REF] Kavitha | µ 1 ; µ 2 )-Pseudo almost automorphic solutions of fractional order neutral integrodifferential equations[END_REF][START_REF] Wang | Pseudo almost automorphic solution of semilinear fractional differential equations with the Caputo derivatives[END_REF][START_REF] Zhao | Asymptotic behavior of mild solutions to semilinear fractional differential equations[END_REF][START_REF] Xia | Pseudo Almost automporphy of two-term fractional functional differential equations[END_REF][START_REF] Cao | Asymptotically Almost Periodicity for a Class of Weyl-Liouville fractional Evolution Equations[END_REF].

Inspired by above questions and works, we consider in this paper the following semilinear fractional integrodifferential equation

(1.2) W -∞ D α t u (t) = Au (t) + t -∞ a (t -s) Au (s) ds + f (t, u (t)) , t ∈ R where W -∞ D α t (•) is Weyl fractional derivative of order 0 < α < 1, A : D (A)
⊂ Ω → Ω is the generator of an α-resolvent family {T α (t)} t≥0 which is uniformly integrable on the Banach space Ω, and f : R × Ω → Ω belongs to a closed subspace of the space of continuous and bounded functions satisfying some Lipschitz type conditions. Here, we will impose that f : R × X → X is Stepanov type µ-pseudo almost automorphic.

Some particular cases of the choice of α and a(t -s) are as follows:

1. Taking α = 1 in Eq.(1.2), we have

u (t) = Au (t) + t -∞ a (t -s) Au (s) ds + f (t, u (t)) .
2. For a(t -s) = e -β(t-s) in Eq.(1.2), we have

W -∞ D α t u (t) = Au (t) + t -∞ e -β(t-s) Au (s) ds + f (t, u (t)) . 3. For a(t -s) = λe -β(t-s) in Eq.(1.2), we have W -∞ D α t u (t) = Au (t) + λ t -∞ e -β(t-s) Au (s) ds + f (t, u (t)) .
4. Taking α = 1 and a(t -s) = λe -β(t-s) in Eq.(1.2), we have

u (t) = Au (t) + λ t -∞ e -β(t-s) Au (s) ds + f (t, u (t)) .
The main objective of this paper is to investigate the existence of a new class of bounded mild solutions (namely the concept of Stepanov type µ-pseudo almost automorphic functions) to a semilinear fractional integro-differential equations given by Eq.(1.2), by means of results of ergodicity and composition theorems of Stepanov type µ-pseudo almost automorphic functions.

In addition to the particular cases presented above, from the choice of a(t -s) and α = 1, as the results are obtained for Stepanov µ-pseudo almost automorphic functions, there are also particular cases, for example, when µ is a Lebesgue measure.

To prove our results, we will make the following assumptions: (T 1 ) Assume that A generates an α-resolvent family {T α (t)} t≥0 such that T α (t) ≤ ϕ α (t) for all t ≥ 0 where ϕ

α (•) ∈ L 1 (R + ) is nonincreasing such that ϕ 0 := ∞ n=0 ϕ n (n) < ∞.
(T 2 ) Assume that f ∈ PAA p (R × Ω, Ω, µ) and there exists a positive number L f such that

(1.3) f (t, ζ) -f (t, η) ≤ L f ζ -η
for all t ∈ R and each ζ, η ∈ Ω. (T 3 ) Suppose that f ∈ PAA p (R × Ω, Ω, µ) and there exists a nonnegative function

L f (•) ∈ BS p (R), with p > 1 such that (1.4) f (t, ζ) -f (t, η) ≤ L f (t) ζ -η , (1.5) lim r→∞ 1 µ ([-r, r]) [-r,r] L f (t) dµ (t) < ∞ for all t ∈ R and each x, y ∈ Ω. (T 4 ) The function f = g + h ∈ PAA p (R × Ω, Ω, µ) with g ∈ AS p (R × Ω, Ω), h b ∈ E (Ω, L p (0, 1; Ω) , µ)
and there exists nonnegative functions

L f (•) , L g (•) ∈ AS κ (R, R) with κ ≥ max p, p p -1
such that for all u, v ∈ Ω and t ∈ R

(1.6) f (s, u) -f (s, v) ≤ L f (t) u -v , (1.7) g (s, u) -g (s, v) ≤ L g (t) u -v . (T 5 ) The functions f = g + h ∈ PAA p (R × Ω, Ω, µ) where g ∈ AS p (R × Ω, Ω) is uniformly continuous in any bounded subset M ⊂ Ω uniformly in t ∈ R and h b ∈ E (Ω, L p (0, 1; Ω) , µ). (T 6 ) f ∈ PAA p (R × Ω, Ω, µ) and f (t, ζ) is uniformly continuous in any bounded subset M ⊂ Ω uniformly for t ∈ R and for every bounded subset M ⊂ Ω, {f (•, ζ) ; ζ ∈ M } is bounded PAA p (R × Ω, Ω, µ).
In the rest, the article is organized as follows. In Section 2, we present some definitions and results that are essential for the development of this paper. In Section 3, the main result of this paper, that is, we investigated the existence of mild bounded solutions for a fractional integro-differential equation class in the sense of the Weyl derivative in the Banach space, by means of necessary and sufficient conditions, of Stepanov concept µ-pseudo almost automorphic functions and results ergodicity.

Preliminaries

In this section, we will present some essential definitions and results throughout the paper. Let (Ω, • ) and (Λ, • ) be two Banach spaces and let BC (R, Ω) denote the Banach space of all bounded continuous functions from R to Ω, equipped with the supremum norm f ∞ = sup t∈R f (t) . The notation B (Ω, Λ) stands for the space of bounded linear operator topology, and we abbreviate to B (Ω), whenever X = Λ. Throughout this work, we denote by B the Lebesgue σ-field of R and by M the set of all positive measures µ on B satisfying µ

(R) = +∞ and µ ([a, b]) < +∞, for all a, b ∈ R (a < b). Definition 2.

[15]

A continuous function f : R → Ω is said to be almost automorphic if for every sequence of real numbers {s n } n∈N there exists a subsequence {s n } n∈N such that g (t) := lim n→∞ f (t + s n ) is well defined for each t ∈ R, and

(2.1) lim n→∞ g (t -s n ) = f (t)
for each t ∈ R. The collection of all such functions will be denoted by AA (R, Ω).

Definition 2.2. [15, 13] A continuous function f : R × Ω → Ω is said to be almost automorphic if f (t, ζ)
is almost automorphic for each t ∈ R uniformly for all ζ ∈ B, where B is any bounded subset of Ω. The collection of all such functions will be denoted by AA (R × Ω, Ω).

Definition 2.3. [14]

The set of all bounded continuous functions with vanishing mean value can be defined as

AA 0 (R, Ω) = φ ∈ BC (R, Ω) : lim T →∞ 1 2T T -T φ (σ) dσ = 0 .
Similarly, we define by AA 0 (R × Λ × Λ, Ω) the set of all continuous functions f : R × Λ × Λ → Ω which belong to BC (R × Λ × Λ, Ω) and satisfy lim

T →∞ 1 2T T -T f (ζ, η) dσ = 0 uniformly for (ζ, η) in any bounded subset of Λ × Λ. Definition 2.4. [4] Let µ ∈ M . A bounded continuous function f : R → Ω is said to be µ-ergodic if lim r→∞ 1 µ ([-r, r]) [-r,r] f (t) d µ (t) = 0.
We denote the space of all such functions by E (R, Ω, µ) (or E (Ω, µ) for abbreviation).

Definition 2.5. [4] Let µ ∈ M . A continuous function f : R × Λ → Ω is said to be µ-ergodic if f (•, η) is bounded for each η ∈ Λ and lim r→∞ 1 µ ([-r, r]) [-r,r] f (t, η) d µ (t) = 0
uniformly in η ∈ Λ. We denote the set of all such functions by

E (R × Λ, Ω, µ) (or E (Λ, Ω, µ) for abbrevia- tion). Definition 2.6. [4] Let µ ∈ M . A continuous function f : R → Ω is said to be µ-pseudo almost automorphic if f is written in the form: f = g + φ where g ∈ AA (R, Ω) and φ ∈ E (R, Ω, µ). Let PAA (R, Ω, µ) denote the space of all such functions. Definition 2.7. [4] Let µ ∈ M . A continuous function f : R × Λ → Ω is said to be µ-pseudo almost automorphic if f is written in the form: f = g + φ where g ∈ AA (R × Λ, Ω) and φ ∈ E (R × Λ, Ω, µ). Definition 2.8. [4] Let µ ∈ M . Then (E (R, Ω, µ) , • ∞ ) is a Banach space.
For µ ∈ M and τ ∈ R, we denote µ c the positive measure on (R, B) defined by

µ τ (A) = µ (a + τ : a ∈ A) for A ∈ B.
From µ ∈ M , we state the following hypothesis. (H 0 ) For all τ ∈ R, there exist γ > 0 and a bounded interval I such that

µ τ (A) ≤ γµ (A) when A ∈ B satisfies A ∩ I = φ. Lemma 2.9. [4] Let µ ∈ M satisfy (H 0 ), then E (R, Ω, µ) is translation invariant, therefore, PAA (R, Ω, µ) is also translation invariant. Lemma 2.10. [4] Let µ ∈ M . Assume that PAA (R, Ω, µ) is translation invariant. Then the decomposition of a µ -pseudo almost automorphic function in the form f = g + φ where g ∈ AA (R, Ω) and φ ∈ E (R, Ω, µ) is unique. Lemma 2.11. [4] Let µ ∈ M . Assume that PAA (R, Ω, µ) is translation invariant. Then PAA (R, Ω, µ; • ∞ ) is a Banach space.
Definition 2.12. [START_REF] Diagana | Existence of weighted pseudo almost periodic solutions to some classes of differential equations with S p -weighted pseudo almost periodic coefficients[END_REF][START_REF] N'guérékata | Stepanov-like almost automorphic functions and monotone evolution equations[END_REF] The Bochner transform

f b (t, s), t ∈ R, s ∈ [0, 1] of a solution f : R → Ω is defined by f b (t, s) := f (t + s) .
Definition 2.13. [START_REF] Diagana | Existence of weighted pseudo almost periodic solutions to some classes of differential equations with S p -weighted pseudo almost periodic coefficients[END_REF][START_REF] N'guérékata | Stepanov-like almost automorphic functions and monotone evolution equations[END_REF] Let p ∈ [1, ∞). The space BS p (Ω) of all Stepanov bounded functions, with the exponent p, consists of all measurable functions f : R → Ω such that f b ∈ L ∞ (R, L p (0, 1; Ω)). This a Banach space with the norm and φ b ∈ E (L p (0, 1; Ω) , µ). In other words, a function f ∈ L p loc (R, Ω) is said to be Stepanov type µ-pseudo almost automorphic relatively to the measure µ, if its Bochner transform f b : R → L p (0, 1; Ω) is µ-pseudo almost automorphic in the sense that there exist two functions g, φ : R → Ω such that f = g + φ where g ∈ AS p (Ω) and φ b ∈ E (L p (0, 1; Ω) , µ), that is, φ b ∈ BC (L p (0, 1; Ω)) and

f S p = f b L ∞ (R,L p (0,1;ζ)) = sup t∈R t+1 t f (τ ) p dτ 1/p
Λ → Ω, (t, u) → f (t, u) with f (•, u) ∈ L p loc (R, Ω), for each u ∈ Λ, is said to be S p -almost automorphic in t ∈ R uniformly in u ∈ Λ if t → f (t, u) is S p -almost automorphic for each u ∈ Λ.
lim r→∞ 1 µ ([-r, r]) [-r,r] t+1 t φ (s) p ds 1/p dµ (t) = 0.
The set of all such functions will be denoted by PAA p (R, Ω, µ). ,Ω) for each u ∈ Λ is said to be Stepanov type µ-pseudo almost automorphic (or S p -µ-pseudo almost automorphic) if it can be expressed as f = g + φ, where g ∈ AS p (R × Λ, Ω) and φ b ∈ E (Λ, L p (0, 1; Ω) , µ). We denote by PAA p (R × Λ, Ω, µ) the set of all such functions. Lemma 2.19. [START_REF] Chang | Stepanov-like weighted pseudo almost automorphic functions via measure theory[END_REF] Let µ ∈ M and I be a bounded interval (eventually

Definition 2.18. [6] Let µ ∈ M . A function f : R × Λ → Ω, (t, u) → f (t, u) with f (•, u) ∈ L p loc (R
I = φ). Assume that f (•) ∈ BS p (R, Ω).
Then the following assertions are equivalent:

1. f b (•) ∈ E (L p (0, 1; Ω) , µ); 2. lim r→+∞ 1 µ ([-r, r] \I) [-r,r]\I t+1 t f (s) p ds 1/p dµ (t).
3. For any ε > 0,

lim r→+∞ µ t ∈ [-r, r] \I : t+1 t f (s) p ds 1/p > ε µ ([-r, r] \I) = 0. Lemma 2.20. [6] Let µ ∈ M. Satisfy (H 0 ). Then E (L p (0, 1; Ω) , µ) is translation invariant, therefore PAA p (R, Ω, µ) is also translation invariant. Lemma 2.21. [6] Let µ ∈ M satisfy (H 0 ). If f ∈ PAA (R, Ω, µ) then f ∈ PAA p (R, Ω, µ) for each 1 ≤ p < ∞.
In other words PAA (R, Ω, µ) ⊆ PAA p (R, Ω, µ). Thus we have AA (R, Ω) ⊂ PAA (R, Ω, µ) ⊂ PAA p (R, Ω, µ). Thus, we have AA(R, X) ⊂ P AA(R, X, µ) ⊂ P AA p (R, X, µ).

Lemma 2.22. [START_REF] Chang | Stepanov-like weighted pseudo almost automorphic functions via measure theory[END_REF] Let µ ∈ M and f ∈ PAA p (R, Ω, µ) be such that f = g + χ, where g ∈ AS p (Ω) and

χ b ∈ E (L p (0, 1; Ω) , µ). If PAA p (R, Ω, µ) is translation invariant, then {g (t) : t ∈ R} ⊆ {f (t) : t ∈ R} (the closure of range f ). Lemma 2.23. [6] Let µ ∈ M . Assume that PAA p (R, Ω, µ) is translation invariant. Then (PAA p (R, Ω, µ) , • S p ) is a Banach space. Lemma 2.24. [6] Let µ ∈ M . Suppose that f = g + h ∈ PAA (R × Ω, Ω, µ) with g ∈ AS p (R × Ω, Ω), h b ∈ E (Ω, L p (0, 1; Ω) , µ
) and satisfies the following condition: (H 1 ) There exists a constant L > 0 such that, for all x, y ∈ Ω and t ∈ R

f (t, ζ) -f (t, η) ≤ L ζ -η . If v = v 1 + v 2 ∈ PAA p (R, Ω, µ) with v 1 ∈ AS p (Ω), v 2 ∈ E (L p (0, 1; Ω) , µ) and K 1 = {v 1 (t) : t ∈ R} is compact. Then f (•, v (•)) ∈ PAA p (R, Ω, µ). Lemma 2.25. [6] Let µ ∈ M and f = g + h ∈ PAA p (R × Ω, Ω, µ) with g ∈ AS p (R × Ω, Ω) , h b ∈ E (L p (0, 1; Ω) , µ).
Assume that the following conditions are satisfied:

1. There exists a nonnegative function

L (•) ∈ BS p (R) with p > 1 such that, for all ζ, η ∈ Ω and t ∈ R, t+1 t f (s, ζ) -f (s, η) ds 1/p < L (t) ζ -η , lim r→∞ 1 µ ([-r, r]) [-r,r] L (t) dµ (t) < ∞. 2. g (t, ζ) is uniformly continuous is any bounded subset K ⊆ Ω uniformly for t ∈ R. If u = u 1 + u 2 ∈ PAA p (R, Ω, µ) with u 1 ∈ AS p (Ω) , u b 2 ∈ E (L p (0, 1; Ω) , µ) and K 2 = {u 1 (t) , t ∈ R} is compact, then f (•, u (•)) belongs to PAA p (R, Ω, µ).
Lemma 2.26. [START_REF] Chang | Stepanov-like weighted pseudo almost automorphic functions via measure theory[END_REF] Let µ ∈ M and f = g + φ ∈ PAA p (R × Ω, Ω, µ) with g ∈ AS p (R × Ω, Ω) , φ b ∈ E (Ω, L p (0, 1; Ω) , µ). Assume that following conditions hold:

1. f (t, ζ) is uniformly conditions in any bounded subset K ⊆ Ω uniformly for t ∈ R.

g (t, ζ

) is uniformly continuous in any bounded subset K ⊆ Ω uniformly for t ∈ R. ,µ). Assume that the following conditions are satisfied:

For any bounded subset

K ⊆ Ω, {f (•, ζ) : ζ ∈ K } is bounded in PAA p (R × Ω, Ω, µ). If ζ = v 1 + v 2 ∈ PAA p (R, Ω, µ), with v 1 ∈ AS p (Ω), v 2 ∈ E (L p (0, 1; Ω) , µ) and Q = {ζ (t) : t ∈ R}, Q 1 = {v 1 (t) : t ∈ R} are compact then f (•, ζ (•)) belongs to PAA p (R, Ω, µ). Theorem 2.27. [6] Let µ ∈ M, p > 1 and f = g + χ ∈ PAA p (R × Ω, Ω, µ) with g ∈ AS p (R × Ω, Ω) , χ b ∈ E (Ω, L p (0, 1; Ω)
1. There exists nonnegative functions L f (•) , L g (•) ∈ AS κ (R, R) with κ ≥ max p, p p-1 such that, for all u, v ∈ Ω and t ∈ R f (s, u) -f (s, v) ≤ L f (t) u -v , g (s, u) -g (s, v) ≤ L g (t) u -v . 2. u = u 1 + u 2 ∈ PAA p (R, Ω, µ) with u 1 ∈ AS p (Ω), u 2 ∈ E (L p (0, 1; ζ) , µ) and K 3 = {u 1 (t) : t ∈ R} is compact in X. Then there exists q ∈ [1, p) such that F : R → Ω defined by F (•) = (f •, u (•)) belongs to PAA q (R, Ω, µ).
Given a function g : R → X, the Weyl fractional integral of order α > 0 is defined by [START_REF] Sousa | On the ψ-Hilfer fractional derivative[END_REF][START_REF] Sousa | Leibniz type rule: ψ-Hilfer fractional operator[END_REF][START_REF] Ponce | Bounded mild solutions to fractional integro-differential equations in Banach spaces[END_REF]]

I α -∞ u(t) := 1 Γ(α) t -∞ (t -s) α-1 u(s)ds, t ∈ R
when this integral is convergent.

On the other hand, the Weyl fractional derivative

W -∞ D α t u of order α > 0 is defined by [1, 2, 17] W -∞ D α t u(t) := d n dt n D -(n-α) u(t), t ∈ R where n = [α] + 1.
Definition 2.28. [START_REF] Lizama | Bounded solutions to a class of semilinear integro-differential equations in Banach spaces[END_REF] Let Ω be a Banach space. A strongly continuous function T : R + → B (Ω) is said to be immediately norm continuous if T : (0, ∞) → B (Ω) is continuous. Definition 2.29. [START_REF] Ponce | Bounded mild solutions to fractional integro-differential equations in Banach spaces[END_REF] Let A be a closed and linear operator with domain D (A) defined on a Banach space Ω and α > 0. Given a ∈ L 1 loc (R + ), we say that A is the generator of an α-resolvent family, if there exists ω ≥ 0 and a strongly continuous function

T α : [0, ∞) → B (Ω) such that λ α 1 + a (λ) : Re (λ) > ⊂ ρ (A) and for all x ∈ Ω (λ α -(1 + a (λ)) A) -1 x = 1 1 + a (λ) λ α 1 + a (λ) -A -1 x = ∞ 0 e -λt T α (t) xdt, Re (λ) > ω
where a denotes the Laplace transform of a. In this case, {T α (t)} t≥0 is called the α-resolvent family generated by A.

Remark 2.30. [START_REF] Ponce | Bounded mild solutions to fractional integro-differential equations in Banach spaces[END_REF] Observe that if b (t) = g α (t) + (g α * a) (t) t ≥ 0, where g α (t) = t α-1 Γ (α) and (g α * a) (t) = t 0 g α (t -s) a (s) ds, then we have that the family α-resolvent {T α (t)} t≥0 is a (b, g α )-regularized family.

In particular, if a = 0, an 1-resolvent family is the same as a c 0 -semigroup, whereas that a 2-resolvent family corresponds to the concepts of sine family. Therefore, if A is the generator of an α-resolvent family {T α (t)} t≥0 then have that the family {T α (t)} t≥0 verifies the following properties: Sufficient conditions implying that {T α (t)} t≥0 ⊂ B (Ω) is an α-resolvent family.

Let (•) : R → R be a continuous function such that (t) ≥ 1 for all t ∈ R and (t) → ∞ as |t| → ∞. We consider the space [START_REF] Henríquez | Compact almost automorphic solutions to integral equations with infinite delay[END_REF]

C (Ω) = u ∈ C (R, Ω) : lim |t|→∞ u (t) (t) = 0 .
Endowed with the norm u = sup

t∈R u (t) (t)
, it is a Banach space.

Lemma 2.31. [START_REF] Henríquez | Compact almost automorphic solutions to integral equations with infinite delay[END_REF] A subset E ⊆ C (Ω) is a relatively compact set if it verifies the following conditions:

(C 1 ) The set E (t) = {u (t) : u ∈ E} is relatively compact in X for each t ∈ R. (C 2 )
The set E is equicontinuous. (C 3 ) For each ε > 0, there exists L > 0 such that u (t) ≤ ε (t) for all u ∈ E and all |t| > L.

Main results

In this section, we will attack the main results of this paper, that is, new results of the existence of mild solutions for a class of fractional integrodifferential equations in the sense of the Weyl fractional derivative, through the concept of Stepanov type µ-pseudo almost automorphic function. Definition 3.1. A function u : R → Ω is said to be a mild solution to Eq.(1.2) if

(3.1) u (t) = t -∞ T α (t -s) f (s, u (s)) ds
for all t ∈ R, where {T α (t)} t≥0 is given by Remark 2.30.

Lemma 3.2. Let µ ∈ M , 0 < α < 1 and β > 0, δ = 0 with δ + β > 0 and condition (T 1 ) holds. If f : R → Ω is Stepanov type µ-pseudo almost automorphic and F α (t) is given by

(3.2) F α (t) = t -∞ T α (t -s) f (s) ds, t ∈ R then F α ∈ PAA (R, Ω, µ).
Proof. Indeed, since f ∈ PAA p (R, Ω, µ), there exists g 1 ∈ AS p (Ω) and g 2 ∈ E (L p (0, 1; Ω) , µ) such that f = g 1 + g 2 (see Definition 2.17). So

F α (t) = t -∞ T α (t -s) g 1 (s) ds + t -∞ T α (t -s) g 2 (s) ds = φ (t) + ψ (t) ,
where

φ (t) = t -∞ T α (t -s) g 1 (s) ds and ψ (t) = t -∞ T α (t -s) g 2 (s) ds.
The proof will be discussed in two steps.

Step 1: φ (t) ∈ AA (Ω). Consider

φ α n (t) = t-n+1 t-n T α (t -s) g 1 (s) ds
for each t ∈ R and n = 1, 2, 3.... Using the condition (T 1 ) and Holder inequality, yields

φ α n (t) ≤ t-n+1 t-n T α (t -s) g 1 (s) ds ≤ t-n+1 t-n ϕ α (t -s) g 1 (s) ds ≤ ϕ α (n -1) t-n+1 t-n g 1 (s) p ds 1/p ≤ ϕ α (n -1) g 1 S p . Since ∞ n=1 ϕ α (n -1) := ∞ n=1 ϕ α (n) < ∞,
we denote that norm the well-known Weierstrass theorem that the series

φ (t) := t -∞ T α (t -s) g 2 (s) ds = ∞ n=1 φ α n (t) . Clearly, x (t) ∈ C (R, Ω) and φ (t) ≤ ∞ n=1 φ α n (t) ≤ ∞ n=0 ϕ α (n) g 1 S p .
Since g ∈ AS p (R, Ω) , then for every sequence {s n } n∈N , there exists a sequence {s n } n∈N and a function

g 1 (•) ∈ L p loc (R, Ω) such that for each t ∈ R lim m→∞ t+1 t g 1 (s + s m ) -g 1 (s) p ds 1/p = 0 and lim m→∞ t-n+1 t-n g 1 (s -s m ) -g 1 (s) p ds 1/p = 0. Now, let φ α n (t) = t-n+1 t-n
T α (s) g (t -s) ds. Then using the Holder inequality, yields

φ α n (t + s m ) -φ α n (t) ≤ t-n+1 t-n T α (s) g 1 (t + s m -s) -g 1 (t -s) ds ≤ ϕ α (n -1) t-n+1 t-n g 1 (t + s m -s) -g 1 (t -s) ds 1/p . Note that, φ α n (t + s m ) -φ α n (t) → 0 as m → ∞.
Analogously, it is proved that φ α n (t + s m ) -φ α n (t) = 0. Thus, we conclude that each φ α n ∈ AA (Ω) and consequently their uniform limit φ ∈ AA (Ω).

Step 2. ψ (t) ∈ E (R, Ω, µ).

Consider ψ α n (t) = t-n+1 t-n
T α (t -s) g 2 (s) ds for each t ∈ R and n = 1, 2, .... Again, using the condition (T 1 ) and Holder inequality, yields

ψ α n (t) ≤ t-n+1 t-n T α (t -s) g 2 (s) ds ≤ t-n+1 t-n ϕ α (t -s) g 2 (s) ds ≤ ϕ α (n -1) t-n+1 t-n g 2 (s) p ds 1/p .
Then, for r > 0, we have

1 µ ([-r, r]) [-r,r] ψ α n (t) dµ (t) ≤ ϕ n (n -1) µ ([-r, r]) [-r,r] t-n+1 t-n g 2 (s) p ds 1/p dµ (t) .
Since g b 2 ∈ E (L p (0, 1; Ω) , µ) the above inequality gives rise to

ψ α n ∈ E (R, Ω, µ) for n = 1, 2, ... Since g 2 S p ∞ n=0 ϕ α (n) < ∞, then we deduce from the Weierstrass M-test that the series ∞ n=0 ψ α n (t) is uniformly convergent on R and ψ (t) = t -∞ T α (t -s) g 2 (s) ds = ∞ n=1 ψ α n (t).
Applying ψ n ∈ E (R, Ω, µ) and the inequality

1 µ ([-r, r]) [-r,r] ψ (t) dµ (t) ≤ 1 µ ([-r, r]) [-r,r] ψ(t) - k n=1 ψ k n (t) dµ (t) + k n=1 1 µ([-r, r]) [-r,r] ψ α n (t) dµ(t) → 0,
we obtain that the uniform

ψ(t) = ∞ n=1 ψ α n (t) ∈ E (R, Ω, µ). Therefore F α (t) = φ (t) + ψ (t) is µ-pseudo almost automorphic. Theorem 3.3. Let µ ∈ M . Assume the conditions (H 0 ), (T 1 )-(T 2 ) are satisfied and the function f = h 1 + h 2 ∈ PAA p (R × Ω, Ω, µ) with h 1 ∈ AS p (R × Ω, Ω) and h b 2 ∈ E (Ω, L p (0, 1; Ω) µ).
Then Eq.( 1) has a unique µ-pseudo almost automorphic mild solution on R, provided that L f ϕ n L 1 (R) < 1.

Proof. Let Θ : PAA (R, Ω, µ) → PAA (R, Ω, µ) be the nonlinear operator defined by

(3.3) Θζ (t) = t -∞ T α (t -s) f (s, ζ(s)) ds, t ∈ R.
Step 1. Θ (PAA (R, Ω, µ)) ⊆ PAA (R, Ω, µ). First, using the fact that ζ ∈ PAA (R, Ω, µ) is relatively compact with the above Lemma 2.21, Lemma 2.24, follows that f (•, ζ (•)) ∈ PAA p (R, Ω, µ). Hence, from Lemma 3.2, we know that Θζ (•) ∈ PAA (R, Ω, µ).

Step 2. Θ has a unique fixed point.

Let t ∈ R, ζ, η ∈ PAA (R, Ω, µ), yields Θζ (t) -Θη (t) ≤ t -∞ T α (t -s) f (s, ζ(s)) -f (s, η(s)) ds ≤ L f t -∞ T α (t -s) ζ(s) -η(s) ds = L f t 0 T α (s) x (t -s) -y (t -s) ds ≤ L f ζ -η ∞ ϕ α L 1 (R) which implies that Θζ (t) -Θη (t) ≤ L f ζ -η ∞ ϕ α L 1 (R) .
Therefore, we concluded that, by means of the Banach fixed point theorem with L f ϕ α L 1 < 1, Θ has a unique fixed point ζ in PAA (R, Ω, µ) which is the µ-pseudo almost automorphic solution to Eq.(1.2). Theorem 3.4. Let µ ∈ M . Assume that (H 0 ), (T 1 ), (T 3 ) and (T 5 ) holds, then Eq.(1.2) admits a unique µ-pseudo almost automorphic mild solution whenever L f S p ϕ 0 < 1.

Proof. To prove this result, we consider the nonlinear operator Γ given by

Θζ (t) = t -∞ T α (t -s) f (s, ζ(s)) ds, t ∈ R.
Step 1. ΘP AA ⊂ P AA. Now, for ζ ∈ PAA (R, X, µ) and using Lemma 2.21 and Lemma 2.25 it follows that the function s → f (s, ζ(s)) is in PAA p (R, Ω, µ). On the other hand, using the Lemma 3.2 we infer that Θζ ∈ PAA (R, Ω, µ), i.e., Θ maps PAA (R, Ω, µ) into itself.

Step 2. Θ has a unique fixed point in PAA (R, Ω, µ).

Indeed, for each t ∈ R, ζ, η ∈ PAA (R, Ω, µ), yields Θζ (t) -Θη (t) ≤ t -∞ T α (t -s) f (s, ζ(s)) -f (s, η(s)) ds ≤ ∞ n=1 t-n+1 t-n ϕ α (t -s) L f (s) x -y ∞ ds ≤ ∞ n=1 ϕ α (n -1) t-n+1 t-n L f (s) p ds 1/p ζ -η ∞ ≤ ϕ 0 L f (s) S p ζ -η ∞ .
In this sense, we have Θζ -

Θη ≤ ϕ 0 L f (s) S p ζ -η ∞ .
Since ϕ 0 L f (s) S p < 1, using the Banach fixed point theorem, Θ has a unique fixed point x ∈ PAA (R, Ω, µ).

Theorem 3.5. Let µ ∈ M . Assume that (H 0 ), (T 1 ) and (T 4 ) are true. Then there exists a unique µ-pseudo almost automorphic mild solution to Eq.(1.2), provided that ϕ 0 L f S r < 1.

Proof. For proof of this result, we will consider the same operator Θ given in Theorem 3.3 Eq.(3.3), given by

Θζ (t) = t -∞ T α (t -s) f (s, ζ(s)) ds, t ∈ R.
Step 1. ΘP AA ⊂ P AA. Now, for x ∈ PAA (R, Ω, µ) and using the Lemma 2.21 and Theorem 2.27 it follows that the functions s → f (s, ζ(s)) is in PAA q (R, Ω, µ) , q ∈ [1, p). On the other hand, using the Lemma 3.2 we infer that Θx ∈ PAA (R, Ω, µ), i.e., Θ maps PAA (R, Ω, µ) into itself.

Step 2. Θ has a unique fixed point in PAA (R, Ω, µ). Indeed, for each t ∈ R, ζ, η ∈ PAA (R, Ω, µ), yields

Θζ (t) -Θη (t) ≤ t -∞ T α (t -s) f (s, ζ(s)) -f (s, η(s)) ds ≤ t -∞ ϕ α (t -s) L f (s) ζ(s) -η(s) ds ≤ ∞ n=1 t-n+1 t-n ϕ α (t -s) L f (s) ds ζ -η ∞ ≤ ∞ n=1 ϕ α (n -1) t-n+1 t-n L f (s) r ds 1/r ζ -η ∞ ≤ ϕ 0 L f S r ζ -η ∞ . Follows that Θζ -Θη ∞ ≤ ϕ 0 L f S r ζ -η ∞ .
In view of the inequality ϕ 0 L f S r < 1, using the Banach fixed point theorem, Θ has a unique fixed point x ∈ PAA (R, Ω, µ). Therefore, we conclude the proof. So far, the results of the existence of µ-pseudo almost automorphic solutions to Eq.(1.2), have been obtained using the fact that f satisfies the Lipschitz condition. Now, let's discuss the next result of this paper, removing this condition and imposing another.

The following existence result is based upon Leray-Schauder nonlinear alternative theorem. Consider the follows conditions:

(T 7 ) There exists a continuous nondecreasing function

Θ : [0, ∞) → (0, ∞) such that f (t, θ) ≤ Θ ( θ )
for all t ∈ R and θ ∈ Ω.

Theorem 3.6. Let µ ∈ M. Assume that conditions H 0 , T 1 are satisfied. Let f : R × Ω → Ω be a function which satisfies assumptions (T 5 )-(T 7 ) and the following additional conditions:

F1. For each κ ≥ 0, the function t → t -∞ ϕ α (t -s) Θ (κ (s)) ds belongs to BC (R). Let (3.4) λ (κ) = t -∞ ϕ α (t -s) Θ (κ (s)) ds .
F2. For each ε > 0, there exists a δ > 0, such that for every

ζ, η ∈ C (Ω), ζ -η ≤ δ implies that t -∞ ϕ α (t -s) f (s, ζ(s)) -f (s, η(s)) ds ≤ ε for all t ∈ R. F3. lim inf ξ→∞ ξ λ(ξ) > 1.
F4. For all a, b ∈ R, a < b and κ > 0, the set f

(s, ζ) , a ≤ s ≤ b, ζ ∈ C (Ω) , ζ ≤ κ is relatively compact in Ω.
Then Eq.(1.2) has a least one µ-pseudo almost automorphic mild solution on t ∈ R.

Proof. Consider the following operator Θ :

C (Ω) → C (Ω) given by Θζ (t) := t -∞ T α (t -s) f (s, ζ(s)) ds, t ∈ R.
The main objective of this test is to ensure that Θ has a fixed point in PAA (R, Ω, µ). For this, it will be investigated in 5 stages.

Step 1. Θ is well defined.

For ζ ∈ C (Ω), yields Θζ (t) ≤ t -∞ T α (t -s) f (s, ζ(s)) ds ≤ t -∞ ϕ α (t -s) Θ ( θ (s) ) ds ≤ t -∞ ϕ α (t -s) Θ θ (s) ds.
In this sense, using the condition (F1), we concluded that Θ is well defined.

Step 2. The operator Θ is continuous.

Indeed, for any ε > 0, we take δ > 0 involved in condition (F2). If ζ, η ∈ C (Ω) and ζ -η ≤ δ, then we have

Θζ (t) -Θη (t) ≤ t -∞ T α (t -s) f (s, ζ(s)) -f (s, η(s)) ds ≤ t -∞ ϕ α (t -s) f (s, ζ(s)) -f (s, η(s)) ds ≤ ε
and we concluded this step.

Step 3. Θ is completely continuous. Let B κ (Ω) is a closed ball with center at 0 and radius κ in the space Ω. Moreover, let

V = Θ (B κ (C (Ω))) and ν = Θ (ζ) for ζ ∈ B κ (C (Ω)).
Note that, using the condition (F1), the function s → ϕ α (s) Θ (κ (t -s)) is integrable on [0, ∞). Hence, for ε > 0 we can choose a ≥ 0 such that Using the strong continuity of T α (•) and condition (F4) on f , then K is a relatively compact set, and V (t) ⊆ ac 0 (K) + B ε (Ω). Therefore V (t) is a relatively subset of Ω for t ∈ R.

To conclude this step, we will prove that the set V is equicontinuous. Indeed, consider the following decomposition ν (t + s) -ν (t) = Finally, using the condition (F1), we have

ν (t) (t) ≤ 1 (t) t -∞
ϕ α (t -s) Θ (κ (s)) ds → 0 as |t| → ∞ and this converge is independent of ζ ∈ B κ (C (Ω)). Therefore, using Lemma 2.31, we concluded that, V is relatively compact set in C (Ω). Using the condition (F3), we conclude that the set P is bounded.

Step 5. It follows from Lemma 2.21, (T 6 -T 7 ) and Lemma 2.26 that the function t → f (t, ζ (t)) belongs to PAA p (R, Ω, µ) whenever ζ ∈ PAA (R, Ω, µ). Moreover, from Lemma 3.2, we infer that Θ (PAA (R, Ω, µ)) ⊆ PAA (R, Ω, µ) and noting that PAA (R, Ω, µ) is a closed subspace of C (Ω) consequently we can consider Θ : PAA (R, Ω, µ) → PAA (R, Ω, µ). Using conditions (F1)-(F3), we deduce that this map is completely continuous. Applying Lemma 2.32, we infer that Θ has a fixed point ζ ∈ PAA (R, Ω, µ).

1 . 0 b. t 0 b 0 b

 1000 T α (0) = g α (0); 2. T α (t) ζ ∈ D (A) and T α (t) Aζ = AT α (t) ζ for all ζ ∈ D (A) and t ≥ 0; 3. T α (t) ζ = g α (t) ζ + t (t -s) AT α (t) ζds, for all x ∈ D (A) and t ≥ 0; 4(t -s) T α (t) ζds ∈ D (A) and T α (t) ζ = g α (t) ζ +A t (t -s) T α (s) ζds, for all ζ ∈ Ω and t ≥ 0.

Lemma 2 . 32 .

 232 [START_REF] Chang | Some new results on bounded solutions to a semilinear integrodifferential equation in Banach spaces[END_REF] (Leray-Schauder alternative theorem) Let D be a closed convex subset of a Banach space Ω such that 0 ∈ D. Let Γ : D → D be a completely continuous map. Then the set {ζ ∈ D : ζ = λΓ (ζ) , 0 < λ < 1} is bounded or the map Γ has a fixed point in D.

  s) Θ (κ (t -s)) ds ≤ ε.Sinceν (t) = a 0 T α (s) f (t -s, ζ (t -s)) ds + ∞ a T α (s) f (t -s, ζ (t -s)) ds and ∞ a T α (s) f (t -s, ζ (t -s)) ds ≤ ∞ a T α (s) f (t -s, ζ (t -s)) ds ≤ ∞ a ϕ α (s) Θ (κ (t -s)) ds we have ν (t) ∈ ac 0 (K) + B ε (Ω), where c 0 (K) denotes the convex hull of K andK = T α (s) f (ξ, ζ) : 0 ≤ s ≤ a, t -a ≤ ξ ≤ t, ζ ≤ κ .

s 0 T 0 ( 0 T

 000 α (σ) f (t + s -σ, ζ (t + s -σ)) dσ + a T α (σ + s) -T α (σ)) f (t -σ, ζ (t -σ)) dσ + ∞ a (T α (σ + s) -T α (σ)) f (t -σ, ζ (t -σ)) dσ. (3.5)For each ε > 0, we can choose a > 0 and δ 1 > 0 such thats 0 T α (σ) f (t + s -σ, ζ (t + s -σ)) dσ + ∞ a (T α (σ + s) -T α (σ)) f (t -σ, ζ (t -σ)) dσ ≤ s α (σ) f (t + s -σ, x (t + s -σ)) dσ + ∞ a (T α (σ + s) -T α (σ)) f (t -σ, ζ (t -σ)) dσ ≤ s 0 ϕ α (σ) Θ (κ (t + s -σ)) dσ + ∞ a (ϕ α (σ + s) + ϕ α (σ)) Θ (κ (t -σ)) dσ ≤ ε 2 (3.6) for s ≤ δ 1 . Moreover, since {f (t -σ, ζ (t -σ)) : 0 ≤ σ ≤ a, ζ ∈ B κ (C (Ω))} is relatively compact and T α (•) is strongly continuous, choose δ 2 > 0 such that (3.7) (T α (σ + s) -T α (σ)) f (t -σ, ζ (t -σ)) ≤ ε 2a for s ≤ δ 2 .Combining the estimates (3.5), (3.6) and (3.7), we get ν (t + s) -ν (t) ≤ ε for s small enough and independent of ζ ∈ B κ (C (Ω)).

Step 4 .T

 4 P = ζ γ : ζ γ = γ Θ (ζ γ ) , γ ∈ (0, 1) is bounded. First, assume that ζ γ (•) is a solution of equation ζ γ = γ Θ (ζ γ ) for some 0 < γ < 1. Note that ζ γ α (t -s) f (s, ζ γ (s)) ds ≤ t -∞ ϕ α (t -s) Θ ( ζ γ (s) ) ds ≤ t -∞ ϕ α (t -s) Θ ζ γ (s) ds ≤ λ ζ γ (s) .Hence, yields ζ γ λ x γ ≤ 1.

.

  Definition 2.14.[START_REF] N'guérékata | Stepanov-like almost automorphic functions and monotone evolution equations[END_REF] The space AS p (Ω) of Stepanov type almost automorphic (or S p -almost automorphic) functions consists of all f ∈ BS p (Ω) such that f b ∈ AA (L p (0, 1; Ω)). In other words, a function f ∈ L p loc (R, Ω) is said to be S p -almost automorphic if its Bochner transform f b : R → L p (0, 1; Ω) is almost automorphic in the sense that, for every sequence of real numbers {s n } n∈N there exist a subsequence {s n } n∈N and a function g ∈ L p loc (R, Ω) such that

		lim n→∞	t	t+1	f (s + s n ) -g (s)	p ds	1 p	= 0
	and	lim n→∞	t	t+1	g (s + s n ) -f (s)	p ds	1 p	= 0
	pointwise on R.

Definition 2.15. [16] A function f : R ×

  That means, for every sequence of real numbers {s n } n∈N there exists a subsequence {s n } n∈N and a function g (•, u) ∈ L p loc (R, Ω) such that

	lim n→∞	t	t+1	p f (s + s 1	= 0
	and
			t+1
	lim n→∞	t		g (s + s

n , u) -g (s, u) p ds n , u) -f (s, u) p ds 1 p = 0 pointwise on R and for each u ∈ Λ. We denote by AS p (R × Λ, Ω) the set of all such functions. Definition 2.16.

[START_REF] Diagana | Existence of pseudo-almost automorphic solutions to some abstract differential equations with S p -pseudo-almost automorphic coefficients[END_REF]

A function f ∈ BS p (Ω) is said to be Stepanov type pseudo almost automorphic if it can be decomposed as f = g + ϕ where g ∈ AS p (Ω) and ϕ b ∈ AA 0 (R, L p (0, 1; Ω)). Denote by PAA p the set of all functions. Definition 2.17.

[START_REF] Chang | Stepanov-like weighted pseudo almost automorphic functions via measure theory[END_REF]

Let µ ∈ M . A function f ∈ BS p (Ω)is said to be Stepanov type µ-pseudo almost automorphic (or S p -µ-pseudo almost automorphic) if it can be expressed as f = g + φ, where g ∈ AS p (Ω)


Optical observations of ultrasound contrast agent destruction

Michiel Postema, Ayache Bouakaz, Chien Ting Chin, Nico de Jong

To cite this version:

34

We conclude that fission is the primary destruction mechanism for the experimental UCA, in case

35

of destruction during the first cycles of ultrasound insonification.

37

This work has been supported by the Foundation STW (RKG.5104), and by the Interuniversity

38

Cardiology Institute of The Netherlands (ICIN).

  demonstrating destruction mechanisms that could not be classified into the categories 33 fusion, fission, and sonic cracking, for example asymmetric collapse without fragmentation.

	1	Optical observations of ultrasound contrast agent destruction
	2	Michiel Postema, Ayache Bouakaz, Chien Ting Chin, Nico de Jong
	3	Erasmus MC (University Medical Center), Rotterdam, The Netherlands
	4
	5	Microbubble destruction has been under investigation for its potential application in drug and
	6	gene delivery, and for the improvement of diagnostic ultrasound contrast agent (UCA) detection
	7	methods. We investigated UCA destruction mechanisms by subjecting two different contrast
	8	agents to high-amplitude ultrasound, and optically recording its behavior with a fast-framing
	9	camera. The fast-framing camera was capable of recording an optical image sequence of eight
	10	frames per experiment and operated at a speed of three million frames per second. Each image
	17
	18	all experiments we transmitted 10 cycles of ultrasound with a center frequency of 0.5 MHz. Peak
	19	negative acoustic pressures applied ranged from 0.64 MPa to 0.85 MPa. We observed the
	20	following microbubble destruction mechanisms:
	21	hal-03189246 Fusion -the coalescence of two or more bubbles. This mechanism was only observed while
	22	microbubbles were expanding. We recorded 133 image sequences showing experimental UCA
	23	microbubble fusion. The fusion mechanism was found to be analog to mechanisms described for
	24	bigger gas bubbles.
	25	Fission -the fragmentation of one or more bubbles into smaller bubbles. This mechanism was
	26	only observed, as suggested from cavitation bubble theory, around maximum compression. We
	27	recorded 83 image sequences showing experimental UCA microbubble fission.
	28	Sonic cracking -the formation of a bubble shell defect causing gas escape. We recorded 17
	29	image sequences showing Quantison™ microbubble cracking. The number of cracking bubbles
	30	compared to the number of bubbles visible in the image sequences is low. Tiny flaws in the

Michiel Postema, Ayache Bouakaz, Chien Ting Chin, Nico de Jong. Optical observations of ultrasound contrast agent destruction. Acta Acustica united with Acustica, 2003, German Acoustical Society Workshops on Physical Acoustics: Bubbles and their application in medical acoustics; Acoustical methods in biomedical technology, 30.9-2.10.2002 in Bad Honnef, Physikzentrum, 89 (4), pp.728. sequence was taken during one cycle of ultrasound. We recorded 482 image sequences with an 11 experimental UCA (Bracco Research SA, Geneva, Switzerland), freely flowing through a 12 capillary cellulose tube. The experimental UCA consists of encapsulated gas bubbles with a 13 phospholipid shell. The microbubbles have a median diameter of 2 µm. We also recorded 57 14 image sequences with Quantison™ (Upperton Ltd., Nottingham, UK), poured into a container 15 and covered with a microscopic slip. Quantison™ consists of air-filled albumin shells. The 16 microbubbles have a mean diameter of 3.2 µm. Shell thicknesses range from 0.2 µm to 0.3 µm. In bubble shell may account for the reason why certain bubbles crack and other bubbles stay intact.

sequences


We investigated UCA destruction mechanisms by subjecting two different contrast agents to high-amplitude ultrasound, and optically recording their behavior with a fast-framing camera.

Optically observing ultrasound insonified UCA microbubble with a high-speed camera is a promising method for analyzing microbubble oscillation and destruction behavior.

Methods:

The fast-framing camera was capable of recording an optical image sequence of eight frames per experiment and operated at a speed of three million frames per second. Each image sequence was taken during one cycle of ultrasound. We recorded 482 image sequences with an experimental UCA (Bracco Research SA, Geneva, Switzerland), freely flowing through a capillary cellulose tube. The microbubbles had a median diameter of 2 µm. The shells were phospholipid monolayers with an estimated thickness of 10 nm. We also recorded 57 image sequences with Quantison™ (Upperton Ltd., Nottingham, UK), poured into a container and covered with a microscopic slip. The microbubbles had a mean diameter of 3.2 µm. The shells consisted of human serum albumin (0.2-0.3 µm).

In all experiments we transmitted 10 cycles of ultrasound with a center frequency of 0.5 MHz.

Peak negative acoustic pressures corresponded to mechanical indices (MI) between 0.9 and 1.2.

Results:

The optical image sequences revealing destructive behavior were classified as follows.

Coalescence -the fusion of two or more bubbles. This mechanism was only observed while microbubbles were expanding. We recorded 133 image sequences showing experimental UCA microbubble coalescence. The coalescence mechanism was found to be analog to mechanisms described for larger free gas bubbles.

We included bubble coalescence as a UCA microbubble destruction mechanism, since the total number of bubbles in the UCA decreases due to coalescence.

Fragmentation -the fragmentation (fission) of one or more bubbles into smaller bubbles. This mechanism was only observed, as suggested from cavitation bubble theory, around maximum compression. We recorded 83 image sequences showing experimental UCA microbubble fragmentation.

We observed successive coalescence and fragmentation in 12 image sequences. Sonic cracking -the formation of a bubble shell defect causing gas escape [START_REF] Takeuchi | A sequence of microscopic image frames collected during insonification of a hard-shell model contrast agent[END_REF].

We recorded 17 image sequences showing Quantison™ microbubble cracking. The number of cracking bubbles compared to the number of bubbles visible in the image sequences is low.

We recorded 15 image sequences demonstrating destruction mechanisms that could not be classified into the categories coalescence, fragmentation, and sonic cracking, for example asymmetric collapse without fragmentation. Jet behavior was clearly observed twice. One of these events was previously published [START_REF] Postema | A sequence of microscopic image frames of a freely flowing contrast agent microbubble[END_REF]. Apparently liquid was propelled into the freely flowing microbubble, in the direction of the capillary wall nearby.

Discussion and conclusions: Coalescence we observed in 28% of our high-MI experimental events with a lipid-shelled UCA, when insonifying at 0.5 MHz. We proposed a coalescence mechanism of UCA microbubbles, based on a mechanism of colliding free gas bubbles and the so-called stalk-pore hypothesis. Bubble coalescence was only clearly observed with the experimental UCA, and not with Quantison™. For the bubble concentration used, the spacing between the Quantison™ bubbles was much larger than a maximal excursion, and therefore bubble collision during the expansion phase may not have occurred.

Bubble fragmentation was only observed with the experimental UCA, and not with Quantison™. We conclude that its bubble wall velocity and acceleration are too low to induce instabilities big enough for fragmentation.

The repetitive coalescing and fragmenting behavior recorded suggests that the destruction behavior of contrast microbubbles is more complex than previously assumed [START_REF] Chomas | Mechanisms of contrast agent destruction[END_REF].

We showed that sonic cracking is feasible for the hard-shelled contrast agent Quantison™. Tiny flaws in the bubble shell may account for the reason why certain bubbles cracked and others stood intact.

At before-mentioned acoustic frequency and pressures, when not taking into account coalescence as a real destruction mechanism, we found that fragmentation is the primary destruction mechanism for the experimental UCA, in case of destruction during the first cycles of ultrasound insonification. Since we were only looking at early cycles of the ultrasonic waves, this limits our conclusion.

The destruction mechanisms presented have potential clinical applications in high-MI imaging, noninvasive blood pressure measurements, neovascularization, and targeted drug and gene delivery.

  . of Experimental Echocardiography, Thoraxcentre, Erasmus MC (University Medical Ctr.), The Netherlands; 2 -Interuniversity Cardiology Institute of The Netherlands (ICIN) Introduction: Ultrasonic contrast agent (UCA) consists of liquid containing gas bubbles with diameters of a few micrometers, either free or encapsulated by a shell. UCA bubble destruction behavior has been under investigation for its potential application in high MI imaging techniques, noninvasive pressure measurements, neovascularization, and in drug and gene delivery. The success of such new diagnostic and therapeutic techniques is directly related to our understanding of UCA microbubble destruction.

Acknowledgment: This work has been supported by the Technology Foundation STW (RKG.5104).