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

. 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 [3] 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 [16] 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 [5,6,7,8,9,12,10,11,18].
Blot et al. [4] 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. [4], 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 [39], 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. [5], 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) 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. [40], 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 [51,52,53,54] 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 [45,46,47,48,49,50] and references therein.
In 2013 Ponce [17] investigated the existence and uniqueness of bounded solutions for the semilinear fractional differential equation 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 [30,31,32,33,34,35,36,37,38,41,42,42,43].
On the other hand, Xia [29] 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 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 [19,24,21,22,23,20,25,26,27,30].
Inspired by above questions and works, we consider in this paper the following semilinear fractional integrodifferential equation 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 2. For a(t − s) = e −β(t−s) in Eq.(1.2), we have 4. Taking α = 1 and a(t − s) = λe −β(t−s) in Eq.(1.2), we have 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 (T 2 ) Assume that f ∈ PAA p (R × Ω, Ω, µ) and there exists a positive number L f such that and there exists nonnegative functions 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

[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 for each t ∈ R. The collection of all such functions will be denoted by AA (R, Ω).
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 Similarly, we define by AA 0 (R × Λ × Λ, Ω) the set of all continuous functions f : R × We denote the space of all such functions by E (R, Ω, µ) (or E (Ω, µ) for abbreviation).

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
Given a function g : R → X, the Weyl fractional integral of order α > 0 is defined by [1,2,17] for all t ∈ R, where {T α (t)} t≥0 is given by Remark 2.30.
Step 1: φ (t) ∈ AA (Ω).Consider for each t ∈ R and n = 1, 2, 3.... Using the condition (T 1 ) and Holder inequality, yields we denote that norm the well-known Weierstrass theorem that the series 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 Then using the Holder inequality, yields 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. Θ has a unique fixed point.
In this sense, we have Θζ 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).
To conclude this step, we will prove that the set V is equicontinuous.Indeed, consider the following decomposition ν (t + s) − ν (t) =

g
automorphic for each u ∈ Λ.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+1 t f (s + s n , u) − g (s, u) (s + s n , u) − f (s, u) p ds 1 p = 0pointwise on R and for each u ∈ Λ.We denote by AS p (R × Λ, Ω) the set of all such functions.Definition 2.16.[7]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.