HAL Id: hal-03189230
https://hal.archives-ouvertes.fr/hal-03189230
Preprint submitted on 2 Apr 2021
HAL
is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire
HAL, estdestinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
STEPANOV TYPE µ-PSEUDO ALMOST AUTOMORPHIC MILD SOLUTIONS OF
SEMILINEAR FRACTIONAL
INTEGRODIFFERENTIAL EQUATIONS
José Vanterler da Costa Sousa, Gaston Mandata N’Guérékata
To cite this version:
José Vanterler da Costa Sousa, Gaston Mandata N’Guérékata. STEPANOV TYPE
µ-PSEUDO AL-MOST AUTOMORPHIC MILD SOLUTIONS OF SEMILINEAR FRACTIONAL INTEGRODIF-
FERENTIAL EQUATIONS. 2021. �hal-03189230�
STEPANOV TYPE µ-PSEUDO ALMOST AUTOMORPHIC MILD SOLUTIONS OF SEMILINEAR FRACTIONAL INTEGRODIFFERENTIAL EQUATIONS
J. VANTERLER DA C. SOUSA∗, G. M. N’GUEREKATA
Abstract. In this paper, using necessary and sufficient conditions, the new concept of Stepanovµ-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.
1. 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’Gu´er´ekata 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 aµ-pseudo almost automorphic function in the particular case where the measureµis 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
u0(t) =Au(t) +α Z t
−∞
e−β(t−s)Au(s)ds+f(t, u(t))
whereα, β∈R,A:D(A)⊂X →X is a closed linear operator defined on a Banach space X, andf belongs to a closed subspace of the space of continuous and bounded functions. The results were discussed when f(·,·) anduare 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 typeµ-pseudo almost automorphic functions to a semilinear integro-differential equations given by
(1.1) u0(t) =Au(t) +α
Z t
−∞
e−β(t−s)Au(s)ds+f(t, u(t))
where α, β∈R withα >0,α6= 0 andα+β >0,A :D(A)⊂X →X is the generator of an immediately norm continuous C0-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,uand 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.
2010Mathematics Subject Classification. 35R11, 34K14, 35B15, 45N05, 58D25..
∗Correspondent author. J. Vanterler da C. Sousa.
Key words and phrases. Stepanov typeµ-pseudo almost automorphic functions, Fractional integro-differential equations, bounded solutions, fixed point theorems.
1
In 2013 Ponce [17] investigated the existence and uniqueness of bounded solutions for the semilinear fractional differential equation
Dαu(t) =Au(t) + Z 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 ∈ L1(R+) is a scalar-valued kernel andf :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 periodicP C-mild solution for the impulsive fractional integro-differential equations involving Caputo fractional derivative in a Banach space given by
cDαu(t) +Au(t) =f(t, u(t)) + (Ku)(t) +
∞
X
k=−∞
Gk(u(t))δ(t−τk)
where (Ku)(t) = Z 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,Gk:D(Gk)⊂X →Xare continuous impulsive operators,δ(·) is the Dirac’s delta-function,τk ∈T, whereTwill 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
(1.2) W−∞Dαtu(t) =Au(t) + Z t
−∞
a(t−s)Au(s)ds+f(t, u(t)), t∈R
where W−∞Dαt (·) is Weyl fractional derivative of order 0 < α < 1, A : D(A) ⊂Ω → Ω is the generator of anα-resolvent family{Tα(t)}t≥0 which is uniformly integrable on the Banach space Ω, andf :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 thatf :R×X →X is Stepanov typeµ-pseudo almost automorphic.
Some particular cases of the choice ofαanda(t−s) are as follows:
1. Takingα= 1 in Eq.(1.2), we have u0(t) =Au(t) +
Z t
−∞
a(t−s)Au(s)ds+f(t, u(t)). 2. Fora(t−s) =e−β(t−s) in Eq.(1.2), we have
W
−∞Dαtu(t) =Au(t) + Z t
−∞
e−β(t−s)Au(s)ds+f(t, u(t)). 3. Fora(t−s) =λe−β(t−s)in Eq.(1.2), we have
W
−∞Dαtu(t) =Au(t) +λ Z t
−∞
e−β(t−s)Au(s)ds+f(t, u(t)). 4. Takingα= 1 anda(t−s) =λe−β(t−s)in Eq.(1.2), we have
u0(t) =Au(t) +λ Z 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 ofa(t−s) andα= 1, as the results are obtained for Stepanovµ-pseudo almost automorphic functions, there are also particular cases, for example, whenµis a Lebesgue measure.
To prove our results, we will make the following assumptions:
(T1) Assume thatAgenerates anα-resolvent family{Tα(t)}t≥0 such thatkTα(t)k ≤ϕα(t) for allt≥0 where ϕα(·)∈L1(R+) is nonincreasing such thatϕ0:=
∞
P
n=0
ϕn(n)<∞.
(T2)Assume thatf ∈ PAAp(R×Ω,Ω, µ) and there exists a positive numberLf such that
(1.3) kf(t, ζ)−f(t, η)k ≤Lfkζ−ηk
for allt∈Rand eachζ, η∈Ω.
(T3)Suppose thatf ∈ PAAp(R×Ω,Ω, µ) and there exists a nonnegative functionLf(·)∈ BSp(R), with p >1 such that
(1.4) kf(t, ζ)−f(t, η)k ≤Lf(t)kζ−ηk,
(1.5) lim
r→∞
1 µ([−r, r])
Z
[−r,r]
Lf(t)dµ(t)<∞ for allt∈Rand eachx, y∈Ω.
(T4)The functionf =g+h∈ PAAp(R×Ω,Ω, µ) with g ∈ ASp(R×Ω,Ω), hb ∈E(Ω, Lp(0,1; Ω), µ) and there exists nonnegative functionsLf(·),Lg(·)∈ ASκ(R,R) withκ≥max
p, p
p−1
such that for all u, v∈Ω andt∈R
(1.6) kf(s, u)−f(s, v)k ≤ Lf(t)ku−vk, (1.7) kg(s, u)−g(s, v)k ≤ Lg(t)ku−vk.
(T5)The functionsf =g+h∈ PAAp(R×Ω,Ω, µ) where g∈ ASp(R×Ω,Ω) is uniformly continuous in any bounded subsetM ⊂Ω uniformly int∈Randhb∈E(Ω, Lp(0,1; Ω), µ).
(T6)f ∈ PAAp(R×Ω,Ω, µ) andf(t, ζ) is uniformly continuous in any bounded subsetM ⊂Ω uniformly for t∈Rand for every bounded subsetM ⊂Ω,{f(·, ζ) ;ζ∈M}is boundedPAAp(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.
2. Preliminaries
In this section, we will present some essential definitions and results throughout the paper.
Let (Ω,k·k) and (Λ,k·k) be two Banach spaces and letBC(R,Ω) denote the Banach space of all bounded continuous functions from R to Ω, equipped with the supremum norm kfk∞ = sup
t∈R
kf(t)k. 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 byB the Lebesgueσ−field ofRand byM the set of all positive measuresµonBsatisfyingµ(R) = +∞andµ([a, b])<+∞, for alla, b∈R(a < b).
Definition 2.1. [15]A continuous functionf :R→Ωis said to be almost automorphic if for every sequence of real numbers {s0n}n∈
N there exists a subsequence{sn}n∈
Nsuch that g(t) := lim
n→∞f(t+sn) is well defined for each t∈R, and
(2.1) lim
n→∞g(t−sn) =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
AA0(R,Ω) = (
φ∈ BC(R,Ω) : lim
T→∞
1 2T
Z T
−T
kφ(σ)kdσ= 0 )
.
Similarly, we define byAA0(R×Λ×Λ,Ω) the set of all continuous functionsf :R×Λ×Λ→Ω which belong toBC(R×Λ×Λ,Ω) and satisfy
lim
T→∞
1 2T
Z T
−T
kf(ζ, η)kdσ= 0 uniformly for (ζ, η) in any bounded subset of Λ×Λ.
Definition 2.4. [4] Let µ∈M. A bounded continuous functionf :R→Ωis said to be µ-ergodic if
r→∞lim 1 µ([−r, r])
Z
[−r,r]
kf(t)kdµ(t) = 0.
We denote the space of all such functions byE (R,Ω, µ) (orE(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
r→∞lim 1 µ([−r, r])
Z
[−r,r]
kf(t, η)kdµ(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]Letµ∈M. A continuous functionf :R→Ωis said to beµ-pseudo almost automorphic if f is written in the form:
f =g+φ
whereg∈ AA(R,Ω)andφ∈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 iff is written in the form:
f =g+φ whereg∈ AA(R×Λ,Ω)andφ∈E(R×Λ,Ω, µ).
Definition 2.8. [4] Let µ∈M. Then(E (R,Ω, µ),k·k∞)is a Banach space.
Forµ∈M andτ ∈R, we denoteµc the positive measure on (R,B) defined by µτ(A) =µ(a+τ:a∈A)
forA∈ B.
From µ∈M, we state the following hypothesis.
(H0) For all τ∈R, there existγ >0 and a bounded intervalI such that µτ(A)≤γµ(A)
whenA∈ BsatisfiesA∩I=φ.
Lemma 2.9. [4]Let µ∈M satisfy (H0), thenE(R,Ω, µ)is translation invariant, therefore,PAA(R,Ω, µ) is also translation invariant.
Lemma 2.10. [4] Letµ∈M. Assume thatPAA(R,Ω, µ)is translation invariant. Then the decomposition of a µ-pseudo almost automorphic function in the form f =g+φwhereg∈ AA(R,Ω)andφ∈E(R,Ω, µ) is unique.
Lemma 2.11. [4]Letµ∈M. Assume thatPAA(R,Ω, µ)is translation invariant. ThenPAA(R,Ω, µ;k·k∞) is a Banach space.
Definition 2.12. [9,16] The Bochner transformfb(t, s),t∈R,s∈[0,1]of a solution f :R→Ωis defined by
fb(t, s) :=f(t+s).
Definition 2.13. [9, 16] Let p ∈ [1,∞). The space BSp(Ω) of all Stepanov bounded functions, with the exponent p, consists of all measurable functions f : R → Ω such that fb ∈ L∞(R, Lp(0,1; Ω)). This a Banach space with the norm
kfkSp= fb
L∞(
R,Lp(0,1;ζ))= sup
t∈R
Z t+1
t
kf(τ)kpdτ 1/p
.
Definition 2.14. [16]The spaceASp(Ω)of Stepanov type almost automorphic (orSp−almost automorphic) functions consists of all f ∈ BSp(Ω) such that fb ∈ AA(Lp(0,1; Ω)). In other words, a function f ∈ Lploc(R,Ω) is said to be Sp−almost automorphic if its Bochner transform fb : R → Lp(0,1; Ω) is almost automorphic in the sense that, for every sequence of real numbers {s0n}n∈
Nthere exist a subsequence{sn}n∈
and a function g∈Lploc(R,Ω)such that N n→∞lim
Z t+1
t
kf(s+sn)−g(s)kpds 1p
= 0 and
n→∞lim Z t+1
t
kg(s+sn)−f(s)kpds 1p
= 0 pointwise on R.
Definition 2.15. [16] A functionf :R×Λ→Ω,(t, u)→f(t, u)with f(·, u)∈Lploc(R,Ω), for eachu∈Λ, is said to be Sp−almost automorphic int∈R uniformly in u∈Λ if t→f(t, u) isSp−almost automorphic for eachu∈Λ. That means, for every sequence of real numbers{s0n}n∈
N there exists a subsequence{sn}n∈
N
and a function g(·, u)∈Lploc(R,Ω)such that
n→∞lim Z t+1
t
kf(s+sn, u)−g(s, u)kpds 1p
= 0 and
n→∞lim Z t+1
t
kg(s+sn, u)−f(s, u)kpds
1 p
= 0
pointwise on Rand for each u∈Λ. We denote by ASp(R×Λ,Ω)the set of all such functions.
Definition 2.16. [7] A function f ∈ BSp(Ω) is said to be Stepanov type pseudo almost automorphic if it can be decomposed as f =g+ϕwhereg∈ ASp(Ω) andϕb ∈ AA0(R, Lp(0,1; Ω)). Denote byPAAp the set of all functions.
Definition 2.17. [6] Let µ ∈ M. A function f ∈ BSp(Ω)is said to be Stepanov type µ-pseudo almost automorphic (orSp−µ-pseudo almost automorphic) if it can be expressed asf =g+φ, whereg∈ ASp(Ω) and φb ∈E (Lp(0,1; Ω), µ). In other words, a function f ∈Lploc(R,Ω)is said to be Stepanov typeµ-pseudo almost automorphic relatively to the measure µ, if its Bochner transform fb :R→Lp(0,1; Ω) isµ−pseudo almost automorphic in the sense that there exist two functions g, φ : R → Ω such that f = g+φ where g∈ ASp(Ω)and φb ∈E(Lp(0,1; Ω), µ), that is,φb∈BC(Lp(0,1; Ω))and
r→∞lim 1 µ([−r, r])
Z
[−r,r]
Z t+1
t
kφ(s)kpds 1/p
dµ(t) = 0.
The set of all such functions will be denoted byPAAp(R,Ω, µ).
Definition 2.18. [6]Let µ∈M. A function f :R×Λ→Ω, (t, u)→f(t, u) withf(·, u)∈Lploc(R,Ω)for each u∈Λ is said to be Stepanov typeµ-pseudo almost automorphic (orSp−µ-pseudo almost automorphic) if it can be expressed as f =g+φ, whereg∈ ASp(R×Λ,Ω)and φb∈E(Λ, Lp(0,1; Ω), µ). We denote by PAAp(R×Λ,Ω, µ)the set of all such functions.
Lemma 2.19. [6]Letµ∈M andIbe a bounded interval (eventuallyI=φ). Assume thatf(·)∈ BSp(R,Ω).
Then the following assertions are equivalent:
1. fb(·)∈E (Lp(0,1; Ω), µ);
2. lim
r→+∞
1 µ([−r, r]\I)
Z
[−r,r]\I
Z t+1
t
kf(s)kpds 1/p
dµ(t).
3. For anyε >0,
r→+∞lim µ
(
t∈[−r, r]\I: Z t+1
t
kf(s)kpds 1/p
> ε )!
µ([−r, r]\I) = 0.
Lemma 2.20. [6] Let µ ∈ M. Satisfy (H0). Then E (Lp(0,1; Ω), µ) is translation invariant, therefore PAAp(R,Ω, µ)is also translation invariant.
Lemma 2.21. [6] Let µ ∈ M satisfy (H0). If f ∈ PAA(R,Ω, µ) then f ∈ PAAp(R,Ω, µ) for each 1≤p <∞. In other words PAA(R,Ω, µ)⊆ PAAp(R,Ω, µ). Thus we have AA(R,Ω)⊂ PAA(R,Ω, µ)⊂ PAAp(R,Ω, µ).
Thus, we haveAA(R, X)⊂P AA(R, X, µ)⊂P AAp(R, X, µ).
Lemma 2.22. [6] Let µ ∈ M and f ∈ PAAp(R,Ω, µ) be such that f = g+χ, where g ∈ ASp(Ω) and χb∈E (Lp(0,1; Ω), µ). IfPAAp(R,Ω, µ)is translation invariant, then
{g(t) :t∈R} ⊆ {f(t) :t∈R} (the closure of range f).
Lemma 2.23. [6]Letµ∈M. Assume thatPAAp(R,Ω, µ)is translation invariant. Then(PAAp(R,Ω, µ),k·kSp) is a Banach space.
Lemma 2.24. [6] Let µ ∈ M. Suppose that f = g+h ∈ PAA(R×Ω,Ω, µ) with g ∈ ASp(R×Ω,Ω), hb∈E (Ω, Lp(0,1; Ω), µ) and satisfies the following condition:
(H1)There exists a constant L >0 such that, for allx, y∈Ω andt∈R kf(t, ζ)−f(t, η)k ≤Lkζ−ηk.
If v =v1+v2 ∈ PAAp(R,Ω, µ) with v1 ∈ ASp(Ω),v2∈E (Lp(0,1; Ω), µ) and K1 ={v1(t) :t∈R} is compact. Then f(·, v(·))∈ PAAp(R,Ω, µ).
Lemma 2.25. [6] Let µ ∈ M and f = g+h ∈ PAAp(R×Ω,Ω, µ) with g ∈ ASp(R×Ω,Ω), hb ∈ E (Lp(0,1; Ω), µ). Assume that the following conditions are satisfied:
1. There exists a nonnegative function L(·)∈ BSp(R)withp >1 such that, for allζ, η∈Ω andt∈R, Z t+1
t
kf(s, ζ)−f(s, η)kds 1/p
< L(t)kζ−ηk,
r→∞lim 1 µ([−r, r])
Z
[−r,r]
L(t)dµ(t)<∞.
2. g(t, ζ) is uniformly continuous is any bounded subset K0 ⊆Ω uniformly for t∈R. If u=u1+u2 ∈ PAAp(R,Ω, µ) with u1 ∈ ASp(Ω), ub2 ∈ E (Lp(0,1; Ω), µ) and K2 = {u1(t), t∈R} is compact, then f(·, u(·))belongs toPAAp(R,Ω, µ).
Lemma 2.26. [6] Let µ ∈ M and f = g +φ ∈ PAAp(R×Ω,Ω, µ) with g ∈ ASp(R×Ω,Ω), φb ∈ E (Ω, Lp(0,1; Ω), µ). Assume that following conditions hold:
1. f(t, ζ)is uniformly conditions in any bounded subset K0 ⊆Ω uniformly fort∈R. 2. g(t, ζ) is uniformly continuous in any bounded subsetK0⊆Ωuniformly fort∈R.
3. For any bounded subsetK0⊆Ω,{f(·, ζ) :ζ∈K0} is bounded inPAAp(R×Ω,Ω, µ). Ifζ=v1+v2∈ PAAp(R,Ω, µ), withv1∈ ASp(Ω),v2∈E (Lp(0,1; Ω), µ)andQ={ζ(t) :t∈R}, Q1={v1(t) :t∈R} are compact thenf(·, ζ(·))belongs toPAAp(R,Ω, µ).
Theorem 2.27. [6] Let µ ∈ M, p > 1 and f = g+χ ∈ PAAp(R×Ω,Ω, µ) with g ∈ ASp(R×Ω,Ω), χb∈E (Ω, Lp(0,1; Ω), µ). Assume that the following conditions are satisfied:
1. There exists nonnegative functionsLf(·),Lg(·)∈ ASκ(R,R)withκ≥maxn p,p−1p o
such that, for all u, v∈Ωandt∈R
kf(s, u)−f(s, v)k ≤ Lf(t)ku−vk, kg(s, u)−g(s, v)k ≤ Lg(t)ku−vk.
2. u =u1+u2 ∈ PAAp(R,Ω, µ) with u1 ∈ ASp(Ω), u2 ∈ E (Lp(0,1;ζ), µ) and K3 ={u1(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 PAAq(R,Ω, µ).
Given a functiong:R→X, the Weyl fractional integral of orderα >0 is defined by [1,2,17]
I−∞α u(t) := 1 Γ(α)
Z t
−∞
(t−s)α−1u(s)ds, t∈R when this integral is convergent.
On the other hand, the Weyl fractional derivativeW−∞Dαtuof orderα >0 is defined by [1,2,17]
W
−∞Dαtu(t) := dn
dtnD−(n−α)u(t), t∈R where n= [α] + 1.
Definition 2.28. [39] Let Ωbe 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. [17] Let A be a closed and linear operator with domainD(A)defined on a Banach space Ω and α >0. Given a∈L1loc(R+), we say that A is the generator of an α-resolvent family, if there exists ω≥0 and a strongly continuous functionTα: [0,∞)→ B(Ω) such that
λα
1 +ba(λ):Re(λ)>
⊂ρ(A)and for all x∈Ω
(λα−(1 +ba(λ))A)−1x = 1 1 +ba(λ)
λα
1 +ba(λ)−A −1
x
= Z ∞
0
e−λtTα(t)xdt, Re(λ)> ω
wherebadenotes the Laplace transform ofa. In this case,{Tα(t)}t≥0is called theα−resolvent family generated by A.
Remark 2.30. [17] Observe that ifb(t) =gα(t) + (gα∗a) (t)t≥0, wheregα(t) = tα−1
Γ (α) and(gα∗a) (t) = Z t
0
gα(t−s)a(s)ds, then we have that the family α-resolvent {Tα(t)}t≥0 is a (b, gα)-regularized family.
In particular, if a = 0, an 1-resolvent family is the same as a c0-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 {Tα(t)}t≥0 then have that the family {Tα(t)}t≥0 verifies the following properties:
1. Tα(0) =gα(0);
2. Tα(t)ζ∈D(A) andTα(t)Aζ=ATα(t)ζ for allζ∈D(A)andt≥0;
3. Tα(t)ζ=gα(t)ζ+ Z t
0
b(t−s)ATα(t)ζds, for allx∈D(A)andt≥0;
4.
Z t
0
b(t−s)Tα(t)ζds∈D(A)andTα(t)ζ=gα(t)ζ+A Z t
0
b(t−s)Tα(s)ζds,for allζ∈Ωandt≥0.
Sufficient conditions implying that {Tα(t)}t≥0⊂B(Ω)is an α-resolvent family.
Let%(·) :R→Rbe a continuous function such that%(t)≥1 for allt∈Rand%(t)→ ∞as|t| → ∞. We consider the space [12]
C%(Ω) =
u∈C(R,Ω) : lim
|t|→∞
u(t)
%(t) = 0
.
Endowed with the normkuk%= sup
t∈R
ku(t)k
%(t) , it is a Banach space.
Lemma 2.31. [12] A subsetE⊆C%(Ω)is a relatively compact set if it verifies the following conditions:
(C1)The setE(t) ={u(t) :u∈E} is relatively compact inX for each t∈R. (C2)The setE is equicontinuous.
(C3)For each ε >0, there existsL >0 such that ku(t)k ≤ε%(t)for allu∈E and all|t|> L.
Lemma 2.32. [5] (Leray-Schauder alternative theorem)LetDbe a closed convex subset of a Banach spaceΩ such that0∈D. LetΓ :D→Dbe a completely continuous map. Then the set{ζ∈D:ζ=λΓ (ζ),0< λ <1}
is bounded or the mapΓ has a fixed point inD.
3. 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 functionu:R→Ωis said to be a mild solution to Eq.(1.2)if
(3.1) u(t) =
Z t
−∞
Tα(t−s)f(s, u(s))ds for all t∈R, where{Tα(t)}t≥0 is given byRemark 2.30.
Lemma 3.2. Letµ∈M,0< α <1 andβ >0,eδ6= 0 withδe+β >0and condition(T1)holds. Iff :R→Ω is Stepanov type µ-pseudo almost automorphic andFα(t)is given by
(3.2) Fα(t) =
Z t
−∞Tα(t−s)f(s)ds, t∈R then Fα∈ PAA(R,Ω, µ).
Proof. Indeed, since f ∈ PAAp(R,Ω, µ), there exists g1 ∈ ASp(Ω) and g2 ∈ E(Lp(0,1; Ω), µ) such that f =g1+g2 (see Definition2.17). So
Fα(t) = Z t
−∞
Tα(t−s)g1(s)ds+ Z t
−∞
Tα(t−s)g2(s)ds=φ(t) +ψ(t), where φ(t) =
Z t
−∞
Tα(t−s)g1(s)dsandψ(t) = Z t
−∞
Tα(t−s)g2(s)ds.
The proof will be discussed in two steps.
Step 1: φ(t)∈ AA(Ω).
Consider
φαn(t) =
Z t−n+1
t−n
Tα(t−s)g1(s)ds
for eacht∈Randn= 1,2,3.... Using the condition (T1) and Holder inequality, yields kφαn(t)k ≤
Z t−n+1
t−n
kTα(t−s)k kg1(s)kds
≤
Z t−n+1
t−n
ϕα(t−s)kg1(s)kds
≤ ϕα(n−1)
Z t−n+1
t−n
kg1(s)kpds 1/p
≤ ϕα(n−1)kg1kSp. Since
∞
P
n=1
ϕα(n−1) :=
∞
P
n=1
ϕα(n) < ∞, we denote that norm the well-known Weierstrass theorem that the series
φ(t) :=
Z t
−∞
Tα(t−s)g2(s)ds=
∞
X
n=1
φαn(t). Clearly,x(t)∈C(R,Ω) andkφ(t)k ≤
∞
P
n=1
kφαn(t)k ≤
∞
P
n=0
ϕα(n)kg1kSp.
Since g ∈ ASp(R,Ω),then for every sequence {sn}n∈N, there exists a sequence{sn}n∈Nand a function ge1(·)∈Lploc(R,Ω) such that for each t∈R
m→∞lim Z t+1
t
kg1(s+sm)−ge1(s)kpds 1/p
= 0 and
m→∞lim
Z t−n+1
t−n
kge1(s−sm)−g1(s)kpds 1/p
= 0.
Now, letφeαn(t) =
Z t−n+1
t−n Tα(s)eg(t−s)ds. Then using the Holder inequality, yields
φαn(t+sm)−φeαn(t) ≤
Z t−n+1
t−n
kTα(s)k kg1(t+sm−s)−eg1(t−s)kds
≤ ϕα(n−1)
Z t−n+1
t−n
kg1(t+sm−s)−eg1(t−s)kds 1/p
. Note that,
φαn(t+sm)−φeαn(t)
→0 asm→ ∞. Analogously, it is proved that
φeαn(t+sm)−φαn(t) = 0. Thus, we conclude that eachφαn ∈ AA(Ω) and consequently their uniform limitφ∈ AA(Ω).
Step 2. ψ(t)∈E(R,Ω, µ).
Consider ψαn(t) =
Z t−n+1
t−n
Tα(t−s)g2(s)dsfor eacht ∈Randn= 1,2, .... Again, using the condition (T1) and Holder inequality, yields
kψαn(t)k ≤
Z t−n+1
t−n
kTα(t−s)k kg2(s)kds
≤
Z t−n+1
t−n
ϕα(t−s)kg2(s)kds
≤ ϕα(n−1)
Z t−n+1
t−n
kg2(s)kpds 1/p
. Then, forr >0, we have
1 µ([−r, r])
Z
[−r,r]
kψαn(t)kdµ(t)≤ϕn(n−1) µ([−r, r]) Z
[−r,r]
Z t−n+1
t−n
kg2(s)kpds 1/p
dµ(t).
Since g2b ∈ E(Lp(0,1; Ω), µ) the above inequality gives rise to ψαn ∈ E (R,Ω, µ) for n = 1,2, ... Since kg2kSp
∞
P
n=0
ϕα(n)< ∞, then we deduce from the Weierstrass M-test that the series
∞
P
n=0
ψnα(t) is uniformly convergent onRandψ(t) =
Z t
−∞
Tα(t−s)g2(s)ds=
∞
X
n=1
ψnα(t).
Applyingψn∈E (R,Ω, µ) and the inequality 1
µ([−r, r]) Z
[−r,r]
kψ(t)kdµ(t)≤ 1 µ([−r, r])
Z
[−r,r]
kψ(t)−
k
X
n=1
ψnk(t)kdµ(t) +
k
X
n=1
1 µ([−r, r])
Z
[−r,r]
kψαn(t)kdµ(t)→0, we obtain that the uniform ψ(t) = P∞
n=1ψnα(t) ∈ E(R,Ω, µ). Therefore Fα(t) = φ(t) +ψ(t) is µ-pseudo
almost automorphic.
Theorem 3.3. Let µ ∈ M. Assume the conditions (H0), (T1)-(T2) are satisfied and the function f = h1+h2 ∈ PAAp(R×Ω,Ω, µ) with h1 ∈ ASp(R×Ω,Ω) and hb2 ∈E (Ω, Lp(0,1; Ω)µ). ThenEq.(1) has a uniqueµ-pseudo almost automorphic mild solution on R, provided thatLfkϕnkL1(R)<1.
Proof. LetΘ :e PAA(R,Ω, µ)→ PAA(R,Ω, µ) be the nonlinear operator defined by
(3.3)
Θζe (t) =
Z t
−∞Tα(t−s)f(s, ζ(s))ds, t∈R. Step 1. Θ (PAAe (R,Ω, µ))⊆ PAA(R,Ω, µ).
First, using the fact thatζ∈ PAA(R,Ω, µ) is relatively compact with the above Lemma2.21, Lemma2.24, follows that f(·, ζ(·))∈ PAAp(R,Ω, µ). Hence, from Lemma3.2, we know that
Θζe
(·)∈ PAA(R,Ω, µ).
Step 2. Θ has a unique fixed point.e Lett∈R, ζ, η∈ PAA(R,Ω, µ), yields
Θζe (t)−
Θηe (t)
≤
Z t
−∞
kTα(t−s)k kf(s, ζ(s))−f(s, η(s))kds
≤ Lf
Z t
−∞
kTα(t−s)k kζ(s)−η(s)kds
= Lf
Z t
0
kTα(s)k kx(t−s)−y(t−s)kds
≤ Lfkζ−ηk∞kϕαkL1(R)
which implies that
Θζe (t)−
Θηe (t)
≤ Lfkζ−ηk∞kϕαkL1(R).
Therefore, we concluded that, by means of the Banach fixed point theorem withLfkϕαkL1 <1,Θ has ae unique fixed pointζ inPAA(R,Ω, µ) which is theµ-pseudo almost automorphic solution to Eq.(1.2).
Theorem 3.4. Let µ∈M. Assume that (H0),(T1), (T3) and (T5) holds, then Eq.(1.2)admits a unique µ-pseudo almost automorphic mild solution wheneverkLfkSpϕ0<1.
Proof. To prove this result, we consider the nonlinear operator Γ given by
Θζe (t) =
Z t
−∞
Tα(t−s)f(s, ζ(s))ds, t∈R. Step 1. ΘP AAe ⊂P AA.
Now, for ζ ∈ PAA(R, X, µ) and using Lemma 2.21and Lemma 2.25 it follows that the function s → f(s, ζ(s)) is inPAAp(R,Ω, µ). On the other hand, using the Lemma 3.2we infer that Θζe ∈ PAA(R,Ω, µ), i.e., Θ mapse PAA(R,Ω, µ) into itself.
Step 2. Θ has a unique fixed point ine PAA(R,Ω, µ).
Indeed, for eacht∈R,ζ, η∈ PAA(R,Ω, µ), yields
Θζe (t)−
Θηe (t)
≤
Z t
−∞
kTα(t−s)k kf(s, ζ(s))−f(s, η(s))kds
≤
∞
X
n=1
Z t−n+1
t−n
ϕα(t−s)Lf(s)kx−yk∞ds
≤
∞
X
n=1
ϕα(n−1)
Z t−n+1
t−n
kLf(s)kpds 1/p
kζ−ηk∞
≤ ϕ0kLf(s)kSpkζ−ηk∞. In this sense, we have
Θζe −Θηe
≤ϕ0kLf(s)kSpkζ−ηk∞.
Sinceϕ0kLf(s)kSp<1, using the Banach fixed point theorem,Θ has a unique fixed pointe x∈ PAA(R,Ω, µ).
Theorem 3.5. Letµ∈M. Assume that(H0),(T1)and(T4)are true. Then there exists a uniqueµ-pseudo almost automorphic mild solution to Eq.(1.2), provided thatϕ0kLfkSr <1.
Proof. For proof of this result, we will consider the same operatorΘ given in Theoreme 3.3Eq.(3.3), given by
Θζe (t) =
Z t
−∞
Tα(t−s)f(s, ζ(s))ds, t∈R. Step 1. ΘP AAe ⊂P AA.
Now, for x ∈ PAA(R,Ω, µ) and using the Lemma2.21 and Theorem 2.27it follows that the functions s → f(s, ζ(s)) is in PAAq(R,Ω, µ), q ∈ [1, p). On the other hand, using the Lemma 3.2 we infer that Θxe ∈ PAA(R,Ω, µ), i.e., Θ mapse PAA(R,Ω, µ) into itself.
Step 2. Θ has a unique fixed point ine PAA(R,Ω, µ).
Indeed, for eacht∈R,ζ, η∈ PAA(R,Ω, µ), yields
Θζe (t)−
Θηe (t)
≤
Z t
−∞
kTα(t−s)k kf(s, ζ(s))−f(s, η(s))kds
≤ Z t
−∞
ϕα(t−s)Lf(s)kζ(s)−η(s)kds
≤
∞
X
n=1
Z t−n+1
t−n
ϕα(t−s)Lf(s)dskζ−ηk∞
≤
∞
X
n=1
ϕα(n−1)
Z t−n+1
t−n
kLf(s)krds 1/r
kζ−ηk∞
≤ ϕ0kLfkSrkζ−ηk∞ .
Follows that
Θζe −Θηe
∞≤ϕ0kLfkSrkζ−ηk∞.
In view of the inequalityϕ0kLfkSr <1, using the Banach fixed point theorem,Θ has a unique fixed pointe 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:
(T7)There exists a continuous nondecreasing functionΘ : [0,e ∞)→(0,∞) such that kf(t, θ)k ≤Θ (kθk)e
for allt∈Randθ∈Ω.
Theorem 3.6. Let µ∈M. Assume that conditionsH0,T1 are satisfied. Let f :R×Ω→Ωbe a function which satisfies assumptions(T5)-(T7)and the following additional conditions:
F1. For each κ≥0, the functiont→ Z t
−∞
ϕα(t−s)Θ (κe %e(s))dsbelongs toBC(R). Let
(3.4) λ(κ) =
Z t
−∞
ϕα(t−s)Θ (κe %e(s))ds
e%
. F2. For each ε >0, there exists aδ >0,such that for everyζ, η∈C
%e(Ω),kζ−ηk
%e≤δimplies that Z t
−∞
ϕα(t−s)kf(s, ζ(s))−f(s, η(s))kds≤ε for all t∈R.
F3. lim inf
ξ→∞
ξ λ(ξ) >1.
F4. For all a, b∈R,a < b andκ >0, the setn
f(s, ζ), a≤s≤b, ζ ∈C
%e(Ω), kζk
%e≤κo
is relatively compact inΩ.
ThenEq.(1.2)has a least oneµ-pseudo almost automorphic mild solution on t∈R. Proof. Consider the following operatorΘ :e C
%e(Ω)→C
%e(Ω) given by
Θζe (t) :=
Z t
−∞
Tα(t−s)f(s, ζ(s))ds, t∈R.
The main objective of this test is to ensure thatΘ has a fixed point ine PAA(R,Ω, µ). For this, it will be investigated in 5 stages.
Step 1. Θ is well defined.e Forζ∈C
e%(Ω), yields
Θζe (t)
≤
Z t
−∞
kTα(t−s)k kf(s, ζ(s))kds
≤ Z t
−∞
ϕα(t−s)Θ (kθe (s)k)ds
≤ Z t
−∞
ϕα(t−s)Θe kθk
%e%e(s) ds.
In this sense, using the condition(F1), we concluded thatΘ is well defined.e Step 2. The operatorΘ is continuous.e
Indeed, for any ε >0, we takeδ >0 involved in condition(F2). Ifζ, η∈C
%e(Ω) andkζ−ηk
%e≤δ, then we have
Θζe (t)−
Θηe (t)
≤
Z t
−∞
kTα(t−s)k kf(s, ζ(s))−f(s, η(s))kds
≤ Z t
−∞
ϕα(t−s)kf(s, ζ(s))−f(s, η(s))kds
≤ ε and we concluded this step.
Step 3. Θ is completely continuous.e
LetBκ(Ω) is a closed ball with center at 0 and radiusκin the space Ω. Moreover, letV =Θ (Be κ(C
%e(Ω))) andν =Θ (ζ) fore ζ∈Bκ(C
e%(Ω)).
Note that, using the condition(F1), the functions→ϕα(s)Θ (κe %e(t−s)) is integrable on [0,∞). Hence, forε >0 we can choosea≥0 such that
Z ∞
a
ϕα(s)Θ (κe %e(t−s))ds≤ε.
Since
ν(t) = Z a
0
Tα(s)f(t−s, ζ(t−s))ds+ Z ∞
a
Tα(s)f(t−s, ζ(t−s))ds and
Z ∞
a
Tα(s)f(t−s, ζ(t−s))ds
≤ Z ∞
a
kTα(s)k kf(t−s, ζ(t−s))kds
≤ Z ∞
a
ϕα(s)Θ (κe %e(t−s))ds we have ν(t)∈ac0(K) +Bε(Ω), wherec0(K) denotes the convex hull ofKand
K=n
Tα(s)f(ξ, ζ) : 0≤s≤a, t−a≤ξ≤t, kζk
e%≤κo .
Using the strong continuity of Tα(·) and condition (F4) onf, then Kis a relatively compact set, and V (t)⊆ac0(K) +Bε(Ω). ThereforeV(t) is a relatively subset of Ω fort∈R.
To conclude this step, we will prove that the set V is equicontinuous. Indeed, consider the following decomposition
ν(t+s)−ν(t) = Z s
0
Tα(σ)f(t+s−σ, ζ(t+s−σ))dσ +
Z a
0
(Tα(σ+s)−Tα(σ))f(t−σ, ζ(t−σ))dσ +
Z ∞
a
(Tα(σ+s)−Tα(σ))f(t−σ, ζ(t−σ))dσ.
(3.5)
For each ε >0, we can choosea >0 andδ1>0 such that
Z s
0
Tα(σ)f(t+s−σ, ζ(t+s−σ))dσ+ Z ∞
a
(Tα(σ+s)−Tα(σ))f(t−σ, ζ(t−σ))dσ
≤ Z s
0
kTα(σ)k kf(t+s−σ, x(t+s−σ))kdσ+ Z ∞
a
k(Tα(σ+s)−Tα(σ))k kf(t−σ, ζ(t−σ))kdσ
≤ Z s
0
ϕα(σ)Θ (κe %e(t+s−σ))dσ+ Z ∞
a
(ϕα(σ+s) +ϕα(σ))Θ (κe %e(t−σ))dσ
≤ ε 2 (3.6)
fors≤δ1. Moreover, since{f(t−σ, ζ(t−σ)) : 0≤σ≤a, ζ ∈Bκ(C
%e(Ω))}is relatively compact andTα(·) is strongly continuous, choose δ2>0 such that
(3.7) k(Tα(σ+s)−Tα(σ))f(t−σ, ζ(t−σ))k ≤ ε 2a
fors≤δ2.
Combining the estimates (3.5), (3.6) and (3.7), we get kν(t+s)−ν(t)k ≤ ε for s small enough and independent ofζ∈Bκ(C
%e(Ω)).
Finally, using the condition(F1), we have kν(t)k
%e(t) ≤ 1 e%(t)
Z t
−∞
ϕα(t−s)Θ (κe %e(s))ds→0 as|t| → ∞and this converge is independent ofζ∈Bκ(C
%e(Ω)). Therefore, using Lemma2.31, we concluded that,V is relatively compact set inC
%e(Ω).
Step 4. P=n
ζγ :ζγ=γΘ (ζe γ), γ ∈(0,1)o
is bounded.
First, assume thatζγ(·) is a solution of equationζγ =γΘ (ζe γ) for some 0< γ <1. Note that kζγ(t)k =
γΘ (ζe γ)
= γ
Z t
−∞
Tα(t−s)f(s, ζγ(s))ds
≤ Z t
−∞
ϕα(t−s)Θ (kζe γ(s)k)ds
≤ Z t
−∞
ϕα(t−s)Θe kζγk
e%%e(s) ds
≤ λkζγk
e%%e(s). Hence, yields
kζγk
%e
λkxγk
%e
≤1.
Using the condition (F3), we conclude that the setP is bounded.
Step 5. It follows from Lemma2.21, (T6-T7) and Lemma2.26that the functiont→f(t, ζ(t)) belongs to PAAp(R,Ω, µ) wheneverζ∈ PAA(R,Ω, µ). Moreover, from Lemma3.2, we infer thatΘ (PAAe (R,Ω, µ))⊆ PAA(R,Ω, µ) and noting that PAA(R,Ω, µ) is a closed subspace of C
e%(Ω) consequently we can consider Θ :e PAA(R,Ω, µ)→ PAA(R,Ω, µ). Using conditions (F1)-(F3), we deduce that this map is completely continuous. Applying Lemma2.32, we infer thatΘ has a fixed pointe ζ∈ PAA(R,Ω, µ).
References
[1] Sousa, J. Vanterler da C., and E. Capelas de Oliveira. ”On the ψ-Hilfer fractional derivative.” Commun. Nonlinear Sci.
Numer. Simul. 60 (2018): 72-91.
[2] Sousa, J. Vanterler da C., and E. Capelas de Oliveira. ”Leibniz type rule: ψ-Hilfer fractional operator.” Commun. Nonlinear Sci. Numer. Simul. 77 (2019): 305-311.
[3] Bochner, S. Continuous mappings of almost automorphic and almost periodic functions. Proceedings of the National Academy of Sciences of the United States of America 52.4 (1964): 907.
[4] Blot, J., P. Cieutat, and K. Ezzinbi. Measure theory and pseudo almost automorphic functions: New developments and applications. Nonlinear Analysis: Theory, Methods & Applications 75.4 (2012): 2426-2447.
[5] Chang, Y.-K., Xue-Yan Wei, and G. M. N’Gu´er´ekata. Some new results on bounded solutions to a semilinear integro- differential equation in Banach spaces. The J. Integral Equ. Appl. 27.2 (2015): 153-178.
[6] Chang, Y.-K., G. M. N’Gu´er´ekata, and R. Zhang. Stepanov-like weighted pseudo almost automorphic functions via measure theory. Bull. Malaysian Math. Sci. Soc. 39.3 (2016): 1005-1041.
[7] Diagana, T. Existence of pseudo-almost automorphic solutions to some abstract differential equations withSp-pseudo-almost automorphic coefficients. Nonlinear Analysis: Theory, Methods & Applications 70.11 (2009): 3781-3790.
[8] Diagana, T. Almost automorphic type and almost periodic type functions in abstract spaces. New York: Springer, 2013.
[9] Diagana, T., G. M. Mophou and G. M. N’Gu´er´ekata. Existence of weighted pseudo almost periodic solutions to some classes of differential equations withSp-weighted pseudo almost periodic coefficients. Nonlin. Anal. 72 (2010), 430–438.
[10] Ding, H.-S., J. Liang, and T.-J. Xiao. Almost automorphic solutions to nonautonomous semilinear evolution equations in Banach spaces. Nonlinear Analysis: Theory, Methods & Applications 73.5 (2010): 1426-1438.
[11] Granas, A., and J. Dugundji. Elementary fixed point theorems. Fixed Point Theory. Springer, New York, NY, 2003. 9-84.
[12] Henr´ıquez, H. R., and C. Lizama. Compact almost automorphic solutions to integral equations with infinite delay. Nonlinear Analysis: Theory, Methods & Applications 71.12 (2009): 6029-6037.
[13] Liang, J., Gaston M. N’Gu´er´ekata, T.-J. Xiao and J. Zhang. Some properties of pseudo-almost automorphic functions and applications to abstract differential equations. Nonlinear Analysis: Theory, Methods & Applications 70.7 (2009): 2731-2735.
[14] Mophou, G. M. Weighted pseudo almost automorphic mild solutions to semilinear fractional differential equations. Appl.
Math. Comput. 217.19 (2011): 7579-7587.
[15] N’Gu´er´ekata, Gaston M. Topics in Almost Automorphy. Springer Science & Business Media, 2007.
[16] N’Gu´er´ekata, Gaston M., and Alexander Pankov. Stepanov-like almost automorphic functions and monotone evolution equations. Nonlinear Analysis: Theory, Methods & Applications 68.9 (2008): 2658-2667.
[17] Ponce, R. Bounded mild solutions to fractional integro-differential equations in Banach spaces. Semigroup Forum. Vol. 87.
No. 2. Springer US, 2013.
[18] Chang, Y.-K., M.-J. Zhang, and R. Ponce. Weighted pseudo almost automorphic solutions to a semilinear fractional differential equation with Stepanov-like weighted pseudo almost automorphic nonlinear term. Appl. Math. Comput. 257 (2015):
158-168.
[19] Chang, Y.-K., and T.-W. Feng. Properties on measure pseudo almost automorphic functions and applications to fractional differential equations in Banach spaces. Electr. J. Diff. Equ. 47 (2018): 1-14.
[20] Kavitha, V., S. Abbas, and R. Murugesu. (µ1;µ2)-Pseudo almost automorphic solutions of fractional order neutral integro- differential equations. Nonlinear Studies 24.3 (2017).
[21] Chang, Y.-K., R. Zhang, and Gaston M. N’Gu´er´ekata. Weighted pseudo almost automorphic mild solutions to semilinear fractional differential equations. Comput. Math. Appl. 64.10 (2012): 3160-3170.
[22] Mophou, G. M. Weighted pseudo almost automorphic mild solutions to semilinear fractional differential equations. Appl.
Math. Comput. 217.19 (2011): 7579-7587.
[23] Alvarez-Pardo, E., and C. Lizama. Weighted pseudo almost automorphic mild solutions for two-term fractional order differential equations. Appl. Math. Comput. 271 (2015): 154-167.
[24] Kavitha, V., P.-Z. Wang, and R. Murugesu. Existence of weighted pseudo almost automorphic mild solutions to fractional integro-differential equations. J. Frac. Cal. Appl. 4.1 (2013): 37-55.
[25] Wang, D., and Z. Xia. Pseudo almost automorphic solution of semilinear fractional differential equations with the Caputo derivatives. Frac. Cal. Appl. Anal. 18.4 (2015): 951-971.
[26] Zhao, J. Q., Yong-Kui Chang, and Gaston M. N’Gu´er´ekata. Asymptotic behavior of mild solutions to semilinear fractional differential equations. J. Opt. Theory Appl. 156.1 (2013): 106-114.
[27] Xia, Z., and J. Chai. Pseudo Almost automporphy of two-term fractional functional differential equations. J. Appl. Anal.
Comput. 8.6 (2018): 1604-1644.
[28] Cao, J., Z. Huang, and Gaston M. N’Gu´er´ekata. Existence of Asymptotically Almost Automorphic Mild Solutions of Semilinear Fractional Differential Equations. Inter. J. Diff. Equ. 2018 (2018).
[29] Xia, Z.. Pseudo almost periodicity of fractional integro-differential equations with impulsive effects in Banach spaces.
Czechoslovak Math. J. 67.1 (2017): 123-141.
[30] Cao, J., A. Debbouche, and Y. Zhou. Asymptotically Almost Periodicity for a Class of Weyl–Liouville fractional Evolution Equations. Mediterr. J. Math. 15.4 (2018): 1-22.
[31] Chen, C., and M. Li. On fractional resolvent operator functions. Semigroup Forum. Vol. 80. No. 1. 2010.
[32] Lizama, C. Regularized solutions for abstract Volterra equations. J. Math. Anal. Appl. 243.2 (2000): 278-292.
[33] Lizama, C. An operator theoretical approach to a class of fractional order differential equations. Appl. Math. Lett. 24.2 (2011): 184-190.
[34] Araya, D., and C. Lizama. Almost automorphic mild solutions to fractional differential equations. Nonlinear Analysis:
Theory, Methods & Applications 69.11 (2008): 3692-3705.
[35] Cuevas, C., and C. Lizama. Almost automorphic solutions to a class of semilinear fractional differential equations. Appl.
Math. Lett. 21.12 (2008): 1315-1319.
[36] Ding, H.-S., J. Liang, and T.-J. Xiao. Almost automorphic solutions to abstract fractional differential equations. Adv. Diff.
Equ. 2010 (2010): 1-9.
[37] Agarwal, R. P., B. Andrade, and C. Cuevas. Weighted pseudo-almost periodic solutions of a class of semilinear fractional differential equations. Nonlinear Analysis: Real World Applications 11.5 (2010): 3532-3554.
[38] Cuevas, C., and J. C´esar de Souza. Existence ofS-asymptoticallyω-periodic solutions for fractional order functional integro- differential equations with infinite delay. Nonlinear Analysis: Theory, Methods & Applications 72.3-4 (2010): 1683-1689.
[39] Lizama, C., and R. Ponce. Bounded solutions to a class of semilinear integro-differential equations in Banach spaces.
Nonlinear Analysis: Theory, Methods & Applications 74.10 (2011): 3397-3406.
[40] Ezzinbi, K., S. Fatajou, and F. Z. Elamrani. Eberlein weak almost periodic solutions for a class of integro-differential equations with infinite delay. Nonautonomous Dyn. Sys. 5.1 (2018): 127-137.
[41] Agarwal, Ravi P., B. De Andrade, and C. Cuevas. Weighted pseudo-almost periodic solutions of a class of semilinear fractional differential equations. Nonlinear Analysis: Real World Applications 11.5 (2010): 3532-3554.
[42] Cuevas, C., and J. C´esar de Souza.S-asymptoticallyω-periodic solutions of semilinear fractional integro-differential equa- tions. Appl. Math. Lett. 22.6 (2009): 865-870.
[43] de Andrade, B., and C. Cuevas. S-asymptoticallyω-periodic and asymptoticallyω-periodic solutions to semi-linear Cauchy problems with non-dense domain. Nonlinear Analysis: Theory, Methods & Applications 72.6 (2010): 3190-3208.
[44] Cuevas, C., A. Sepulveda, and H. Soto. Almost periodic and pseudo-almost periodic solutions to fractional differential and integro-differential equations. Appl. Math. Comput. 218.5 (2011): 1735-1745.
[45] Abbas, S., Mouffak Benchohra, and M. A. Darwish. Fractional differential inclusions of Hilfer type under weak topologies in Banach spaces. Asian-European J. Math. 13.01 (2020): 2050015.
[46] Sousa, J. Vanterler da C., Mouffak Benchohra, and Gaston M. N’Gu´er´ekata. Attractivity for differential equations of fractional order andψ-Hilfer type. Fract. Cal. Appl. Anal. 23.4 (2020): 1188-1207.
[47] Sousa, J. Vanterler da C., Fahd Jarad, and Thabet Abdeljawad. Existence of mild solutions to Hilfer fractional evolution equations in Banach space. Annals Funct. Anal. 12.1 (2021): 1-16.
[48] You, Z., Michal Feˇckan, and JinRong Wang. Relative controllability of fractional delay differential equations via delayed perturbation of Mittag-Leffler functions. J. Comput. Appl. Math. 378 (2020): 112939.
[49] Abbas, S., M. Benchohra, G. M. N’Gu´er´ekata. Darboux problem for fractional-order discontinuous hyperbolic partial differential equations in Banach algebras. Complex Variables Elliptic Equ. 57.2-4 (2012): 337-350.
[50] Kilbas, A. A., Hari M. Srivastava, and Juan J. Trujillo. Theory and applications of fractional differential equations. Vol.
204. elsevier, 2006.
[51] Chaudhary, N. I., M. A. Z. Raja, Y. He, Z. A. Khan and J. A. Tenreito Machado. Design of multi innovation fractional LMS algorithm for parameter estimation of input nonlinear control autoregressive systems. Appl. Math. Modell. 93 (2020):
412-425
[52] Luo, Dahui, JinRong Wang, D. Shen, Michal Feˇckan. Iterative learning control for fractional-order multi-agent systems. J.
Franklin Inst. 356.12 (2019): 6328-6351.
[53] Aghayan, S., A. Alfi, and J. A. Tenreiro Machado. Robust stability analysis of uncertain fractional order neutral-type delay nonlinear systems with actuator saturation. Appl. Math. Modell. 90 (2021): 1035-1048.
[54] Sweilam, N. H., S. M. AL-Mekhlafi, A. O. Albalawi and J. A. Tenreiro Machado. Optimal control of variable-order fractional model for delay cancer treatments. Appl. Math. Modell. 89 (2021): 1557-1574.
(J. Vanterler da C. Sousa)Center for Mathematics, Computing and Cognition, Federal University of ABC, Avenida dos Estados, 5001, Bairro Bangu, 09.210-580, Santo Andre, SP - Brazil
Email address:[email protected],[email protected]
(G. M. N’Gu´er´ekata) NEERLab, Department of Mathematics, Morgan State University, Baltimore, MD 21251, USA
Email address:[email protected]
