Generalized Asymptotically Almost Periodic and Generalized Asymptotically Almost Automorphic Solutions of Abstract Multiterm Fractional Differential Inclusions

and Applied Analysis 3 Ifα ∈ (0, 2)\{1},β > 0, andN ∈ N\{1}, then the following special cases of Lemma 1 hold good: Eα,β (z) = 1 αz(1−β)/αez + εα,β (z) , 󵄨󵄨󵄨󵄨arg (z)󵄨󵄨󵄨󵄨 < απ 2 , Eα,β (z) = εα,β (z) , 󵄨󵄨󵄨󵄨arg (−z)󵄨󵄨󵄨󵄨 < π − απ 2 , (8) where εα,β (z) = N−1 ∑ n=1 z−n Γ (β − αn) + O (|z|−N) , |z| 󳨀→ ∞. (9) For further information about the Mittag-Leffler functions, compare [11, 12] and the references cited there. 2. Stepanov and Weyl Generalizations of (Asymptotically) Almost Periodic and Almost Automorphic Functions The class of almost periodic functions was introduced by H. Bohr in 1925 and later generalized bymany other mathematicians. Let I = R or I = [0,∞), and let f : I → X be continuous, where X is a Banach space with the norm ‖ ⋅ ‖. For any number ε > 0 given in advance, we say that a number τ > 0 is an ε-period forf(⋅) iff ‖f(t+τ)−f(t)‖ ≤ ε, t ∈ I.The set consisting of all ε-periods for f(⋅) is denoted by θ(f, ε). We say that f(⋅) is almost periodic, a.p. for short, iff for each ε > 0 the set θ(f, ε) is relatively dense in I, which means that there exists l > 0 such that any subinterval of I of length l meets θ(f, ε). For basic information about various classes of almost periodic functions and their generalizations, we refer the reader to [4–8, 10, 12, 13, 16, 19, 21, 26–34].The space consisting of all almost periodic functions from the interval I intoX will be denoted by AP(I : X). It is well known that the vector space PT([0,∞) : X) consisting of all bounded continuous T-periodic functions, denoted by PT([0,∞) : X), PT([0,∞) : X) fl {f ∈ Cb([0,∞)) : f(t + T) = f(t), t ≥ 0}, is a vector subspace of AP([0,∞) : X). Set APT([0,∞) : X) fl PT([0,∞) : X) ⊕ C0([0,∞) : X). Suppose that 1 ≤ p < ∞, l > 0, and f, g ∈ Lploc(I : X), where I = R or I = [0,∞). Define the Stepanov “metric” by Dp Sl [f (⋅) , g (⋅)] fl sup x∈I [1l ∫ x 󵄩󵄩󵄩󵄩f (t) − g (t)󵄩󵄩󵄩󵄩p dt] 1/p . (10) Then, in scalar-valued case, there exists Dp W [f (⋅) , g (⋅)] fl lim l→∞ Dp Sl [f (⋅) , g (⋅)] (11) in [0,∞]. The distance appearing in (11) is called the Weyl distance of f(⋅) and g(⋅).The Stepanov and Weyl “norm” of f(⋅) are introduced by 󵄩󵄩󵄩󵄩f󵄩󵄩󵄩󵄩Sp l fl Dp Sl [f (⋅) , 0] , 󵄩󵄩󵄩󵄩f󵄩󵄩󵄩󵄩Wp fl Dp W [f (⋅) , 0] , (12) respectively. We say that a function f ∈ Lploc(I : X) is Stepanov p-bounded, Sp-bounded shortly, iff 󵄩󵄩󵄩󵄩f󵄩󵄩󵄩󵄩Sp fl sup t∈I (∫ t 󵄩󵄩󵄩󵄩f (s)󵄩󵄩󵄩󵄩p ds) < ∞. (13) The space LpS(I : X) consisting of all Sp-bounded functions becomes a Banach space when equipped with the above norm. A function f ∈ LpS(I : X) is called Stepanov palmost periodic, Sp-almost periodic shortly, iff the function f : I → Lp([0, 1] : X), defined by f(t)(s) fl f(t + s), t ∈ I, s ∈ [0, 1] is almost periodic. It is said that f ∈ LpS([0,∞) : X) is asymptotically Stepanov p-almost periodic, asymptotically Sp-almost periodic for short, iff f : [0,∞) → Lp([0, 1] : X) is asymptotically almost periodic. It is a well-known fact that if f(⋅) is an almost periodic (resp., a.a.p.) function then f(⋅) is also Sp-almost periodic (resp., asymptotically Sp-a.a.p.) for 1 ≤ p < ∞.The converse statement is not true, in general. By APSp(I : X) we denote the space consisted of all Sp-almost periodic functions I 󳨃→ X. A function f ∈ LpS([0,∞) : X) is said to be asymptotically Stepanov palmost periodic, asymptotically Sp-almost periodic for short, iff f : [0,∞) → Lp([0, 1] : X) is asymptotically almost periodic. By APSp([0,∞) : X) and AAPSp([0,∞) : X) we denote the vector spaces consisting of all Stepanov palmost periodic functions and asymptotically Stepanov palmost periodic functions, respectively. Let us recall that any asymptotically almost periodic function is also asymptotically Stepanov p-almost periodic (1 ≤ p < ∞). The converse statement is clearly not true because an asymptotically Stepanov p-almost periodic function need not be continuous. We are continuing by explaining the basic definitions and results about the (asymptotically) Weyl-almost periodic functions. Definition 2 (see [35]). Assume that I = R or I = [0,∞). Let 1 ≤ p < ∞ and f ∈ Lploc(I : X). (i) It is said that the function f(⋅) is equi-Weyl-p-almost periodic,f ∈ e−Wp ap(I : X) for short, iff for each ε > 0 we can find two real numbers l > 0 and L > 0 such that any interval I󸀠 ⊆ I of length L contains a point τ ∈ I󸀠 such that sup x∈I [1l ∫ x 󵄩󵄩󵄩󵄩f (t + τ) − f (t)󵄩󵄩󵄩󵄩p dt] 1/p ≤ ε, i.e., Dp Sl [f (⋅ + τ) , f (⋅)] ≤ ε. (14) (ii) It is said that the function f(⋅) is Weyl-p-almost periodic, f ∈ Wp ap(I : X) for short, iff for each ε > 0 4 Abstract and Applied Analysis we can find a real number L > 0 such that any interval I󸀠 ⊆ I of length L contains a point τ ∈ I󸀠 such that lim l→∞ sup x∈I [1l ∫ x 󵄩󵄩󵄩󵄩f (t + τ) − f (t)󵄩󵄩󵄩󵄩p dt] 1/p ≤ ε, i.e., lim l→∞ Dp Sl [f (⋅ + τ) , f (⋅)] ≤ ε. (15) We know that APSp(I : X) ⊆ e − Wp ap(I : X) ⊆ Wp ap(I : X) in the set theoretical sense and that any of these two inclusions can be strict ([26]). We refer the reader to [35] for basic definitions and results about asymptotically Weyl-almost periodic functions. Definition 3. We say that q ∈ Lploc([0,∞) : X) is Weyl-pvanishing iff lim t→∞ lim l→∞ sup x≥0 [1l ∫ x 󵄩󵄩󵄩󵄩q (t + s)󵄩󵄩󵄩󵄩p ds] 1/p = 0. (16) It is clear that for any function q ∈ Lploc([0,∞) : X)we can replace the limits in (16). It is said that q ∈ Lploc([0,∞) : X) is equi-Weyl-p-vanishing iff lim l→∞ lim t→∞ sup x≥0 [1l ∫ x 󵄩󵄩󵄩󵄩q (t + s)󵄩󵄩󵄩󵄩p ds] 1/p = 0. (17) If q ∈ Lploc([0,∞) : X) and q(⋅) is equi-Weyl-p-vanishing, then q(⋅) is Weyl-p-vanishing. The converse statement does not hold, in general ([35]). By Wp 0 ([0,∞) : X) and e − Wp 0 ([0,∞) : X) we denote the vector spaces consisting of all Weyl-p-vanishing functions and equi-Weyl-p-vanishing functions, respectively. It can be simply proved that the limit of any uniformly convergent sequence of bounded continuous functions that are (asymptotically) almost periodic or automorphic, respectively (asymptotically), Stepanov almost periodic or automorphic, has again this property. The following result holds for the Weyl class. Proposition 4. Let (fn) be a uniformly convergent sequence of functions from e − Wp([0,∞) : X) ∩ Cb([0,∞) : X), respectively, Wp([0,∞) : X) ∩ Cb([0,∞) : X), where 1 ≤ p < ∞. If f(⋅) is the corresponding limit function, then f ∈ e − Wp([0,∞) : X) ∩ Cb([0,∞) : X), respectively, f ∈ Wp([0,∞) : X) ∩ Cb([0,∞) : X). Proof. We will prove the part (i) only for the equi-Weyl-palmost periodic functions. It is clear that f ∈ Cb([0,∞) : X). Let ε > 0 be given in advance. Then there exists an integer n0(ε) such that for each n ≥ n0(ε) we have that 󵄩󵄩󵄩󵄩fn (t) − f (t)󵄩󵄩󵄩󵄩 ≤ ε, t ≥ 0. (18) By definition, we know that there exist two real numbers ln0 > 0 and Ln0 > 0 such that any interval I󸀠 ⊆ I of length Ln0 contains a point τn0 ∈ I󸀠 such that


Introduction and Preliminaries
Almost periodic and asymptotically almost periodic solutions of differential equations in Banach spaces have been considered by many authors so far (for the basic information on the subject, we refer the reader to the monographs [1][2][3][4][5][6][7][8][9][10]).Concerning almost automorphic and asymptotically almost automorphic solutions of abstract differential equations, one may refer, for example, to the monographs by Diagana [4], N'Guérékata [5], and references cited therein.
Of concern is the following abstract multiterm fractional differential inclusion: where  ∈ N \ {1},  1 , . . .,  −1 are bounded linear operators on a Banach space , A is a closed multivalued linear operator on , 0 ≤  1 < ⋅ ⋅ ⋅ <   , 0 ≤  <   , (⋅) is an -valued function, and D   denotes the Caputo fractional derivative of order  ( [11,12]).In this paper, we provide the notions of -regularized ( 1 ,  2 )-existence and uniqueness propagation families for (1) and -regularized -propagation families for (1).In Section 4, we profile these solution operator families in terms of vector-valued Laplace transform, while in Section 5 we consider asymptotical behaviour of analytic integrated solution operator families for (1).The main result of paper, Theorem 18, enables one to consider asymptotically periodic solutions, asymptotically almost periodic solutions, and asymptotically almost automorphic solutions of certain classes of abstract integrodifferential equations in Banach spaces.In a similar way, we can give the basic information about the following abstract semilinear multiterm fractional differential inclusion: where  ∈ N \ {1},  1 , . . .,  −1 are bounded linear operators on a Banach space , A is a closed multivalued linear operator on , 0 ≤  1 < ⋅ ⋅ ⋅ <   , 0 ≤  <   , and (⋅, ⋅) is an -valued function satisfying certain assumptions.Since we essentially follow the method proposed by Kostić et al. [13] (see also [12, Subsection 2.10.1]), the 2 Abstract and Applied Analysis boundedness of linear operator  1 , . . .,  −1 is crucial for applications of vector-valued Laplace transform and therefore will be the starting point in our work.
The organization and main ideas of this paper can be briefly described as follows.In Section 2, we present the basic information about Stepanov and Weyl generalizations of asymptotically almost periodic functions and asymptotically almost automorphic functions (Proposition 4 is the only new contribution in this section).The main aim of third section is to give a brief recollection of results and definitions about multivalued linear operators in Banach spaces; in a separate Section 3.1, we analyze degenerate (, )regularized -resolvent families subgenerated by multivalued linear operators.Section 4, which is written almost in an expository manner, is devoted to the study of -regularized -propagation families for (1).The main result of fifth section is Theorem 18, where we investigate the asymptotic behaviour of   -regularized -propagation families for (1).In the proof of this theorem, we use the well-known results on analytical properties of vector-valued Laplace transform established by Sova in [14] (see, e.g., [2,Theorem 2.6.1]) in place of Cuesta's method established in the proof of [15,Theorem 2.1].The proof of Theorem 18 is much simpler and transparent than that of [15,Theorem 2.1] because of the simplicity of contour Γ in our approach.We will essentially use this fact for improvement of some known results on the asymptotic behaviour of solution operator families governing solutions of abstract two-term fractional differential equations, established recently by Keyantuo et al. [16] and Luong [17].Contrary to a great number of papers from the existing literature, Theorem 18 is applicable to the almost sectorial operators, generators of integrated or -regularized semigroups, and multivalued linear operators employing in the analysis of (fractional) Poisson heat equations in  spaces ( [18,19]).For more details, see Section 6.
We use the standard notation throughout the paper.By  we denote a complex Banach space.If  is also such a space, then by (, ) we denote the space of all continuous linear mappings from  into ; () ≡ (, ).If  is a linear operator acting on , then the domain, kernel space, and range of  will be denoted by (), (), and (), respectively.The symbol  denotes the identity operator on .By   ([0, ∞) : ) we denote the space consisted of all bounded continuous functions from [0, ∞) into ; the symbol  0 ([0, ∞) : ) denotes the closed subspace of   ([0, ∞) : ) consisting of functions vanishing at infinity.By BUC([0, ∞) : ) we denote the space consisted of all bounded uniformly continuous functions from [0, ∞) to .This space becomes one of Banach's spaces when equipped with the sup-norm.Let us recall that a subset   of  is said to be total in  iff its linear span is dense in .
Let  ∈  1 loc ([0, ∞) : ).Consider the Laplace integral During the past few decades, considerable interest in fractional calculus and fractional differential equations has been stimulated due to their numerous applications in many areas of physics and engineering.A great number of important phenomena in electromagnetics, acoustics, viscoelasticity, aerodynamics, electrochemistry, and cosmology are well described and modelled by fractional differential equations.For basic information about fractional calculus and nondegenerate fractional differential equations, one may refer, for example, to [11,12,[20][21][22][23][24][25] and the references cited therein.

Stepanov and Weyl Generalizations of (Asymptotically) Almost Periodic and Almost Automorphic Functions
The class of almost periodic functions was introduced by H. Bohr in 1925 and later generalized by many other mathematicians.Let  = R or  = [0, ∞), and let  :  →  be continuous, where  is a Banach space with the norm ‖ ⋅ ‖.
We say that (⋅) is almost periodic, a.p. for short, iff for each  > 0 the set (, ) is relatively dense in , which means that there exists  > 0 such that any subinterval of  of length  meets (, ).For basic information about various classes of almost periodic functions and their generalizations, we refer the reader to [4-8, 10, 12, 13, 16, 19, 21, 26-34].The space consisting of all almost periodic functions from the interval  into  will be denoted by AP( : ).
We are continuing by explaining the basic definitions and results about the (asymptotically) Weyl-almost periodic functions.
We refer the reader to [35] for basic definitions and results about asymptotically Weyl-almost periodic functions.
It can be simply proved that the limit of any uniformly convergent sequence of bounded continuous functions that are (asymptotically) almost periodic or automorphic, respectively (asymptotically), Stepanov almost periodic or automorphic, has again this property.The following result holds for the Weyl class.

Proposition 4. Let (𝑓 𝑛 ) be a uniformly convergent sequence of functions from
Proof.We will prove the part (i) only for the equi-Weyl-almost periodic functions.It is clear that  ∈   ([0, ∞) : ).Let  > 0 be given in advance.Then there exists an integer  0 () such that for each  ≥  0 () we have that By definition, we know that there exist two real numbers   0 > 0 and   0 > 0 such that any interval Then, for the proof of equi-Weyl--almost periodicity of function (⋅), we can choose the same  fl   0 > 0 and  =   0 > 0, and the same  fl   0 from any subinterval   ⊆ [0, ∞); speaking-matter-of-factly, we have for all  ≥ 0, so that a simple calculation involving (18) gives the existence of a finite constant Then the final result simply follows from (19).
And, just a few words about (generalized) automorphic extensions of introduced classes, where our results clearly apply.Let  : R →  be continuous.As it is well known, (⋅) is called almost automorphic, a.a.for short, iff for every real sequence (  ) there exist a subsequence (  ) of (  ) and a map  : R →  such that lim pointwise for  ∈ R. If this is the case, then it is well known that  ∈   (R : ) and that the limit function (⋅) must be bounded on R but not necessarily continuous on R. Furthermore, it is clear that the uniform convergence of one of the limits appearing in (22) implies the convergence of the second one in this equation and that, in this case, the function (⋅) has to be almost periodic and the function (⋅) has to be continuous.If the convergence of limits appearing in ( 22) is uniform on compact subsets of R, then we say that (⋅) is compactly almost automorphic, c.a.a. for short.The vector space consisting of all almost automorphic, respectively, compactly almost automorphic functions, is denoted by AA(R : ), respectively, AA  (R : ).By Bochner's criterion [4], any almost periodic function has to be compactly almost automorphic.The converse statement is not true, however [36].It is also worth noting that P. Bender proved in doctoral dissertation that that a.a.function (⋅) is c.a.a.iff it is uniformly continuous (1966, Iowa State University).
It is well-known that the reflexion at zero keeps the spaces AA(R : ) and AA  (R : ) unchanged and that the function (⋅) from ( 22) satisfies ‖‖ ∞ = ‖‖ ∞ and () ⊆ (), later needed to be a compact subset of .An interesting example of an almost automorphic function that is not almost periodic has been constructed by W. A. Veech A continuous function  : R →  is called asymptotically (compact) almost automorphic, a.(c.)a.a. for short, iff there exist a function ℎ ∈  0 ([0, ∞) : ) and a (compact) almost automorphic function  : R →  such that () = ℎ() + (),  ≥ 0. Using Bochner's criterion again, it readily follows that any asymptotically almost periodic function [0, ∞)  →  is asymptotically (compact) almost automorphic.It is well known that the spaces of almost periodic, almost automorphic, compactly almost automorphic functions and asymptotically (compact) almost automorphic functions are closed subspaces of   (R : ) when equipped with the sup-norm.
We refer the reader to [28] for the notion of Stepanovlike almost automorphic functions.The concepts of Weylalmost automorphy and Weyl pseudo almost automorphy, more general than those of Stepanov almost automorphy and Stepanov pseudo almost automorphy, were introduced by Abbas [37] in 2012.Besides the concepts of Stepanovlike almost automorphic functions, our results apply also to the classes of Weyl-almost automorphic functions and Besicovitch almost automorphic functions, introduced in [38] (cf.[7,39] for more details).

Multivalued Linear Operators in Banach Spaces
In this section, we will present some necessary definitions and auxiliary results from the theory of multivalued linear operators in Banach spaces.For further information in this direction, the reader may consult the monographs by Cross [40] and Favini and Yagi [18].Let  and  be two Banach spaces over the field of complex numbers.A multivalued mapping A :  → () is said to be a multivalued linear operator (MLO) iff the following two conditions hold: In the case that  = , then we say that A is an MLO in .It is well-known that the equality A+A = A(+) holds for every ,  ∈ (A) and for every ,  ∈ C with || + || ̸ = 0.If A is an MLO, then A0 is always a linear subspace of  and A =  + A0 for any  ∈ (A) and  ∈ A.Put (A) fl {A :  ∈ (A)}.Then the set (A) fl A −1 0 = { ∈ (A) : 0 ∈ A} is called the kernel of A. The inverse A −1 of an MLO is generally defined by (A −1 ) fl (A) and It is said that an MLO A :  → () is closed iff for any two sequences (  ) in (A) and (  ) in  such that   ∈ A  ; for all  ∈ N we have that lim →∞   =  and lim →∞   =  imply  ∈ (A) and  ∈ A.
We need the following lemma from [19].
Henceforward, Ω will always be an appropriate subspace of R and  will always be the Lebesgue measure defined on Ω.
Suppose that A is an MLO in  and that  ∈ () is possibly noninjective operator satisfying A ⊆ A.Then the -resolvent set of A,   (A) for short, is defined as the union of those complex numbers  ∈ C for which We will use the following extension of [19, Theorem 1.2.4(i)], whose proof can be left to the reader as an easy exercise (see also the proof of [18,Theorem 1.7,p. 24]).Lemma 7. Let ,  ∈ () and let A be an MLO.If  = , A ⊆ A, A ⊆ A, and ( − A) −1  ∈ (), then one has Suppose that A is an MLO in .Then  ∈ C is said to be an eigenvalue of A iff there exists an element  ∈  \ {0} such that  ∈ A; we call  an eigenvector of operator A corresponding to the eigenvalue .Let us recall that, in purely multivalued case, an element  ∈ \{0} can be an eigenvector of operator A corresponding to different values of scalars .The point spectrum of A,   (A) for short, is defined as the union of all eigenvalues of A.
Unless stated otherwise, we will always assume henceforth that the function (⋅) is a scalar-valued kernel on [0, ) and that the operator  ∈ () is injective.For more details about abstract degenerate differential equations, the reader may consult the monographs [18,[41][42][43].
We will use the following definition.
Hereafter, the following equality will play an important role in our analysis: for any  = 0, . . .,   − 1.The basic properties of subgenerators and integral generators continue to hold, with appropriate changes, in degenerate case; compare [12] A function  ∈ ([0, ∞) : ) is said to be (i) a strong solution of (33) iff there exists a continuous function  A ∈ ([0, ∞) : ) such that  A () ∈ A() for all  ≥ 0 and (ii) a mild solution of (33) iff Clearly, every strong solution of ( 33) is also a mild solution of the same problem while the converse statement is not true, in general.We similarly define the notion of a strong (mild) solution of problem (28).
We have the following:   () −1 2   ,  ≥ 0, is a strong solution of (28), provided   ∈  2 (⋂ −1 =0 (  )) for 0 ≤  ≤   − 1.For our later purposes, it will be sufficient to characterize the introduced classes of -regularized propagation families by vector-valued Laplace transform; keeping in mind Lemmas 5-7, the proofs are almost the same as in nondegenerate case and we will only notify some details of the proof of Theorem 14 below because the formulation of [12, Theorem 2.10.9] is slightly misleading since the injectivity of operator   for  ∈ C with R >  has not been clarified in a proper way and property (ii) in the formulation of this theorem is required to hold for all  ∈ N 0   −1 .
Proof.Concerning assertion (I), we will only sketch the main details of the proof of the injectivity of operator   for every  ∈ C with R >  and k() ̸ = 0 (we know that ( 37)-( 38) hold on account of Theorem 13).Observe that we do not need the condition (I)(ii) for the proof of (II), where we only use an elementary argumentation as well as Lemmas 5-7 (the composition property (31) follows by applying the Laplace transform and Lemma 7, while the commutation of operator families   (⋅) with the operators  and   for 0 ≤  ≤  − 1 is much simpler to show).The consideration is quite similar in the case that the condition (II) holds and, because of that, we will consider only the first case.Let  0 ∈ C with R 0 >  and k( 0 ) ̸ = 0 be fixed, and let 0 ∈   0  for some  ∈ .Using the fact that (( 0 ()) ≥0 , . . ., (   −1 ()) ≥0 ) is a global regularized -uniqueness propagation family for (1), we can simply prove that by performing the Laplace transform at the both sides of the composition property (31).By the injectivity of the operator    − + ∑ ∈     −   for  =  0 , we obtain that  = 0 and the claimed assertion follows.
For our purposes, the following result will be sufficiently enough (cf.Theorem 14 and [2, Theorem 2.6.1,Proposition 2.6.3 b]); we feel duty bound to say that the small inconsistencies in the formulation of [12,Theorem 2.10.11] have been made; see also [34].
Let the following three conditions hold: (ii) The operator   is injective for all  + Σ +/2 .

Asymptotical Behaviour of 𝑘 𝑖 -Regularized 𝐶-Propagation Families for (1)
The main aim of this section is to investigate polynomial decaying of   -regularized -propagation families for (1) as time goes to infinity.Applications of Theorem 16 (see also Remark 17) will be crucial in our work and we start by observing that it is not clear how one can prove the injectivity of operator   , given by (36), in general case.Because of that, we will first focus our attention to the case that   =   , where   ∈ C for 1 ≤  ≤  − 1, by exploring the generation of fractionally integrated -propagation families for (1) only.Moreover, we will assume that the numbers   are nonnegative for 1 ≤  ≤  − 1 and that  − 1 <  (the case  − 1 ≥  can be analyzed similarly) and the multivalued linear operator A under our consideration is possibly not densely defined.
(ii) Denote by  the set consisting of all eigenvectors  of operator A which corresponds to eigenvalues  ∈ C of operator A for which the mapping belongs to the space F  .Then the mapping   →   (),  ≥ 0, belongs to the space F  for all  ∈ span(); furthermore, the mapping   →   (),  ≥ 0, belongs to the space F  for all  ∈ span() provided additionally that (  ()) ≥0 is bounded.
Proof.We will examine the case  − 1 <  only.The proof of (i) can be given following the lines of the proof of [9, Theorem 1.1.11],with appropriate changes briefly described as follows.Since the operator (1−/   − )+∑ −1 =1 (  /   −  ) is invertible in () for all  ∈ C with || sufficiently large, because the norm of bounded linear operator (/   − ) − ∑ −1 =1 (  /   −  ) for such values of  is strictly less than 1, we get that the term is well-defined for all  ∈ C with R >  0 , for some  0 > max(0, abs()).Set Then it is clear that and that there exists a finite constant  > 0 such that where  *  0 (⋅) denotes the th convolution power of  0 (⋅).The function () fl ∑ ∞ =1  *  0 (),  > 0, is well-defined since there exists a finite constant   > 0 such that and, due to Lemma 1, For the remaining part of proof of (i), it suffices to repeat literally the arguments from the proof of [9,Theorem 1.1.11].
For the proof of (ii), observe first that, if  ∈ A for some  ∈ C, then performing the Laplace transform at the both sides of the composition property (31), as it has been done as in our previous examinations, immediately yields that for R > 0 suff.large, and therefore   () =  , (),  ≥ 0.
As a consequence, we have that the mapping   →   (),  ≥ 0, belongs to the space F  for all  ∈ span().The boundedness of (  ()) ≥0 implies the uniform convergence of   ()  to   () ( ≥ 0) for any sequence (  ) ∈ span() converging to some element  ∈ span(); then the final result follows by combining the previously proved statement and the fact that the limit of a uniform convergent sequence of bounded continuous functions belonging to any space from F  belongs to this space again (see Proposition 4 for the class of (equi-) Weyl-almost periodic functions).
for  ∈  satisfying  ∈ A ( ∈ C).To the best knowledge of the authors, in the handbooks containing tables of Laplace transforms, the explicit forms of functions like  , (⋅) are not known, with the exception of some very special cases of the coefficients   ,   (see, e.g., [12,Remark 3.3.10(vi)]).
The following theorem is motivated by some pioneering results of Ruess and Summers concerning integration of asymptotically almost periodic functions [24].
Proof.From our previous considerations of nondegenerate case, it is well known that any mild solution of the abstract Cauchy inclusion (33) has to satisfy the following equality: With the help of Laplace transform and a simple calculation, it can be simply verified that the function (⋅), whose Laplace transform is given by (69), is a mild solution of abstract fractional inclusion (33).
A, B :  → () are two MLOs, then we define its sum A+B by (A+B) fl (A)∩(B) and (A+B) fl A + B,  ∈ (A + B).It is evident that A + B is likewise an MLO.We write A ⊆ B iff (A) ⊆ (B) and A ⊆ B for all  ∈ (A).Let A :  → () and B :  → () be two MLOs, where  is a complex Banach space.The product of A and B is defined by (BA) fl { ∈ (A) : (B) ∩ A ̸ = 0} and BA fl B((B) ∩ A).A simple proof shows that BA :  → () is an MLO and (BA) −1 = A −1 B −1 .The scalar multiplication of an MLO A :  → () with the number  ∈ C, A for short, is defined by (A) fl (A) and (A)() fl A,  ∈ (A).Then A :  → () is an MLO and ()A = (A) = (A), ,  ∈ C.