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

Authors: G. M. N’Guérékata; Marko Kostić
Journal: Abstract and Applied Analysis
 (2018)

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## Abstract

The main aim of this paper is to investigate generalized asymptotical almost periodicity and generalized asymptotical almost automorphy of solutions to a class of abstract (semilinear) multiterm fractional differential inclusions with Caputo derivatives. We illustrate our abstract results with several examples and possible applications.


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## Body

## 1. 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 [1–10]). 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:(1)Dtαnut+∑i=1n-1AiDtαiut∈ADtαut+ft,t≥0,uk0=uk,k=0,…,αn-1,where n∈N∖1, A1,…,An-1 are bounded linear operators on a Banach space X, A is a closed multivalued linear operator on X, 0≤α1<⋯<αn, 0≤α<αn, f(·) is an X-valued function, and Dtα denotes the Caputo fractional derivative of order α ([11, 12]). In this paper, we provide the notions of k-regularized (C1,C2)-existence and uniqueness propagation families for (1) and k-regularized C-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:(2)Dtαnut+∑i=1n-1AiDtαiut∈ADtαut+ft,ut,t≥0,uk0=uk,k=0,…,αn-1,where n∈N∖1, A1,…,An-1 are bounded linear operators on a Banach space X, A is a closed multivalued linear operator on X, 0≤α1<⋯<αn, 0≤α<αn, and f(·,·) is an X-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 boundedness of linear operator A1,…,An-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 Section2, 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 (a,k)-regularized C-resolvent families subgenerated by multivalued linear operators. Section 4, which is written almost in an expository manner, is devoted to the study of k-regularized C-propagation families for (1). The main result of fifth section is Theorem 18, where we investigate the asymptotic behaviour of ki-regularized C-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 C-regularized semigroups, and multivalued linear operators employing in the analysis of (fractional) Poisson heat equations in Lp-spaces ([18, 19]). For more details, see Section 6.We use the standard notation throughout the paper. ByX we denote a complex Banach space. If Y is also such a space, then by L(X,Y) we denote the space of all continuous linear mappings from X into Y; L(X)≡L(X,X). If A is a linear operator acting on X, then the domain, kernel space, and range of A will be denoted by D(A), N(A), and R(A), respectively. The symbol I denotes the identity operator on X. By Cb([0,∞):X) we denote the space consisted of all bounded continuous functions from [0,∞) into X; the symbol C0([0,∞):X) denotes the closed subspace of Cb([0,∞):X) consisting of functions vanishing at infinity. By BUC([0,∞):X) we denote the space consisted of all bounded uniformly continuous functions from [0,∞) to X. This space becomes one of Banach’s spaces when equipped with the sup-norm. Let us recall that a subset X′ of X is said to be total in X iff its linear span is dense in X.Letf∈Lloc1([0,∞):X). Consider the Laplace integral(3)Lfλ≔f~λ≔∫0∞e-λtftdt≔limτ→∞∫0τe-λtftdt,for λ∈C. If f~(λ0) exists for some λ0∈C, then we define the abscissa of convergence of f~(·) by(4)absXf≔inf⁡Rλ:f~λexists;otherwise, absXf≔+∞. It is said that f(·) is Laplace transformable or equivalently that f(·) belongs to the class (P1)-X, iff absXf<∞; in scalar-valued case, we write (P1)≔(P1)-C and abs(f)≔absC(f).Ifζ>0, then we define gζ(t)≔tζ-1/Γ(ζ), t>0; g0(t)≡ the Dirac delta distribution. Here, Γ(·) denotes the Gamma function. Set Σβ≔{z∈C∖0:argz<β} (β∈(0,π]), Nn≔{1,…,n}, and Nn0≔0,1,…,n  (n∈N).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, 20–25] and the references cited therein.We will use only the Caputo fractional derivatives. Letζ>0. Then the Caputo fractional derivative Dtζu ([11, 12]) is defined for those functions u∈C⌈ζ⌉-1([0,∞):X) for which g⌈ζ⌉-ζ∗(u-∑j=0⌈ζ⌉-1u(j)(0)gj+1)∈C⌈ζ⌉([0,∞):X), by(5)Dtζut≔dζdtζgζ-ζ∗u-∑j=0ζ-1uj0gj+1.Assuming that the Caputo fractional derivative Dtζu(t) exists, then for each number ν∈(0,ζ) the Caputo fractional derivative Dtνu(t) exists, as well.The Mittag-Leffler functionEβ,γ(z) (β>0, γ∈R) is defined by(6)Eβ,γz≔∑k=0∞zkΓβk+γ,z∈C.Set Eβ(z)≔Eβ,1(z), z∈C.The asymptotic behaviour of the Mittag-Leffler functionEα,β(z) is given in the following lemma (see, e.g., [12]):Lemma 1.
Let0<σ<1/2π. Then, for every z∈C∖{0} and m∈N∖{1},(7)Eα,βz=1α∑sZs1-βeZs-∑j=1m-1z-jΓβ-αj+Oz-m,where Zs is defined by Zs≔z1/αe2πis/α and the first summation is taken over all those integers s satisfying argz+2πs<απ/2+σ.Ifα∈(0,2)∖{1}, β>0, and N∈N∖{1}, then the following special cases of Lemma 1 hold good:(8)Eα,βz=1αz1-β/αez1/α+εα,βz,argz<απ2,Eα,βz=εα,βz,arg-z<π-απ2,where(9)εα,βz=∑n=1N-1z-nΓβ-αn+Oz-N,z⟶∞.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 by many other mathematicians. LetI=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 for f(·) 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 into X will be denoted by AP(I:X).It is well known that the vector spacePT([0,∞):X) consisting of all bounded continuous T-periodic functions, denoted by PT([0,∞):X), PT([0,∞):X)≔{f∈Cb([0,∞)):f(t+T)=f(t),t≥0}, is a vector subspace of AP([0,∞):X). Set APT([0,∞):X)≔PT([0,∞):X)⊕C0([0,∞):X).Suppose that1≤p<∞, l>0, and f,g∈Llocp(I:X), where I=R or I=[0,∞). Define the Stepanov “metric” by(10)DSlpf·,g·≔supx∈I1l∫xx+lft-gtpdt1/p.Then, in scalar-valued case, there exists(11)DWpf·,g·≔liml→∞DSlpf·,g·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(12)fSlp≔DSlpf·,0,fWp≔DWpf·,0,respectively. We say that a function f∈Llocp(I:X) is Stepanov p-bounded, Sp-bounded shortly, iff(13)fSp≔supt∈I∫tt+1fspds1/p<∞.The space LSp(I:X) consisting of all Sp-bounded functions becomes a Banach space when equipped with the above norm. A function f∈LSp(I:X) is called Stepanov p-almost periodic, Sp-almost periodic shortly, iff the function f^:I→Lp([0,1]:X), defined by f^(t)(s)≔f(t+s), t∈I, s∈[0,1] is almost periodic. It is said that f∈LSp([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 iff(·) 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.ByAPSp(I:X) we denote the space consisted of all Sp-almost periodic functions I↦X. A function f∈LSp([0,∞):X) is said to be asymptotically Stepanov p-almost 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 p-almost periodic functions and asymptotically Stepanov p-almost periodic functions, respectively.Let us recall that any asymptotically almost periodic function is also asymptotically Stepanovp-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 thatI=R or I=[0,∞). Let 1≤p<∞ and f∈Llocp(I:X).(i)
It is said that the functionf(·) is equi-Weyl-p-almost periodic, f∈e-Wapp(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(14)supx∈I1l∫xx+lft+τ-ftpdt1/p≤ϵ,i.e.,DSlpf·+τ,f·≤ϵ.(ii)
It is said that the functionf(·) is Weyl-p-almost periodic, f∈Wapp(I:X) for short, iff for each ϵ>0 we can find a real number L>0 such that any interval I′⊆I of length L contains a point τ∈I′ such that(15)liml→∞supx∈I1l∫xx+lft+τ-ftpdt1/p≤ϵ,i.e.,liml→∞DSlpf·+τ,f·≤ϵ.We know thatAPSp(I:X)⊆e-Wapp(I:X)⊆Wapp(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 thatq∈Llocp([0,∞):X) is Weyl-p-vanishing iff(16)limt→∞liml→∞supx≥01l∫xx+lqt+spds1/p=0.It is clear that for any functionq∈Llocp([0,∞):X) we can replace the limits in (16). It is said that q∈Llocp([0,∞):X) is equi-Weyl-p-vanishing iff(17)liml→∞limt→∞supx≥01l∫xx+lqt+spds1/p=0.Ifq∈Llocp([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 W0p([0,∞):X) and e-W0p([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-p-almost 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(18)fnt-ft≤ϵ,t≥0.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(19)supx∈I1ln0∫xx+ln0fn0t+τ-fn0tpdt1/p≤ϵ.Then, for the proof of equi-Weyl-p-almost periodicity of function f(·), we can choose the same l≔ln0>0 and L=Ln0>0, and the same τ≔τn0 from any subinterval I′⊆[0,∞); speaking-matter-of-factly, we have(20)ft+τ-ft≤ft+τ-fn0t+τ+fn0t+τ-fn0t+ft-fn0tfor all t≥0, so that a simple calculation involving (18) gives the existence of a finite constant cp>0 such that(21)supx∈I1l∫xx+lft+τ-ftpdt1/p≤cpϵ+supx∈I1ln0∫xx+ln0fn0t+τ-fn0tpdt1/p≤2cpϵ.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. Letf:R→X be continuous. As it is well known, f(·) is called almost automorphic, a.a. for short, iff for every real sequence (bn) there exist a subsequence (an) of (bn) and a map g:R→X such that(22)limn→∞ft+an=gt,limn→∞gt-an=ft,pointwise for t∈R. If this is the case, then it is well known that f∈Cb(R:X) and that the limit function g(·) 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 f(·) has to be almost periodic and the function g(·) has to be continuous. If the convergence of limits appearing in (22) is uniform on compact subsets of R, then we say that f(·) 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:X), respectively, AAc(R:X). 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 f(·) is c.a.a. iff it is uniformly continuous (1966, Iowa State University).It is well-known that the reflexion at zero keeps the spacesAA(R:X) and AAc(R:X) unchanged and that the function g(·) from (22) satisfies f∞=g∞ and R(g)⊆R(f)¯, later needed to be a compact subset of X. An interesting example of an almost automorphic function that is not almost periodic has been constructed by W. A. Veech(23)ft≔2+eit+eit22+eit+eit2,t∈R.A continuous functionf:R→X is called asymptotically (compact) almost automorphic, a.(c.)a.a. for short, iff there exist a function h∈C0([0,∞):X) and a (compact) almost automorphic function q:R→X such that f(t)=h(t)+q(t), t≥0. Using Bochner’s criterion again, it readily follows that any asymptotically almost periodic function [0,∞)↦X 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 Cb(R:X) when equipped with the sup-norm.We refer the reader to [28] for the notion of Stepanov-like almost automorphic functions. The concepts of Weyl-almost 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 Stepanov-like 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).


## 3. 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].LetX and Y be two Banach spaces over the field of complex numbers. A multivalued mapping A:X→P(Y) is said to be a multivalued linear operator (MLO) iff the following two conditions hold:(i)
D(A)≔{x∈X:Ax≠∅} is a linear subspace of X;(ii)
Ax+Ay⊆A(x+y), x,y∈D(A), and λAx⊆A(λx), λ∈C, x∈D(A).In the case thatX=Y, then we say that A is an MLO in X. It is well-known that the equality λAx+ηAy=A(λx+ηy) holds for every x,y∈D(A) and for every λ,η∈C with λ+η≠0. If A is an MLO, then A0 is always a linear subspace of Y and Ax=f+A0 for any x∈D(A) and f∈Ax. Put R(A)≔Ax:x∈DA. Then the set N(A)≔A-10=x∈DA:0∈Ax is called the kernel of A. The inverse A-1 of an MLO is generally defined by D(A-1)≔R(A) and A-1y≔x∈DA:y∈Ax. It is checked at once that A-1 is an MLO in X, and that N(A-1)=A0 and (A-1)-1=A. If N(A)={0}, that is, if A-1 is single-valued, then A is called injective. If A,B:X→P(Y) are two MLOs, then we define its sum A+B by D(A+B)≔D(A)∩D(B) and (A+B)x≔Ax+Bx, x∈D(A+B). It is evident that A+B is likewise an MLO. We write A⊆B iff D(A)⊆D(B) and Ax⊆Bx for all x∈D(A).LetA:X→P(Y) and B:Y→P(Z) be two MLOs, where Z is a complex Banach space. The product of A and B is defined by D(BA)≔{x∈D(A):D(B)∩Ax≠∅} and BAx≔B(D(B)∩Ax). A simple proof shows that BA:X→P(Z) is an MLO and (BA)-1=A-1B-1. The scalar multiplication of an MLO A:X→P(Y) with the number z∈C, zA for short, is defined by D(zA)≔D(A) and (zA)(x)≔zAx, x∈D(A). Then zA:X→P(Y) is an MLO and (ωz)A=ω(zA)=z(ωA), z,ω∈C.It is said that an MLOA:X→P(Y) is closed iff for any two sequences (xn) in D(A) and (yn) in Y such that yn∈Axn; for all n∈N we have that limn→∞xn=x and limn→∞yn=y imply x∈D(A) and y∈Ax.We need the following lemma from [19].Lemma 5.
LetΩ be a locally compact, separable metric space, and let μ be a locally finite Borel measure defined on Ω. Suppose that A:X→P(Y) is a closed MLO. Let f:Ω→X and g:Ω→Y be μ-integrable, and let g(x)∈Af(x), x∈Ω. Then ∫Ωfdμ∈D(A) and ∫Ωgdμ∈A∫Ωfdμ.Henceforward,Ω will always be an appropriate subspace of R and μ will always be the Lebesgue measure defined on Ω.Denote by (P1)-X the vector space consisting of all Laplace transformable functions f:[0,∞)→X; by f~(·) we denote the Laplace transform of f(·), defined as in [2]. We need also the following lemma from [19].Lemma 6.
Assume thatA:X→P(Y) is a closed MLO and that f∈ (P1)-X, l∈ (P1)-Y and (f~(λ),l~(λ))∈A, λ∈C for Rλ>max⁡(abs(f),abs(l)). Then l(t)∈Af(t) for any t≥0 which is a point of continuity of both functions f(t) and l(t).Suppose thatA is an MLO in X and that C∈L(X) is possibly noninjective operator satisfying CA⊆AC. Then the C-resolvent set of A, ρC(A) for short, is defined as the union of those complex numbers λ∈C for which(i)
R(C)⊆R(λ-A);(ii)
(λ-A)-1C is a single-valued linear continuous operator on X.The operatorλ↦(λ-A)-1C is called the C-resolvent of A  (λ∈ρC(A)); the resolvent set of A is defined by ρA≔ρIA, R(λ:A)≡(λ-A)-1  (λ∈ρ(A)).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.
LetB,D∈L(X) and let A be an MLO. If BD=DB, BA⊆AB, DA⊆AD, and (B-A)-1D∈L(X), then one has(24)B-A-1DA⊆BB-A-1D-D⊆AB-A-1D.Suppose thatA is an MLO in X. Then λ∈C is said to be an eigenvalue of A iff there exists an element x∈X∖{0} such that λx∈Ax; we call x an eigenvector of operator A corresponding to the eigenvalue λ. Let us recall that, in purely multivalued case, an element x∈X∖{0} can be an eigenvector of operator A corresponding to different values of scalars λ. The point spectrum of A, σp(A) for short, is defined as the union of all eigenvalues of A.


### 3.1. Degenerate(a,k)-Regularized C-Resolvent Operator Families

If it is not stated otherwise, we assume that0<τ≤∞, k∈C([0,τ)), k≠0, a∈Lloc1([0,τ)), a≠0, A:X→P(X) is an MLO, C1∈L(Y,X), C2∈L(X) is injective, C∈L(X) is injective, and CA⊆AC.We need the following notions from [19].Definition 8.
Suppose0<τ≤∞, k∈C([0,τ)), k≠0, a∈Lloc1([0,τ)), a≠0, A:X→P(X) is an MLO, C1∈L(Y,X), and C2∈L(X) is injective.(i)
Then it is said thatA is a subgenerator of a (local, if τ<∞) mild (a,k)-regularized (C1,C2)-existence and uniqueness family (R1(t),R2(t))t∈[0,τ)⊆L(Y,X)×L(X) iff the mappings t↦R1(t)y, t≥0, and t↦R2(t)x, t∈[0,τ), are continuous for every fixed x∈X and y∈Y, and the following conditions hold:(25)∫0tat-sR1syds,R1ty-ktC1y∈A,t∈0,τ,y∈Y,(26)∫0tat-sR2syds=R2tx-ktC2x,whenevert∈0,τ,x,y∈A.(ii)
Let(R1(t))t∈[0,τ)⊆L(Y,X) be strongly continuous. Then it is said that A is a subgenerator of a (local, if τ<∞) mild (a,k)-regularized C1-existence family (R1(t))t∈[0,τ) iff (25) holds.(iii)
Let(R2(t))t∈[0,τ)⊆L(X) be strongly continuous. Then it is said that A is a subgenerator of a (local, if τ<∞) mild (a,k)-regularized C2-uniqueness family (R2(t))t∈[0,τ) iff (26) holds.Definition 9.
Suppose that0<τ≤∞, k∈C([0,τ)), k≠0, a∈Lloc1([0,τ)), a≠0, A:X→P(X) is an MLO, C∈L(X) is injective, and CA⊆AC. Then it is said that a strongly continuous operator family (R(t))t∈[0,τ)⊆L(X) is an (a,k)-regularized C-resolvent family with a subgenerator A iff (R(t))t∈[0,τ) is a mild (a,k)-regularized C-uniqueness family having A as subgenerator, R(t)C=CR(t), and R(t)A⊆AR(t)  (t∈[0,τ)).Ifτ=∞, (R(t))t≥0 is said to be exponentially bounded (bounded) iff there exists ω∈R (ω=0) such that the family e-ωtRt:t≥0⊆L(X) is bounded. If k(t)=gα+1(t), where α≥0, then it is also said that (R(t))t∈[0,τ) is an α-times integrated (a,C)-resolvent family; 0-times integrated (a,C)-resolvent family is further abbreviated to (a,C)-resolvent family. We accept a similar terminology for the classes of mild (a,k)-regularized C1-existence families and mild (a,k)-regularized C2-uniqueness families.The integral generator of a mild(a,k)-regularized C2-uniqueness family (R2(t))t∈[0,τ) (mild (a,k)-regularized (C1,C2)-existence and uniqueness family (R1(t),R2(t))t∈[0,τ)) is defined through(27)Aint≔x,y∈X×X:R2tx-ktC2x=∫0tat-sR2syds,t∈0,τ;the integral generator of an (a,k)-regularized C-regularized family (R(t))t∈[0,τ) is defined in a similar fashion. The integral generator Aint is a closed MLO in X which is, in fact, the maximal subgenerator of (R2(t))t∈[0,τ) ((R(t))t∈[0,τ)) with respect to the set inclusion. We refer the reader to [19] for the notion of an exponentially bounded, analytic (a,k)-regularized C-resolvent operator family.Unless stated otherwise, we will always assume henceforth that the functionk(·) is a scalar-valued kernel on [0,τ) and that the operator C∈L(X) is injective. For more details about abstract degenerate differential equations, the reader may consult the monographs [18, 41–43].


## 3.1. Degenerate(a,k)-Regularized C-Resolvent Operator Families

If it is not stated otherwise, we assume that0<τ≤∞, k∈C([0,τ)), k≠0, a∈Lloc1([0,τ)), a≠0, A:X→P(X) is an MLO, C1∈L(Y,X), C2∈L(X) is injective, C∈L(X) is injective, and CA⊆AC.We need the following notions from [19].Definition 8.
Suppose0<τ≤∞, k∈C([0,τ)), k≠0, a∈Lloc1([0,τ)), a≠0, A:X→P(X) is an MLO, C1∈L(Y,X), and C2∈L(X) is injective.(i)
Then it is said thatA is a subgenerator of a (local, if τ<∞) mild (a,k)-regularized (C1,C2)-existence and uniqueness family (R1(t),R2(t))t∈[0,τ)⊆L(Y,X)×L(X) iff the mappings t↦R1(t)y, t≥0, and t↦R2(t)x, t∈[0,τ), are continuous for every fixed x∈X and y∈Y, and the following conditions hold:(25)∫0tat-sR1syds,R1ty-ktC1y∈A,t∈0,τ,y∈Y,(26)∫0tat-sR2syds=R2tx-ktC2x,whenevert∈0,τ,x,y∈A.(ii)
Let(R1(t))t∈[0,τ)⊆L(Y,X) be strongly continuous. Then it is said that A is a subgenerator of a (local, if τ<∞) mild (a,k)-regularized C1-existence family (R1(t))t∈[0,τ) iff (25) holds.(iii)
Let(R2(t))t∈[0,τ)⊆L(X) be strongly continuous. Then it is said that A is a subgenerator of a (local, if τ<∞) mild (a,k)-regularized C2-uniqueness family (R2(t))t∈[0,τ) iff (26) holds.Definition 9.
Suppose that0<τ≤∞, k∈C([0,τ)), k≠0, a∈Lloc1([0,τ)), a≠0, A:X→P(X) is an MLO, C∈L(X) is injective, and CA⊆AC. Then it is said that a strongly continuous operator family (R(t))t∈[0,τ)⊆L(X) is an (a,k)-regularized C-resolvent family with a subgenerator A iff (R(t))t∈[0,τ) is a mild (a,k)-regularized C-uniqueness family having A as subgenerator, R(t)C=CR(t), and R(t)A⊆AR(t)  (t∈[0,τ)).Ifτ=∞, (R(t))t≥0 is said to be exponentially bounded (bounded) iff there exists ω∈R (ω=0) such that the family e-ωtRt:t≥0⊆L(X) is bounded. If k(t)=gα+1(t), where α≥0, then it is also said that (R(t))t∈[0,τ) is an α-times integrated (a,C)-resolvent family; 0-times integrated (a,C)-resolvent family is further abbreviated to (a,C)-resolvent family. We accept a similar terminology for the classes of mild (a,k)-regularized C1-existence families and mild (a,k)-regularized C2-uniqueness families.The integral generator of a mild(a,k)-regularized C2-uniqueness family (R2(t))t∈[0,τ) (mild (a,k)-regularized (C1,C2)-existence and uniqueness family (R1(t),R2(t))t∈[0,τ)) is defined through(27)Aint≔x,y∈X×X:R2tx-ktC2x=∫0tat-sR2syds,t∈0,τ;the integral generator of an (a,k)-regularized C-regularized family (R(t))t∈[0,τ) is defined in a similar fashion. The integral generator Aint is a closed MLO in X which is, in fact, the maximal subgenerator of (R2(t))t∈[0,τ) ((R(t))t∈[0,τ)) with respect to the set inclusion. We refer the reader to [19] for the notion of an exponentially bounded, analytic (a,k)-regularized C-resolvent operator family.Unless stated otherwise, we will always assume henceforth that the functionk(·) is a scalar-valued kernel on [0,τ) and that the operator C∈L(X) is injective. For more details about abstract degenerate differential equations, the reader may consult the monographs [18, 41–43].


## 4.k-Regularized C-Propagation Families for (1)

Recall thatn∈N∖{1}, A1,…,An-1 are bounded linear operators on a Banach space X, A is a closed multivalued linear operator on X, 0≤α1<⋯<αn, 0≤α<αn, and f(t) is an X-valued function. Henceforth, we always assume that k,k1,k2,… are scalar-valued kernels and a≠0 in Lloc1([0,τ)). Set mj≔⌈αj⌉, 1≤j≤n, m≔m0≔⌈α⌉, A0≔A, and α0≔α.We will use the following definition.Definition 10.
A functionu∈Cmn-1([0,∞):X) is called a (strong) solution of (1) iff AiDtαiu∈C([0,∞):X) for 0≤i≤n-1, gmn-αn∗(u-∑k=0mn-1ukgk+1)∈Cmn([0,∞):X), and (1) holds.Integrating both sides of (1) αn-times and employing the closedness of A, Lemma 5, and the equality [11, (1.21)], it readily follows that any strong solution u(t), t≥0 of (1) satisfies the following:(28)u·-∑k=0mn-1ukgk+1·+∑j=1n-1gαn-αj∗Aju·-∑k=0mj-1ukgk+1·∈gαn-α∗Au·-∑k=0m-1ukgk+1·.Ifi∈Nmn-10, then we define Di≔{j∈Nn-1:mj-1≥i}. Plugging uj=0, 0≤j≤mn-1, j≠i, in (28), we get(29)u·;0,…,ui,…,0-uigi+1·+∑j∈Digαn-αj∗Aju·;0,…,ui,…,0-uigi+1·+∑j∈Nn-1∖Digαn-αj∗Aju·;0,…,ui,…,0∈gαn-α*Au·;0,…,ui,…,0,m-1<i,gαn-α*Au·;0,…,ui,…,0-uigi+1·,m-1≥i,where ui appears in the ith place (0≤i≤mn-1) starting from 0. Proceeding as in nondegenerate case [12], this inclusion motivates us to introduce the following extension of [12, Definition  2.10.2] (cf. also [34, Definition 2.1] and [32, Definitions 3.6 and 3.7] for similar notions).Definition 11.
Suppose that0<τ≤∞, k∈C([0,τ)), C,C1,C2∈L(X), and  C and C2 are injective. A sequence ((R0(t))t∈[0,τ),…,(Rmn-1(t))t∈[0,τ)) of strongly continuous operator families in L(X) is called a (local, if τ<∞):(i)
k-regularized C1-existence propagation family for (1) iff the following holds:(30)Ri·x-k∗gi·C1x+∑j∈DiAjgαn-αj∗Ri·x-k∗gi·C1x+∑j∈Nn-1∖DiAjgαn-αj∗Ri·x∈Agαn-α∗Ri·x,m-1<i,x∈X,Agαn-α∗Ri·x-k∗gi·C1x·,m-1≥i,x∈X,for anyi=0,…,mn-1.(ii)
k-regularized C2-uniqueness propagation family for (1) iff the following holds:(31)Ri·x-k∗gi·C2x+∑j∈Digαn-αj∗Ri·Ajx-k∗gi·C2Ajx+∑j∈Nn-1∖Digαn-αj∗Ri·Ajx·=gαn-α∗Ri·y·,m-1<i,gαn-α∗Ri·y-k∗gi·C2y·,m-1≥i,provided(x,y)∈A and i∈Nmn-10.(iii)
k-regularized C-resolvent propagation family for (1), in short k-regularized C-propagation family for (1), iff ((R0(t))t∈[0,τ),…,(Rmn-1(t))t∈[0,τ)) is a k-regularized C-uniqueness propagation family for (1), and if for every t∈[0,τ), i∈Nmn-10, and j∈Nn-10, one has Ri(t)Aj⊆AjRi(t), Ri(t)C=CRi(t), and CAj⊆AjC.In the case thatk(t)=gζ+1(t), where ζ≥0, then we also say that ((R0(t))t∈[0,τ),…,(Rmn-1(t))t∈[0,τ)) is a ζ-times integrated C-resolvent propagation family for (1); 0-times integrated C-resolvent propagation family for (1) is simply called C-resolvent propagation family for (1). For a k-regularized (C1,C2)-existence and uniqueness family ((R0(t))t∈[0,τ),…,(Rmn-1(t))t∈[0,τ)), it is said that is exponentially bounded iff each single operator family (R0(t))t∈[0,τ),…,(Rmn-1(t))t∈[0,τ) is. The above terminological agreement is accepted for all other classes of k-regularized C-propagation families introduced so far.IfAj=cjI, where cj∈C for 1≤j≤n-1, then it is also said that A is a subgenerator of ((R0(t))t∈[0,τ),…,(Rmn-1(t))t∈[0,τ)). The notion of integral generator of ((R0(t))t∈[0,τ),…,(Rmn-1(t))t∈[0,τ)) is introduced as in nondegenerate case [12].Hereafter, the following equality will play an important role in our analysis:(32)Ri·x-k∗gi·Cx+∑j=1n-1Ajgαn-αj∗Ri·x-k∗gi·Cx+∑j∈Nn-1∖DiAjgαn-αj+i∗k·Cx∈Agαn-α∗Ri·x,m-1<i,x∈X,Agαn-α∗Ri·x-k∗gi·Cx,m-1≥i,x∈X,for any i=0,…,mn-1. The basic properties of subgenerators and integral generators continue to hold, with appropriate changes, in degenerate case; compare [12] and [19, Section 3.2] for more details. We leave to the interested reader the problem of transferring the assertions of [12, Propositions  2.10.3–2.10.5, Theorem  2.10.7] to degenerate case.The following is a degenerate version of [12, Definition  2.10.6].Definition 12.
Letf∈C([0,∞):X). Consider the following inhomogeneous Cauchy inclusion:(33)ut+∑j=1n-1gαn-αj∗Ajut∈ft+gαn-α∗Aut,t≥0.A function u∈C([0,∞):X) is said to be(i)
a strong solution of (33) iff there exists a continuous function uA∈C([0,∞):X) such that uA(t)∈Au(t) for all t≥0 and(34)ut+∑j=1n-1gαn-αj∗Ajut=ft+gαn-α∗uAt,t≥0,(ii)
a mild solution of (33) iff(35)ut+∑j=1n-1Ajgαn-αj∗ut∈ft+Agαn-α∗ut,t∈0,T.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:(a)
If((R0(t))t≥0,…,(Rmn-1(t))t≥0) is a C1-existence propagation family for (1), then the function u(t)≔∑i=0mn-1Ri(t)xi, t≥0, is a mild solution of (28) with ui=C1xi for 0≤i≤mn-1.(b)
If((R0(t))t≥0,…,(Rmn-1(t))t≥0) is a C2-uniqueness propagation family for (1), and Ri(t)Aj⊆AjRi(t)x, t≥0, x∈⋂j=0n-1D(Aj), i∈Nmn-10, and j∈Nn-10, then the function u(t)≔∑i=0mn-1Ri(t)C2-1ui, t≥0, is a strong solution of (28), provided ui∈C2(⋂j=0n-1D(Aj)) for 0≤i≤mn-1.For our later purposes, it will be sufficient to characterize the introduced classes ofk-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 Pλ for λ∈C with Rλ>ω has not been clarified in a proper way and property (ii) in the formulation of this theorem is required to hold for all i∈Nmn-10.Theorem 13.
Supposek(t) satisfies (P1), ω≥max(0,abs(k)),(Ri(t))t≥0 is strongly continuous, and the family {e-ωtRi(t):t≥0}⊆L(X) is bounded, provided 0≤i≤mn-1. Let A be a closed MLO on X, let C1,C2∈L(X), and let C2 be injective. Set(36)Pλ≔λαn-α+∑j=1n-1λαj-αAj-A,λ∈C∖0.(i)
SupposeAj∈L(X), j∈Nn-1. Then ((R0(t))t≥0,…,(Rmn-1(t))t≥0) is a global k-regularized C1-existence propagation family for (1) iff the following conditions hold:
(a)
The inclusion(37)Pλ∫0∞e-λtRitxdt∋λαn-α-ik~λC1x+∑j∈Diλαj-α-ik~λAjC1xholds providedx∈X, i∈Nmn-10, m-1<i, and Rλ>ω.(b)
The inclusion(38)Pλ∫0∞e-λtRitx-k∗gitC1xdt∋-∑j∈Nn-1∖Diλαj-α-ik~λAjC1xholds providedx∈X, i∈Nmn-10, m-1≥i, and Rλ>ω.(ii)
SupposeAj∈L(X), j∈Nn-1. Then ((R0(t))t≥0,…,(Rmn-1(t))t≥0) is a global k-regularized C2-uniqueness propagation family for (1) iff, for every λ∈C with Rλ>ω, x∈D(A), and y∈Ax, the following equality holds:(39)∫0∞e-λtRitx-k∗gitC2xdt+∑j∈Diλαj-αn∫0∞e-λtRitAjx-k∗gitC2Ajxdt+∑j∈Nn-1∖Diλαj-αn∫0∞e-λtRitAjxdt=λα-αn∫0∞e-λtRitydt,m-1<i,λα-αn∫0∞e-λtRity-k∗gitC2ydt,m-1≥i.Theorem 14.
Supposek(t) satisfies (P1), ω≥max(0,abs(k)),(Ri(t))t≥0 is strongly continuous, and the family {e-ωtRi(t):t≥0}⊆L(X) is bounded, provided 0≤i≤mn-1.(I)
Let the following two conditions hold:
(i)
CAj⊆AjC, j∈Nn-10, Aj∈L(X), j∈Nn-1, AiAj=AjAi, i,j∈Nn-1, and AjA⊆AAj, j∈Nn-1.(ii)
There exists an indexi∈Nmn-10 satisfying exactly one of the following two conditions:
(a)
m-1<i and the operator λαn-i+∑j∈Diλαj-iAj is injective for every λ∈C with Rλ>ω and k~(λ)≠0,(b)
m-1≥i, Nn-1∖Di≠∅, and the operator ∑j∈Nn-1∖Diλαj-iAj is injective for every λ∈C with Rλ>ω and k~(λ)≠0.If((R0(t))t≥0,…,(Rmn-1(t))t≥0) is a global k-regularized C-resolvent propagation family for (1) and (30) holds, then Pλ is injective for every λ∈C with Rλ>ω and k~(λ)≠0 and equalities (37)-(38) are fulfilled.(II)
Suppose thatPλ is injective for every λ∈C with Rλ>ω and k~(λ)≠0 and equalities (37)-(38) are fulfilled. If condition (I)(i) holds, then ((R0(t))t≥0,…,(Rmn-1(t))t≥0) is a global k-regularized C-resolvent propagation family for (1).Proof.
Concerning assertion (I), we will only sketch the main details of the proof of the injectivity of operatorPλ for every λ∈C with Rλ>ω and k~(λ)≠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 Ri(·) with the operators C and Aj for 0≤j≤n-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 λ0∈C with Rλ0>ω and k~(λ0)≠0 be fixed, and let 0∈Pλ0x for some x∈X. Using the fact that ((R0(t))t≥0,…,(Rmn-1(t))t≥0) is a global k-regularized C-uniqueness propagation family for (1), we can simply prove that(40)λ0αn-ix+∑j∈Diλ0αj-iAjx=0by performing the Laplace transform at the both sides of the composition property (31). By the injectivity of the operator λαn-i+∑j∈Diλαj-iAj for λ=λ0, we obtain that x=0 and the claimed assertion follows.These results enable one to simply clarify the Hille-Yosida type theorems for exponentially boundedk-regularized C-resolvent propagation families. The analytical properties of k-regularized C-resolvent propagation families can be analyzed similarly as in nondegenerate case [12]. We will use the following definition.Definition 15.
(i) Letβ∈(0,π], and let ((R0(t))t≥0,…,(Rmn-1(t))t≥0) be a k-regularized C-resolvent propagation family for (1). Then it is said that ((R0(t))t≥0,…,(Rmn-1(t))t≥0) is an analytic k-regularized C-resolvent propagation family of angle β, iff for each i∈Nmn-10 there exists a function Ri:Σα→L(X) which satisfies that, for every x∈X, the mapping z↦Ri(z)x, z∈Σβ is analytic and that(a)
Ri(t)=Ri(t), t>0, and(b)
limz→0,z∈ΣγRi(z)x=Ri(0)x for all γ∈(0,β) and x∈X.
(ii) Suppose that((R0(t))t≥0,…,(Rmn-1(t))t≥0) is an analytic k-regularized C-resolvent propagation family of angle β. Then it is said that ((R0(t))t≥0,…,(Rmn-1(t))t≥0) is an exponentially bounded, analytic k-regularized C-resolvent propagation family of angle β, respectively, bounded k-regularized C-resolvent propagation family, iff for every γ∈(0,β), there exists ωγ≥0, respectively, ωγ=0, such that the family {e-ωγRzRi(z):z∈Σγ}⊆L(X) is bounded for all i∈Nmn-10. Since there is no risk for confusion, we will identify in the sequel Ri(·) and Ri(·).For our purposes, the following result will be sufficiently enough (cf. Theorem14 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].Theorem 16.
Assumek(t) satisfies (P1), ω≥max(0,abs(k)), β∈(0,π/2], and, for every i∈Nmn-10, the function (k∗gi)(t) can be analytically extended to a function ki:Σβ→C satisfying that, for every γ∈(0,β), the set {e-ωzki(z):z∈Σγ} is bounded.
Let the following three conditions hold:(i)
CAj⊆AjC, j∈Nn-10, Aj∈L(X), j∈Nn-1, AiAj=AjAi, i,j∈Nn-1, and AjA⊆AAj, j∈Nn-1.(ii)
The operatorPλ is injective for all ω+Σβ+π/2.(iii)
Letqi:ω+Σπ/2+β→L(X)  (0≤i≤mn-1) satisfy that, for every x∈X, the mapping λ↦qi(λ)x, λ∈ω+Σπ/2+β is analytic and that for each i∈Nmn-10 there exists an operator Di∈L(X) such that(41)qiλx=ki~λPλ-1λαn-αC+∑j∈Diλαj-αAjCx,x∈X,λ∈Vi,providedm-1<i,(42)qiλx=-ki~λPλ-1∑j∈Nn-1∖Diλαj-αAjCx,x∈X,λ∈Vi,providedm-1≥i, (43)thefamilyλ-ωqiλ:λ∈ω+Σπ/2+γisequicontinuous∀γ∈0,β,and, in the caseD(A)¯≠X,(44)limλ→+∞λqiλx=Dix,x∉DA¯.Then there exists an exponentially bounded, analytic k-regularized C-resolvent propagation family ((R0(t))t≥0,…,(Rmn-1(t))t≥0) for (1), of angle β. Furthermore, (30) holds, the family {e-ωzRi(z):z∈Σγ} is bounded for all i∈Nmn-10 and γ∈(0,β), and Ri(z)Aj⊆AjRi(z), z∈Σβ, and j∈Nn-10.Remark 17.
For the sequel, it will be very important to note that the notion introduced in Definition11(iii) and Definition 15 can be introduced for any single operator family (Ri(t))t≥0 of the tuple ((R0(t))t≥0,…,(Rmn-1(t))t≥0). The assertions of Theorems 14 and 16 can be simply reformulated for (Ri(t))t≥0; for example, if the index i∈Nmn-10 is given in advance, then in the formulation of Theorem 16 it suffices to assume that the function (k∗gi)(·) can be analytically extended to a function ki:Σβ→C satisfying that, for every γ∈(0,β), the set {e-ωzki(z):z∈Σγ} is bounded, and that (i)-(ii) hold and (iii) holds only for this specified index i. It will be said that (Ri(t))t≥0 is an (exponentially bounded, analytic/analytic) ki-regularized C-resolvent propagation family. All terminological agreements explained before will be accepted for ki-regularized C-resolvent propagation families; the classes of ki-regularized C1-existence propagation families and ki-regularized C2-uniqueness propagation families are introduced similarly.


## 5. Asymptotical Behaviour ofki-Regularized C-Propagation Families for (1)

The main aim of this section is to investigate polynomial decaying ofki-regularized C-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 Pλ, given by (36), in general case. Because of that, we will first focus our attention to the case that Aj=cjI, where cj∈C for 1≤j≤n-1, by exploring the generation of fractionally integrated C-propagation families for (1) only. Moreover, we will assume that the numbers cj are nonnegative for 1≤j≤n-1 and that m-1<i (the case m-1≥i can be analyzed similarly) and the multivalued linear operator A under our consideration is possibly not densely defined.Theorem 18.
Suppose thatcj≥0 for 1≤j≤n-1, ζ′≥0, A:X→P(X) is a closed MLO, C∈L(X) is injective, CA⊆AC, and the following condition holds:(H)
There exist finite constantsc<0, M>0, 0<θ<π, and β∈(0,1] such that(45)c+Σπ-θ¯⊆ρCA,λ-A-1C≤Mλ-cβ,λ∈c+Σπ-θ¯.Assume that the mapping λ↦(λ-A)-1C, λ∈c+Σπ-θ¯ is strongly continuous. Assume also that i∈Nmn-10 satisfies m-1<i, and(46)αn-α-i-ζ′-αn-αβ≤0,(47)ν′≔π-θαn-α-π2>0.Set ζ≔ζ′ if A is densely defined, ζ>ζ′ otherwise, and ki(·)≔gζ+1(·). Then there exists an exponentially bounded, analytic ki-regularized C-propagation family (Ri(t))t≥0 for (1), of angle ν≔min(ν′,π/2). Moreover, (30) holds and there exists a finite constant M′>0 such that(48)Rit≤M′tα+ζ+i-αn+∑j∈Di,cj≠0tα+ζ+i-αj,t>0.Proof.
Since we have assumed that the mappingλ↦(λ-A)-1C, λ∈c+Σπ-θ¯ is strongly continuous, and its restriction to c+Σπ-θ is strongly analytic on this region; see [19, Proposition  1.2.6(iii)]. Taking into account (47) and cj≥0 for 1≤j≤n-1, we get(49)λαn-α+∑j∈Dicjλαj-α∈Σαn-απ/2+ν⊆Σαn-απ/2+ν⊆Σπ-θ,λ∈Σν+π/2.It is clear that (46) implies(50)αj-α-i-ζ′-αn-αβ≤0∀j∈Diwithcj≠0.Using an elementary argumentation, (46), (49), and (50), we can simply verify that the conditions of Theorem 16 hold with ω>0 sufficiently large, k(t)=gζ+1(t), t≥0, and Di=0, in the case that the operator A is not densely defined. Hence, A is a subgenerator of an exponentially bounded, analytic ζ-times integrated C-propagation family (Ri(t))t≥0 for (1), of angle ν≔min(ν′,π/2), as claimed. Estimate (48) remains to be proved. Fix the numbers t>0 and 0<γ<ν. By the proof of [2, Theorem 2.6.1] and Cauchy theorem, we have that, for every x∈X,(51)Ritx=12πi∫Γeλtλ-ζ-1Pλ-1λαn-α-iCx+∑j∈Di,cj≠0cjλαj-α-iCxdλ,where Γ is oriented counterclockwise and consists of Γ±≔{rei(π/2+γ):r≥t-1} and Γ0≔{t-1eiθ:|θ|≤π/2+γ}. Keeping in mind (49) and the estimate from the condition (H), it readily follows that for each λ∈Γ we have that Pλ-1C≤M/|csin⁡θ|β, so that(52)Rit≤M/csin⁡θβ2π×∫ΓeRλtλ-ζ-1λαn-α-i+∑j∈Di,cj≠0cjλαj-α-idλ.But, then estimate (48) follows from a simple integral computation that is very similar to that appearing in the proof of [2, Theorem 2.6.1].Remark 19.
(i) It is worth noting that the value of exponentβ in (H) does not depend on the final estimate (48). In our proof, we only use the estimate Pλ-1C≤M/|csin⁡θ|β, λ∈Γ.
(ii) As mentioned in the introductory part, the proof of important result of Cuesta [15, Theorem 2.1] for the classical fractional oscillation resolvent families generated by densely defined linear operators [11], satisfying the condition (H) with β=1, C=I, and ω=c<0, follows completely different lines. Furthermore, in our approach, the case in which π/2<θ<π or αn-α<1 can occur, Theorem 18 is applicable in the qualitative analysis of fractional relaxation multiterm differential inclusions (but not in the analysis of generalized asymptotical almost periodicity and generalized asymptotical almost automorphy of solutions; see (53)).LetF∈{APT([0,∞):C),AAP([0,∞):C),AAA([0,∞):C)}, where the symbol AAA([0,∞):C) denotes the space of scalar-valued asymptotically almost automorphic functions, and let ki(·) be defined as above. If (Ri(t))t≥0 is the ki-regularized C-propagation family for (1), constructed with the help of Theorem 18, and f∈F, then it can be easily checked that ((Ri∗f)(t))t≥0 is a ki-regularized C-propagation family (Ri(t))t≥0 for (1), satisfying additionally (30), where ki(·)=(gζ+1∗f)(·). By Theorem 18, some known assertions concerning inheritance of asymptotical periodicity, almost asymptotical almost periodicity and asymptotical almost automorphy under the action of finite convolution products ([5, 16]), and the assertions (a)-(b) clarified above, this yields the following result (the uniqueness of solutions follows from the fact that [12, Theorem 2.10.7] holds in degenerate case and that the condition (⋄) stated in the formulation of [12, Proposition 2.10.3] holds true, which can be verified by performing the Laplace transform).Corollary 20.
Let the requirements of Theorem18 hold, let f∈F, and let ki(·)=(gζ+1∗f)(·). Set ux(t)≔(Ri∗f)(t)x, t≥0, x∈X. Assume that(53)α+ζ+i-αn<-1,α+ζ+i-αj<-1∀j∈Diwithcj≠0.Then, for every x∈X, ux(·)∈F is a unique mild solution of the abstract Cauchy inclusion(54)u·-gζ+1+i∗f·Cx+∑j=1n-1cjgαn-αj∗u·-gζ+1+i∗f·Cx+∑j∈Nn-1∖Dicjgαn-αj+i+ζ+1∗f·Cx∈Agαn-α∗u·.Furthermore, ux(·) is a strong solution of (54) provided that x∈D(A).Concerning the Stepanov asymptotical almost periodicity and (equi-) Weyl asymptotical almost periodicity, some extra conditions on the vanishing part of functionf(·) must be imposed if we want that the solution ux(·) defined above belongs to the same class of functions as f(·). For example, we have the following:(i)
Stepanov class: suppose that1≤p<∞ and f:[0,∞)→X is asymptotically Sp-almost periodic and that the locally p-integrable functions g:R→X, q:[0,∞)→X satisfy the conditions from [30, Lemma 1.1], and limt→+∞∫tt+1(∫s/2sq(r)dr)pds=0. Then the function (Ri∗f)(·) is asymptotically Sp-almost periodic (see [33, Proposition 2.13, Remark 2.14]).(ii)
Weyl classes: ifg:R→X is bounded and Weyl-almost periodic and q∈e-W01([0,∞):X) satisfies the following conditions:
(i)
the mappingt↦∫0tq(r)dr, t>0, is bounded as t→+∞,(ii)
limt→+∞∫0t(t+1-r)-νq(r)dr=0.then the function(Ri∗(g+q))(·) is in class Weaap1([0,∞):X), where Weaap1([0,∞):X)≔Wap1([0,∞):X)+e-W01([0,∞):X); see [35, Proposition  2.3(i), Example 5.5]. Here, ν>1 is chosen so that Ri(t)=O(t-ν), t≥1.Denote byF′ the set consisting of all generalized (asymptotically) almost periodic function spaces and all generalized (asymptotically) almost automorphic function spaces considered so far. Let F′′ denote the collection of all spaces from F′ that are not in any class of functions obtained as a sum of some spaces of (equi-) Weyl almost periodic (automorphic) functions and some space of (equi-) Weyl almost vanishing functions.The second part of the following proposition is very similar to [7, Proposition 2.5.1].Proposition 21.
Suppose thatk(t) satisfies (P1) and (Ri(t))t≥0 is a strongly Laplace transformable ki-regularized C-propagation family for (1).(i)
For everyλ∈C, there exists a function fλ(·) satisfying (P1)-L(X) and(55)fλt≔L-11-λzαn-αI+∑j=1n-1Ajzαn-αj-1∑j∈Dik~zziCAjt,t≥0,provided thatm-1<i, respectively,(56)fλt≔L-11-λzαn-αI+∑j=1n-1Ajzαn-αj-1×∑j∈Dik~zziCAj-λk~zzαn-α+iCt,t≥0,provided thatm-1≥i.(ii)
Denote byD the set consisting of all eigenvectors x of operator A which corresponds to eigenvalues λ∈C of operator A for which the mapping(57)fλ,xt≔fλtx,t≥0,belongs to the space F′. Then the mapping t↦Ri(t)x, t≥0, belongs to the space F′ for all x∈span(D); furthermore, the mapping t↦Ri(t)x, t≥0, belongs to the space F′′ for all x∈span(D)¯ provided additionally that (Ri(t))t≥0 is bounded.Proof.
We will examine the casem-1<i 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-λ/zαn-α)I+∑j=1n-1Aj/zαn-αj is invertible in L(X) for all z∈C with z sufficiently large, because the norm of bounded linear operator λ/zαn-αI-∑j=1n-1Aj/zαn-αj for such values of z is strictly less than 1, we get that the term(58)1-λzαn-αI+∑j=1n-1Ajzαn-αj-1∑j∈Dik~zziCAjis well-defined for all z∈C with Rz>ω0, for some ω0>max(0,abs(k)). Set(59)H0t≔λgαn-αjt+∑j=1n-1gαn-αjtAj,t>0,and(60)a≔min⁡αn-α0,…,αn-αn-1,b≔max⁡αn-α0,…,αn-αn-1.Then it is clear that(61)∫0∞e-ztH0tdt=λzαn-αI-∑j=1n-1Ajzαn-αj,Rz>ω0,and that there exists a finite constant c>0 such that(62)H0∗kt≤cngkαn-bt,t∈0,1,H0∗kt≤cngkαn-at,t≥1,where H0∗k(·) denotes the kth convolution power of H0(·). The function H(t)≔∑k=1∞H0∗k(t), t>0, is well-defined since there exists a finite constant c′>0 such that(63)Ht≤∑k=1∞cktkαn-b-1Γkαn-b≤ctαn-b-1∑k=0∞ctαn-bkΓkαn-b+αn-b≤ctαn-b-1Eαn-b,αn-bctαn-b≤c′tαn-b-1,t∈0,1and, due to Lemma 1,(64)Ht≤∑k=1∞cktkαn-a-1Γkαn-a≤ctαn-a-1∑k=0∞ctαn-akΓkαn-a+αn-a≤ctαn-a-1Eαn-a,αn-actαn-a≤c′tαn-a-11+ctαn-a1-αn-a/αn-aec1/αn-at,t≥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 λx∈Ax for some λ∈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(65)1-λzαn-αI+∑j=1n-1Ajzαn-αj∫0∞e-ztRitxdt=∑j∈Dik~zziCAjx,for Rz>0 suff. large, and therefore Ri(t)x=fλ,x(t), t≥0. As a consequence, we have that the mapping t↦Ri(t)x, t≥0, belongs to the space F′ for all x∈span(D). The boundedness of (Ri(t))t≥0 implies the uniform convergence of Ri(t)xn to Ri(t)x  (t≥0) for any sequence (xn)∈span(D) converging to some element x∈span(D)¯; 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).Remark 22.
Ifm-1<i and Aj=cjI for some cj∈C  (1≤i≤n-1), then a simple calculation shows that(66)fλ,xt=∑j∈Dik~zzαn-izαn-λz-α+∑j=1n-1cjz-αjCAjx,for x∈X satisfying λx∈Ax  (λ∈C). To the best knowledge of the authors, in the handbooks containing tables of Laplace transforms, the explicit forms of functions like fλ,x(·) are not known, with the exception of some very special cases of the coefficients αj, cj (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].Theorem 23.
Assume that(Ri(t))t≥0 is an exponentially bounded ki-regularized C-propagation family (Ri(t))t≥0 for (1). Let m-1<i, and let there exist a number ν>1 such that Ri(t)=O(t-ν), t≥1. Assume, further, that the following conditions hold:(i)
Letf∈C([0,∞):X) satisfy that there exists a function g∈AAP([0,∞):X) such that C-1ft=gi∗k∗gt, t≥0.(ii)
Assume thatg(t)=gap(t)+g0(t), t≥0, where gap∈AP([0,∞):X) and g0∈C0([0,∞):X).(iii)
Letαn-αj∈N for all j∈Di.(iv)
Assume thatgk∗g∈L∞([0,∞):X) for all k∈N and c0⊈X (i.e., X does not contain an isomorphic copy of c0), or R(gk∗g) is weakly relatively compact in X for all k∈N.(v)
For everyk∈N, we have(67)∫0∞g0tdt<∞,∫0∞∫t∞g0sdsdt<∞,⋮∫0∞∫t∞∫xk∞⋯∫x1∞g0sdsdx1⋯dxkdt<∞.Then there exists a unique exponentially bounded mild solution u(·) of the abstract Cauchy inclusion (33). Furthermore, u∈AAP([0,∞):X).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:(68)Ri∗ft=k∗gi∗Cut+∑j∈Digαn-αj+i∗k∗CAjut,t≥0;See, for example, [12, Theorem 2.10.7]. Taking the Laplace transform, we get(69)u~z=I+∑j∈Dizαj-αnAj-1Ri~zC-1f~zzik~z-1,Rz>ω,k~z≠0.Since (C-1f)(t)=(gi∗k∗g)(t), t≥0, we get(70)u~z=I+∑j∈Dizαj-αnAj-1Ri~zg~z,Rz>ω,k~z≠0.By the proofs of Proposition 21 and [9, Theorem 1.1.11], the right-hand side of above equality is really the Laplace transform of a continuous exponentially bounded function u(·) given by(71)ut=∑k=1∞Ri∗-∑j∈Digαn-αj·Aj∗k∗gt+Ri∗gt,t≥0.With the help of Laplace transform and a simple calculation, it can be simply verified that the function u·, whose Laplace transform is given by (69), is a mild solution of abstract fractional inclusion (33). The growth order of Ri· implies that the function t↦Ri∗gt, t≥0, is in AAP0,∞:X. Since AAP0,∞:X is closed in Cb0,∞:X and {Ri∗[-∑j∈Digαn-αj·Aj]∗k∗g}(t) converges uniformly to ∑k=1∞{Ri∗[-∑j∈Digαn-αj(·)Aj]∗k∗g}(t) for t≥0, the asymptotical almost periodicity of u· immediately follows if we prove that the function Gkt=gk∗gt, t≥0, is asymptotically almost periodic for all k∈N; see (iii). But, this can be shown by making use of (iv)-(v) and applying successively [24, Theorem 2.2.2]; here, we only want to observe that the vanishing term of function G1(t) is given by ∫t∞g0sds  (t≥0), since ∫0∞g0(s)ds<∞. The uniqueness of exponentially bounded mild solutions of (33) can be proved as follows. Let u(·) be such a solution. Taking the Laplace transform and multiplying after that with zαn-α, we get(72)u~z,zαn-α+∑j=1n-1zαj-αAju~z-zαn-αf~z∈A,Rz>ω,k~z≠0.Hence, -zαn-αf~(z)∈Pzu~(z) and u~(z)=-Pz-1zαn-αf~(z) for Rz>ω, k~(z)≠0. By the uniqueness theorem for the Laplace transform, u(·) must be uniquely determined. The proof of the theorem is thereby complete.Remark 24.
Concerning Theorem23, the case in which m-1≥i is a little bit complicated: it seems that the assumption Aj=cj for some complex numbers cj∈C  (1≤j≤n-1) has to be imposed for establishing of any relevant result. Details can be left to the interested reader.


## 6. Examples and Applications

We have already noted that the method established in the proof of Theorem18 will be further employed for reconsideration and improving some known results recently established by Keyantuo et al. [16] and Luong [17]. The main aim of the following example is to explain how we can do this.Example 1.
In [16], the authors have considered the abstract two-term fractional differential equation(73)Dtα′+1ut+c1Dtβ′ut=Aut+ft,t≥0,uk0=uk,k=0,1,where c1>0, A is a densely defined linear operator satisfying the condition (H) with β=1, C=I and c<0, 0<α′≤β′≤1, and f(t) is a given X-valued function; here, we have been forced to slightly change the notation used in [16]. For this, the notion of an α′,β′c1-regularized family generated by A, which is a special case of the notion introduced in Definition 9 with a(t)=L-1(1/λα′+1+c1λβ′)(λ), t≥0, and k(t)=L-1(λα′/λα′+1+c1λβ′)(λ), t≥0, has been introduced; compare [16, Lemma 2.5]. Our main contributions will be given in the case that 0<α′<β′≤1, which is used in the formulations and proofs of [16, Theorems 4.3 and 5.3], the main results of afore-mentioned paper (although possible applications can be given in the study of two-term fractional Poisson heat equations on the space H-1(Ω), where Ω is a bounded domain in Rn with smooth boundary [18], we will pay our attention to the case that A=A is a single-valued linear operator).
Let us define an exponentially bounded, analyticα′,β′,Cc1-regularized family (Sα′,β′(t))t≥0 of angle ν∈(0,π/2], subgenerated by A, as the exponentially bounded, analytic (a,k)-regularized C-resolvent family of angle ν, subgenerated by the same operator, with a(t) and k(t) being defined as above. Then we have(74)∫0∞e-λtSα′,β′txdt=λα′λα′+1+c1λβ′-A-1Cx,x∈X,λ>0suff.  largeand the assertion of [16, Theorem 3.1] holds with the initial values x,y∈X and the inhomogeneity f(t) replaced therein with the initial values C-1x,C-1y∈X and the inhomogeneity C-1f(t), respectively, with the meaning clear.
Assume that the condition (H) holds withβ=1, c<0, and θ=π/2-γ′π/2, where α′<γ′<β′. Then we can argue as follows. Similarly as in the proof of Theorem 18, we have that the operator A is a subgenerator of an exponentially bounded, analytic α′,β′,Cc1-regularized family (Sα′,β′(t))t≥0 of angle ν≔π/2(γ′+1)/α′+1-π/2; compare also [12, Theorem 2.2.5]. Estimating the term (λα′+1+c1λβ′-A)-1C uniformly, as in the proof of Theorem 18, and using the integral computation given in the final part of this theorem and [2, Theorem 2.6.1], we get(75)Sα′,β′t=O11+tα′+1=O11+tα′+1+tβ′,t≥0;here we would like to note that the arguments used in proof of [16, Theorem 4.1] are much more complicated than ours and that our estimate is better even in the case that C=I because then, with the notion introduced in [16] accepted, we only require that the operator A is of sectorial angle γ′π/2, not of β′π/2, as the stronger estimate in [16] requires. Using the estimate (75), we can repeat literally the proofs of [16, Theorems 4.3 and 5.3] in order to see that their validity hold with C-sectorial operators of smaller angle γ′π/2, with the meaning clear. In the case that C≠I, we can make applications to generators of analytic C-regularized semigroups generated by nonelliptic differential operators in Lp-spaces [44].
In [17], Luong has investigated the following abstract two-term fractional differential equation with nonlocal conditions in a Banach space X:(76)Dtα′+1ut+c1Dtβ′ut=Aut+ft,t≥0,u0+gu=u0,ut0+hu=y0,where 0<α′≤β′≤1, the functions g(·) and h(·) satisfy certain conditions, and A is of sectorial angle β′π/2. The improvements of main results of this paper, Theorem 13, to C-sectorial operators of angle γ′π/2, with γ′ being clarified above, can be proved straightforwardly (0<α′<β′≤1).As mentioned before, Corollary20 can be applied to the almost sectorial operators and multivalued linear operators used in the analysis of Poisson heat equations.Example 2.
(i) ([45]) Assume that η∈(0,1), q∈N, Ω is a bounded domain in Rn with boundary of class C4q, and E≔Cη(Ω¯). Consider the operator A:D(A)⊆Cζ(Ω¯)→Cη(Ω¯) given by(77)Aux≔∑β≤2qaβxDβux∀x∈Ω¯with domain D(A)≔{u∈C2q+η(Ω¯):Dβu∂Ω=0forallβ≤q-1}. In this place, β∈N0n, β=∑i=1nβj, and Dβ=∏i=1n(1/i∂/∂xi)βi. Let aβ:Ω¯→C satisfy the following:(i)
aβ(x)∈R for all x∈Ω¯ and β=2q.(ii)
aβ∈Cη(Ω¯) for all β≤2q, and(iii)
there exists a constantM>0 such that(78)M-1ξ2q≤∑β=2qaβxξβ≤Mξ2q∀ξ∈Rn,x∈Ω¯.Then there exists a sufficiently large number σ>0 such that the operator -Aσ≡-(A+σ) satisfies the condition (H) with β=1-η/2q, C=I, c<0, and some θ∈(0,π/2). Let us remind ourselves that A is not densely defined and that the value of exponent β=1-η/2q is sharp. Applications of Corollary 20 are clear and here we would like to illustrate just one of them: cj=0 for j<n-1, αn-1=αn-, α=0+, i=0, and αnη/2q<ζ<αn-1.
(ii) ([18]) Let X≔Lp(Ω), where Ω is a bounded domain in Rn, b>0, m(x)≥0 a.e. x∈Ω, m∈L∞(Ω), and 1<p<∞. Suppose that the operator A≔Δ-b acts on X with the Dirichlet boundary conditions and that B is the multiplication operator by the function m(x). By the analysis contained in [18, Example 3.6], the condition (H) is satisfied for the multivalued linear operator A≔AB-1 with β=1/p, C=I, and some numbers c<0 and θ∈(0,π/2); here it is worth noting that the validity of additional condition [18, (3.42)] on the function m(x) enables us to get the better exponent β in (H), provided that p>2. Applications in the study of the existence and uniqueness of asymptotically almost periodic and asymptotically almost automorphic solutions of multiterm fractional integrodifferential Poisson heat equation(79)ut,x-gζ+1+i∗ft,xφx+∑j=1n-1cjgαn-αj∗ut,x-gζ+1+i∗ft,xφx+∑j∈Nn-1∖Dicjgαn-αj+i+ζ+1∗ft,x=Δ-bvt,x;vt,x=0,t,x∈0,∞×∂Ω,where m(x)v(t,x)=∫0tgαn-α(t-s)u(s,x)ds for all t≥0 and x∈Ω, are immediate (φ∈X).Example 3.
In [12, Subsection 3.3.2], we have analyzed hypercyclic and topologically mixing properties of solutions of abstract multiterm fractional Cauchy problem (1) with A=A being single-valued and Aj=cjI for some cj∈C  (1≤i≤n-1). Let it be the case. With the help of Proposition 21, we can reconsider a great number of examples given in the above-mentioned part of [12] and provide several interesting applications in the investigation of problem about the existence of a dense linear subspace X′ of X such that the mapping t↦Ri(t)x, t≥0, is asymptotically almost periodic for all x∈X′; here, (Ri(t))t≥0 is a given ki-regularized C-propagation family for (1) with a subgenerator A (cf. [7] for further information in this direction). For the sake of illustration, we will consider only the situation of [12, Example  3.3.12(ii)]; see also Ji and Weber [31]. Suppose that X is a symmetric space of noncompact type and rank one, p>2, and the parabolic domain Pp and the positive real number cp possess the same meaning as in [31]. Suppose, further, that P(z)=∑j=0kajzj, z∈C is a nonconstant complex polynomial with ak>0, n=2, 0<a<2, α2=2a, α1=0, α=a, c1>0, i=0, and θ<min(π/2-narctan⁡p-2/2p-1,π/2-narctan⁡p-2/2p-1-π/2a). Then we already know that the operator -eiθP(ΔX,p♮) is the integral generator of an exponentially bounded, analytic resolvent propagation family ((Rθ,P,0(t))t≥0,…,(Rθ,P,⌈2a⌉-1(t))t≥0) of angle min(π-narctan⁡p-2/2p-1-|θ|/a-π/2,π/2), and that (Rθ,P,0(t))t≥0 is topologically mixing provided the condition(80)-eiθPintPp∩ita+c1it-a:t∈R∖0≠∅;compare [12] for the notion. Our new assumption will be, instead of (80), that there exist a number λ0∈C∖(-∞,0] and a sufficiently small number ϵ>0 such that λa+c1λ-a∈-eiθP(int(Pp)) and that λa∉Σaπ/2 for λ-λ0≤ϵ. Let λ-λ0≤ϵ. Since there exists an x≠0 such that -eiθP(ΔX,p♮)x=(λa+c1λ-a)x due to our assumption λa+c1λ-a∈-eiθP(int(Pp))⊆σp(-eiθP(ΔX,p♮)), we can employ [12, Lemma 3.3.7] in order to see that(81)Rθ,P,0tx=λat-aλ2a-c1Ea,2-aλata-Ea,2-ac1λ-atax+λaλ2a-c1λaEaλata+a-1λaEa,2λata-c1λ-aEac1λ-ata-a-1c1λ-aEa,2c1λ-atax+λa+c1λ-aλaλ2a-c1Eaλata-Eac1λ-atax,t>0;see also [12, p. 418]. Due to our assumption λa∉Σaπ/2, the asymptotic expansion formulae (8)-(9), and the fact that the first term in the above expression can be continuously extended to the nonnegative real axis, we get the mapping t↦Rθ,P,0(t)x, t≥0, is asymptotically almost periodic. Since the set {x∈X:∃λ∈C∖(-∞,0] s.t. |λ-λ0|≤ϵ and -eiθP(ΔX,p♮)x=(λa+c1λ-a)x} is total in X, Proposition 21 implies that there exists a dense linear subspace X′ of X such that the mapping t↦Rθ,P,0(t)x, t≥0, is asymptotically almost periodic for all x∈X′.



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*Source: 5947393-2018-01-22.xml*