# On a Class of Abstract Time-Fractional Equations on Locally Convex Spaces

Authors: Marko Kostić; Cheng-Gang Li; Miao Li
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

This paper is devoted to the study of abstract time-fractional equations of the following form:Dtαnu(t)+∑i=1n−1AiDtαiu(t)=ADtαu(t)+f(t), t>0, u(k)(0)=uk, k=0,...,⌈αn⌉−1, where n∈ℕ∖{1}, A and A1,...,An−1 are closed linear operators on a sequentially complete locally convex space E,0≤α1<⋯<αn, 0≤α<αn, f(t) is an E-valued function, and Dtα denotes the Caputo fractional derivative of order α (Bazhlekova (2001)). We introduce and systematically analyze various classes of k-regularized (C1,C2)-existence and uniqueness (propagation) families, continuing in such a way the researches raised in (de Laubenfels (1999, 1991), Kostić (Preprint), and Xiao and Liang (2003, 2002). The obtained results are illustrated with several examples.


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## Body

## 1. Introduction and Preliminaries

A great number of abstract time-fractional equations appearing in engineering, mathematical physics, and chemistry can be modeled through the abstract Cauchy problem(1.1)Dtαnu(t)+∑i=1n-1AiDtαiu(t)=ADtαu(t)+f(t),t>0,u(k)(0)=uk,k=0,…,⌈αn⌉-1.
For further information about the applications of fractional calculus, the interested reader may consult the monographs by Baleanu et al. [1], Klafter et al. (Eds.) [2], Kilbas et al. [3], Mainardi [4], Podlubny [5], and Samko et al. [6]; we also refer to the references [7–19].The aim of this paper is to develop some operator theoretical methods for solving the abstract time-fractional equations of the form (1.1). We start by quoting some special cases. The study of qualitative properties of the abstract Basset-Boussinesq-Oseen equation:
(1.2)u′(t)-ADtαu(t)+u(t)=f(t),t≥0,u(0)=0(α∈(0,1)),
describing the unsteady motion of a particle accelerating in a viscous fluid under the action of the gravity, has been initiated by Lizama and Prado in [17]. For further results concerning the C-wellposedness of (1.2), [20, 21] are of importance. In [12], Karczewska and Lizama have recently analyzed the following stochastic fractional oscillation equation:
(1.3)u(t)+∫0t‍(t-s)[ADsαu(s)+u(s)]ds=W(t),t>0,
where 1<α<2, A is the generator of a bounded analytic C0-semigroup on a Hilbert space H and W(t) denotes an H-valued Wiener process defined on a stochastic basis (Ω,ℱ,P). The theory of (a,k)-regularized resolvent families (cf. [12, Theorems  3.1 and 3.2]) can be applied in the study of deterministic counterpart of (1.3) in integrated form:
(1.4)u(t)+∫0t‍(t-s)1-αΓ(2-α)Au(s)ds+∫0t‍(t-s)u(s)ds=∫0t‍(t-s)f(s)ds,t>0,
where Γ(·) denotes the Gamma function and f∈Lloc1([0,∞):E). Equation (1.4) generalizes the so-called Bagley-Torvik equation, which can be obtained by plugging α=3/2 in (1.4), and models an oscillation process with fractional damping term (cf. [21] for the analysis of C-wellposedness and perturbation properties of (1.4)). After differentiation, (1.4) becomes, in some sense,
(1.5)u′′(t)+ADtαu(t)+u(t)=f(t),t≥0;u(0)=u′(0)=0.
Notice also that the periodic solutions for the equation
(1.6)Dαu(t)+BDβu(t)+Au(t)=f(t),t∈[0,2π],
where A and B are closed linear operators defined on a complex Banach space X, 0≤β<α≤2, f∈C([0,2π]:X) and Dα denotes the Liouville-Grünwald fractional derivative of order α, have been studied by Keyantuo and Lizama in [13]. Observe also that Diethelm analyzed in [22, Chapter 8] scalar-valued multiterm Caputo fractional differential equations. Consider, for illustration purposes, the following abstract time-fractional equation:
(1.7)Dtαu(t)+Dtβu(t)=au(t),t>0;u(0)=u0,u'(0)=0,
where 1<α<2, 0<β<α and A=a is a certain complex constant. Applying the Laplace transform (see, e.g., [10, (1.23)]), we get:
(1.8)(λα+λβ)u~(λ)-(λα-1+λβ-1)u0=au~(λ).
Therefore,
(1.9)u~(λ)=λα-1+λβ-1λα+λβ-au0.
By (24) and (26) in [19], it readily follows that:
(1.10)u(t)=∑n=0∞‍(-1)nt(α-β)n[Eα,(α-β)n+1n+1(atα)+tα-βEα,(α-β)(n+1)+1n+1(atα)]u0,
where
(1.11)Eα,βγ(z)=∑n=0∞‍(γ)nznΓ(nα+β)n!
is the generalized Mittag-Leffler function. Here (γ)n=γ(γ+1)⋯(γ+n-1) (n∈ℕ) and (γ)0=1. The formula (1.10) shows that it is quite complicated to apply Fourier multiplier theorems to the abstract time-fractional equations of the form (1.1); for some basic references in this direction, the reader may consult [16, 23]. Before going any further, we would also like to observe that Atanacković et al. considered in [8], among many other authors, the following fractional generalization of the telegraph equation:
(1.12)τDtαu(t)+Dtβu(t)=Duxx,x∈(0,l),t>0,
where 0<β≤α≤2, τ>0 and D>0. In that paper, solutions to signalling and Cauchy problems in terms of a series and integral representation are given.In the second section, we continue the analysis from our recent paper [15], where it has been assumed that Aj=cjI for some complex constants cj∈ℂ (1≤j≤n-1); here, and in the sequel of the second section, I denotes the identity operator on E. We introduce and clarify the basic structural properties of various types of k-regularized (C1,C2)-existence and uniqueness propagation families. This is probably the best concept for the investigation of integral solutions of the abstract time-fractional equation (1.1) with Aj∈L(E), 1≤j≤n-1. If there exists an index j∈ℕn-1 such that Aj∉L(E), then the vector-valued Laplace transform cannot be so easily applied (cf. Theorems 2.10–2.11), which implies, however, that there exist some limitations to the introduced classes of propagation families. The notion of a strong solution of (1.1) is introduced in Definition 2.1, and the notions of strong and mild solutions of inhomogeneous equations of the form (2.15) below are introduced in Definition 2.7. The generalized variation of parameters formula is proved in Theorem 2.8.On the other hand, the notions ofC1-existence families and C2-uniqueness families for the higher order abstract Cauchy problem (ACPn) were introduced by Xiao and Liang in [24, Definition  2.1]. In the third section, we will introduce more general classes of (local) k-regularized C1-existence families for (1.1), k-regularized C2-uniqueness families for (1.1), and k-regularized C-resolvent families for (1.1). Our intention in this section is to transfer results of [24] to abstract time-fractional equations. In addition, various adjoint type theorems for k-regularized C-resolvent families are considered in Theorem 3.6.Throughout this paper, we will always assume thatE is a Hausdorff sequentially complete locally convex space over the field of complex numbers, SCLCS for short, and that the abbreviation ⊛ stands for the fundamental system of seminorms which defines the topology of E; in this place, we would like to mention in passing that the locally convex spaces are very important to describe a set of mixed states in quantum theory [2]. The completeness of E, if needed, will be explicitly emphasized. By L(E) is denoted the space of all continuous linear mappings from E into E. Let ℬ be the family of bounded subsets of E and let pB(T):=supx∈Bp(Tx), p∈⊛, B∈ℬ, T∈L(E). Then pB(·) is a seminorm on L(E) and the system (pB)(p,B)∈⊛×ℬ induces the Hausdorff locally convex topology on L(E). Recall that L(E) is sequentially complete provided that E is barreled. Henceforth A is a closed linear operator acting on E, L(E)∋C is an injective operator, and the convolution like mapping * is given by f*g(t):=∫0t‍f(t-s)g(s)ds. The domain, resolvent set and range of A are denoted by D(A), ρ(A) and R(A), respectively. Since it makes no misunderstanding, we will identify A with its graph. Recall that the C-resolvent set of A, denoted by ρC(A), is defined by
(1.13)ρC(A):={λ∈ℂ;λ-Aisinjectiveand(λ-A)-1C∈L(E)}.
Suppose F is a linear subspace of E. Then the part of A in F, denoted by A|F, is a linear operator defined by D(A|F):={x∈D(A)∩F:Ax∈F} and A|Fx:=Ax,x∈D(A|F).DefineEp:=E/p-1(0) (p∈⊛). Then the norm of a class x+p-1(0) is defined by ||x+p-1(0)||Ep:=p(x) (x∈E). The canonical mapping Ψp:E→Ep is continuous and the completion of Ep under the norm ||·||Ep is denoted by Ep¯. Since no confusion seems likely, we will also denote the norms on Ep and L(Ep) (Ep¯ and L(Ep¯)) by ||·||; L⊛(E) denotes the subspace of L(E) which consists of those bounded linear operators T on E such that, for every p∈⊛, there exists cp>0 satisfying p(Tx)≤cpp(x), x∈E. If T∈L⊛(E) and p∈⊛, then the operator Tp:Ep→Ep, defined by Tp(Ψp(x)):=Ψp(Tx), x∈E, belongs to L(Ep). This operator is uniquely extensible to a bounded linear operator Tp¯ on Ep¯, and the following holds: ||Tp||=||Tp¯||. The function πqp:Ep→Eq, defined by πqp(Ψp(x)):=Ψq(x), x∈E, is a continuous homomorphism of Ep onto Eq, and extends therefore, to a continuous linear homomorphism πqp of Ep¯ onto Eq¯. The reader may consult [25] for the basic facts about projective limits of Banach spaces (closed linear operators acting on Banach spaces) and their projective limits. Recall, a closed linear operator A acting on E is said to be compartmentalized (w.r.t. ⊛) if, for every p∈⊛, Ap:={(Ψp(x),Ψp(Ax)):x∈D(A)} is a function. Therefore, T∈L⊛(E) is a compartmentalized operator.Givens∈ℝ in advance, set ⌊s⌋:=sup{l∈ℤ:s≥l} and ⌈s⌉:=inf{l∈ℤ:s≤l}. The principal branch is always used to take the powers. Set ℕl:={1,…,l}, ℕl0:={0,1,…,l}, 0ζ:=0, gζ(t):=tζ-1/Γ(ζ) (ζ>0, t>0) and g0:= the Dirac δ-distribution. If γ∈(0,π], then we define Σγ:={λ∈ℂ:λ≠0,|arg(λ)|<γ}. We refer the reader to [26] and references cited there for the basic material concerning integration in sequentially complete locally convex spaces and vector-valued analytic functions.Letα>0, let β∈ℝ, and let the Mittag-Leffler function Eα,β(z) be defined by Eα,β(z):=∑n=0∞‍zn/Γ(αn+β), z∈ℂ. In this place, we assume that 1/Γ(αn+β)=0 if αn+β∈-ℕ0. Set, for short, Eα(z):=Eα,1(z), z∈ℂ. The Wright function Φγ(t) is defined by Φγ(t):=ℒ-1(Eγ(-λ))(t), t≥0, where ℒ-1 denotes the inverse Laplace transform. For further information concerning Mittag-Leffler and Wright functions, we refer the reader to [10, Section  1.3].The following definition has been recently introduced in [27].Definition 1.1.
Suppose0<τ≤∞, k∈C([0,τ)), k≠0, a∈Lloc1([0,τ)), a≠0 and A is a closed linear operator on E. (i)
Then it is said thatA is a subgenerator of a (local, if τ<∞) (a,k)-regularized (C1,C2)-existence and uniqueness family (R1(t),R2(t))t∈[0,τ)⊆L(E)×L(E) if and only if the mapping t↦(R1(t)x,R2(t)x), t∈[0,τ) is continuous for every fixed x∈E and if the following conditions hold:
(a)
Ri(0)=k(0)Ci, i=1,2,(b)
C2 is injective,(c)
(1.14)A∫0t‍a(t-s)R1(s)xds=R1(t)x-k(t)C1x,t∈[0,τ),x∈E,(1.15)∫0t‍a(t-s)R2(s)Axds=R2(t)x-k(t)C2x,t∈[0,τ),x∈D(A).(ii)
Let(R1(t))t∈[0,τ)⊆L(E) be strongly continuous. Then it is said that A is a subgenerator of a (local, if τ<∞) (a,k)-regularized C1-existence family (R1(t))t∈[0,τ) if and only if R1(0)=k(0)C1 and (1.14) holds.(iii)
Let(R2(t))t∈[0,τ)⊆L(E) be strongly continuous. Then it is said that A is a subgenerator of a (local, if τ<∞) (a,k)-regularized C2-uniqueness family (R2(t))t∈[0,τ) if and only if R2(0)=k(0)C2, C2 is injective and (1.15) holds.It will be convenient to remind us of the following definitions from [14, 20, 26].Definition 1.2.
(i) Let0<τ≤∞, k∈C([0,τ)), k≠0 and let a∈Lloc1([0,τ)), a≠0. A strongly continuous operator family (R(t))t∈[0,τ) is called a (local, if τ<∞) (a,k)-regularized C-resolvent family having A as a subgenerator if and only if the following holds: (a)
R(t)A⊆AR(t), t∈[0,τ), R(0)=k(0)C and CA⊆AC,(b)
R(t)C=CR(t), t∈[0,τ),(c)
R(t)x=k(t)Cx+∫0t‍a(t-s)AR(s)xds, t∈[0,τ), x∈D(A),(R(t))t∈[0,τ) is said to be nondegenerate if the condition R(t)x=0,t∈[0,τ) implies x=0, and (R(t))t∈[0,τ) is said to be locally equicontinuous if, for every t∈(0,τ), the family {R(s):s∈[0,t]} is equicontinuous. In the case τ=∞,(R(t))t≥0 is said to be exponentially equicontinuous (equicontinuous) if there exists ω∈ℝ (ω=0) such that the family {e-ωtR(t):t≥0} is equicontinuous.
(ii) Letβ∈(0,π] and let (R(t))t≥0 be an (a,k)-regularized C-resolvent family. Then it is said that (R(t))t≥0 is an analytic (a,k)-regularized C-resolvent family of angle β, if there exists a function R:Σβ→L(E) satisfying that, for every x∈E, the mapping z↦R(z)x, z∈Σβ is analytic as well as that (a)
R(t)=R(t), t>0 and(b)
limz→0,z∈ΣγR(z)x=k(0)Cx for all γ∈(0,β) and x∈E,(R(t))t≥0 is said to be an exponentially equicontinuous, analytic (a,k)-regularized C-resolvent family, respectively, equicontinuous analytic (a,k)-regularized C-resolvent family of angle β, if for every γ∈(0,β), there exists ωγ≥0, respectively, ωγ=0, such that the set {e-ωγ|z|R(z):z∈Σγ} is equicontinuous. Since there is no risk for confusion, we will identify in the sequel R(·) and R(·).Definition 1.3.
(i) Letk∈C([0,∞)) and a∈Lloc1([0,∞)). Suppose that (R(t))t≥0 is a global (a,k)-regularized C-resolvent family having A as a subgenerator. Then it is said that (R(t))t≥0 is a quasi-exponentially equicontinuous (q-exponentially equicontinuous, for short) (a,k)-regularized C-resolvent family having A as subgenerator if and only if, for every p∈⊛, there exist Mp≥1, ωp≥0 and qp∈⊛ such that:
(1.16)p(R(t)x)≤Mpeωptqp(x),t≥0,x∈E.
(ii) Letβ∈(0,π], and let A be a subgenerator of an analytic (a,k)-regularized C-resolvent family (R(t))t≥0 of angle β. Then it is said that (R(t))t≥0 is a q-exponentially equicontinuous, analytic (a,k)-regularized C-resolvent family of angle β, if for every p∈⊛ and ε∈(0,β), there exist Mp,ε≥1,ωp,ε≥0 and qp,ε∈⊛ such that
(1.17)p(R(z)x)≤Mp,εeωp,ε|z|qp,ε(x),z∈Σβ-ε,x∈E.For a global(a,k)-regularized (C1,C2)-existence and uniqueness family (R1(t),R2(t))t≥0 having A as subgenerator, it is said that is locally equicontinuous (exponentially equicontinuous, (q-)exponentially equicontinuous, analytic, (q-)exponentially analytic,…) if and only if both (R1(t))t≥0 and (R2(t))t≥0 are.The reader may consult [26, Theorems  2.7 and 2.8] for the basic Hille-Yosida type theorems for exponentially equicontinuous (a,k)-regularized C-resolvent families. The characterizations of exponentially equicontinuous, analytic (a,k)-regularized C-resolvent families in terms of spectral properties of their subgenerators are given in [26, Theorems  3.6 and 3.7]. For further information concerning q-exponentially equicontinuous (a,k)-regularized C-resolvent families, we refer the reader to [20, 25].Henceforth, we assume thatk,k1,k2,… are scalar-valued kernels and that a≠0 in Lloc1([0,τ)). All considered operator families will be nondegenerate.The following conditions will be used in the sequel:(H1)
A is densely defined and (R(t))t∈[0,τ) is locally equicontinuous.(H2)
ρ(A)≠∅.(H3)
ρC(A)≠∅, R(C)¯=E and (R(t))t∈[0,τ) is locally equicontinuous.(H3)’
ρC(A)≠∅ and C-1AC=A.(H4)
A is densely defined and (R(t))t∈[0,τ) is locally equicontinuous, or ρC(A)≠∅.(H5)
(H1)∨ (H2) ∨ (H3) ∨ (H3)’.(P1)
k(t) is Laplace transformable, that is, it is locally integrable on [0,∞) and there exists β∈ℝ so that k~(λ)=ℒ(k)(λ):=limb→∞∫0b‍e-λtk(t)dt:=∫0∞‍e-λtk(t)dt exists for all λ∈ℂ with ℜλ>β. Put abs(k):=inf{ℜλ:k~(λ)exists}.


## 2. The Main Structural Properties ofk-Regularized (C1,C2)-Existence and Uniqueness Propagation Families

In this section, we will always assume thatE is a SCLCS, A and A1,…,An-1 are closed linear operators acting on E, n∈ℕ∖{1}, 0≤α1<⋯<αn and 0≤α<αn. Our intention is to clarify the most important results concerning the C-wellposedness of (1.1). Set mj:=⌈αj⌉, 1≤j≤n, m:=m0:=⌈α⌉, A0:=A and α0:=α.Definition 2.1.
A functionu∈Cmn-1([0,∞):E) is called a (strong) solution of (1.1) if and only if AiDtαiu∈C([0,∞):E) for 0≤i≤n-1, gmn-αn*(u-∑k=0mn-1‍ukgk+1)∈Cmn([0,∞):E) and (1.1) holds. The abstract Cauchy problem (1.1) is said to be (strongly) C-wellposed if: (i)
for everyu0,…,umn-1∈⋂0≤j≤n-1‍C(D(Aj)), there exists a unique solution u(t;u0,…,umn-1) of (1.1);(ii)
for everyT>0 and q∈⊛, there exist c>0 and r∈⊛ such that, for every u0,…,umn-1∈⋂0≤j≤n-1‍C(D(Aj)), the following holds:
(2.1)q(u(t;u0,…,umn-1))≤c∑k=0mn-1‍r(C-1uk),t∈[0,T].In the case of abstract Cauchy problem (ACPn), the definition of C-wellposedness introduced above is slightly different from the corresponding definition introduced by Xiao and Liang [28, Definition  5.2, page 116] in the Banach space setting (cf. also [28, Definition  1.2, page 46] for the case C=I). Recall that the notion of a strong C-propagation family is important in the study of existence and uniqueness of strong solutions of the abstract Cauchy problem (ACPn); compare [28, Section  3.5, pages 115–130] for further information in this direction. Suppose now that u(t)≡u(t;u0,…,umn-1), t≥0 is a strong solution of (1.1), with f(t)≡0 and initial values u0,…,umn-1∈R(C). Convoluting both sides of (1.1) with gαn(t), and making use of the equality [10, (1.21)], it readily follows that u(t), t≥0 satisfies the following:
(2.2)u(·)-∑k=0mn-1‍ukgk+1(·)+∑j=1n-1‍gαn-αj*Aj[u(·)-∑k=0mj-1‍ukgk+1(·)]=gαn-α*A[u(·)-∑k=0m-1‍ukgk+1(·)].
In the sequel of this section, we will primarily consider various types of solutions of the integral equation (2.2).Giveni∈ℕmn-10 in advance, set Di:={j∈ℕn-1:mj-1≥i}. Then it is clear that Dmn-1⊆⋯⊆D0. Plugging uj=0, 0≤j≤mn-1, j≠i, in (2.2), one gets:
(2.3)[u(·;0,…,ui,…,0)-uigi+1(·)]+∑j∈Di‍gαn-αj*Aj[u(·;0,…,ui,…,0)-uigi+1(·)]+∑j∈ℕn-1∖Di‍[gαn-αj*Aju(·;0,…,ui,…,0)]={gαn-α*Au(·;0,…,ui,…,0),m-1<i,gαn-α*A[u(·;0,…,ui,…,0)-uigi+1(·)],m-1≥i,
where ui appears in the ith place (0≤i≤mn-1) starting from 0. Suppose now 0<τ≤∞, 0≠K∈Lloc1([0,τ)) and k(t)=∫0t‍K(s)ds, t∈[0,τ). Denote Ri(t)C-1ui=(K*u(·;0,…,ui,…,0))(t), t∈[0,τ), 0≤i≤mn-1. Convoluting formally both sides of (2.3) with K(t), t∈[0,τ), one obtains that, for 0≤i≤mn-1:
(2.4)[Ri(·)C-1ui-(k*gi)(·)ui]+∑j∈Di‍gαn-αj*Aj[Ri(·)C-1ui-(k*gi)(·)ui]+∑j∈ℕn-1∖Di‍[gαn-αj*AjRi(·)C-1ui]={(gαn-α*ARi)(·)C-1ui,m-1<i,gαn-α*A[Ri(·)C-1ui-(k*gi)(·)ui],m-1≥i.
Motivated by the above analysis, we introduce the following definition.Definition 2.2.
Suppose0<τ≤∞, k∈C([0,τ)), C, C1, C2∈L(E), 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(E) is called a (local, if τ<∞): (i)
k-regularized C1-existence propagation family for (1.1) if and only if Ri(0)=(k*gi)(0)C1 and the following holds:
(2.5)[Ri(·)x-(k*gi)(·)C1x]+∑j∈Di‍Aj[gαn-αj*(Ri(·)x-(k*gi)(·)C1x)]+∑j∈ℕn-1∖Di‍Aj(gαn-αj*Ri)(·)x={A(gαn-α*Ri)(·)x,m-1<i,x∈E,A[gαn-α*(Ri(·)x-(k*gi)(·)C1x)](·),m-1≥i,x∈E,
for any i=0,…,mn-1.(ii)
k-regularized C2-uniqueness propagation family for (1.1) if and only if Ri(0)=(k*gi)(0)C2 and
(2.6)[Ri(·)x-(k*gi)(·)C2x]+∑j∈Di‍gαn-αj*[Ri(·)Ajx-(k*gi)(·)C2Ajx]+∑j∈ℕn-1∖Di‍(gαn-αj*Ri(·)Ajx)(·)={(gαn-α*Ri(·)Ax)(·),m-1<i,gαn-α*[Ri(·)Ax-(k*gi)(·)C2Ax](·),m-1≥i,
for any x∈⋂0≤j≤n-1‍D(Aj) and i∈ℕmn-10.(iii)
k-regularized C-resolvent propagation family for (1.1), in short k-regularized C-propagation family for (1.1), if ((R0(t))t∈[0,τ),…,(Rmn-1(t))t∈[0,τ)) is a k-regularized C-uniqueness propagation family for (1.1), 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.The above classes of propagation families can be defined by purely algebraic equations (cf. [11, 15, 27]). We will not go into further details about this topic here.As indicated before, we will consider only nondegeneratek-regularized C-resolvent propagation families for (1.1). In case k(t)=gζ+1(t), where ζ≥0, it is also said that ((R0(t))t∈[0,τ),…,(Rmn-1(t))t∈[0,τ)) is a ζ-times integrated C-resolvent propagation family for (1.1); 0-times integrated C-resolvent propagation family for (1.1) is simply called C-resolvent propagation family for (1.1). 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 locally equicontinuous (exponentially equi-continuous, (q-)exponentially equicontinuous, analytic, (q-)exponentially analytic,…) if and only if all single operator families (R0(t))t∈[0,τ),…,(Rmn-1(t))t∈[0,τ) are. The above terminological agreements and abbreviations can be simply understood for the classes of k-regularized C1-existence propagation families and k-regularized C2-uniqueness propagation families. The class of k-regularized (C1,C2)-existence and uniqueness propagation families for (1.1) can be also introduced (cf. Definitions 1.1 and 3.1 below).In case thatAj=cjI, where cj∈ℂ for 1≤j≤n-1, it is also said that the operator A is a subgenerator of ((R0(t))t∈[0,τ),…,(Rmn-1(t))t∈[0,τ)). Now we would like to notice the following: if A is a subgenerator of a k-regularized C-resolvent propagation family ((R0(t))t∈[0,τ),…,(Rmn-1(t))t∈[0,τ)) for (1.1), then, in general, there do not exist ai∈Lloc1([0,τ)), i∈ℕmn-10 and ki∈C([0,τ)) such that (Ri(t))t∈[0,τ) is an (ai,ki)-regularized C-resolvent family with subgenerator A; the same observation holds for the classes of k-regularized C1-existence propagation families and k-regularized C2-uniqueness propagation families. Despite this fact, the structural results for k-regularized C-resolvent propagation families can be derived by using appropriate modifications of the proofs of corresponding results for (a,k)-regularized C-resolvent families. Furthermore, these results can be clarified for any single operator family (Ri(t))t∈[0,τ) of the tuple ((R0(t))t∈[0,τ),…,(Rmn-1(t))t∈[0,τ)).Let((R0(t))t∈[0,τ),…,(Rmn-1(t))t∈[0,τ)) be a k-regularized C-resolvent propagation family with subgenerator A. Then one can simply prove that the validity of condition (H5) implies the following functional equation:
(2.7)[Ri(·)x-(k*gi)(·)Cx]+∑j=1n-1‍cjgαn-αj*[Ri(·)x-(k*gi)(·)Cx]+∑j∈ℕn-1∖Di‍cj[gαn-αj+i*k](·)Cx={A[gαn-α*Ri](·)x,m-1<i,x∈E,A[gαn-α*(Ri(·)x-(k*gi)(·)Cx)],m-1≥i,x∈E,
for any i=0,…,mn-1. The set consisted of all subgenerators of ((R0(t))t∈[0,τ),…,(Rmn-1(t))t∈[0,τ)), denoted by χ(R), need not to be finite. Notice that the supposition A∈χ(R) obviously implies C-1AC∈χ(R). The integral generator A^ of ((R0(t))t∈[0,τ),…,(Rmn-1(t))t∈[0,τ)) is defined as the set of all pairs (x,y)∈E×E such that, for every i=0,…,mn-1 and t∈[0,τ), the following holds:
(2.8)[Ri(·)x-(k*gi)(·)Cx]+∑j=1n-1‍cjgαn-αj*[Ri(·)x-(k*gi)(·)Cx]+∑j∈ℕn-1∖Di‍cj[gαn-αj+i*k](·)Cx={[gαn-α*Ri](·)y,m-1<i,gαn-α*[Ri(·)y-(k*gi)(·)Cy],m-1≥i.
It is a linear operator on E which extends any subgenerator A∈χ(R) and satisfies A^=C-1A^C. We have the following. (i)
Ri(t)(λ-A)-1C=(λ-A)-1CRi(t), t∈[0,τ), provided A∈χ(R), λ∈ρC(A) and 0≤i≤mn-1.(ii)
Let((R0(t))t∈[0,τ),…,(Rmn-1(t))t∈[0,τ)) be locally equicontinuous. Then:
(a)
A^ is a closed linear operator.(b)
A^∈χ(R), if Ri(t)Ri(s)=Ri(s)Ri(t), 0≤t, s<τ, i∈ℕmn-10.(c)
A^=C-1AC, if A∈χ(R) and (H5) holds. Furthermore, the condition (H5) can be replaced by (2.7).(iii)
Let{A,B}⊆χ(R). Then Ax=Bx, x∈D(A)∩D(B), and A⊆B⇔D(A)⊆D(B). Assume that (2.7) holds for A, and that (2.7) holds for A replaced by B. Then we have the following:
(a)
C-1AC=C-1BC and C(D(A))⊆D(B).(b)
A and B have the same eigenvalues.(c)
A⊆B⇒ρC(A)⊆ρC(B).Albeit the similar assertions can be considered in general case, we will omit the corresponding discussion even in the case that Aj∈L(E) for 1≤j≤n-1.Proposition 2.3.
Leti∈ℕmn-10, and let ((R0(t))t∈[0,τ),…,(Rmn-1(t))t∈[0,τ)) be a locally equicontinuous k-regularized C-resolvent propagation family for (1.1). If (2.5) holds with C1=C, then the following holds:(i)
the equality(2.9)Ri(t)Ri(s)=Ri(s)Ri(t),0≤t,s<τholds providedm-1<i and the following condition:
(⋄)
any of the assumptionsf(t)+∑j∈Di‍Aj(gαn-αj*f)(t)=0, t∈[0,τ), or A(gαn-α*f)(t)=0, for some f∈C([0,τ):E), implies f(t)=0, t∈[0,τ);(ii)
the equality (2.9) holds provided m-1≥i,ℕn-1∖Di≠∅, and the following condition:
(⋄⋄)
if∑j∈ℕn-1∖Di‍Aj(gαn-αj*f)(t)=0, t∈[0,τ), for some f∈C([0,τ):E), then f(t)=0, t∈[0,τ).Proof.
Letx∈E and s∈[0,τ) be fixed. Define ui(t):=Ri(t)Ri(s)x-Ri(s)Ri(t)x, t∈[0,τ). Using (2.5), it is not difficult to prove that
(2.10)A∫0t‍gαn-α(t-r)u(r)dr=u(t)+∑j=1n-1‍∫0t‍Aj(gαn-αj*u)(r)dr=0,t∈[0,τ).
Let m-1<i. Convoluting both sides of (2.10) with Ri(·), we easily infer that u(t)+∑j=1n-1‍Aj(gαn-αj*u)(t)=0, t∈[0,τ) and A(gαn-α*u)(t)=0, t∈[0,τ). Now the equality (2.9) follows from (⋄). The proof is quite similar in the case m-1≥i.Remark 2.4.
The equations (1.1) with α=0 are much easier to deal with, since in this case, m=0 and m-1<i for all i∈ℕmn-10. In general, (1.1) with α>0 cannot be reduced to an equivalent equation of the previously considered form.Proposition 2.5.
Suppose((Rj,0(t))t∈[0,τ),…,(Rj,mn-1(t))t∈[0,τ)) is a locally equicontinuous kj-regularized C-resolvent propagation family for (1.1), j=1,2, and 0≤i≤mn-1. Then we have the following.(i)
Ifm-1<i and (⋄) holds, then
(2.11)(k1*R2,i)(t)x=(k2*R1,i)(t)x,x∈⋂j=0n-1‍D(Aj),t∈[0,τ).If, additionally,(2.12)⋂j=0n-1‍D(Aj)isdenseinE,then (2.11) holds for all x∈E.(ii)
The equality (2.11) holds provided m-1≥i, ℕn-1∖Di≠∅ and (⋄⋄); assuming additionally (2.12), we have the validity of (2.11) for all x∈E.Proof.
We will only prove the second part of proposition. Letx∈⋂j=0n-1‍D(Aj). Then the functional equation of (Rj,i(t))t∈[0,τ) (j=1,2) implies:
(2.13)[(k2*gi)*(R1,i(·)x-(k1*gi)(·)Cx)](·)={R2,i(·)+∑j∈Di‍gαn-αj*[R2,i(·)Aj-(k*gi)(·)CAj]+∑j∉Di‍gαn-αj*R2,i(·)Aj-gαn-α*[R2,i(·)A-(k*gi)(·)CA]}*[R1,i(·)x-(k*gi)(·)Cx](·)={R2,i(·)+∑j∈Di‍gαn-αj*[R2,i(·)Aj-(k*gi)(·)CAj]+∑j∉Di‍gαn-αj*R2,i(·)Aj}*[R1,i(·)-(k1*gi)(·)Cx](·)-[R2,i(·)x-(k2*gi)(·)C]*A(gαn-α*[R1,i(·)x-(k1*gi)(·)Cx])(·),
which yields after a tedious computation:
(2.14)∑j∉Di‍gαn-αj*Aj[(k2*R1,i)(·)-(k1*R2,i)(·)]≡0.
In view of (⋄⋄), the above equality shows that (k2*R1,i)(t)x=(k1*R2,i)(t)x, t∈[0,τ). It can be simply verified that the condition (2.12) implies that (2.9) holds for all x∈E.Proposition 2.6.
Let((R0(t))t∈[0,τ),…,(Rmn-1(t))t∈[0,τ)) be a locally equicontinuous k-regularized C1-existence propagation family (k-regularized C2-unique-ness propagation family, k-regularized C-resolvent propagation family) for (1.1), and let b∈Lloc1([0,τ)) be a kernel. Then the tuple(((b*R0)(t))t∈[0,τ),…,((b*Rmn-1)(t))t∈[0,τ)) is a locally equicontinuous (k*b)-regularized C1-existence propagation family ((k*b)-regularized C2-uniqueness propagation family, (k*b)-regularized C-resolvent propagation family) for (1.1).Suppose nowE is complete, (1.1) is C-wellposed, ⋂j=0n-1‍D(Aj) is dense in E and 0≤i≤mn-1. Set Ri(t)x:=u(t;0,…,Cx,…,0)(t), t≥0, x∈⋂j=0n-1‍D(Aj), where 0≤i≤mn-1 and Cx appears in the ith place in the preceding expression. Since we have assumed that E is complete, the operator Ri(t) (t≥0) can be uniquely extended (cf. also (ii) of Definition 2.1) to a bounded linear operator on E. It can be easily proved that ((R0(t))t∈[0,τ),…,(Rmn-1(t))t∈[0,τ)) is a locally equicontinuous C-uniqueness propagation family for (1.1), and that the assumption CAj⊆AjC, j∈ℕn-10 implies Ri(t)C=CRi(t), t≥0. In case that Aj=cjI, where cj∈ℂ for 1≤j≤n-1, one can apply the arguments given in the proof of [29, Proposition  1.1, page 32] in order to see that ((R0(t))t∈[0,τ),…,(Rmn-1(t))t∈[0,τ)) is a locally equicontinuous C-resolvent propagation family for (1.1). Regrettably, it is not clear how one can prove in general case that Ri(t)Aj⊆AjRi(t),  j∈ℕn-10, t≥0.The following definition also appears in [15].Definition 2.7.
LetT>0 and f∈C([0,T]:E). Consider the following inhomogeneous equation:
(2.15)u(t)+∑j=1n-1‍(gαn-αj*Aju)(t)=f(t)+(gαn-α*Au)(t),t∈[0,T].
A function u∈C([0,T]:E) is said to be (i)
a strong solution of (2.15) if and only if Aju∈C([0,T]:E), j∈ℕn-10 and (2.15) holds for every t∈[0,T];(ii)
a mild solution of (2.15) if and only if (gαn-αj*u)(t)∈D(Aj), t∈[0,T], j∈ℕn-10 and
(2.16)u(t)+∑j=1n-1‍Aj(gαn-αj*u)(t)=f(t)+A(gαn-α*u)(t),t∈[0,T].It is clear that every strong solution of (2.15) is also a mild solution of the same problem. The converse statement is not true, in general. One can similarly define the notion of a strong (mild) solution of the problem (2.2).Let0<τ≤∞, and let T∈(0,τ). Then the following holds: (a)
if((R0(t))t∈[0,τ),…,(Rmn-1(t))t∈[0,τ)) is a C1-existence propagation family for (1.1), then the function u(t)=∑i=0mn-1‍Ri(t)xi, t∈[0,T], is a mild solution of (2.2) with ui=C1xi for 0≤i≤mn-1;(b)
if((R0(t))t∈[0,τ),…,(Rmn-1(t))t∈[0,τ)) is a C2-uniqueness propagation family for (1.1), and AjRi(t)x=Ri(t)Ajx, t∈[0,T], x∈⋂j=0n-1‍D(Aj), i∈ℕmn-10, j∈ℕn-10, then the function u(t)=∑i=0mn-1‍Ri(t)C2-1ui, t∈[0,T], is a strong solution of (2.2), provided ui∈C2(⋂j=0n-1‍D(Aj)) for 0≤i≤mn-1.Theorem 2.8.
Suppose((R0(t))t∈[0,τ),…,(Rmn-1(t))t∈[0,τ)) is a locally equicontinuous k-regularized C2-uniqueness propagation family for (1.1), (2.5) holds, T∈(0,τ) and f∈C([0,T]:E). Then the following holds: (i)
ifm-1<i, then any strong solution u(t) of (2.15) satisfies the equality:
(2.17)(Ri*f)(t)=(k*gi*C2u)(t)+∑j∈Di‍(gαn-αj+i*k*C2Aju)(t),for anyt∈[0,T]. Therefore, there is at most one strong (mild) solution for (2.15), provided that (⋄) holds,(ii)
ifm-1≥i, then any strong solution u(t) of (2.15) satisfies the equality:
(2.18)(Ri*f)(t)=-∑j∈ℕn-1∖Di‍(gαn-αj+i*k*C2Aju)(t),t∈[0,T].Therefore, there is at most one strong (mild) solution for (2.15), provided that ℕn-1∖Di≠∅ and that (⋄⋄) holds.Proof.
We will only prove the second part of theorem. Letm-1≥i. Taking into account (2.6), we get:
(2.19)[Ri-(k*giC)]*f=[Ri-(k*giC)]*{u+∑j=1n-1‍(gαn-αj*Aju)-(gαn-α*Au)}=[Ri-(k*giC)]*(u+∑j=1n-1‍(gαn-αj*Aju))-{[Ri-(k*giC)]+∑j∈Di‍[gαn-αj*(Ri(·)Ajx-(k*gi)(·)C2Ajx)]+∑j∉Di‍(gαn-αj*Ri(·)Ajx)}*u=-∑ℕn-1∖Di‍(gαn-αj+i*k*C2Aju)(t),t∈[0,T].
This implies the uniqueness of strong solutions to (2.15), provided that ℕn-1∖Di≠∅ and that (⋄⋄) holds. The uniqueness of mild solutions in the above case follows from the fact that, for every such a solution u(t), there exists a sufficiently large ζ>0 such that the function (gζ*u)(·) is a strong solution of (2.15), with f(·) replaced by (gζ*f)(·) therein.If((R0(t))t≥0,…,(Rmn-1(t))t≥0) is a (local) k-regularized C-resolvent propagation family for (1.1), then Theorem 2.8 shows that there exist certain relations between single operator families (R0(t))t≥0,…, and (Rmn-1(t))t≥0 (cf. also [15] and [28, page 116]). It would take too long to analyze such relations in detail.The subsequent theorems can be shown by modifying the arguments given in the proof of [30, Theorem  2.2.1].Theorem 2.9.
Supposek(t) satisfies (P1), ω≥max(0,abs(k)), (Ri(t))t≥0 is strongly continuous, and the family {e-ωtRi(t):t≥0} is equicontinuous, provided 0≤i≤mn-1. Let A be a closed linear operator on E, let C1,C2∈L(E), and let C2 be injective. Set Pλ:=λαn-α+∑j=1n-1‍λαj-αAj-A, λ∈ℂ∖{0}.(i)
SupposeAj∈L(E), j∈ℕn-1. Then ((R0(t))t≥0,…,(Rmn-1(t))t≥0) is a global k-regularized C1-existence propagation family for (1.1) if and only if the following conditions hold.
(a)
The equality(2.20)Pλ∫0∞‍e-λtRi(t)xdt=λαn-α-ik~(λ)C1x+∑j∈Di‍λαj-α-ik~(λ)AjC1x,holds providedx∈E, i∈ℕmn-10, m-1<i and ℜλ>ω.(b)
The equality(2.21)Pλ∫0∞‍e-λt[Ri(t)x-(k*gi)(t)C1x]dt=-∑j∈ℕn-1∖Di‍λαj-α-ik~(λ)AjC1x,holds providedx∈E, i∈ℕmn-10, m-1≥i and ℜλ>ω.(ii)
SupposeRi(0)=(k*gi)(0)C2x, x∈E∖⋂0≤j≤n-1‍D(Aj)¯, i∈ℕmn-10. Then ((R0(t))t≥0,…,(Rmn-1(t))t≥0) is a global k-regularized C2-uniqueness propagation family for (1.1) if and only if, for every λ∈ℂ with ℜλ>ω, and for every x∈⋂0≤j≤n-1‍D(Aj), the following equality holds:
(2.22)∫0∞‍e-λt[Ri(t)x-(k*gi)(t)C2x]dt+∑j∈Di‍λαj-αn∫0∞‍e-λt[Ri(t)x-(k*gi)(t)C2Ajx]dt+∑j∈ℕn-1∖Di‍λαj-αn∫0∞‍e-λtRi(t)Ajxdt={λα-αn∫0∞‍e-λtRi(t)Axdt,m-1<i,λα-αn∫0∞‍e-λt[Ri(t)Ax-(k*gi)(t)C2Ax]dt,m-1≥i.Theorem 2.10.
Supposek(t) satisfies (P1), ω≥max(0,abs(k)), (Ri(t))t≥0 is strongly continuous, and the family {e-ωtRi(t):t≥0} is equicontinuous, provided 0≤i≤mn-1. Let CAj⊆AjC, j∈ℕn-10, Aj∈L(E), j∈ℕn-1, AiAj=AjAi, i,j∈ℕn-1 and AjA⊆AAj, j∈ℕn-1. Assume, additionally, that the operator λαn-i+∑j∈Di‍λαj-iAj is injective for every i∈ℕmn-10 with m-1<i and for every λ∈ℂ with ℜλ>ω and k~(λ)≠0, and that the operator ∑j∈ℕn-1∖Di‍λαj-iAj is injective for every i∈ℕmn-10 with m-1≥i and for every λ∈ℂ with ℜλ>ω and k~(λ)≠0. Then ((R0(t))t≥0,…,(Rmn-1(t))t≥0) is a global k-regularized C-resolvent propagation family for (1.1), and (2.5) holds, if and only if the equalities (2.20)-(2.21) are fulfilled.Keeping in mind Theorem2.10, one can simply clarify the most important Hille-Yosida type theorems for exponentially equicontinuous k-regularized C-resolvent propagation families (cf. also [15] and [26, Theorem  2.8] for further information in this direction). Notice also that the preceding theorem can be slightly reformulated for k-regularized (C1,C2)-existence and uniqueness resolvent propagation families.The analytical properties ofk-regularized C-resolvent propagation families are stated in the following two theorems whose proofs are omitted (cf. [14, Theorems  2.16-2.17] and [26, Lemma  3.3, Theorems  3.4, 3.6, and 3.7]).Theorem 2.11.
Supposeβ∈(0,π/2], ((R0(t))t≥0,…,(Rmn-1(t))t≥0) is an analytic k-regularized C-resolvent propagation family for (1.1), k(t) satisfies (P1), (2.5) holds, and k~(λ) can be analytically continued to a function k^:ω+Σ(π/2)+β→ℂ, where ω≥max(0,abs(k)). Suppose CAj⊆AjC, j∈ℕn-10, Aj∈L(E), j∈ℕn-1, AiAj=AjAi, i,j∈ℕn-1 and AjA⊆AAj, j∈ℕn-1. Let the family
(2.23){e-ωzRi(z):z∈Σγ}beequicontinuous,providedi∈ℕmn-10andγ∈(0,β),
and let the set
(2.24){(λ-ω)k^(λ)λ-i:λ∈ω+Σ(π/2)+γ}
be bounded provided γ∈(0,β) and m-1≥i. Set
(2.25)Ni:={λ∈ω+Σ(π/2)+β:k^(λ)(λαn+∑j∈Di‍λαjAj)isinjective},
provided m-1<i, and
(2.26)Ni:={λ∈ω+Σ(π/2)+β:k^(λ)(λαn+∑j∈ℕn-1∖Di‍λαjAj)isinjective},
provided m-1≥i. Suppose Ni is an open connected subset of ℂ, and the set Ni∩{λ∈ℂ:ℜλ>ω} has a limit point in {λ∈ℂ:ℜλ>ω}, for any i∈ℕmn-10. Then the operator Pλ is injective for every λ∈Ni and i∈ℕmn-10,
(2.27)limλ→+∞,λ∈Niλk~(λ)Pλ-1(λαn-α-i+∑j∈Di‍λαj-α-iAj)Cx=(k*gi)(0)Cx,
provided m-1<i and x∈E, and
(2.28)limλ→+∞,λ∈Niλk~(λ)Pλ-1∑j∈ℕn-1∖Di‍λαj-α-iAjCx=0,
provided m-1≥i and x∈E. Suppose, additionally, that there exists μ∈ℂ such that Pμ-1C∈L(E). Then the family
(2.29){(λ-ω)k^(λ)(λαn-α+∑j=1n-1‍λαj-αAj-C-1AC)-1×(λαn-α-iC+∑j∈Di‍λαj-α-iAjC):λ∈Ni∩(ω+Σ(π/2)+γ)}isequicontinuous,
provided m-1<i and γ∈(0,β), respectively, the family
(2.30){(λ-ω)k^(λ)(λαn-α+∑j=1n-1‍λαj-αAj-C-1AC)-1∑j∈ℕn-1∖Di‍λαj-α-iAjC:λ∈Ni∩(ω+Σ(π/2)+γ)(λαn-α+∑j=1n-1‍λαj-αAj-C-1AC)-1}isequicontinuous,
provided m-1≥i and γ∈(0,β), the mapping
(2.31)λ↦(λαn-α+∑j=1n-1‍λαj-αAj-C-1AC)-1(λαn-α-iC+∑j∈Di‍λαj-α-iAjC)x,
defined for λ∈Ni, is analytic, provided m-1<i and x∈E, and the mapping
(2.32)λ↦(λαn-α+∑j=1n-1‍λαj-αAj-C-1AC)-1∑j∈ℕn-1∖Di‍λαj-α-iAjCx,λ∈Ni,
is analytic, provided m-1≥i and x∈E.Theorem 2.12.
Assumek(t) satisfies (P1), ω≥max(0,abs(k)), β∈(0,π/2] and, for every i∈ℕmn-10 with m-1≥i, 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 CAj⊆AjC, j∈ℕn-10, Aj∈L(E), j∈ℕn-1, AiAj=AjAi, i,j∈ℕn-1 and AjA⊆AAj, j∈ℕn-1. Assume, additionally, that for each i∈ℕmn-10 the set Vi:=Ni∩{λ∈ℂ:ℜλ>ω} contains the set {λ∈ℂ:ℜλ>ω,k~(λ)≠0}, and that R(λαnC+∑j∈Di‍λαjAjC)⊆R(Pλ), provided m-1<i and λ∈Vi, respectively, R(λαnC+∑j∈ℕn-1∖Di‍λαjAjC)⊆R(Pλ), provided m-1≥i and λ∈Vi (cf. the formulation of preceding theorem). Suppose also that the operator λαnI+∑j∈Di‍λαjAj is injective, provided m-1<i and λ∈Vi, and that the operator λαnI+∑j∈ℕn-1∖Di‍λαjAj is injective, provided m-1≥i and λ∈Vi. Let qi:ω+Σ(π/2)+β→L(E) (0≤i≤mn-1) satisfy that, for every x∈E, the mapping λ↦qi(λ)x, λ∈ω+Σ(π/2)+β is analytic as well as that:
(2.33)qi(λ)x=k~(λ)Pλ-1(λαn-α-iC+∑j∈Di‍λαj-α-iAjC)x,x∈E,λ∈Vi,
provided m-1<i,
(2.34)qi(λ)x=-k~(λ)Pλ-1∑j∈ℕn-1∖Di‍λαj-α-iAjCx,x∈E,λ∈Vi,
provided m-1≥i,
(2.35)thefamily{(λ-ω)qi(λ):λ∈ω+Σ(π/2)+γ}isequicontinuous∀γ∈(0,β),
and, in the case D(A)¯≠E,
(2.36)limλ→+∞λqi(λ)x={(k*gi)(0)Cx,x∉D(A)¯,m-1<i,0,x∉D(A)¯,m-1≥i.
Then there exists an exponentially equicontinuous, analytic k-regularized C-resolvent propagation family ((R0(t))t≥0,…,(Rmn-1(t))t≥0) for (1.1). Furthermore, the family {e-ωzRi(z):z∈Σγ} is equicontinuous for all i∈ℕmn-10 and γ∈(0,β), (2.5) holds, and Ri(z)Aj⊆AjRi(z), z∈Σβ, j∈ℕn-10.In this paper, we will not consider differential properties ofk-regularized C-resolvent (propagation) families. For more details, the interested reader may consult [30], and especially, [26, Theorems  3.18–3.20]. Notice also that the assertion of [26, Proposition  3.12] can be reformulated for k-regularized C-resolvent (propagation) families.In the following theorem, which possesses several obvious consequences, we considerq-exponentially equicontinuous k-regularized I-resolvent propagation families in complete locally convex spaces.Theorem 2.13.
(i) Supposek(0)≠0, ((R0(t))t≥0,…,(Rmn-1(t))t≥0) is a q-exponentially equicontinuous k-regularized I-resolvent propagation family for (1.1), Aj∈L⊛(E), j∈ℕn-1, and for every p∈⊛, there exist Mp≥1 and ωp≥0 such that
(2.37)p(Ri(t)x)≤Mpeωptp(x),t≥0,x∈E,0≤i≤mn-1.
Then A is a compartmentalized operator and, for every seminorm p∈⊛, ((R0,p(t)¯)t≥0,…,(Rmn-1,p(t)¯)t≥0) is an exponentially bounded k-regularized Ip¯-resolvent propagation family for (1.1), in Ep¯, with Aj replaced by Aj,p¯ (0≤j≤n-1). Furthermore,
(2.38)∥Ri,p(t)¯∥≤Mpeωpt,t≥0,0≤i≤mn-1,
and ((R0,p(t)¯)t≥0,…,(Rmn-1,p(t)¯)t≥0) is a q-exponentially equicontinuous, analytic k-regularized Ip¯-resolvent propagation family of angle β∈(0,π], provided that ((R0(t))t≥0,…,(Rmn-1(t))t≥0) is. Assume additionally that (2.5) holds. Then, for every p∈⊛, (2.5) holds with Aj and ((R0(t))t≥0,…,(Rmn-1(t))t≥0) replaced by Aj,p¯ and ((R0,p(t)¯)t≥0,…,(Rmn-1,p(t)¯)t≥0).
(ii) Supposek(t) satisfies (P1), E is complete, A is a compartmentalized operator in E, Aj=cjI for some cj∈ℂ (1≤j≤n-1) and, for every p∈⊛, Ap¯ is a subgenerator (the integral generator, in fact) of an exponentially bounded k-regularized Ip¯-resolvent propagation family ((R0,p(t)¯)t≥0,…,(Rmn-1,p(t)¯)t≥0) in Ep¯ satisfying (2.38), and (2.5) with A and ((R0(t))t≥0,…,(Rmn-1(t))t≥0) replaced, respectively, by Ap¯ and ((R0,p(t)¯)t≥0,…,(Rmn-1,p(t)¯)t≥0). Suppose, additionally, that ℕn-1∖Di≠∅ and ∑j∈ℕn-1∖Di‍|cj|2>0, provided m-1≥i. Then, for every p∈⊛, (2.37) holds (0≤i≤mn-1) and A is a subgenerator (the integral generator, in fact) of a q-exponentially equicontinuous k-regularized I-resolvent propagation family ((R0(t))t≥0,…,(Rmn-1(t))t≥0) satisfying (2.5). Furthermore, ((R0(t))t≥0,…,(Rmn-1(t))t≥0) is a q-exponentially equicontinuous, analytic k-regularized I-resolvent propagation family of angle β∈(0,π] provided that, for every p∈⊛, ((R0,p(t)¯)t≥0,…,(Rmn-1,p(t)¯)t≥0) is a q-exponentially bounded, analytic k-regularized Ip¯-resolvent propagation family of angle β.Proof.
The proof is almost completely similar to that of [20, Theorem  3.1], and we will only outline a few relevant facts needed for the proof of (i). Suppose x,y∈D(A) and p(x)=p(y) for some p∈⊛. Then (2.6) in combination with (2.37) implies that Ψp(Ri(t)A(x-y))=0, t≥0, provided m-1<i, and Ψp(Ri(t)A(x-y)-(k*gi)(t)(x-y))=0, t≥0, provided m-1≥i. In any case, Ψp(Ri(t)A(x-y))=0,t≥0, which implies p(Ri(t)A(x-y))=0, t≥0, and in particular p(k(0)A(x-y))=0. Since k(0)≠0, we obtain p(Ax-Ay)=0 and p(Ax)=p(Ay). Therefore, A is a compartmentalized operator. It is clear that (2.38) holds and that the mapping t↦Ri,p(t)¯xp, t≥0 is continuous for any xp∈Ep. This implies by the standard limit procedure that the mapping t↦Ri,p(t)¯xp¯, t≥0 is continuous for any xp¯∈Ep¯. Now we will prove that, for every p∈⊛, the operator Ap is closable for the topology of Ep¯. In order to do that, suppose (xn) is a sequence in D(A) with limn→∞Ψp(xn)=0 and limn→∞Ψp(Axn)=y, in Ep¯. Using the dominated convergence theorem, (2.6) and (2.37), we get that ∫0t‍gαn-α(t-s)Ri,p(s)¯yds=limn→∞∫0t‍gαn-α(t-s)Ri,p(s)¯Ψp(Axn)ds=0, for any t≥0. Taking the Laplace transform, one obtains Ri,p(t)¯y=0,t≥0. Since Ri,p(0)¯=k(0)Ip¯, we get that y=0 and that Ap is closable, as claimed. Suppose 0≤i≤mn-1. It is checked at once that Ri,p(t)¯Aj,p¯⊆Aj,p¯Ri,p(t)¯, t≥0, i∈ℕmn-10, j∈ℕn-1. The functional equation (2.6) for the operators Aj,p¯, 0≤j≤n-1 and ((R0,p(t)¯)t≥0,…,(Rmn-1,p(t)¯)t≥0) can be trivially verified, which also holds for the functional equation (2.6) in case of its validity for the operators Aj, 0≤j≤n-1, and ((R0(t))t≥0,…,(Rmn-1(t))t≥0). The remaining part of the proof can be obtained by copying the final part of the proof of [20, Theorem  3.1(i)].Remark 2.14.
In the second part of Theorem2.13, we must restrict ourselves to the case in which Aj=cjI for some cj∈ℂ (1≤j≤n-1). As a matter of fact, it is not clear how one can prove that the operator λαnIp¯+∑j∈Di‍λαjAj,p¯ is injective, provided m-1<i, ℜλ>ω and k~(λ)≠0, as well as that the operator ∑j∈ℕn-1∖Di‍λαjAj,p¯ is injective, provided m-1≥i, ℜλ>ω and k~(λ)≠0. Then Theorem 2.10 is inapplicable, which implies that the argumentation used in the proof of [20, Theorem  3.1(ii)] does not work for the proof of fact that, for every i∈ℕmn-10 and t>0, {Ri,p(t)¯:p∈⊛} is a projective family of operators.


## 3.k-Regularized (C1,C2)-Existence and Uniqueness Families for (1.1)

Throughout this section, we will always assume thatX and Y are sequentially complete locally convex spaces. By L(Y,X) is denoted the space which consists of all bounded linear operators from Y into X. The fundamental system of seminorms which defines the topology on X, respectively, Y, is denoted by ⊛X, respectively, ⊛Y. The symbol I designates the identity operator on X.Let0<τ≤∞. A strongly continuous operator family (W(t))t∈[0,τ)⊆L(Y,X) is said to be locally equicontinuous if and only if, for every T∈(0,τ) and for every p∈⊛X, there exist qp∈⊛Y and cp>0 such that p(W(t)y)≤cpqp(y), y∈Y, t∈[0,T]; the notion of equicontinuity of (W(t))t∈[0,τ) is defined similarly. Notice that (W(t))t∈[0,τ) is automatically locally equicontinuous in case that the space Y is barreled.Following Xiao and Liang [24], we introduce the following definition.Definition 3.1.
Suppose0<τ≤∞, k∈C([0,τ)), C1∈L(Y,X), and C2∈L(X) is injective. (i)
A strongly continuous operator family(E(t))t∈[0,τ)⊆L(Y,X) is said to be a (local, if τ<∞) k-regularized C1-existence family for (1.1) if and only if, for every y∈Y, the following holds: E(·)y∈Cmn-1([0,τ):X), E(i)(0)y=0 for every i∈ℕ0 with i<mn-1, Aj(gαn-αj*E(mn-1))(·)y∈C([0,τ):X) for 0≤j≤n-1, and
(3.1)E(mn-1)(t)y+∑j=1n-1‍Aj(gαn-αj*E(mn-1))(t)y-A(gαn-α*E(mn-1))(t)y=k(t)C1y,
for any t∈[0,τ).(ii)
A strongly continuous operator family(U(t))t∈[0,τ)⊆L(X) is said to be a (local, if τ<∞) k-regularized C2-uniqueness family for (1.1) if and only if, for every τ∈[0,τ) and x∈⋂0≤j≤n-1‍D(Aj), the following holds:
(3.2)U(t)x+∑j=1n-1‍(gαn-αj*U(·)Ajx)(t)-(gαn-α*U(·)Ax)(t)y=(k*gmn-1)(t)C2x.(iii)
A strongly continuous family((E(t))t∈[0,τ),(U(t))t∈[0,τ))⊆L(Y,X)×L(X) is said to be a (local, if τ<∞) k-regularized (C1,C2)-existence and uniqueness family for (1.1) if and only if (E(t))t∈[0,τ) is a k-regularized C1-existence family for (1.1), and (U(t))t∈[0,τ) is a k-regularized C2-uniqueness family for (1.1).(iv)
SupposeY=X and C=C1=C2. Then a strongly continuous operator family (R(t))t∈[0,τ)⊆L(X) is said to be a (local, if τ<∞) k-regularized C-resolvent family for (1.1) if and only if (R(t))t∈[0,τ) is a k-regularized C-uniqueness family for (1.1), R(t)Aj⊆AjR(t), for 0≤j≤n-1 and t∈[0,τ), as well as R(t)C=CR(t), t∈[0,τ), and CAj⊆AjC, for 0≤j≤n-1.In casek(t)=gζ+1(t), where ζ≥0, it is also said that (E(t))t∈[0,τ) is a ζ-times integrated C1-existence family for (1.1); 0-times integrated C1-existence family for (1.1) is also said to be a C1-existence family for (1.1). The notion of (exponential) analyticity of C1-existence families for (1.1) is taken in the sense of Definition 1.2(ii); the above terminological agreement can be simply understood for all other classes of uniqueness and resolvent families introduced in Definition 3.1.Integrating both sides of (3.1) sufficiently many times, we easily infer that (cf. [24, Definition 2.1, page 151; and (2.8), page 153]):
(3.3)E(l)(t)y+∑j=1n-1‍Aj(gαn-αj*E(l))(t)y-A(gαn-α*E(l))(t)y=(k*gmn-1-l)(t)C1y,
for any t∈[0,τ), y∈Y and l∈ℕmn-10. In this place, it is worth noting that the identity (3.3), with k(t)=1, l=0, τ=∞ and αj=j (0≤j≤n-1), has been used in [24] for the definition of a C1-existence family for (ACPn). It can be simply proved that this definition is equivalent with the corresponding one given by Definition 3.1.Proposition 3.2.
Let((E(t))t∈[0,τ),(U(t))t∈[0,τ)) be a k-regularized (C1,C2)-existence and uniqueness family for (1.1), and let (U(t))t∈[0,τ) be locally equicontinuous. If Aj∈L(X), j∈ℕn-1 or α≤min(α1,…,αn-1), then C2E(t)y=U(t)C1y, t∈[0,τ), y∈Y.Proof.
Lety∈Y be fixed. Using the local equicontinuity of (U(t))t∈[0,τ), we easily infer that the mappings t↦((gαn-α*U)*E(·)y)(t), t∈[0,τ) and t↦(U*(gαn-α*E(·)y))(t), t∈[0,τ) are continuous and coincide. The prescribed assumptions also imply that, for every j∈ℕn-1, t∈[0,τ) and y∈Y,
(3.4)(gαn-α*U*Aj(gαn-αj*E(·)y))(t)y=(gαn-α*UAj*gαn-α*E(·)y)(t)y.
Keeping in mind (3.2)-(3.3) and the foregoing arguments, we get that
(3.5)gαn-α*U*[E(·)y+∑j=1n-1‍Aj(gαn-αj*E)(·)y-k(·)C1y]=gαn-α*UA*[gαn-α*E](·)y=[U(·)+∑j=1n-1‍(gαn-αj*U(·)Aj)-k(·)C2]*gαn-α*E(·)y.
This, in turn, implies the required equality C2E(t)y=U(t)C1y, t∈[0,τ).Definition 3.3.
Suppose0≤i≤mn-1. Then we define Di′:={j∈ℕn-10:mj-1≥i}, Di′′:=ℕn-10∖Di′ and
(3.6)Di:={x∈⋂j∈Di′′‍D(Aj):Ajui∈R(C1),j∈Di′′}.In the first part of subsequent theorem (cf. also [24, Remark  2.2, Example  2.5, Remark  2.6]), we will consider the most important case k(t)=1. The analysis is similar if k(t)=gn+1(t) for some n∈ℕ.Theorem 3.4.
(i) Suppose(E(t))t∈[0,τ) is a C1-existence family for (1.1), T∈(0,τ), and ui∈Di for 0≤i≤mn-1. Then the function
(3.7)u(t)=∑i=0mn-1‍uigi+1(t)-∑i=0mn-1‍∑j∈ℕn-1∖Di‍(gαn-αj*E(mn-1-i))(t)vi,j+∑i=mmn-1‍(gαn-α*E(mn-1-i))(t)vi,0,0≤t≤T,
is a strong solution of the problem (2.2) on [0,T], where vi,j∈Y satisfy Ajui=C1vi,j for 0≤j≤n-1.
(ii) Suppose(U(t))t∈[0,τ) is a locally equicontinuous k-regularized C2-uniqueness family for (1.1), and T∈(0,τ). Then there exists at most one strong (mild) solution of (2.2) on [0,T], with ui=0, i∈ℕmn-10.Proof.
A straightforward computation involving (3.3) shows that
(3.8)u(·)-∑i=0mn-1‍uigi+1(·)+∑j=1n-1‍Aj(gαn-αj*[u(·)-∑i=0mj-1‍uigi+1(·)])=-∑i=0mn-1‍∑j∈ℕn-1∖Di‍(gαn-αj*R(mn-1-i))(·)vi,j+∑i=mmn-1‍(gαn-α*R(mn-1-i))(·)vi,0+∑j=1n-1‍Aj(gαn-αj*{∑i=mjmn-1‍gi+1(·)ui-∑i=0mn-1‍∑l∈ℕn-1∖Di‍(gαn-αl*R(mn-1-i))(·)vi,l+∑i=mmn-1‍(gαn-α*R(mn-1-i))(·)vi,0∑i=0mn-1‍})=-∑i=0mn-1‍∑j∈ℕn-1∖Di‍(gαn-αj*R(mn-1-i))(·)vi,j+∑i=mmn-1‍(gαn-α*R(mn-1-i))(·)vi,0+∑j=1n-1‍∑i=mjmn-1‍C1vi,jgαn-αj+i+1(·)-∑i=0mn-1‍∑l∈ℕn-1∖Di‍gαn-αl*[-R(mn-1-i)(·)vi,l+A(gαn-α*R(mn-1-i))(·)vi,l+gi+1(·)C1vi,l]+∑i=mmn-1‍gαn-α*[-R(mn-1-i)(·)vi,0+A(gαn-α*R(mn-1-i))(·)vi,0+gi+1(·)C1vi,0]=gαn-α*A[u(·)-∑i=0m-1‍uigi+1(·)],
since
(3.9)∑j=1n-1‍∑i=mjmn-1‍C1vi,jgαn-αj+i+1(·)=∑i=0mn-1‍∑j∈ℕn-1∖Di‍C1vi,jgαn-αj+i+1(·).
This implies that u(t) is a mild solution of (2.2) on [0,T]. In order to complete the proof of (i), it suffices to show that Dtαnu(t)∈C([0,T]:X) and AiDtαiu∈C([0,T]:X) for all i∈ℕn-10. Towards this end, notice that the partial integration implies that, for every t∈[0,T],
(3.10)gmn-αn*[u(·)-∑i=0mn-1‍uigi+1(·)](t)=∑i=mmn-1‍(gmn-α+i*E(mn-1))(t)vi,0-∑i=0mn-1‍∑j∈ℕn-1∖Di‍(gmn-αj+i*E(mn-1))(t)vi,j.
Therefore, Dtαnu∈C([0,T]:X) and
(3.11)Dtαnu(t)=dmndtmn{gmn-αn*[u(·)-∑i=0mn-1‍uigi+1(·)](t)}=∑i=mmn-1‍(gi-α*E(mn-1))(t)vi,0-∑i=0mn-1‍∑j∈ℕn-1∖Di‍(gi-αj*E(mn-1))(t)vi,j.
Suppose, for the time being, i∈ℕn-10. Then Aiuj∈R(C1) for j≥mi. Moreover, the inequality l≥αj holds provided 0≤l≤mn-1 and j∈ℕn-1∖Dl, and Aj(gαn-αj*E(mn-1))(·)y∈C([0,T]:X) for 0≤j≤n-1 and y∈Y. Now it is not difficult to prove that
(3.12)AiDtαiu(·)=∑j=mimn-1‍gj+1-αi(·)Aiuj-∑l=0mn-1‍∑j∈ℕn-1∖Dl‍[gl-αj*Ai(gαn-αi*E(mn-1))](·)vl,j+∑l=mmn-1‍[gl-α*Ai(gαn-αi*E(mn-1))](·)vl,0∈C([0,T]:X),
finishing the proof of (i). The second part of theorem can be proved as follows. Suppose u(t) is a strong solution of (2.2) on [0,T], with ui=0, i∈ℕmn-10. Using this fact and the equality
(3.13)∫0t‍∫0t-s‍gαn-αj(r)U(t-s-r)Aju(s)drds=∫0t‍∫0s‍gαn-αj(r)U(t-s)Aju(s-r)drds,
for any t∈[0,T] and j∈ℕn-10, we easily infer that (for more general results, see [31, Proposition  2.4(i)], and [29, page 155]):
(3.14)(U*u)(t)=(k*gmn-1C2*u)(t)+∫0t‍∫0t-s‍[gαn-αj(r)U(t-s-r)Aju(s)-gαn-α(r)U(t-s-r)Au(s)]drds=(k*gmn-1C2*u)(t)+(U*u)(t),t∈[0,T].
Therefore, (k*gmn-1C2*u)(t)=0, t∈[0,T] and u(t)=0, t∈[0,T].Before proceeding further, we would like to notice that the solutionu(t), given by (3.7), need not to be of class C1([0,T]:X), in general. Using integration by parts, it is checked at once that (3.7) is an extension of the formula [24, (2.5); Theorem  2.4, page 152]. Notice, finally, that the proof of Theorem 3.4(ii) is much simpler than that of [24, Theorem  2.4(ii)].The standard proof of following theorem is omitted (cf. also [24, Theorem  2.7, Remark  2.8, Theorem  2.9] and [28, Chapter 1]).Theorem 3.5.
Supposek(t) satisfies (P1), (E(t))t≥0⊆L(Y,X), (U(t))t≥0⊆L(X), ω≥max(0,abs(k)), C1∈L(Y,X) and C2∈L(X) is injective. Set Pλ:=I+∑j=1n-1‍λαj-αnAj-λα-αnA, ℜλ>0. (i)
(a) Let(E(t))t≥0 be a k-regularized C1-existence family for (1.1), let the family {e-ωtE(t):t≥0} be equicontinuous, and let the family {e-ωtAj(gαn-αj*E)(t):t≥0} be equicontinuous (0≤j≤n-1). Then the following holds:
(3.15)Pλ∫0∞‍e-λtE(t)ydt=k~(λ)λ1-mnC1y,y∈Y,ℜλ>ω.
(b) Let the operator Pλ be injective for every λ>ω with k~(λ)≠0. Suppose, additionally, that there exist strongly continuous operator families (W(t))t≥0⊆L(Y,X) and (Wj(t))t≥0⊆L(Y,X) such that {e-ωtW(t):t≥0} and {e-ωtWj(t):t≥0} are equicontinuous (0≤j≤n-1) as well as that
(3.16)∫0∞‍e-λtW(t)ydt=k~(λ)Pλ-1C1y,∫0∞‍e-λtWj(t)ydt=k~(λ)λαj-αnAjPλ-1C1y,
for every λ∈ℂ with ℜλ>ω and k~(λ)≠0, y∈Y and j∈ℕn-10. Then there exists a k-regularized C1-existence family for (1.1), denoted by (E(t))t≥0. Furthermore, E(mn-1)(t)y=W(t)y, t≥0, y∈Y and Aj(gαn-αj*E(mn-1))(t)y=Wj(t)y, t≥0, y∈Y, j∈ℕn-10.(ii)
Let the assumptions of (i) hold withk(t)=1. If mn>1, then one suppose additionally that, for every j∈ℕn-10, there exists a strongly continuous operator family (Vj(t))t≥0⊆L(Y,X) such that {e-ωtVj(t):t≥0} is equicontinuous as well as that
(3.17)∫0∞‍e-λtVj(t)ydt=λαj-αn-1Pλ-1AjC1y,
for every λ∈ℂ with ℜλ>ω, and y∈D(AjC1). Let ui∈Di, and let C1vi=ui for some vi∈Y (0≤i≤mn-1). Then, for every p∈⊛X, there exist cp>0 and qp∈⊛Y such that the corresponding solution u(t) satisfies the following estimate:
(3.18)p(u(t))≤cpeωt∑i=0mn-1‍qp(vi),t≥0,ifω>0,(3.19)p(u(t))≤cpgmn(t)∑i=0mn-1‍qp(vi),t≥0,ifω=0.(iii)
Suppose(U(t))t≥0 is strongly continuous and the operator family {e-ωtU(t):t≥0} is equicontinuous. Then (U(t))t≥0 is a k-regularized C2-uniqueness family for (1.1) if and only if, for every x∈⋂j=0n-1‍D(Aj), the following holds:
(3.20)∫0∞‍e-λtU(t)Pλxdt=k~(λ)λ1-mnC2x,ℜλ>ω.The Hausdorff locally convex topology onE* defines the system (|·|B)B∈ℬ of seminorms on E*, where |x*|B:=supx∈B|〈x*,x〉|, x*∈E*, B∈ℬ. Let us recall that E* is sequentially complete provided that E is barreled. Following Wu and Zhang [32], we also define on E* the topology of uniform convergence on compacts of E, denoted by 𝒞(E*,E); more precisely, given a functional x0*∈E*, the basis of open neighborhoods of x0* with respect to 𝒞(E*,E) is given by N(x0*:K,ε):={x*∈E*:supx∈K|〈x*-x0*,x〉|<ε}, where K runs over all compacts of E and ε>0. Then (E*,𝒞(E*,E)) is locally convex, complete and the topology 𝒞(E*,E) is finer than the topology induced by the calibration (|·|B)B∈ℬ.Now we focus our attention to the adjoint type theorems for (local)k-regularized C-resolvent families. The proof of following theorem follows from the arguments given in the proofs of [26, Theorems  2.14 and 2.15]; because of that, we will omit it.Theorem 3.6.
(i) SupposeX is barreled, ζ>0, (R(t))t∈[0,τ) is a k-regularized C-resolvent family for (1.1), and ⋂j=0n-1‍D(Aj)¯=R(C)¯=X. Then ((gζ*R(·)*)(t))t∈[0,τ) is a k-regularized C*-resolvent family for (1.1), with Aj replaced by Aj* (0≤j≤n-1).
(ii) SupposeX is barreled, (R(t))t∈[0,τ) is a (local, global exponentially equicontinuous) k-regularized C-resolvent family for (1.1), and ⋂j=0n-1‍D(Aj)¯=R(C)¯=X. Put Z:=⋂j=0n-1‍D(Aj*)¯. Then (R(t)|Z*)t∈[0,τ), is a (local, global exponentially equicontinuous) k-regularized C|Z*-resolvent family for (1.1), in Z.
(iii) Suppose(R(t))t∈[0,τ) is a locally equicontinuous k-regularized C-resolvent family for (1.1), and ⋂j=0n-1‍D(Aj)¯=R(C)¯=X. Then (R(t)*)t∈[0,τ) is a locally equicontinuous k-regularized C*-resolvent family for (1.1), in (X*,𝒞(X*,X)), with Aj replaced by Aj* (0≤j≤n-1). Furthermore, if (R(t))t≥0 is exponentially equicontinuous, then (R(t)*)t≥0 is also exponentially equicontinuous.Notice here that a similar theorem can be proved for the class ofk-regularized C-resolvent propagation families.Letf∈C([0,T]:X). Convoluting both sides of (1.1) with gαn(t), we get that
(3.21)u(·)-∑k=0mn-1‍ukgk+1(·)+∑j=1n-1‍gαn-αj*Aj[u(·)-∑k=0mj-1‍ukgk+1(·)]=gαn-α*A[u(·)-∑k=0m-1‍ukgk+1(·)]+(gαn*f)(·),t∈[0,T].In the subsequent theorem, whose proof follows from a slight modification of the proof of [24, Theorem  3.1(i)], we will analyze inhomogeneous Cauchy problem (3.21) in more detail.Theorem 3.7.
Suppose(E(t))t∈[0,τ) is a locally equicontinuous C1-existence family for (1.1), T∈(0,τ), and ui∈Di for 0≤i≤mn-1. Let f∈C([0,T]:X), let g∈C([0,T]:Y) satisfy C1g(t)=f(t), t∈[0,T], and let G∈C([0,T]:Y) satisfy (gαn-mn+1*g)(t)=(g1*G)(t), t∈[0,T]. Then the function
(3.22)u(t)=∑i=0mn-1‍uigi+1(t)-∑i=0mn-1‍∑j∈ℕn-1∖Di‍(gαn-αj*E(mn-1-i))(t)vi,j+∑i=mmn-1‍(gαn-α*E(mn-1-i))(t)vi,0+∫0t‍E(t-s)G(s)ds,0≤t≤T
is a mild solution of the problem (3.21) on [0,T], where vi,j∈Y satisfy Ajui=C1vi,j for 0≤j≤n-1. If, additionally, g∈C1([0,T]:Y) and (E(mn-1)(t))t∈[0,τ)⊆L(Y,X) is locally equicontinuous, then the solution u(t), given by (3.22), is a strong solution of (1.1) on [0,T].Remark 3.8.
Suppose that all conditions quoted in the first part of the above theorem hold, and the family(E(mn-1)(t))t∈[0,τ)⊆L(Y,X) is locally equicontinuous. We assume, instead of condition g∈C1([0,T]:Y), that there exists a locally equicontinuous C2-uniqueness family for (1.1) on [0,τ), as well as that there exist functions hj∈L1([0,T]:Y) such that Ajf(t)=C1hj(t), t∈[0,T], 0≤j≤n-1 (cf. also the formulation of [24, Theorem  3.1(ii)]). Using the functional equation for (E(t))t∈[0,τ), one can simply prove that, for every σ∈[0,T], the function
(3.23)rσ(·)=E(·)g(σ)-gmn(·)f(σ)+∑j=1n-1‍(gαn-αj*E(·)hj(σ))(·)-(gαn-α*E(·)h0(σ))(·)
is a mild solution of the problem
(3.24)u(t)+∑j=1n-1‍Aj(gαn-αj*u)(t)-A(gαn-α*u)(t)=0,t∈[0,T].
By the uniqueness of solutions, we have that the following holds:
(3.25)E(t-σ)g(σ)-gmn(t-σ)f(σ)+∑l=1n-1‍(gαn-αl*E(·)hl(σ))(t-σ)(gαn-α*E(·)h0(σ))(t-σ)=0,
provided 0≤t, σ≤T and σ≤t. Fix i∈ℕn-10. Then the above equality implies that, for every j∈ℕmn-10 with j≤min(⌊αi+mn-αn-1-1⌋,⌊αi+mn-α-1⌋), one has:
(3.26)AiE(j)(t-σ)g(σ)-gmn-j(t-σ)C1hi(σ)+∑l=1n-1‍Ai(gαn-αl*E(j)(·)hl(σ))(t-σ)-Ai(gαn-α*E(j)(·)h0(σ))(t-σ)=0,
provided 0≤t, σ≤T and σ≤t. For such an index j, we conclude from (3.26) that the mapping t↦∫0t‍AiE(j)(t-σ)g(σ)dσ, t∈[0,T] is continuous. Observe now that the condition
(3.27)αn-αi-mn+min(⌊αi+mn-αn-1-1⌋,⌊αi+mn-α-1⌋)≥0,i∈ℕn-10,
which holds in the case of abstract Cauchy problem (ACPn), shows that the mapping t↦Ai[gαn-αi-mn+j*E(j)*g](t), t∈[0,T] is continuous as well as that the mapping t↦(d/dt)[E(mn-1)*g](t), t∈[0,T] is continuous. Hence, the validity of condition (3.27) implies that the function u(t), given by (3.22), is a strong solution of (1.1) on [0,T].


## 4. Subordination Principles

The proof of following theorem can be derived by using Theorem3.5 and the argumentation given in [10, Section 3].Theorem 4.1.
SupposeC1∈L(Y,X), C2∈L(X) is injective and γ∈(0,1).  (i)
Letω≥max(0,abs(k)), and let the assumptions of Theorem 3.5(i)-(b) hold. Put
(4.1)Wγ(t):=∫0∞‍t-γΦγ(t-γs)W(s)yds,t>0,y∈Y,Wγ(0):=W(0).Define, for everyj∈ℕn-10 and t≥0, Wj,γ(t) by replacing W(t) in (4.1) with Wj(t). Suppose that there exist a number ν>0 and a continuous kernel kγ(t) satisfying (P1) and kγ~(λ)=λγ-1k~(λγ), λ>ν. Then there exists an exponentially bounded kγ-regularized C1-existence family (Eγ(t))t≥0 for (1.1), with αj replaced by αjγ therein (0≤j≤n-1). Furthermore, the family {(1+t⌈αnγ⌉-2)-1e-ω1/γtEγ(t):t≥0} is equicontinuous.(ii)
Letω≥0, let the assumptions of Theorem 3.5(ii) hold, and let k(t)=kγ(t)=1. Define, for every j∈ℕn-10 and t≥0, Vj,γ(t) by replacing W(t) in (4.1) with Vj(t). Then, for every j∈ℕn-10, the family {e-ω1/γtVj,γ(t):t≥0} is equicontinuous,
(4.2)∫0∞‍e-λtVj,γ(t)ydt=λαjγ-αnγ-1Pλγ-1AjC1y,for everyλ∈ℂ with ℜ(λγ)>ω, and y∈D(AjC1). Let ui∈Di,γ (defined in the obvious way), and let C1vi=ui for some vi∈Y (0≤i≤⌈αnγ⌉-1). Then, for every p∈⊛X, there exist cp>0 and qp∈⊛Y such that the corresponding solution u(t) satisfies the following estimate:
(4.3)p(u(t))≤cpeω1/γt∑i=0⌈αnγ⌉-1‍qp(vi),t≥0,ifω>0,p(u(t))≤cpg⌈αnγ⌉(t)∑i=0⌈αnγ⌉-1‍qp(vi),t≥0,ifω=0.(iii)
Suppose(U(t))t≥0 is a k-regularized C2-uniqueness family for (1.1), and the family {e-ωtU(t):t≥0} is equicontinuous. Define, for every t≥0, Uγ(t) by replacing W(t) in (4.1) with U(t). Suppose that there exist a number ν>0 and a continuous kernel kγ(t) satisfying (P1) and kγ~(λ)=λγ(2-mn)-2+⌈αnγ⌉k~(λγ), λ>ν. Then there exists a kγ-regularized C2-uniqueness family for (1.1), with αj replaced by αjγ therein (0≤j≤n-1). Furthermore, the family {e-ω1/γtUγ(t):t≥0} is equicontinuous.Remark 4.2.
(i) Consider the situation of Theorem4.1(iii). Then we have the obvious equality (k*gmn-1)(0)=(kγ*g⌈αnγ⌉-1)(0). If σ≥1,k(t)=gσ(t) and (σ-1+mn-1)γ+1-⌈αnγ⌉≥0 (this inequality holds provided σ≥2), then kγ(t)=g(σ-1+mn-1)γ+2-⌈αnγ⌉(t).
(ii) Letb∈Lloc1([0,∞)) be a kernel, and let (U(t))t∈[0,τ) be a (local) k-regularized C2-uniqueness family for (1.1). Then ((b*U)(t))t∈[0,τ) is a (b*k)-regularized C2-uniqueness family for (1.1).
(iii) Concerning the analytical properties ofkγ-regularized C1-existence families in Theorem 4.1(i), the following facts should be stated. (a)
The mappingt↦Eγ(t), t>0 admits an extension to Σmin(((1/γ)-1)(π/2),π) and, for every y∈Y, the mapping z↦Eγ(z)y, z∈Σmin(((1/γ)-1)(π/2),π) is analytic.(b)
Letε∈(0,min(((1/γ)-1)(π/2),π)), and let (W(t))t≥0 be equicontinuous. Then (Eγ(t))t≥0 is an exponentially equicontinuous, analytic kγ-regularized C1-existence family of angle min(((1/γ)-1)(π/2),π), and for every p∈⊛X, there exist Mp,ε>0 and qp,ε∈⊛Y such that
(4.4)p(Eγ(z)y)≤Mp,εqp,ε(y)(1+|z|⌈αnγ⌉-1),z∈Σmin(((1/γ)-1)(π/2),π)-ε.(c)
(Eγ(t))t≥0 is an exponentially equicontinuous, analytic kγ-regularized C1-existence family of angle min(((1/γ)-1)(π/2),π/2).The similar statements hold for thekγ-regularized C2-uniqueness family (Uγ(t))t≥0 in Theorem 4.1(iii).The results onk-regularized (C1,C2)-existence and uniqueness families can be applied in the study of following abstract Volterra equation:
(4.5)u(t)=f(t)+∑j=0n-1‍(aj*Aju)(t),t∈[0,τ),
where 0<τ≤∞, f∈C([0,τ):X), a0,…,an-1∈Lloc1([0,τ)), and A0,…,An-1 are closed linear operators on X. As in Definition 2.7, by a mild solution, respectively, strong solution, of (4.5), we mean any function u∈C([0,τ):X) such that Aj(aj*u)(t)∈C([0,τ):X), j∈ℕn-10 and that
(4.6)u(t)=f(t)+∑j=0n-1‍Aj(aj*u)(t),t∈[0,τ),
respectively, any function u∈C([0,τ):X) such that u(t)∈⋂j=0n-1‍D(Aj), t∈[0,τ) and that (4.5) holds.We need the following definition.Definition 4.3.
Suppose0<τ≤∞, k∈C([0,τ)), C1∈L(Y,X), and C2∈L(X) is injective. (i)
A strongly continuous operator family(E(t))t∈[0,τ)⊆L(Y,X) is said to be a (local, if τ<∞) k-regularized C1-existence family for (4.5) if and only if
(4.7)E(t)y=k(t)C1y+∑j=0n-1‍Aj(aj*E)(t)y,t∈[0,τ),y∈Y.(ii)
A strongly continuous operator family(U(t))t∈[0,τ)⊆L(X) is said to be a (local, if τ<∞) k-regularized C2-uniqueness family for (4.5) if and only if
(4.8)U(t)x=k(t)C2x+∑j=0n-1‍(aj*AjU)(t)x,t∈[0,τ),x∈⋂j=0n-1‍D(Aj).Notice also that one can introduce the classes ofk-regularized (C1,C2)-existence and uniqueness families as well as k-regularized C-resolvent families for (4.5); compare Definition 3.1. The full analysis of k-regularized (C1,C2)-existence and uniqueness families for (4.5) falls out from the framework of this paper.The following facts are clear.(i)
Suppose(E(t))t∈[0,τ) is a k-regularized C1-existence family for (4.5). Then, for every y∈Y, the function u(t)=E(t)y, t∈[0,τ), is a mild solution of (4.5) with f(t)=k(t)C1y, t∈[0,τ).(ii)
Let(U(t))t∈[0,τ) be a locally equicontinuous k-regularized C2-uniqueness family for (4.5). Then there exists at most one mild (strong) solution of (4.5).The proof of following subordination principle is standard and therefore omitted (cf. the proofs of [29, Theorem  4.1, page 101] and [24, Theorem  2.7]).Theorem 4.4.
(i) Suppose there is an exponentially equicontinuousk-regularized C1-existence family for (1.1). Let c(t) be completely positive, let c(t), k(t) and k1(t) satisfy (P1), and let ω0>0 be such that, for every λ>ω0 with c~(λ)≠0 and k~(1/c~(λ))≠0, the following holds:
(4.9)a~j(λ)=-k1~(λ)c~(λ)1+αn-αjλk~(1/c~(λ)),j∈ℕn-1,a~0(λ)=-k1~(λ)c~(λ)1+αn-αλk~(1/c~(λ)).
Assume, additionally, that there exist a number z∈ℂ and a function k2(t) satisfying (P1) so that, for every λ>ω0 with c~(λ)≠0 and k~(1/c~(λ))≠0, one has:
(4.10)k~1(λ)k~(1/c~(λ))=z+k~2(λ).
Then there exists an exponentially equicontinuous k1-regularized C1-existence family for (4.5).
(ii) Suppose there is an exponentially equicontinuousk-regularized C2-uniqueness family for (1.1). Let c(t) be completely positive, let c(t), k(t) and k1(t) satisfy (P1), and let ω0>0 be such that, for every λ>ω0 with c~(λ)≠0 and k~(1/c~(λ))≠0, the following holds:
(4.11)a~j(λ)=c~(λ)αn-αj,j∈ℕn-10,k~1(λ)=λ-1c~(λ)mn-2k~(1c~(λ)).
Then there exists an exponentially equicontinuous k1-regularized C2-uniqueness family for (4.5).It is not difficult to reformulate Theorem4.4 for the class of strong C-propagation families (cf. also Example 5.3 below).Although our analysis tends to be exhaustive, we cannot cover, in this limited space, many interested subjects. For example, the characterizations of some special classes ofq-exponentially equicontinuous k-regularized (C1,C2)-existence and uniqueness families in complete locally convex spaces. We also leave to the interested reader the problem of clarifying the Trotter-Kato type theorems for introduced classes.


## 5. Examples and Applications

We start this section with the following example.Example 5.1.
Supposecj∈ℂ (1≤j≤n-1) and, for every i∈ℕmn-10 with m-1≥i, one has ℕn-1∖Di≠∅ and ∑j∈ℕn-1∖Di‍|cj|2>0. Let Aj=cjI for 1≤j≤n-1. (i)
(a)
Suppose0<δ≤2, σ≥1, (πδ/2(αn-α))-(π/2)>0, and A is a subgenerator of an exponentially equicontinuous (gδ,gσ)-regularized C-resolvent family (Rδ(t))t≥0 which satisfies the following equality:
(5.1)A∫0t‍gδ(t-s)Rδ(s)xds=Rδ(t)x-gσ(t)Cx,x∈E,t≥0.Putσ':=max(1,1+(αn-α)(σ-1)δ-1) and θ:=min(π/2,πδ/2(αn-α)-(π/2)). By [26, Theorem  2.7], we have that, for every sufficiently small ε>0, there exists ωε>0 such that ωε+Σ(π/2)δ-ε⊆ρC(A) and the family {|λ|(δ-σ)/δ(1+|λ|1/δ)(λ-A)-1C:λ∈ωε+Σ(π/2)α-ε} is equicontinuous. Notice also that
(5.2)arg(λαn-α+∑j=1n-1‍cjλαj-α)=arg(λαn-α|λ|α-((αn-1+αn)/2)+∑j=1n-1‍cjλαj-α|λ|α-((αn-1+αn)/2))≈arg(λαn-α|λ|α-((αn-1+αn)/2))=(αn-α)arg(λ),λ→∞,arg(λ)<παn-α.Due to the choice ofθ, we have that, for every sufficiently small ε>0, there exists ωε>0 such that, for every λ∈ωε+Σ(π/2)+θ-ε, one has:
(5.3)arg(λαn-α+∑j=1n-1‍cjλαj-α)<π2δ-ε.Therefore, we have the following: if the operatorA is densely defined, then the above inequality in combination with Theorem 2.12 indicates that A is a subgenerator of an exponentially equicontinuous, analytic (σ′-1)-times integrated C-resolvent propagation family ((R0(t))t≥0,…,(Rmn-1(t))t≥0) for (1.1), with θ being the angle of analyticity; if the operator A is not densely defined, then the above conclusion continues to hold with σ' replaced by any number σ′′>σ′.(a′)
Suppose0<δ≤2, σ≥1, (δ((π/2)+γ)/(αn-α))-(π/2)>0, A is a subgenerator of an exponentially equicontinuous, analytic (gδ,gσ)-regularized C-resolvent family (Rδ(t))t≥0 of angle γ∈(0,π/2], and (5.1) holds. Put σ1:=σ' and θ1:=min(π/2,(δ((π/2)+γ)/(αn-α))-(π/2)). If the operator A is densely defined, then it follows from [26, Theorem  3.6] and the above analysis that the operator C-1AC is the integral generator of an exponentially equicontinuous, analytic (σ1-1)-times integrated C-resolvent propagation family ((R0(t))t≥0,…,(Rmn-1(t))t≥0) for (1.1), with θ1 being the angle of analyticity; if the operator A is not densely defined, then the above conclusion continues to hold with σ1 replaced by any number σ2>σ1. Now we will apply this result to the following fractional analogue of the telegraph equation:
(5.4)Dtα2u(t,x)+c1Dtα1u(t,x)=DΔxu(t,x),t>0,x∈ℝn,wherec1>0, D>0 and 0<α1≤α2<2. Let E be one of the spaces Lp(ℝn) (1≤p≤∞), C0(ℝn), Cb(ℝn), BUC(ℝn) and 0≤l≤n. Put ℕ0l:={α∈ℕ0n:αl+1=⋯=αn=0} and recall that the space El (0≤l≤n) is defined by El:={f∈E:f(α)∈Eforallα∈ℕ0l}. The totality of seminorms (qα(f):=||f(α)||E,f∈El;α∈ℕ0l) induces a Fréchet topology on El. Let Tl possess the same meaning as in [33], and let A:=DΔ act with its maximal distributional domain. Suppose first E≠L∞(ℝn) and E≠Cb(ℝn). Then the operator A is the integral generator of an exponentially equicontinuous, analytic C0-semigroup of angle π/2, which implies that A is the integral generator of an exponentially equicontinuous, analytic I-regularized resolvent propagation family (R0(t))t≥0, if α2≤1, respectively, ((R0(t))t≥0,(R1(t))t≥0) if α2>1, of angle ζ=min(π/2,(π/α2)-(π/2)); the established conclusion also holds in the Fréchet nuclear space Ξ which consists of those smooth functions on ℝn with period 1 along each coordinate axis [26]. In this place, we would like to observe that it is not clear whether the angle of analyticity of constructed I-regularized resolvent propagation families, in the case that α1<α2<1, can be improved by allowing that ζ takes the value min(π,(π/α2)-(π/2)). Suppose now E=L∞(ℝn) or E=Cb(ℝn). Then, for every σ′>1, the operator A is the integral generator of an exponentially equicontinuous, analytic (σ′-1)-times integrated I-regularized resolvent propagation family (R0(t))t≥0, if α2≤1, respectively, ((R0(t))t≥0,(R1(t))t≥0) if α2>1, of angle min(π/2,(π/α2)-(π/2)).(b)
Suppose0<δ≤2, σ≥1, (πδ/2(αn-α))-(π/2)>0, a>0, b∈(0,1), ka,b(t):=ℒ-1(exp(-aλb))(t), t≥0 and A is a subgenerator of an exponentially equicontinuous (gδ,ka,b)-regularized C-resolvent family (Ra,b(t))t≥0 which satisfies the following equality:
(5.5)A∫0t‍gδ(t-s)Ra,b(s)xds=Ra,k(t)x-ka,b(t)Cx,x∈E,t≥0.Letθ be defined as in (a). Then it is checked at once that (αn-α)bδ-1<1 and (αn-α)bδ-1((π/2)+θ)<π/2. Put k1(t):=ka1,b1(t), t≥0, where b1:=(αn-α)bδ-1 and a1>a(cos((αn-α)bδ-1((π/2)+θ)))-1. It is clear that, for every θ'∈(0,θ), there exists a sufficiently large ωθ'>0 such that, for every λ∈ωθ'+Σ(π/2)+θ',
(5.6)|k1~(λ)||k~((λαn-α+∑j=1n-1‍cjλαj-α)1/δ)|≤|k1~(λ)|exp(a|λ|b1+∑j=1n-1‍|cj||λ|(αj-α)b/δ).Arguing as in (a), we reveal thatA is a subgenerator of an exponentially equicontinuous, analytic k1-regularized C-resolvent propagation family ((R0(t))t≥0,…,(Rmn-1(t))t≥0) for (1.1), with θ being the angle of analyticity.(b′)
Suppose0<δ≤2, σ≥1, δ(((π/2)+γ)/(αn-α))-(π/2)>0, Aj=cjI (1≤j≤n-1), a>0,  b∈(0,1), A is a subgenerator of an exponentially equicontinuous, analytic (gδ,ka,b)-regularized C-resolvent family (Ra,b(t))t≥0 of angle γ∈(0,π/2], and (5.5) holds. Assume, additionally, that b(1+(2γ/π))≤1. Define θ1 as in (a)’, and k2(t):=ka2,b2(t), t≥0, where b2:=(αn-α)bδ-1 and a2>a(cos((αn-α)bδ-1((π/2)+θ1)))-1. Then one can simply verify that (αn-α)b<δ and (αn-α)bδ-1((π/2)+γ)≤π/2. Making use of [26, Theorem  3.6] and the foregoing arguments, we obtain that the operator C-1AC is the integral generator of an exponentially equicontinuous, analytic k2-regularized C-resolvent propagation family ((R0(t))t≥0,…,(Rmn-1(t))t≥0) for (1.1), with θ being the angle of analyticity. Before proceeding further, we would like to recommend for the reader [14, 20, 21, 26, 30, 34] for some examples of (nondensely defined, in general) differential operators generating various types of (gσ,ka,b)-regularized C-resolvent families.(ii)
SupposeE is complete, 0<δ≤2, (πδ/2(αn-α))-(π/2)>0, and A is the densely defined generator of a q-exponentially equicontinuous (gδ,g1)-regularized I-resolvent family (Rδ(t))t≥0 which satisfies that, for every p∈⊛, there exist Mp≥1 and ωp≥0 such that p(Rδ(t)x)≤Mpeωptp(x), t≥0, x∈E. By [20, Theorem  3.1], we infer that A is a compartmentalized operator and that, for every p∈⊛, the operator Ap¯ is the integral generator of an exponentially bounded (gδ,g1)-regularized Ip¯-resolvent family in Ep¯. Then the first part of this example shows that Ap¯ is the integral generator of an exponentially bounded, analytic Ip¯-resolvent propagation family, with min(π/2,(πδ/2(αn-α))-(π/2)) being the angle of analyticity. By Theorem 2.13(ii), we obtain that A is the integral generator of a q-exponentially equicontinuous, analytic I-resolvent propagation family ((R0(t))t∈[0,τ),…,(Rmn-1(t))t∈[0,τ)) for (1.1), and that the corresponding angle of analyticity is min(π/2,(πδ/2(αn-α))-(π/2)). It can be simply shown that, for every p∈⊛ and i∈ℕmn-10, there exist Mp,i≥1 and ωp,i≥0 such that p(Ri(t)x)≤Mp,ieωp,itp(x), t≥0, x∈E. In the continuation, we will also present some other applications of (a,k)-regularized C-resolvent families in the analysis of some special cases of (1.1); as already mentioned, this theory is inapplicable if some of initial values u0,…,umn-1 is a non-zero element of E. Consider the abstract Basset-Boussinesq-Oseen equation (1.2) and assume that E is complete. Set aα(t):=ℒ-1(λα/(λ+1))(t), t≥0, kα(t):=e-t,  t≥0 and δα:=min(π/2,(πα/2(1-α))). Suppose A is the integral generator of a q-exponentially equicontinuous (g1,g1)-regularized I-resolvent family (R(t))t≥0 satisfying (2.37); cf. [20, 25] for important examples of differential operators generating q-exponentially equicontinuous (gδ,g1)-regularized I-resolvent families. Then it has been proved in [20] that A is the integral generator of a q-exponentially equicontinuous, analytic (aα,kα)-regularized resolvent family of angle δα. Notice, finally, that the choice of function aα(t) instead of g1(t) has some advantages.Example 5.2.
Suppose1≤p≤∞, E:=Lp(ℝ), m:ℝ→ℂ is measurable, aj∈L∞(ℝ), (Ajf)(x):=aj(x)f(x), x∈ℝ, f∈E (1≤j≤n-1) and (Af)(x):=m(x)f(x), x∈ℝ, with maximal domain. Assume s∈(1,2), δ=1/s, Mp=p!s and kδ(t)=ℒ-1(e-λδ)(t), t≥0. Denote by M(t) the associated function of the sequence (Mp) [30] and put Λα',β',γ':={λ∈ℂ:Reλ≥γ'-1M(α'λ)+β'}, α'>0, β'>0, γ'>0. Clearly, there exists a constant Cs>0 such that M(λ)≤Cs|λ|1/s, λ∈ℂ. Hereafter we assume that the following condition holds: (H)
for everyτ>0, there exist α′>0, β'>0 and d>0 such that τ≤cos(δπ/2)/Csα'1/s and
(5.7)|λαn-α+∑j=1n-1‍λαj-αaj(x)-m(x)|≥d,x∈ℝ,λ∈Λα',β',1.Notice that the above condition holds provided n=2, α2-α=2, α2-α1=1 and m(x)=(1/4)a12(x)-(1/16)a14(x)-1, x∈ℝ (cf. [31]), and that the validity of condition (H) does not imply, in general, the essential boundedness of the function m(·). We will prove that A is the integral generator of a global (not exponentially bounded, in general) kδ-regularized I-resolvent propagation family ((R0(t))t≥0,…,(Rmn-1(t))t≥0) for (1.1). Clearly, it suffices to show that, for every τ∈(0,∞), A is the integral generator of a local kδ-regularized I-resolvent propagation family for (1.1) on [0,τ). Suppose that τ>0 is given in advance, and that α′>0, β′>0 and d>0 satisfy (H), for this τ. Let Γ denote the upwards oriented boundary of ultralogarithmic region Λα′,β′,1. Put, for every t∈[0,τ), f∈E and x∈ℝ,
(5.8)(Ri(t)f)(x):=12πi∫Γ‍eλt-λδ[λαn-α-i+∑j∈Di‍λαj-α-iaj(x)]f(x)λαn-α+∑j=1n-1‍λαj-αaj(x)-m(x)dλ,
if m-1<i, and
(5.9)(Ri(t)f)(x):=(-1)2πi∫Γ‍eλt-λδλαj-α-iaj(x)λαn-α+∑j=1n-1‍λαj-αaj(x)-m(x)dλ+(kδ*gi)(t)f(x),
if m-1≥i. It is clear that, for every i∈ℕmn-10, Ri(t)Aj⊆AjRi(t), t∈[0,τ), j∈ℕn-10 and that (Ri(t))t∈[0,τ)⊆L(E) is strongly continuous. Furthermore, the Cauchy theorem implies that Ri(0)=0=kδ(0), i∈ℕmn-10. Now we will prove that the identity (2.6) holds provided m-1<i and C2=I. Let f∈D(A). Then a straightforward computation involving Cauchy theorem shows that (2.6) holds, with x replaced by f(·) therein, if and only if:
(5.10)12πi∫Γ‍eλt-λδ[λαn-α-i+∑j∈Di‍λαj-α-iaj(x)]f(x)λαn-α+∑j=1n-1‍λαj-αaj(x)-m(x)dλ+∑j=1n-1‍12πi∫Γ‍(∫0t‍gαn-αj(t-s)eλsds)e-λδ[λαn-α-i+∑l∈Di‍λαl-α-igl(x)]f(x)λαn-α+∑l=1n-1‍λαl-αal(x)-m(x)dλ-12πi∫Γ‍(∫0t‍gαn-α(t-s)eλsds)e-λδ[λαn-α-i+∑j∈Di‍λαj-α-iaj(x)]m(x)f(x)λαn-α+∑j=1n-1‍λαj-αaj(x)-m(x)dλ=12πi∫Γ‍eλt-λδ[λ-if(x)+∑j∈Di‍λαj-αn-iaj(x)f(x)]dλ.
Using [28, Lemma  5.5, page 23] and the Cauchy theorem, the above equality is equivalent with:
(5.11)12πi∫Γ‍eλt-λδ[λαn-α-i+∑j∈Di‍λαj-α-iaj(x)]f(x)λαn-α+∑j=1n-1‍λαj-αaj(x)-m(x)dλ+∑j=1n-1‍12πi∫Γ‍eλt-λδλαn-αj[λαn-α-i+∑l∈Di‍λαl-α-igl(x)]f(x)λαn-α+∑l=1n-1‍λαl-αal(x)-m(x)dλ-12πi∫Γ‍eλt-λδλαn-α[λαn-α-i+∑j∈Di‍λαj-α-iaj(x)]m(x)f(x)λαn-α+∑j=1n-1‍λαj-αaj(x)-m(x)dλ=12πi∫Γ‍eλt-λδ[λ-if(x)+∑j∈Di‍λαj-αn-iaj(x)f(x)]dλ,
which is true because the integrands appearing on both sides of this equality are equal identically. One can similarly prove that the identity (2.6) holds provided m-1≥i and C2=I, so that ((R0(t))t≥0,…,(Rmn-1(t))t≥0), defined in the obvious way, is a kδ-regularized I-resolvent propagation family for (1.1), with subgenerator A. Notice that the condition (H) implies m(·)/(λαn-α+∑j=1n-1‍λαj-αaj(·)-m(·))∈L∞(ℝ) for all λ∈Λα',β',1, which has as a further consequence that R(Ri(t))⊆D(A), provided t≥0 and m-1<i, and that R(Ri(t)-(kδ*gi)(t))⊆D(A), provided t≥0 and m-1≥i. The equality (2.5) holds for ((R0(t))t≥0,…,(Rmn-1(t))t≥0), the integral generator of ((R0(t))t≥0,…,(Rmn-1(t))t≥0), defined similarly as in the second section, coincides with the operator A, which is the unique subgenerator of ((R0(t))t≥0,…,(Rmn-1(t))t≥0). Notice that, for every compact set K⊆[0,∞), there exists hK>0 such that
(5.12)supt∈K,p∈ℕ0,i∈ℕmn-10∥hkp(dp/dtp)Ri(t)∥p!s<∞,
and that one can similarly consider the generation of local k1/2-regularized I-resolvent propagation families which oblige a modification of the property stated above with s=2. Now we would like to give an example of kδ-regularized I-resolvent propagation family for (1.1) in which Aj∉L(E) for some j∈ℕn-1. Assume n=2, α2-α=2, α2-α1=1, a1(x)=-2x, x∈ℝ and m(x)=x2-x4-1, x∈ℝ. Define A1, A and Ri(·) as before (i=0,1). Then the established conclusions continue to hold since, for every τ>0, there exist α′>0, β'>0 and d>0 such that (H) holds as well as that:
(5.13)x2+(x4-x2+1)|λ|-2|λ2-2xλ+(x4-x2+1)|≤d,x∈ℝ,λ∈Λα',β',1.
Notice, finally, that it is not so difficult to construct examples of local k-regularized C-resolvent propagation families which cannot be extended beyond its maximal interval of existence.Example 5.3.
Suppose1≤p≤∞, X:=Lp(ℝ), a∈ℝ, r>0, ϑ(·)∈W1,∞(ℝ), 1/2<γ≤1, T>0, f∈C([0,T]:X), and (d/dt)(g2γ-1*(d/dx)f(t,·))∈C([0,T]:X). Put A1:=ad/dx and Au:=rΔu-ϑ(·)u with maximal distributional domain. Now we will focus our attention to the following fractional analogue of damped Klein-Gordon equation:
(5.14)Dt2γu(t,x)+a∂∂xDtγu(t,x)-rΔxu(t,x)+ϑ(x)u(t,x)=f(t,x),t>0,x∈ℝ,u(0,x)=ϕ(x),ut(0,x)=ψ(x),x∈ℝ.
The case γ=1 has been analyzed in [24, Example  4.1], showing that there exists an exponentially bounded I-uniqueness family for (5.14) and that, for every μ0∈ρ(A1), there exists an exponentially bounded (μ0-A1)-1-existence family for (5.14) with Y=X. It is worth noting that the estimates obtained in cited example enables one to simply verify that the conditions of Theorem 4.1(i)-(ii) hold with k(t)=1 and C1=(μ0-A1)-1, and that the conditions of Theorem 4.1(iii) hold with k(t)=t and C2=I. This implies that there exists an exponentially bounded g2γ-regularized I-uniqueness family (Uγ(t))t≥0 for (5.14) with αj=jγ, j=0,1,2, and that there exists an exponentially bounded (μ0-A1)-1-existence family (Eγ(t))t≥0 for (5.14) with αj=jγ, j=0,1,2. Applying Theorem 3.7, we obtain that, for every ϕ∈W3,p(ℝ) and ψ∈W3,p(ℝ), there exists a unique mild solution u(t,x) of the corresponding problem (3.21) as well as that there exist M≥1 and ω≥0 such that the following estimate holds for each t≥0:
(5.15)∥u(t,x)∥Lp(ℝ)≤Meωt[∥dds(g2γ-1*ddxf(s,·))∥Lp(ℝ)∥ϕ∥W1,p(ℝ)+∥ψ∥W1,p(ℝ)+∫0t‍(t-s)2γ-2∥f(s,·)∥Lp(ℝ)ds+∫0t‍∥dds(g2γ-1*ddxf(s,·))∥Lp(ℝ)ds].
It is checked at once that the solution u(t,x) is analytically extensible to the sector Σ((1/γ)-1)(π/2), provided that f(t,x)≡0. Suppose now ϑ(x)≡ϑ>0, κ≥|1/2-1/p|, provided 1<p<∞, respectively, κ>1/2, provided p∈{1,∞}, C:=(1-Δ)-(1/2)κ and f(t,x)≡0. Then there exists a strong C-propagation family {(S0(t))t≥0,(S1(t))t≥0} for the problem (5.14) with γ=1 (cf. [28, Example  5.8, page 130]). Using [10, (1.23), page 12; Theorems  3.1–3.3, pages 40–42] and [28, Proposition  5.3(iii), page 116], it readily follows that, for every ϕ∈Wp,2(ℝ) and ψ∈Wp,2(ℝ), the function uγ(t,·), t>0, given by
(5.16)uγ(t,·):=∫0∞‍t-γΦ(st-γ)[S1(s)ϕ+S1′(s)ϕ]ds+∫0t‍g1-γ(t-s)∫0∞‍s-γΦ(rs-γ)S1(r)ψdrds,
is a unique strong solution of the corresponding integral equation (3.21) with u0=Cϕ and u1=Cψ; obviously, this solution is analytically extensible to the sector Σ((1/γ)-1)(π/2). Notice also that one can similarly consider (cf. [24, Example  4.2] for more details) the results concerning the existence and uniqueness of mild solutions of the following time-fractional equation:
(5.17)Dt2γu(t,x)+(ρ1∂3∂x3-ρ2∂2∂x2)Dtγu(t,x)+(c∂2∂x2+a(x))u(t,x)=f(t,x),(5.18)u(0,x)=ϕ(x),ut(0,x)=ψ(x),
and that Theorem 4.4 can be applied in the analysis of the following integral equation:
(5.19)u(t,x)=a∫0t‍a1(t-s)∂∂xu(s,x)ds+∫0t‍a2(t-s)[rΔxu(s,x)-ϑ(x)u(s,x)]ds+f(t,x),
for certain kernels a1(t) and a2(t). We leave details to the interested reader.
Consider now the following slight modification of (5.14):
(5.20)Dt2γu(t,x)+a∂∂xDtγu(t,x)-rei(2-2γ)(π/2)Δxu(t,x)+ϑ(x)u(t,x)=f(t,x),t>0,x∈ℝ,u(0,x)=ϕ(x),(Dtγu(t,x))|t=0=ψ(x),x∈ℝ.
Suppose now that a≠0 (for further information concerning the case a=0, [21, 23] may be of some importance). Although the equality Dt2γu(t,x)=Dtγu(t,x)Dtγu(t,x) does not hold in general, we would like to point out that the existence and uniqueness of mild solutions to the homogeneous counterpart of (5.20) cannot be so easily proved for initial values belonging to the Sobolev space Wk,p(ℝ), for some k∈ℕ. In order to better explain this, we will introduce the new function v(t,x) by v(t,x):=Dtγu(t,x). Then (5.20) can be rewritten in the following equivalent matricial form:
(5.21)Dtγ[u(t,x)v(t,x)]T=[01-rei(2-2γ)π/2-aix](A)[u(t,x)v(t,x)]T,t≥0,
where A=-id/dx; see, for example, [35, 36]. The characteristic values of associated polynomial matrix P(x) := [01-rei(2-2γ)(π/2)-aix] are λ1,2(x)=(1/2)(-aix±a2+4rei(2-2γ)(π/2)), x∈ℝ, which implies that the condition of Petrovskii for systems of abstract time-fractional equations, that is, supx∈ℝℜ((λ1,2(x))1/γ)<∞, is not satisfied [36]. Notice, finally, that (1.1) cannot be converted to an equivalent matrix form, except for some very special values of α0,…,αn.
Before proceeding further, we would like to observe that several examples ofk-times integrated (C1,C2)-existence and uniqueness families, acting on products of possibly different Banach spaces (k∈ℕ), can be constructed following the consideration given in [37, Section  7].Example 5.4.
Lets'>1,
(5.22)E:={f∈C∞[0,1];∥f∥:=supp≥0∥f(p)∥∞p!s′<∞},A:=-dds,D(A):={f∈E;f′∈E,f(0)=0}.
Then ρ(A)=ℂ, and for every η>1, ||R(λ:A)||=O(eη|λ|), λ∈ℂ [21]. Consider now the complex non-zero polynomials Pj(z)=∑l=0nj‍aj,lzl, z∈ℂ, aj,nj≠0 (0≤j≤n-1), and define, for every λ∈ℂ and j∈ℕn-10, the operator Pj(A) by D(Pj(A)):=D(Anj) and Pj(A)f:=∑l=0nj‍aj,lAlf, f∈D(Pj(A)). Our intention is to analyze the smoothing properties of solutions of the equation (3.21) with Aj:=pj(A), j∈ℕn-10, uk=0, k∈ℕmn-10, and a suitable chosen function f(t). In order to do that, set N:=max(dg(P0),…,dg(Pn-1)), 𝒫λ(z):=1+∑j=1n-1‍λαj-αnPj(z)-λα-αnP0(z) (λ∈ℂ∖{0}, z∈ℂ), and after that, Φ:={λ∈ℂ∖{0}:dg(𝒫λ(·))=N,𝒫λ(0)≠0}. Then it is not difficult to prove (cf. [21, Example  2.10]) that, for every λ∈ℂ∖{0}, Pλ-1=(I+∑j=1n-1‍λαj-αnAj-λα-αnA)-1∈L(E) and that
(5.23)Pλ-1=(-1)Ng(λ)-1R(z1,λ:A)⋯R(zN,λ:A),λ∈Φ,
where z1,λ,…,zN,λ denote the zeroes of 𝒫λ(z) and g(λ):=N!-1𝒫λ(N)(0), λ∈Φ. Suppose now that the following condition holds: (H)
there existσ∈(0,1), ω>0 and m>0 such that, for every j∈ℕn-10, one has: |zj,λ|≤m|λ|σ, λ∈Φ, ℜλ>ω.It is well known from the elementary courses of numerical analysis [38] that the condition:(H1)
there existσ∈(0,1),  ω>0 and m>0 such that, for every j∈ℕn-10, one has:
(5.24)|N!𝒫λ(j)(0)j!𝒫λ(N)(0)|1/(N-j)≤12m|λ|σ,λ∈Φ,ℜλ>ω, implies (H). The validity of last condition can be simply verified in many concrete situations, and it seems that slightly better estimates can be obtained only in the case of very special equations of the form (1.1). We would also like to point out that the condition (H) need not to be satisfied, in general. Using (5.23), the inequality ||AlR(μ1:A)⋯R(μl:A)||≤(1+|μ1|||R(μ1:A)||)⋯(1+|μl|||R(μl:A)||) (l∈ℕ,μ1,…,μl∈ℂ), as well as the continuity of mappings λ↦Pλ-1, ℜλ>ω and λ↦AjPλ-1, ℜλ>ω, for 0≤j≤n-1, we obtain the existence of a positive polynomial p(·) such that
(5.25)∥Pλ-1∥+∑j=0n-1‍∥AjPλ-1∥≤p(|λ|)emN|λ|σ,ℜλ>ω.
In what follows, we will use the following family of kernels. Define, for every l>0, the entire function ωl(·) by ωl(λ):=∏p=1∞‍(1+(lλ/ps)), λ∈ℂ, where s:=σ-1. Then it is clear that |ωl(λ)|≥supk∈ℕ∏p=1k‍|1+(lλ/ps)|≥supk∈ℕ∏p=1k‍l|λ|/ps≥supk∈ℕ(l|λ|)k/p!s, λ∈ℂ, ℜλ≥0. Hence, |ωl(λ)|≥eM(l|λ|), λ∈ℂ, ℜλ≥0, where M(λ):=supp∈ℕ0ln|λ|p/p!s, λ∈ℂ∖{0} and M(0):=0. It is also worth noting that, for every ζ∈(0,π/2), p∈ℕ0 and λ∈Σ(π/2)+ζ, we have |1+(lλ/ps)|≥l|ℑλ|/ps≥l(1+tanζ)-1|λ|/ps, and
(5.26)|ωl(λ)|≥eM(l(1+tanζ)-1|λ|),ζ∈(0,π2),l>0,λ∈Σ(π/2)+ζ.
Put now Kl(t):=ℒ-1(1/ωl(λ))(t), t≥0, l>0. Then, for every l>0, 0∈suppKl, Kl(0)=0 and Kl(t) is infinitely differentiable for t≥0. By Theorem 3.5(i)-(b) and (iii), we easily infer from (5.25) that there exists k>0 such that, for every l>k, there exists an exponentially bounded Kl-regularized I-resolvent family (El(t))t≥0 for (1.1), with Y=X=E. Furthermore, the mapping t↦El(t), t≥0 is infinitely differentiable in the uniform operator topology of L(E) and, for every compact set K⊆[0,∞) and for every l>k, there exists hK,l>0 such that
(5.27)supp≥0,t∈KhK,lp∥El(p)(t)∥p!s<∞.
One can similarly construct examples of exponentially bounded, analytic Kl-regularized I-resolvent families.



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