# (a,k)-Regularized C-Resolvent Families: Regularity and Local Properties

Authors: Marko Kostić
Journal: Abstract and Applied Analysis
 (2009)

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## Abstract

We introduce the class of (local)(a,k)-regularized C-resolvent families and
discuss its basic structural properties. In particular, our analysis covers subjects
like regularity, perturbations, duality, spectral properties and subordination
principles. We apply our results in the study of the backwards fractional
diffusion-wave equation and provide several illustrative examples of differentiable
(a,k)-regularized C-resolvent families.


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## Body

## 1. Introduction and Preliminaries

In this review, we will report how a large number of known results concerning(a,k)-regularized resolvents [1–6], C-regularized resolvents [7], and (local) convoluted C-semigroups and cosine functions [8, 9] can be formulated in the case of general (a,k)-regularized C-resolvent families.The paper is organized as follows. In Theorem2.2, Remark 2.3, and Theorems 2.5, 2.6, and 2.7, we analyze the properties of subgenerators of (a,k)-regularized C-resolvent families and slightly improve results from [1]. With a view to further study the problem describing heat conduction in materials with memory and the Rayleigh problem of viscoelasticity in L∞ type spaces, we prove in Theorem 2.8 several different forms of subordination principles [10]. The main objective in Theorems 2.9–2.12, 2.26, 2.28, and  2.32 is to continue the researches raised in [3] and [5, 6]. Our main contributions are Theorems 2.16–2.17, 2.20–2.25, 2.27, and 2.30 clarifying the basic regularity properties of (a,k)-regularized C-resolvent families and a fairly general form of the abstract Weierstrass formula.It is noteworthy that the complete spectral characterization of subgenerators of(a,k)-regularized C-resolvent families exists only in the exponential case and that it is not clear, with exception of various types of local convoluted C-semigroups and cosine functions [9, 11], in what way one can prove a satisfactory Hille-Yosida theorem for local (a,k)-regularized C-resolvent families.Throughout this paperE denotes a nontrivial complex Banach space, L(E) denotes the space of boundedlinear operators from E into E,E* denotes the dual space of E, and A denotes a closed linear operator acting on E. The range and the resolvent set of Aaredenoted by Rang(A) and ρ(A), respectively; [D(A)] denotes the Banach space D(A) equipped with the graph norm. From now on, we assume that L(E)∋C is an injective operator which satisfies CA⊆AC and employ the convolution like mapping * which is given by f*g(t):=∫0tf(t-s)g(s)ds.  Recall, the C-resolvent set of A, denoted by ρC(A), is defined to be the set of all complex numbers λ satisfying that the operator λ-A is injective and that Rang(C)⊆Rang(λ-A). Let us recall that a linear subspace Y⊆D(A) is called a core for A if Y is dense in D(A) with respect to the graph norm. Henceforth we identify a closed linear operator A with its graph G(A); given two closed linear operators A and B on E, the inclusion A⊆B means G(A)⊆G(B). If X is a closed subspace of E, then AX denotes the part of A in X, that is, AX:={(x,y)∈A:x∈X,y∈X}.We mainly use the following conditions.(H1):
A is densely defined.(H2):
ρ(A)≠∅.(H3):
ρC(A)≠∅ and Rang(C)¯=E.(H4):
Ais densely defined orρC(A)≠∅.(H5):
(H1)∨ (H2) ∨ (H3).(P1):
k(t) is Laplace transformable, that is, it is locally integrable on [0,∞) and there exists β∈ℝ so that k̃(λ)=ℒ(k)(λ):=limb→∞∫0be-λtk(t)dt:=∫0∞e-λtk(t)dt exists for all λ∈ℂ with Reλ>β. Put abs(k):=inf{Reλ:k̃(λ)exists}.Let us remind that a functionk∈Lloc1([0,τ)) is called a kernel, if for every ϕ∈C([0,τ)), the supposition ∫0tk(t-s)ϕ(s)ds=0,t∈[0,τ), implies ϕ≡0; due to the famous Titchmarsh's theorem [12], the condition 0∈ supp k implies that k(t) is a kernel. Set Θ(t):=∫0tk(s)ds,t∈[0,τ).


## 2.(a,k)-Regularized C-Resolvent Families

We start with the following definition.Definition 2.1.
Let0<τ≤∞,k∈C([0,τ)),k≠0, and let a∈Lloc1([0,τ)),a≠0. A strongly continuousoperator family (R(t))t∈[0,τ) is called a (local, if τ<∞) (a,k)-regularized C-resolventfamily having A as a subgeneratorif and only ifthe following holds:  (i)
R(t)A⊆AR(t),t∈[0,τ),R(0)=k(0)C, and CA⊆AC,(ii)
R(t)C=CR(t),t∈[0,τ),(iii)
R(t)x=k(t)Cx+∫0ta(t-s)AR(s)xds,t∈[0,τ),x∈D(A).In the case τ=∞,(R(t))t≥0 is said to be exponentially bounded if, additionally, there exist M>0 and ω≥0 such that ∥R(t)∥≤Meωt,t≥0;(R(t))t∈[0,τ) is said to be nondegenerate if the condition R(t)x=0,t∈[0,τ) implies x=0.
From now on, we consider only nondegenerate(a,k)-regularized C-resolvent families. Notice that (R(t))t∈[0,τ) is nondegenerate provided that k(0)≠0 or that (H5) holds for a subgenerator A of (R(t))t∈[0,τ).
In the casek(t)=tα/Γ(α+1), where α>0, and Γ(·) denotes the Gamma function, it is also said that (R(t))t∈[0,τ) is an α-times integrated (a,C)-resolvent family; in such a way, we unify the notion of (local) α-times integrated C-semigroups (a(t)≡1) and cosine functions (a(t)≡t) [1, 13, 14]. Furthermore, in the case k(t):=∫0tK(s)ds,t∈[0,τ), where K∈Lloc1([0,τ)) and K≠0, we obtain the unification concept for (local) K-convoluted C-semigroups and cosine functions [15]. In the case k(t)≡1,(R(t))t∈[0,τ) is said to be a (local) (a,C)-regularized resolvent family with a subgenerator A (cf. also [16] for the definition which does not include the condition (ii) of Definition 2.1).
Designate by℘(R) the set which consists of all subgenerators of (R(t))t∈[0,τ).
Then the following holds.
(i)
A∈℘(R) implies C-1AC∈℘(R).(ii)
IfA∈℘(R) and λ∈ρC(A), then
(2.1)R(t)(λ-A)-1C=(λ-A)-1CR(t),t∈[0,τ).(iii)
Assume, additionally, thata(t) is a kernel. Then one can define the integral generator Â of (R(t))t∈[0,τ) by setting
(2.2)Â:={(x,y)∈E×E:R(t)x-k(t)Cx=∫0ta(t-s)R(s)yds,t∈[0,τ)}.
The integral generator Â of (R(t))t∈[0,τ) is a closed linear operator which satisfies C-1ÂC=Â and extends an arbitrary subgenerator of (R(t))t∈[0,τ). Furthermore, Â∈℘(R), if R(t)R(s)=R(s)R(t),0≤t,s<τ.
Recall that in the case of convolutedC-semigroups and cosine functions, the set ℘(R) becomes a complete lattice under suitable algebraic operations and that induced partial ordering coincides with the usual set inclusion. In general, ℘(R) needs not to be finite [9].
Henceforth we assume that the scalar-valued kernelsk,k1,k2,… are continuous on [0,τ), and that a≠0 in Lloc1([0,τ)).
Assume temporarilyλ∈ρC(A),x∈Rang(C),t∈[0,τ), and put z=(a*R)(t)x.
Following the proof of [1, Lemma  2.2], we have z=λ(a*R)(t)(λ-A)-1x-(a*R)(t)A(λ-A)-1x=λ(a*R)(t)(λ-A)-1x-(R(t)(λ-A)-1x-k(t)C(λ-A)-1x)=λ(λ-A)-1C(a*R)(t)C-1x-((λ-A)-1R(t)x-k(t)(λ-A)-1Cx), where the last two equalities follow on account of CA⊆AC,R(s)A⊆AR(s) and R(s)(λ-A)-1C=(λ-A)-1CR(s),s∈[0,τ). Hence, (λ-A)z=λz-(R(t)x-Cx),
(2.3)∫0ta(t-s)R(s)xds∈D(A),A∫0ta(t-s)R(s)xds=R(t)x-k(t)Cx.
The closedness ofA implies that (2.3) holds for every t∈[0,τ) and x∈Rang(C)¯.Theorem 2.2 (see [1]).
(i) LetA be a subgenerator of an (a,k)-regularized C-resolvent family (R(t))t∈[0,τ), and let (H5) hold. Then (2.3)  holds for every t∈[0,τ) and x∈E. If ρC(A)≠∅, then (2.3)  holds for every t∈[0,τ) and x∈Rang(C)¯.
(ii) LetA be a subgenerator of an (a,ki)-regularized C-resolvent family (Ri(t))t∈[0,τ),i=1,2. Then (k2*R1)(t)=(k1*R2)(t),t∈[0,τ), whenever (H4) holds.
(iii) Let(R1(t))t∈[0,τ) and (R2(t))t∈[0,τ) be two (a,k)-regularized C-resolvent families having A as a subgenerator. Then R1(t)x=R2(t)x,t∈[0,τ),x∈D(A)¯, and R1(t)=R2(t),t∈[0,τ), if (H4) holds.
(iv) LetA be a subgenerator of an (a,k)-regularized C-resolvent family (R(t))t∈[0,τ). If k(t) is absolutely continuous and k(0)≠0, then A is a subgenerator of an (a,C)-regularized resolvent family on [0,τ).Remark 2.3.
(i) Let(Ri(t))t∈[0,τ) be an (a,ki)-regularized C-resolvent family with a subgenerator A,   i=1,2, and let D(A)≠{0}. Then k1=k2.
(ii) Let(Ri(t))t∈[0,τ) be an (a,ki)-regularized C-resolvent family with a subgenerator A,i=1,2. Then, for every α∈ℂ and β∈ℂ,(αR1(t)+βR2(t))t∈[0,τ) is an (a,αk1+βk2)-regularized C-resolvent family with a subgenerator A.
(iii) Let(R(t))t∈[0,τ) be an (a,k)-regularized C-resolvent family with a subgenerator A, and let Lloc1([0,τ))∋b be a kernel. Then A is a subgenerator of an (a,k*b)-regularized C-resolvent family ((b*R)(t))t∈[0,τ).
(iv) Let(R(t))t∈[0,τ) be an (a,C)-regularized resolvent family having A as a subgenerator. Then ((k*R)(t))t∈[0,τ) is an (a,Θ)-regularized C-resolvent family with a subgenerator A.
(v) Suppose(R(t))t∈[0,τ) is an (a,k)-regularized C-resolvent family with a subgenerator A, (H1) or (H3) holds, and a(t) is a kernel. Then the integral generator Â of (R(t))t∈[0,τ) satisfies Â=C-1AC. Toward this end, let (x,y)∈Â. Then ∫0ta(t-s)[k(s)Cx+∫0sa(s-r)R(r)ydr]ds=∫0ta(t-s)R(s)xds∈D(A),t∈[0,τ), and A∫0ta(t-s)[k(s)Cx+∫0sa(s-r)R(r)ydr]ds=A∫0ta(t-s)R(s)xds=R(t)x-k(t)Cx=∫0ta(t-s)R(s)yds,t∈[0,τ). Since (a*R)(t)y∈D(A),(a*a*R)(t)y∈D(A),A(a*a*R)(t)y=(a*(R-kC))(t)y,t∈[0,τ), and a*k≠0 in C([0,τ)), it follows that Cx∈D(A),ACx=Cy,x∈D(C-1AC), and C-1ACx=Âx=y. On the other hand, C-1AC is a subgenerator of (R(t))t∈[0,τ) whenever A is; this implies C-1AC⊆Â and proves the claim. If (H2) holds, then Â=C-1AC=A. In what follows, we also assume that B∈℘(R) and that (H5) holds for B and C. Proceeding as in the proof of [9, Proposition  2.1.1.6], one gets what follows.
(v.1)
C-1AC=C-1BC and C(D(A))⊆D(B).(v.2)
A and B have the same eigenvalues.(v.3)
The assumptionA⊆B implies ρC(A)⊆ρC(B).(v.4)
card(℘(R))=1, if C(D(Â)) is a core for D(Â).(v.5)
A⊆B⇔D(A)⊆D(B) and Ax=Bx,x∈D(A)∩D(B); furthermore, the property (v.5) holds whenever {A,B}⊆℘(R) and a(t) is a kernel.
We refer the reader to [1, page 283] for the definition of (weak) solutions of the problem
(2.4)u(t)=f(t)+∫0ta(t-s)Au(s)ds,t∈[0,τ),
where f∈C([0,τ):E), and to [1, page 285] for the notion of spaces Cn,k([0,τ):E),n∈ℕ,k∈ℕ0 and C0n([0,τ):E),andn∈ℕ.
Define a subsetA* of E*×E* (the use of symbol * is clear from the context) by A*:={(x*,y*)∈E*×E*:x*(Ax)=y*(x)for allx∈D(A)}. In the case when A is densely defined, A* is a linear mapping from E* into E*.Lemma 2.4 (see [17]).
LetA be a closed linear operator. Assume x0∈E,y0∈E, and x*(y0)=y*(x0) for all (x*,y*)∈A*. Then x0∈D(A), and Ax0=y0.Define the mappingKC:C([0,τ):E)→C([0,τ):E) by KCu:=k*Cu,u∈C([0,τ):E). Then KC is linear, bounded, and injective.Keeping in mind Lemma2.4 and the proofs of [1, Theorem  2.7, Corollary  2.9, Remark  2.10, Corollary  2.11, and Corollary   2.13], we have the following.Theorem 2.5.
(i) Supposef∈C([0,τ):E),A is a subgenerator of a (local) (a,k)-regularized C-resolvent family (R(t))t∈[0,τ), and (H5) holds. Then (2.4)  has a unique solution if and only if R*f∈Rang(KC).
(ii) (cf. also [18])  Assume n∈ℕ,f∈C([0,τ):E),A is a subgenerator of a (local) n-times integrated (a,C)-resolvent family (R(t))t∈[0,τ), and (H5) holds. Then (2.4)  has a unique solution if and only if C-1(R*f)∈C0n+1([0,τ):E).
(iii) Let the assumptions of the item (i) of this theorem hold, and letk∈C0n([0,τ):E). Then C-1(R*f)∈C(n+1)([0,τ):E) if and only if C-1(R*f)∈C0n+1([0,τ):E).
(iv) Let (H5) hold. Assume thatn∈ℕ,A is a subgenerator of an n-times integrated (a,C)-regularized resolvent, and a∈BVloc([0,τ):E), respectively  A is a subgenerator of an (a,C)-regularized resolvent family. Assume, further, that C-1f∈C(n+1)([0,τ):E),f(k-1)(0)∈D(An+1-k) and An+1-kf(k-1)(0)∈Rang(C),1≤k≤n+1,  respectively  C-1f∈ACloc([0,τ):E). Then (2.4) has a unique solution.
(v) Assume that (H5) holds,A is a subgenerator of an (a,k)-regularized C-resolvent family, k(t) is absolutely continuous, and k(0)≠0. If  C-1f∈C1([0,τ):E), then there exists a unique solution of (2.4).The proof of following theorem follows from a standard application of Laplace transform techniques.Theorem 2.6.
Letk(t) and a(t) satisfy (P1), and let (R(t))t≥0 be a strongly continuous operator family satisfying ∥R(t)∥≤Meωt,t≥0, for some M>0 and ω≥0. Put ω0:=max(ω,abs(a),abs(k)).(i)
SupposeA is a subgenerator of the exponentially bounded (a,k)-regularized C-resolvent family (R(t))t≥0,  and (H5) holds. Then, for every λ∈ℂ with  Reλ>ω0  and k̃(λ)≠0, the operator I-ã(λ)A is injective,  Rang(C)⊆Rang(I-ã(λ)A),(2.5)k̃(λ)(I-ã(λ)A)-1Cx=∫0∞e-λtR(t)xdt,x∈E,Reλ>ω0,k̃(λ)≠0,(2.6){1ã(λ):λ∈ℂ,Reλ>ω0,k̃(λ)ã(λ)≠0}⊆ρC(A)
and R(s)R(t)=R(t)R(s),t,s≥0.(ii)
Assume that (2.5)-(2.6)  hold. Then A is a subgenerator of the exponentially bounded (a,k)-regularized C-resolvent family (R(t))t≥0.The preceding theorem enables one to establish the real and complex characterization of subgenerators of (locally Lipschitz continuous) exponentially bounded(a,k)-regularized C-resolvent families [1, 9, 12]:Theorem 2.7.
(i) Letk(t) and a(t) satisfy (P1), and let ω0≥max(0,abs(a),abs(k)).  Assume that, for every λ∈ℂ with Reλ>ω0  and k̃(λ)≠0, the operator I-ã(λ)A is injective and that Rang(C)⊆Rang(I-ã(λ)A).  If there exists an analytic function Υ:{λ∈ℂ:Reλ>ω0}→L(E)  with:
(i.1)
Υ(λ)=k̃(λ)(I-ã(λ)A)-1C,λ∈ℂ,Reλ>ω0,(i.2)
∥Υ(λ)∥≤M|λ|r,λ∈ℂ,Reλ>ω0,  for some M>0 and r≥-1, then, for every α>1,A is a subgenerator of a norm continuous, exponentially bounded (a,k*tα+r-1/Γ(α+r))-regularized C-resolvent family.
(ii) Supposek(t) and a(t) satisfy (P1) and (H2) or (H3) holds, and A is a subgenerator of an exponentially bounded (a,Θ)-regularized C-resolvent family (R(t))t≥0 which satisfies the next condition:
(2.7)∥R(t+h)-R(t)∥≤Mheω(t+h),t≥0,h≥0,forsomeM>0,ω≥0.
Then there exists a≥ω0 such that
(2.8){1ã(λ):λ>a,k̃(λ)ã(λ)≠0}⊆ρC(A),(2.9)themappingλ↦H(λ):=k̃(λ)(I-ã(λ)A)-1C,λ>a,k̃(λ)ã(λ)≠0
is infinitely differentiable and
(2.10)∥dkdλkH(λ)∥≤Mk!(λ-ω)k+1,k∈ℕ0,λ>a,k̃(λ)ã(λ)≠0.
(iii) Supposek(t) and a(t) satisfy (P1)  and (2.8)–(2.10) holds. Then A is a subgenerator of an exponentially bounded (a,Θ)-regularized C-resolvent family (R(t))t≥0 which satisfies (2.7).
(iv) SupposeM>0,  ω≥0,k(t) and a(t) satisfy (P1),  and A is densely defined. Then A is a subgenerator of an exponentially bounded (a,k)-regularized C-resolvent family (R(t))t≥0 which satisfies ∥R(t)∥≤Meωt,t≥0 if and only if there exists a≥max(0,abs(a),abs(k))  such that (2.8)–(2.10) hold.Denote bya*n the nth convolution power of the kernel a(t),n∈ℕ, and see [10] for the definition of completely positive functions and the notion used in the subsequent theorem and examples. An insignificant technical modification of the proofs of [1, Theorem  3.7] and [10, Theorems  4.1,  4.3,  4.5] (cf. also [7, Lemma  4.2]) implies the next subordination principles.Theorem 2.8.
(i) Leta(t),b(t), and c(t) satisfy (P1), and let ∫0∞e-βt|b(t)|dt<∞ for some β≥0. Let
(2.11)α=c̃-1(1β)if∫0∞c(t)dt>1β,α=0otherwise,
and let ã(λ)=b̃(1/c̃(λ)),λ≥α. Let A be a subgenerator of a (b,k)-regularized C-resolvent family (Rb(t))t≥0 satisfying ∥Rb(t)∥≤Meωbt,t≥0, for some M>0 and ωb≥0, and let (H2)  or  (H3) hold. Assume, further, that c(t) is completely positive and that there exists a function k1(t) satisfying (P1)  and
(2.12)k̃1(λ)=1λc̃(λ)k̃(1λc̃(λ)),λ>ω0,k̃(1λc̃(λ))≠0,forsomeω0>0.
Let(2.13)ωa=c̃-1(1ωb)if∫0∞c(t)dt>1ωb,ωa=0otherwise.
Then A is a subgenerator of an exponentially bounded, locally Lipschitz continuous (a,1*k1)-regularized C-resolvent (Ra(t))t≥0, and there exists Ma≥1 such that
(2.14)∥Sa(t)∥≤Maeωat,t≥0,ifωb=0orωbc̃(0)≠1,
respectively, for every ε>0, there exists Mε≥1 such that
(2.15)∥Sa(t)∥≤Mεeεt,t≥0,ifωb>0,ωbc̃(0)=1.
Furthermore, if A is densely defined, then A is a subgenerator of an exponentially bounded (a,k1)-regularized C-resolvent (Ra(t))t≥0 which fulfills (2.14), respectively, (2.15).
(ii) Supposeα≥0,A is a subgenerator of an exponentially bounded α-times integrated C-semigroup, a(t) is completely positive and satisfies (P1),  and k(t) satisfies (P1)  and k̃(λ)=ã(λ)α,λ sufficiently large. Then A is a subgenerator of a locally Lipschitz continuous, exponentially bounded (a,t*k)-regularized C-resolvent family ((a,t*a*n)-regularized C-resolvent family if α=n∈ℕ, respectively, (a,t)-regularized C-resolvent family if α=0). If, additionally, A is densely defined, then A is a subgenerator of an exponentially bounded (a,1*k)-regularized C-resolvent family ((a,1*a*n)-regularized C-resolvent family if α=n∈ℕ, respectively, (a,C)-regularized resolvent family if α=0).
(iii) Supposeα≥0 and A is a subgenerator of an exponentially bounded α-times integrated C-cosine function. Let  Lloc1([0,∞))∋c be completely positive, and let a(t)=(c*c)(t),t≥0. (Given Lloc1([0,∞))∋a in advance, such a function c(t) always exists provided a(t) is completely positive or a(t)≠0 is a creep function and a1(t) is log-convex.) Assume that k(t) satisfies (P1), and k̃(λ)=c̃(λ)α/λ,λ sufficiently large. Then A is a subgenerator of a locally Lipschitz continuous, exponentially bounded (a,t*k)-regularized C-resolvent family ((a,t*c*n)-regularized C-resolvent family if α=n∈ℕ, resp. (a,t)-regularized C-resolvent family if α=0). If, additionally, A is densely defined, then A is a subgenerator of an exponentially bounded (a,1*k)-regularized C-resolvent family ((a,1*c*n)-regularized C-resolvent family if α=n∈ℕ, resp. (a,C)-regularized resolvent family if α=0).Denote byAp the realization of the Laplacian with Dirichlet or Neumann boundary conditions on Lp([0,π]n),1≤p<∞. By [19, Theorem  4.2], Ap generates an exponentially bounded α-times integrated cosine function for every α≥(n-1)|(1/2)-(1/p)|. Assume further that c∈BVloc([0,∞)) and that m(t) is a bounded creep function with m0=m(0+)>0. Thanks to [10, Proposition   4.4, page 94], we have that there exists a completely positive function b(t) such that dm*b=1. After the usual procedure, the problem [10, (5.34)] describing heat conduction in materials with memory is equivalent to(2.16)u(t)=(a*Ap)(t)+f(t),t≥0,
where a(t)=(b*dc)(t),t≥0, and f(t) contains r*b as well as the temperature history. In what follows, we assume that(i)
p≠2,(ii)
Γb=∅ or Γf=∅,(iii)
there exists a completely positive functionc1(t) such that a(t)=(c1*c1)(t),t≥0.We refer the reader to [10, pages 140-141] for the analysis of the problem (2.16) in the case: p=2 and m,c∈ℬℱ. Applying Theorem 2.8(iii), one gets that Ap is the integral generator of an exponentially bounded (a,1*ℒ-1(c̃1(λ)(n-1)|(1/2)-(1/p)|/λ)(t))-regularized resolvent family, where ℒ-1 denotes the inverse Laplace transform. Notice also that [10, Lemma  4.3, page 105] implies that, for every β∈[0,1], the function λ↦(c̃1(λ)β/λ) is the Laplace transform of a Bernstein function and that the function k(t) appearing in the formulations of Theorem 2.8(ii)-(iii) always exists. On the other hand, an application of [9, Proposition  2.1.3.12] gives that there exists ω>0 such that Ap is the integral generator of an exponentially bounded (ω-Ap)-⌈(1/2)(n-1)|(1/2)-(1/p)|⌉-regularized cosine function; herein ⌈s⌉=inf{k∈ℤ:s≤k},s∈ℝ. Using Theorem 2.8(iii) again, we have that Ap is the integral generator of an exponentially bounded (a,(ω-Ap)-⌈(1/2)(n-1)|(1/2)-(1/p)|⌉)-regularized resolvent family, and Theorem 2.5(iv) can be applied. In both approaches, regrettably, we must restrict ourselves to the study of pure Dirichlet or Neumann problem. It is also worthwhile to note that Theorem 2.8(iii) can be applied in the analysis of the Rayleigh problem of viscoelasticity in L∞ type spaces; as a matter of fact, the operator A defined on [10, page 136] generates an exponentially bounded α-times integrated cosine function in L∞((0,∞)) for all α>0.Approximation type theorem for exponentially bounded(a,k)-regularized C-resolvent families follows from Theorem 2.6 and [12, Theorem  1.7.5, page 42], and the representation formulae for exponentially bounded (a,k)-regularized C-resolvent families are consequences of the Post-Widder inversion (the Phragamén-Doetsch inversion). For further information, see [2, 12, 20, 21].Using the argumentation given in [3, 5], one can prove the following assertions.Theorem 2.9.
(i) Suppose that the next conditions hold.(i.1)
The mappingt↦|k(t)|,t∈[0,τ), is nondecreasing.(i.2)
There existεa,k>0 and ta,k∈[0,τ) such that
(2.17)|∫0ta(t-s)k(s)ds|≥εa,k∫0t|a(t-s)k(s)|ds,t∈[0,ta,k).(i.3)
A is a subgenerator of an (a,k)-regularized C-resolvent family(R(t))t∈[0,τ), and (H5)  holds.(i.4)
lim supt→0+∥R(t)∥/|k(t)|<∞.Then
(2.18)D(AD(A)¯)={x∈D(A)¯:limt→0+R(t)x-k(t)Cx(a*k)(t)exists},(2.19)Ax=limt→0+R(t)x-k(t)Cx(a*k)(t),x∈D(AD(A)¯).
(ii) SupposeA is a subgenerator of an (a,k)-regularized C-resolvent family (R(t))t∈[0,τ) satisfying ∥R(t)∥=O(k(t)),t→0+,min(a(t),k(t))>0,t∈(0,τ), and (H5) holds. Then (2.18)-(2.19) hold.Theorem 2.10.
(i) SupposeA is a subgenerator of an (a,k)-regularized C-resolvent family (R(t))t∈[0,τ) satisfying ∥R(t)∥=O(k(t)),t→0+ and min(a(t),k(t))>0,t∈(0,τ). Then limt→0+(a*R)(t)x/(a*k)(t)=Cx,x∈D(A)¯.
(ii) SupposeA is a subgenerator of an (a,k)-regularized C-resolvent family (R(t))t∈[0,τ) satisfying ∥R(t)∥=O(k(t)),t→0+,min(a(t),k(t))>0,t∈(0,τ) and (H5) holds. If x∈D(A)¯,y∈E and limt→0+(R(t)x-k(t)Cx)/(a*k)(t)=y, then x∈D(A) and y=Ax.
(iii) SupposeE is reflexive, A is a subgenerator of an (a,k)-regularized C-resolvent family (R(t))t∈[0,τ) satisfying ∥R(t)∥=O(k(t)),t→0+,R(s)R(t)=R(t)R(s),0≤t,s<τ,min(a(t),k(t))>0,t∈(0,τ), and (H5) holds. If x∈D(A)¯ and limt→0+∥(R(t)x-k(t)Cx)/(a*k)(t)∥<∞, then x∈D(A).Theorem 2.11 (cf. also [22]).
Supposeα>0 and A is a subgenerator of an α-times integrated C-semigroup (Sα(t))t∈[0,τ),  respectively, an α-times integrated C-cosine function (Cα(t))t∈[0,τ), which satisfies lim supt→0+∥Sα(t)∥/tα<∞, respectively,  lim supt→0+∥Cα(t)∥/tα<∞. Then, for every x∈D(A) such that Ax∈D(A)¯:(2.20)Ax=limt→0+Γ(α+2)Γ(α+1)Γ(α+1)Sα(t)x-tαCxtα+1(resp.),(2.21)Ax=limt→0+Γ(α+3)Γ(α+1)Γ(α+1)Cα(t)x-tαCxtα+2.Theorem 2.12.
SupposeM>0,ω≥0,A is a densely defined subgenerator of an (a,k)-regularized C-resolvent family (R(t))t≥0 which satisfies ∥R(t)∥≤Meωt,t≥0;B∈L(E),Rang(B)⊆Rang(C), and BCx=CBx,x∈D(A).       Suppose, further, that there exist a function b(t) satisfying (P1) and a number ω0≥ω such that b̃(λ)=ã(λ)/k̃(λ),λ>ω0,k̃(λ)≠0. Then the operator A+B is a subgenerator of an(a,k)-regularized C-resolvent family (RB(t))t≥0 which satisfies ∥RB(t)∥≤M/1-γeμt,t≥0,
(2.22)RB(t)=R(t)+∫0tRB(t-r)C-1B∫0rb(r-s)R(s)xdsdr,t≥0,x∈D(A).Remark 2.13.
In order to prove Theorem2.9(i) and (2.20)-(2.21) in the case of nondensely subgenerators, it is enough to notice that [3, (2.1), page 219] holds for every z∈D(A)¯ and that [3, (2.2), page 219] holds for every x∈D(A) such that Ax∈D(A)¯. On the other hand, if (2.20), resp. (2.21), holds for some x∈D(A), then it is obvious that Ax∈D(A)¯. This implies that the representation formulae (2.20) and (2.21) are best possible in some sense.
Givenα∈(0,π], set Σα:={λ∈ℂ:λ≠0,|argλ|<α}.Definition 2.14 (cf. also [23, Definition  5.1]).
Let0<α≤π/2, 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 α, if there exists an analytic function R:Σα→L(E) which satisfies(i)
R(t)=R(t),t>0,(ii)
limz→0,z∈ΣγR(z)x=k(0)Cx for all γ∈(0,α) and x∈E.
It is said that(R(t))t≥0 is an exponentially bounded, analytic (a,k)-regularized C-resolvent family, respectively, bounded analytic (a,k)-regularized C-resolvent family, of angle α, if for every γ∈(0,α), there exist Mγ>0 and ωγ≥0, resp. ωγ=0, such that ∥R(z)∥≤MγeωγRez,z∈Σγ.Since no confusion seems likely, we also writeR(·) for R(·).The next proposition can be proved by means of the arguments given in [7, Section  3] and [10, Chapter 2].Proposition 2.15.
Supposek(t) and a(t)satisfy (P1), limλ→+∞λk̃(λ)=k(0)≠0,A is densely bounded,  A∉L(E), and there exists ω0≥max(0,abs(k),abs(a)) such that  ∫0∞e-ωt|a(t)|dt<∞. Assume that A is a subgenerator of an exponentially bounded, analytic (a,k)-regularized C-resolvent family (R(t))t≥0 of angle α∈(0,π/2] and that there exists ω≥ω0 such that
(2.23)supz∈Σγ∥e-ωzR(z)∥<∞,∀γ∈(0,α).
Then the function ã(λ) can be extended to a meromorphic function defined on the sector ω+Σπ/2+α.It is worthwhile to mention that it is not clear, all assumptions of Proposition2.15 being satisfied, whether A must be a subgenerator of an (a,C)-regularized resolvent family on [0,τ) (cf. Theorem  2.2(iv). Further on, let us notice that the assertions (i) and (ii) of [10, Theorem  2.2, page 57] still hold in the case of exponentially bounded, analytic (a,C)-regularized resolvent families.The subsequent theorem clarifies the basic analytical properties of(a,k)-regularized C-resolvent families. Notice only that the assertion which naturally corresponds to [7, Lemma  3.7] (cf. also [10, Corollary   2.2, page 53]) does not seem attainable in the case of a general (a,k)-regularized C-resolvent family.Theorem 2.16.
Supposeα∈(0,π/2],k(t)  and a(t) satisfy (P1), (H5) holds, and k̃(λ)can be analytically continued to a function g:ω+Σπ/2+α→ℂ,where ω≥max(0,abs(k),abs(a)). Suppose, further, that A is a subgenerator of an analytic (a,k)-regularized C-resolvent family (R(t))t≥0 of angle α and that (2.23)  holds. Set
(2.24)N:={λ∈ω+Σπ/2+α:g(λ)≠0}.
Then N is an open connected subset of ℂ. Assume that there exists an analytic function â:N→ℂ such that â(λ)=ã(λ),λ∈ℂ,Reλ>ω. Then the operatorI-â(λ)Ais injective for every λ∈N,Rang(C)⊆Rang(I-â(λ)C-1AC) for every λ∈N1:={λ∈N:â(λ)≠0},(2.25)supλ∈N1∩(ω+Σπ/2+γ1)∥(λ-ω)g(λ)(I-â(λ)C-1AC)-1C∥<∞,γ1∈(0,α),(2.26)themappingλ↦(I-â(λ)C-1AC)-1C,λ∈N1isanalytic,(2.27)limλ→+∞,k̃(λ)≠0λk̃(λ)(I-ã(λ)A)-1Cx=k(0)Cx,x∈E.Proof.
By Theorem2.6(i), it follows that, for every λ∈ℂ with Reλ>ω, the operator I-ã(λ)Ais injective  and that Rang(C)⊆Rang(I-ã(λ)A). Since (2.23) holds, one yields that the function q:{λ∈ℂ:Reλ>ω}→L(E) given by q(λ)=∫0∞e-λtR(t)dt,λ∈ℂ,Reλ>ω has an analytic extension q̃(λ):ω+Σπ/2+α→L(E) such that supλ∈ω+Σπ/2+γ∥(λ-ω)q̃(λ)∥<∞ for all γ∈(0,α) [12]. The set N is open and connected ([9], Subsection  2.1.4), and clearly, the mapping F(λ):=q̃(λ)/g(λ),λ∈N, is analytic. Denote by V the set which consists of all complex numbers λ∈N such that I-â(λ)A is injective, Rang(C)⊆Rang(I-â(λ)A), and F(λ)=(I-â(λ)A)-1C. Let ρC(A)∋μ satisfy â(μ)≠0. Then
(2.28)F(λ)(I-â(λ)A)x=Cx,λ∈V,x∈D(A),(2.29)F(λ)Cy=CF(λ)y,λ∈V,y∈E,(2.30)F(λ)Cy=1â(λ)(1â(λ)-A)-1C2y=1â(λ)(1â(μ)-A)-1C2y-(1â(λ)-1â(μ))(1â(μ)-A)-1CF(λ)y,λ∈V,â(λ)≠0,
and the uniqueness theorem for analytic functions implies that (2.28)-(2.29) hold for every λ∈N and that (2.30) holds for every λ∈N such that â(λ)≠0. Let (I-â(λ)A)x=0 for some λ∈N and x∈D(A). Thanks to (2.28), Cx=0,x=0, and I-â(λ)A is injective. Assume, for the time being, λ∈N and â(λ)≠0. Then (2.29)-(2.30) hold, and one gets (I-â(λ)A)CF(λ)y=(I-â(λ)A)F(λ)Cy=C2y-(1/â(λ)-1/â(μ))(1/â(μ)-A)-1CF(λ)y+â(λ)(1/â(λ)-1/â(μ))[-CF(λ)y+1/â(μ)(1/â(μ)-A)-1CF(λ)y]-1/â(μ)(1/â(μ)-A)-1C2y. Then (2.30) implies (I-â(λ)A)CF(λ)y=C2y-1/â(μ)(1/â(μ)-A)-1C2y+1/â(μ)(1/â(μ)-A)-1CF(λ)y-â(λ)/â(μ)[-CF(λ)y+1/â(μ)(1/â(μ)-A)-1CF(λ)y]:=C2y+Rλ,μ. Clearly, Rλ,μ=0 if and only if C2y-CF(λ)y-â(λ)(1/â(μ)-A)CF(λ)y+(â(λ)/â(μ))CF(λ)y=0. In order to see that the last equality is true, one can again apply (2.30). Thereby, (I-â(λ)A)CF(λ)y=C2y,λ∈N,â(λ)≠0, and as an outcome, we obtain that the operator I-â(λ)A is injective for all λ∈N and that Rang(C)⊆Rang(I-â(λ)C-1AC) for all λ∈N with â(λ)≠0, as required. The estimates (2.25)–(2.27) follow by using the argumentation given in the proof of [9, Theorem  2.1.4.4].Vice versa, we have the following theorem which can be proved as in the case of convolutedC-semigroups [9].Theorem 2.17.
Assumek(t) and a(t) satisfy (P1), ω≥max(0,abs(k),abs(a))  and α∈(0,π/2]. Assume, further, that A is a closed linear operator and that, for every λ∈ℂ with Reλ>ω  and k̃(λ)≠0, we have that the operator I-ã(λ)A is injective and that Rang(C)⊆Rang(I-ã(λ)A).  If there exists an analytic function q:ω+Σπ/2+α→L(E)such that
(2.31)q(λ)=k̃(λ)(I-ã(λ)A)-1C,λ∈ℂ,Reλ>ω,k̃(λ)≠0,(2.32)supλ∈ω+Σπ/2+γ∥(λ-ω)q(λ)∥<∞,∀γ∈(0,α),(2.33)limλ→+∞λq(λ)x=k(0)Cx,x∈E,ifD(A)¯≠E,
then A is a subgenerator of an exponentially bounded, analytic (a,k)-regularized C-resolvent family of angle α.Example 2.18 (cf. also [9, Theorem  2.1.4.7]).
Letβ∈(0,2),α>0,k(t)=tα/Γ(α+1), and let a(t)=tβ-1/Γ(β). Let A be densely defined. Then A is a subgenerator of an exponentially bounded, analytic (a,k)-regularized C-resolvent family of angle γ if and only if for every δ∈(0,γ), there exist Mδ>0 and ωδ≥0 such that
(2.34)(ωδ+Σπ/2+δ)1/β⊆ρC(A),∥(λβ-A)-1C∥≤Mδ(1+|λ|)α-β,λ∈(ωδ+Σπ/2+δ)1/β,
the mapping λ↦(λβ-A)-1C,λ∈(ωδ+Σπ/2+δ)1/β is analytic (continuous).
Let(Mp) be a sequence of positive real numbers such that M0=1 and that
(M.1)
Mp2≤Mp+1Mp-1,p∈ℕ,(M.2)
Mn≤AHnminp,q∈ℕ,p+q=nMpMq,n∈ℕ,for someA>1,andH>1,(M.3)
∑p=1∞(Mp-1/Mp)<∞.
The Gevrey sequences(p!s),(pps), and (Γ(1+ps)) satisfy the above conditions, where s>1. Put mp:=(Mp/Mp-1),p∈ℕ; by (M.1), (mp) is increasing, and (M.3)′ implies ∑p=1∞(1/mp)<∞. The associated function of (Mp) is defined by M(λ):=supp∈ℕ0ln(|λ|p/Mp),λ∈ℂ∖{0},M(0):=0. As is known, the function t↦M(t),t≥0, is increasing, absolutely continuous, limt→∞M(t)=+∞ and limt→∞(M(t)/t)=0. For consistency of terminology with [24], we also employ the sequence (Lp:=Mp1/p) and set ωL(t):=∑p=0∞(tp/Lpp),t≥0.
We need the following family of kernels. Define, for everyl>0, the next entire function of exponential type zero ωl(λ):=∏p=1∞1+lλ/mp,λ∈ℂ. Then
(2.35)|ωl(λ)|≥supk∈ℕ∏p=1k|1+(lλ/mp)|≥supk∈ℕ∏p=1kl|λ|mp≥supk∈ℕ(l|λ|)kMp,λ∈ℂ,Reλ≥0,
and this implies that |ωl(λ)|≥eM(l|λ|),λ∈ℂ,Reλ≥0. It is noteworthy that, for every α∈(0,π/2),p∈ℕ0 and λ∈Σπ/2+α,|1+lλ/mp|≥l|Imλ|/mp≥l(1+tanα)-1|λ|/mp.
This yields
(2.36)|ωl(λ)|≥eM(l(1+tanα)-1|λ|),α∈(0,π2),l>0,λ∈Σ(π/2)+α.
Put now
(2.37)kl(t):=ℒ-1(1ωl(λ))(t),t≥0,l>0.
Then, for every l>0,0∈suppkl and kl is infinitely differentiable in t≥0.Definition 2.19.
Let(R(t))t∈[0,τ) be a (local) (a,k)-regularized C-resolvent family having A as a subgenerator, and let the mapping t↦R(t),t∈(0,τ), be infinitely differentiable (in the uniform operator topology). Then it is said that (R(t))t∈[0,τ) is of class CL, resp. of class CL, if and only if for every compact set K⊆(0,τ) there exists hK>0, resp. for every compact set K⊆(0,τ) and for every h>0:
(2.38)supt∈K,p∈ℕ0∥hKp(dp/dtp)R(t)Lpp∥<∞,(resp.,)supt∈K,p∈ℕ0∥hp(dp/dtp)R(t)Lpp∥<∞,(R(t))t∈[0,τ) is said to be ρ-hypoanalytic,1≤ρ<∞, if (R(t))t∈[0,τ) is of class CL with Lp=p!ρ/p.By the proof of the scalar-valued version of the Pringsheim theorem, it follows that the mappingt↦R(t),t∈(0,τ) is real analytic if and only if (R(t))t∈[0,τ) is ρ-hypoanalytic with ρ=1.The main objective in Theorems 2.20–2.24 is to enquire into the basic differential properties of(a,k)-regularized C-resolvent families.Theorem 2.20 ([25]).
SupposeA is a closed linear operator, k(t) and a(t) satisfy (P1), r≥-1,  and there exists ω≥max(0,abs(k),abs(a)) such that, for every z∈{λ∈ℂ:Reλ>ω,k̃(λ)≠0},  we have that the operator I-ã(z)A is injective and that Rang(C)⊆Rang(I-ã(z)A).  If, additionally, for every σ>0, there exist Cσ>0,Mσ>0 and an open neighborhood Ωσ,ω of the region
(2.39)Λσ,ω:={λ∈ℂ:Reλ≤ω,Reλ≥-σln|Imλ|+Cσ}∪{λ∈ℂ:Reλ≥ω},
and an analytic mapping hσ:Ωσ,ω→L(E) such that hσ(λ)=k̃(λ)(I-ã(λ)A)-1C,Reλ>ω,k̃(λ)≠0  and that ∥hσ(λ)∥≤Mσ|λ|rλ∈Λσ,ω, then, for every ζ>1,A is a subgenerator of a norm continuous, exponentially bounded (a,k*tζ+r-1/Γ(ζ+r))-regularized C-resolvent family (R(t))t≥0 satisfying that the mapping t↦R(t),t>0 is infinitely differentiable.Theorem 2.21.
Supposek(t) and a(t) satisfy (P1), (H5) hold and A is a subgenerator of an (a,k)-regularized C-resolvent family (R(t))t≥0 satisfying ∥R(t)∥≤Meω′t,t≥0 for appropriate constants ω′≥max(0,abs(k),abs(a)),  and M>0. If there exists ω>ω′ such that, for every σ>0, there exist Cσ>0 and Mσ>0 so that  (i)
there exist an open neighborhoodΩσ,ω of the region Λσ,ω, and the analytic mappings fσ:Ωσ,ω→ℂ,gσ:Ωσ,ω→ℂ, and hσ:Ωσ,ω→L(E) such that fσ(λ)=k̃(λ),λ∈ℂ,Reλ≥ω  and gσ(λ)=ã(λ),λ∈ℂ,Reλ≥ω,(ii)
for everyλ∈Λσ,ω with Reλ≤ω,  the operator I-ã(λ)A is injective and Rang(C)⊆Rang(I-ã(λ)A),(iii)
hσ(λ)=fσ(λ)(I-gσ(λ)A)-1C,λ∈Λσ,ω,(iv)
∥hσ(λ)∥≤Mσ|Imλ|,λ∈Λσ,ω,Reλ≤ω,  and max(|fσ(λ)|,|gσ(λ)|)≤Mσ,λ∈Λσ,ω,then the mapping t↦R(t)x,t>0 is infinitely differentiable for every fixed x∈D(A2). Furthermore, if D(A2) is dense in E, then the mapping t↦R(t),t>0, is infinitely differentiable.Proof.
Assumeσ>0,ς>0,ω0>ω, and put Γσ1:={λ∈ℂ:Reλ=2Cσ-σln(-Imλ),-∞<Imλ≤-e(2Cσ/σ)},Γσ2:={λ∈ℂ:Reλ=ω0,-e(2Cσ/σ)≤Imλ≤e(2Cσ/σ)},Γσ3:={λ∈ℂ:Reλ=2Cσ-σln(Imλ),e(2Cσ/σ)≤Imλ<+∞},Γσ:=Γσ1∪Γσ2∪Γσ3, and Γk,σ:={λ∈Γσ:|λ|≤k},k∈ℕ. The curves Γσ and Γk,σ are oriented so that Imλ increases along Γσ and Γk,σ,k∈ℕ.  Set, for a sufficiently large k0∈ℕ,Sσk(t):=(1/2πi)∫Γk,σeλt(hσ(λ)/λ2)dλ,t≥0,k≥k0. One can simply prove that (dj/dtj)Sσk(t)=(1/2πi)∫Γk,σeλtλj-2hσ(λ)dλ,t≥0,k≥k0,j∈ℕ. Let k0<k<l. Then (iv) implies
(2.40)∥djdtjSσk(t)-djdtjSσl(t)∥=12π∥∫Γl,σ∩{λ∈ℂ:k≤|λ|≤l}eλtλj-2hσ(λ)dλ∥≤Mσ2πe2Cσt∫Γl,σ∩{λ∈ℂ:k≤|λ|≤l}|Imλ|1-σt|λ|j-2|dλ|,
for all j∈ℕ0. Since |Imλ|1-σt|λ|j-2~|Imλ|j-1-σt,|λ|→∞,λ∈Γσ, one gets that, for every j∈ℕ0 and t>j/σ, the sequence ((dj/dtj)Sσk(t))k is convergent in L(E) and that the convergence is uniform on every compact subset of [j/σ+ς,∞). Put Sj,σ(t):=limk→∞(dj/dtj)Sσk(t),j∈ℕ0,t>j/σ. Then it is obvious that (d/dt)Sj,σ(t)=S(j+1),σ(t),j∈ℕ0,t>(j+1)/σ+ς. This implies that the mapping t↦S0,σ(t),t>(j+1/σ)+ς is j-times differentiable and that (dj/dtj)S0,σ(t)=Sj,σ(t),t>(j+1)/σ)+ς. On the other hand, it is clear that, for every σ>0,x∈D(A2), and λ∈{z∈Ωσ,ω:gσ(z)≠0},(2.41)(I-gσ(λ)A)-1Cx=Cx+gσ(λ)CAx+gσ(λ)(I-gσ(λ)A)-1CA2x.
By (2.41), we get that, for every x∈D(A2) and t>0,(2.42)S0,σ(t)x=12πi∫Γσeλtfσ(λ)Cx+gσ(λ)CAx+gσ(λ)(I-gσ(λ)A)-1CA2xλ2dλ.
With (iv) and the residue theorem in view, it follows that, for every t>0 and x∈D(A2),(2.43)S0,σ(t)x=∫0t(t-s)k(s)dsCx+(t*k*a)(t)CAx+12πi∫ω-i∞ω+i∞eλtfσ(λ)gσ(λ)(I-gσ(λ)A)-1CA2xλ2dλ.
Put R2(t):=∫0t(t-s)R(s)xds,x∈E,t≥0. By Theorem 2.6(i), we get that
(2.44)k̃(λ)λ2(I-ã(λ)A)-1Cx=∫0∞e-λtR2(t)xdt,x∈E,Reλ>ω,k̃(λ)≠0.
This implies that the function λ↦hσ(λ)/λ2 is bounded on some right half plane. Taking into account (2.41), we have that, for every t≥0 and x∈D(A2),(2.45)R2(t)=ℒ-1(k̃(λ)λ2(I-ã(λ)A)-1Cx)(t)=∫0t(t-s)k(s)dsCx+(t*k*a)(t)CAx+12πi∫ω-i∞ω+i∞eλtfσ(λ)gσ(λ)(I-gσ(λ)A)-1CA2xλ2dλ.
By (2.43)-(2.45), S0,σ(t)=R2(t),t>0. The arbitrariness of σ implies that the mapping t↦R2(t),t>0 is infinitely differentiable, finishing the proof.Using the argumentation given in [25], one can prove the following theorems.Theorem 2.22.
Supposek(t) and a(t) satisfy (P1), A  is a subgenerator of a (local) (a,k)-regularized C-resolvent family (R(t))t∈[0,τ),ω≥max(0,abs(k),abs(a)), and m∈ℕ. Denote, for every ε∈(0,1) and a corresponding Kε>0,(2.46)Fε,ω=:{λ∈ℂ:Reλ≥-lnωL(Kε|Imλ|)+ω}.
Assume that, for every ε∈(0,1), there exist Cε>0,Mε>0, an open neighborhood Oε,ω of the region Gε,ω:={λ∈ℂ:Reλ≥ω,k̃(λ)≠0}∪{λ∈Fε,ω:Reλ≤ω},  and analytic mappings fε:Oε,ω→ℂ,gε:Oε,ω→ℂ and hε:Oε,ω→L(E) such that (i)
fε(λ)=k̃(λ),Reλ>ω;gε(λ)=ã(λ),Reλ>ω,(ii)
for everyλ∈Fε,ω, the operator I-gε(λ)A is injective and Rang(C)⊆Rang(I-gε(λ)A),(iii)
hε(λ)=fε(λ)(I-gε(λ)A)-1C,λ∈Gε,ω,(iv)
∥hε(λ)∥≤Mε(1+|λ|)meε|Reλ|,λ∈Fε,ω,Reλ≤ω  and ∥hε(λ)∥≤Mε(1+|λ|)m,λ∈ℂ,Reλ≥ω.Then (R(t))t∈[0,τ)is of class CL.Theorem 2.23.
Supposek(t) and a(t) satisfy (P1), A is a subgenerator of a (local) (a,k)-regularized C-resolvent family (R(t))t∈[0,τ),ω≥max(0,abs(k),abs(a)), and m∈ℕ. Denote, for every ε∈(0,1),ρ∈[1,∞) and a corresponding Kε>0,(2.47)Fε,ω,ρ=:{λ∈ℂ:Reλ≥-Kε|Imλ|1/ρ+ω}.
Assume that, for every ε∈(0,1), there exist Cε>0,Mε>0, an open neighborhood Oε,ω of the region Gε,ω,ρ:={λ∈ℂ:Reλ≥ω,k̃(λ)≠0}∪{λ∈Fε,ω,ρ:Reλ≤ω},  and analytic mappings fε:Oε,ω→ℂ,gε:Oε,ω→ℂ and hε:Oε,ω→L(E) such that the conditions (i)–(iv)  of Theorem 2.22 hold with Fε,ω, resp. Gε,ω, replaced by Fε,ω,ρ, respectively, Gε,ω,ρ. Then (R(t))t∈[0,τ) is of class CL.Theorem 2.24.
Supposeα>0,j∈ℕ, and (R(t))t∈[0,τ) is a (local) (a,k)-regularized C-resolvent family with a subgenerator A. Set
(2.48)Rα(t)x:=∫0t(t-s)α-1Γ(α)R(s)xds,t∈[0,τ),x∈E.
Then(R(t))t∈[0,τ) is an (a,k*(tα-1/Γ(α)))-regularized C-resolvent family with a subgenerator A. Furthermore, if the mapping t↦R(t),t∈(0,τ) is j-times differentiable, then the mapping t↦Rα(t),t∈(0,τ) is likewise j-times differentiable. If this isthe case, then we have, for every t∈[0,τ),b∈(0,t), and x∈E:
(2.49)djdtjRα(t)x=∫0b(t-s)α-1-jΓ(α)∏i=1j(α-i)R(s)xds+∑i=0j(t-b)α+i-jΓ(α+i+1)×∏k=0j-1(α+i-k)R(i)(b)x+∫bt(t-s)αΓ(α+1)djdsjR(s)xds,
and we have the following. (i)
If(R(t))t∈[0,τ) is of class CL, resp. of class CL, then (Rα(t))t∈[0,τ) is likewise of class CL, resptivley, of class CL.(ii)
If(R(t))t∈[0,τ) is ρ-hypoanalytic, 1≤ρ<∞, then (Rα(t))t∈[0,τ) is likewise ρ-hypoanalytic.Before going further, notice that we can slightly reformulate Theorem2.16 and Theorems 2.21–2.23 in the case when the functions k̃(λ) and ã(λ) possess the meromorphic extensions on the corresponding regions defined in formulation of mentioned theorems. Having in mind [9, Theorem  2.1.1.11, Theorem   2.1.1.14], we have the following interesting analogue of [25, Theorem  2.8] which cannot be so easily interpreted in the case of general (a,k)-regularized C-resolvent families.Theorem 2.25.
(i) LetA be a subgenerator of a local K-convoluted C-cosine function (CK(t))t∈[0,τ),0∈suppK,K∈C∞((0,τ)) (K∈Cj((0,τ)),j∈ℕ)  resp. K is of class CL (CL), and let K=K1|[0,τ) for an appropriate complex-valued function K1∈Lloc1([0,2τ)). (Put Θ1(t)=∫0tK1(s)ds; since it makes no misunderstanding, we will also write K and Θ,  for K1 and Θ1, respectively, and denote by K*K the restriction of this function to any subinterval of [0,2τ).) Let the mapping t↦CK(t),t∈(0,τ), be infinitely differentiable (j-times differentiable, j∈ℕ), respectively, and let (CK(t))t∈[0,τ) be of class CL (CL). Then A is a subgenerator of a local (K*K)-convoluted C2-cosine function (CK*K(t))t∈[0,2τ) satisfying that the mapping t↦CK*K(t),t∈(0,2τ), is infinitely differentiable ((j-1)-times differentiable), resp. (CK*K(t))t∈[0,2τ) is of class CL (CL). Furthermore, the suppositions j∈ℕ and K∈Cj((0,τ))∩Cj-1([0,τ)) imply the following: if the mapping t↦CK(t),t∈(0,τ), is j-times differentiable, then the mapping t↦CK*K(t),t∈(0,2τ) is likewise j-times differentiable.
(ii) Supposeα≥0,    j∈ℕ, and A is a subgenerator of a local α-times integrated C-cosine function (Cα(t))t∈[0,τ). Then A is a subgenerator of a local (2α)-times integrated C2-cosine function (C2α(t))t∈[0,2τ) and the following holds.
(ii.1)
If the mappingt↦Cα(t),t∈(0,τ) is infinitely differentiable (j-times differentiable, j∈ℕ), then the mapping t↦C2α(t),t∈(0,2τ) is infinitely differentiable ((j-1)-times differentiable; j-times differentiable, provided α≥j).(ii.2)
If(Cα(t))t∈[0,τ) is of class CL, resp. CL, then (C2α(t))t∈[0,2τ) is likewise of class CL, resp. CL.(ii.3)
Assumeα∈ℕ0,j∈ℕ, and the mapping t↦Cα(t),t∈(0,τ) is infinitely differentiable (j-times differentiable). Then the mapping t↦C2α(t),t∈(0,2τ), is j-times differentiable.Proof.
The first part of (i) can be proved by passing to the theory of semigroups (see [9, Theorem  2.1.1.11] and [25, Theorem  2.8]). So, let us assume j∈ℕ,K∈Cj((0,τ))∩Cj-1([0,τ)),τ0∈(0,τ), and let the mapping t↦CK(t),t∈(0,τ) be j-times differentiable. By [9, Theorem  2.1.1.14], A is a subgenerator of a local (K*K)-convoluted C2-cosine function (CK*K(t))t∈[0,2τ), which is given by
(2.50)CK*K(t)x={∫0tK(t-s)CK(s)Cxds,t∈[0,τ0],2CK(τ0)CK(t-τ0)x+(∫0t-τ0+∫0τ0)K(t-r)CK(r)Cxdr-∫2τ0-tτ0K(r+t-2τ0)CK(r)Cxdr-∫0t-τ0K(r+2τ0-t)CK(r)Cxdr,t∈(τ0,2τ0),x∈E.
Since the mapping t↦CK(t),t∈(0,τ) is j-times differentiable and K∈Cj((0,∞)), we have that the mapping t↦CK*K(t),t∈(0,τ) is also j-times differentiable. Arguing as in [25, Theorem  2.8], one gets that the mappings t↦CK(τ0)CK(t-τ0),t∈(τ0,2τ0),t↦(∫0t-τ0+∫0τ0)K(t-r)CK(r)Cdr,t∈(τ0,2τ0) and t↦∫0t-τ0K(r+2τ0-t)CK(r)Cdr,t∈(τ0,2τ0) are j-times differentiable.
Letf(t):=∫2τ0-tτ0K(r+t-2τ0)CK(r)Cdr,t∈(τ0,2τ0). Using the fact that K∈C1((0,τ))∩C([0,τ)), we have f′(t)=∫2τ0-tτ0K′(r+t-2τ0)CK(r)Cdr+K(0)CK(2τ0-t)C,t∈(τ0,2τ0). Repeating this procedure leads us to the fact that the mapping t↦f(t),t∈(τ0,2τ0) is j-times differentiable, and this completes the proof of (i). The proof of (ii) in the case α∈ℕ follows immediately from (i) with K(t)=(tα-1/Γ(α)) while the proof of (ii) in the case α=0 is much easier [16].Suppose thatmin(a(t),k(t))>0,t∈(0,τ) and that A is a subgenerator of an (a,k)-regularized C-resolvent family (R(t))t∈[0,τ). We define the Favard class Fa,k by setting
(2.51)Fa,k:={x∈E:supt∈(0,τ)∥R(t)x-k(t)Cx∥(a*k)(t)<∞}.
Equipped with the norm |·|a,k:=∥·∥+supt∈(0,τ)(∥R(t)·-k(t)C·∥/(a*k)(t)),Fa,k becomes a Banach space, and in the case when ∥R(t)∥=O(k(t)),t∈[0,τ), we have D(A)⊆Fa,k. The proof of [5, Theorem  3.4] immediately implies the following assertion.Theorem 2.26.
Assumemin(a(t),k(t))>0,t∈(0,τ),abs(k)=abs(a)=0,A  is a subgenerator of an (a,k)-regularized C-resolvent family (R(t))t≥0 satisfying ∥R(t)∥=O(1),t≥0 and (H5) holds. (i)
Letx∈Fa,k. Then
(2.52)supλ>0,k̃(λ)≠0∥A(I-ã(λ)A)-1Cx∥<∞.(ii)
Assume, in addition, that the mappingã:(0,∞)→(0,∞) is surjective and that supt>0(1*a)(t)/(a*k)(t)<∞. Then (2.52) implies Cx∈Fa,k.The assertion (ii) of the next theorem improves [5, Theorem  4.6].Theorem 2.27 (cf. [26, Theorem  4.2] and Proposition  2.12.7).
(i) SupposeA is a subgenerator of a (local, global exponentially bounded) (a,k)-regularized C-resolvent family (R(t))t∈[0,τ),D(A) and Rang(C) are dense in E and α>0. Then A* is a subgenerator of a (local, global exponentially bounded) (a,k*0(tα-1/Γ(α)))-regularized C*-resolvent family (Rα*(t))t∈[0,τ), which is given by
(2.53)Rα(t)x*:=∫0t(t-s)α-1Γ(α)R(s)*x*ds,t∈[0,τ),x*∈E*.
(ii) SupposeA is a subgenerator of a (local, global exponentially bounded) (a,k)-regularized C-resolvent family (R(t))t∈[0,τ), and D(A) and Rang(C) are dense in E. Then the part of A* in D(A*)¯ is a subgenerator of a (local, global exponentially bounded) (a,k)-regularized CD(A*)¯*-resolvent family in E*.
(iii) SupposeE is reflexive, D(A) and Rang(C) are dense in E,k(t) and a(t) satisfy (P1), and A is a subgenerator of a (local, global exponentially bounded) (a,k)-regularized C-resolvent family (R(t))t∈[0,τ). Then A* is a subgenerator of a (local, global exponentially bounded) (a,k)- regularized C*-resolvent family (of the same exponential type, in the second case).Suppose, for the time being, thata∈C([0,τ)) and denote, for every λ∈ℂ, by s(t,λ) the unique continuous solution of the equation
(2.54)s(t,λ)=a(t)+λ∫0ta(t-v)s(v,λ)dv,t∈[0,τ).
Put r(t,λ):=k(t)+λ∫0ts(t-v,λ)k(v)dv,t∈[0,τ). Arguing as in [5, Section  5], one can simply verify the validity of the next theorem.Theorem 2.28.
(i) LetA be a subgenerator of an (a,k)- regularized C-resolvent family(R(t))t∈[0,τ), and let (H5) hold. If the operator r(t,λ)C-R(t) is bijective for some t∈[0,τ) and λ∈ℂ, then λ∈ρ(A).
(ii) LetA be a densely defined subgenerator of an (a,k)-regularized C-resolvent family (R(t))t∈[0,τ), and let (H5) hold. If  Rang(λ-A)¯≠E  for some λ∈ℂ, then, for every t∈[0,τ),Rang(r(t,λ)C-R(t))¯≠E.
(iii) LetA be a subgenerator of an (a,k)-regularized C-resolvent family (R(t))t∈[0,τ). Then the assumption Ax=λx,  for some x∈E and λ∈ℂ, implies r(t,λ)Cx=R(t)x,t∈[0,τ).For further information concerning duality and spectral properties of(a,k)-regularized resolvent families, we refer to [5].Proposition 2.29 (cf. [9, Proposition  2.1.1.17]).
Suppose±A  are subgenerators of (local, global exponentially bounded) (a,k)- regularized C- resolvent families (R±(t))t∈[0,τ), and A2 is closed. Then A2 is a subgenerator of a (local, global exponentially bounded) (a*a,k)-regularized C-resolvent family (R(t))t∈[0,τ), which is given by R(t)x:=1/2(R+(t)x+R-(t)x),x∈E,t∈[0,τ).Proof.
Clearly,(R(t))t∈[0,τ) is a strongly continuous operator family, CA2⊆A2C,R(0)=k(0)C,R(t)C=CR(t) and R(t)A2⊆A2R(t),t∈[0,τ). Let x∈D(A2). Then we have
(2.55)(a*a*R)(t)A2x=12(a*a*R+A)(t)Ax+12(a*a*R-A)(t)Ax=12(a*(R+-kC))(t)Ax+12(a*(R--kC))(t)Ax=12(a*R+)(t)Ax+12(a*R-)(t)Ax=12(R+(t)x-k(t)Cx)+12(R-(t)x-k(t)Cx)=R(t)x-k(t)Cx,t∈[0,τ).
This completes the proof.The next version of the abstract Weierstrass formula extends [15, Theorem  11].Theorem 2.30.
(i) Assume thatk(t) and a(t) satisfy (P1), and there exist M>0 and ω>0 such that |k(t)|≤Meωt,t≥0. Assume, further, that there exist a number ω′≥ω and a function a1(t) satisfying (P1) and ã1(λ)=ã(λ),λ∈ℂ,Reλ>ω′.  Let A be a subgenerator of an exponentially bounded (a,k)-regularized C-resolvent family (C(t))t≥0, and let (H5) hold. Then A is a subgenerator of an exponentially bounded, analytic (a1,k1)-regularized C-resolvent family (R(t))t≥0 of angle (π/2), where
(2.56)k1(t)≔∫0∞e-s2/4tπtk(s)ds,t>0,k1(0)≔k(0),(2.57)R(t)x≔∫0∞e-s2/4tπtC(s)xds,t>0,x∈E,R(0)≔k(0)C.
(ii) Assumek(t) satisfy (P1), β>0, and there exist M>0 and ω>0 such that |k(t)|≤Meωt,t≥0. Let A be a subgenerator of an exponentially bounded (t2β-1/Γ(2β),k)-regularized C-resolvent family (C(t))t≥0, and let (H5) hold. Then A is a subgenerator of an exponentially bounded, analytic (tβ-1/Γ(β),k1)-regularized C-resolvent family (R(t))t≥0 of angle π/2, where k1(t) and R(t) are defined through (2.56) and (2.57).Proof.
Sincek(t) is continuous and exponentially bounded, one can use the substitution r=s/t and the dominated convergence theorem after that to deduce that, for every s≥0,(2.58)k1(t)=∫0∞e-r2/4πk(rt)dr→k1(s),t→s.
This implies k1∈C([0,∞)). Moreover, k1(t) is a kernel since
(2.59)limsupλ→+∞ln|k̃1(λ)|λ=limsupλ→+∞ln|k̃(λ)/λ|λ=0.
Let x∈E be fixed. Then, for every s≥0,(2.60)R(t)x=∫0∞e-r2/4πC(rt)xdr→R(s)x,t→s.
By (2.60), (R(t))t≥0 is a strongly continuous, exponentially bounded operator family. Furthermore, one can employ Theorem 2.6(i) and [12, Proposition   1.6.8] to obtain that, for every  λ∈ℂ with Reλ>β2 and k̃1(λ)≠0,(2.61)∫0∞e-λtR(t)xdt=∫0∞e-λt1πt∫0∞e-s2/4tC(s)xdsdt=1λ∫0∞e-λsC(s)xds=1λk̃(λ)(I-ã(λ)A)-1Cx=k̃1(λ)(I-ã1(λ)A)-1Cx.
By Theorem 2.6(ii), we get that (R(t))t≥0 is an exponentially bounded (a1,k1)-regularized C-resolvent family with a subgenerator A, and the remnant of the proof of (i) may be carried out by modifying the corresponding part of the proof of [15, Theorem  11]; the assertion (ii) follows from (i) with a(t)=t2β-1/Γ(2β).Notice thata1(t)=∫0∞s(e-s2/4t/2πt3/2)a(s)ds,t>0, whenever the function a(t) is exponentially bounded.Example 2.31.
(i)(Reference[15]) Let E=Lp(ℝ),1≤p≤∞,α∈(-1,1), and a(t)=(tα/Γ(α+1)).  Consider the next multiplication operator with maximal domain in E:
(2.62)Af(x):=(1+x+ix2)α+1f(x),x∈ℝ,f∈E.
Assume s∈(1,2),δ=1/s,Mp=p!s, and Kδ(t):=ℒ-1(e-λδ)(t),t≥0. Then A generates a global (not exponentially bounded) (a,Kδ)-regularized resolvent family since, for every τ∈(0,∞),A generates a local (a,Kδ)-regularized resolvent family on [0,τ). In order to show this, designate by M(t) the associated function of the sequence (Mp) and put Λα,β,γ:={λ∈ℂ:Reλ≥(M(αλ)/γ)+β},α>0,β>0,γ>0. Clearly, there exists a constant Cs>0 such that M(λ)≤Cs|λ|1/s,λ∈ℂ. Given τ>0, choose α>0 and β>0 such that τ≤cos(δπ/2)/Csα1/s as well as that Λα,β,1⊆ρ(A) and that the resolvent of A is bounded on the set {λα+1:λ∈Λα,β,1}. Put Γ:=∂(Λα,β,1), and assume that the curve Γ is upward oriented. Define, for every f∈E,x∈ℝ and t∈[0,cos(δπ/2)/Csα1/s),(2.63)(Rδ(t)f)(x):=12πi∫Γλα+1eλt-λδλα+1-(1+x+ix2)α+1dλf(x).
Then one can straightforwardly check that (Rδ(t))t∈[0,τ) is a local (a,Kδ)-regularized resolvent family generated by A. Arguing in the same way, we get that there exists τ0>0 such that A generates a local (a,K1/2)-regularized resolvent family on [0,τ0), where K1/2(t):=ℒ-1(e-λ1/2)(t),t≥0.
(ii) [References [9, 27]] Let A(p!s) and E(p!s) be as in [27, Example  1.6] with Mp=p!s(s>1). Let β∈(0,1), and let, for every l>0,kl(t)=ℒ-1(1/∏p=1∞(1+lλ/ps/β))(t),t≥0, (see (2.36)-(2.37) and a(t)=tβ-1/Γ(β). Then it is obvious that there exist l′>0 and K>0 such that ∥λk̃l′(λ)(I-ã(λ)A)-1∥≤K,λ∈Σπ/2β. This in combination with Theorem 2.17 implies that, for every l>l′, the operator A(p!s) generates an analytic (a,kl)-regularized resolvent of angle (π/2)((1/β)-1). In the meantime, A(p!s) does not generate an exponentially bounded (a,tα/Γ(α+1))-regularized resolvent (α≥0) since A(p!s) is not stationary dense.
(iii) [References [9, 28]; cf. also [29, Example  2.20]] Suppose E:=L2[0,π] and A:=-Δ with the Dirichlet or Neumann boundary conditions, β∈[1/2,1),α>1+β,a(t)=(tβ-1/Γ(β)), and
(2.64)hα,β(λ):=1λα∏n=0∞n2-λβn2+λβ,Reλ>0,λ≠n2/β,n∈ℕ.
Definehα,β:Σπ/2β→ℂ by setting: hα,β(λ)=hα,β(λ),λ∈Σπ/2β,λ≠n2/β,n∈ℕ, and hα,β(n2/β)=0,n∈ℕ. Then the function  hα,β(λ) is analytic, and there exists a constant C>0 such that
(2.65)∥hα,β(λ)(I-ã(λ)A)-1∥≤1+C|λ|β|λ|α,λ∈Σπ/2β.
Let k(t)=ℒ-1(hα,β(λ))(t),t≥0. By Theorem 2.17, it follows that A generates an exponentially bounded, analytic (a,k)-regularized resolvent (R(t))t≥0 of angle  (π/2)(1/β-1). Using the inverse Laplace transform, one can simply prove that ∥R(t)∥=O(tα-1+tα+β-1),t≥0. Since Δ generates a cosine function, we are in a position to apply Theorem 2.17 to deduce that Δ generates an exponentially bounded, analytic (a,k)-regularized resolvent of angle  (π/2)(1/β-1). By Proposition 2.29, we have that the biharmonic operator Δ2, equipped with the suitable boundary conditions, generates an exponentially bounded, analytic (a*a,k)-regularized resolvent of angle  (π/2)(1/β-1). Then the use of Theorem 2.30(ii) enables one to see that there exists a continuous kernel  k1(t) such that Δ2 generates an exponentially bounded, analytic (a,k1)-regularized resolvent family of angle  π/2. Keeping in mind the fact that -Δ2n generates an analytic C0-semigroup of angle  π/2 (cf. for example [30, page 215]), one can prove that, for every n∈ℕ, there exists an exponentially bounded kernel kn(t) such that the polyharmonic operator Δ2n generates an exponentially bounded, (a,kn)-regularized resolvent family of angle  π/2 [15].
It has recently been proved that, in the caseβ=1, there exists an exponentially bounded continuous kernel K(t) such that A generates an exponentially bounded, analytic K-convoluted semigroup of angle π/2 [25]. Let us consider now the case β∈(1,2) and a(t)=(tβ-1/Γ(β)). Choose a number a∈(1/2,1/β) and after that a number s∈(1,1/βa). Put kl(t):=ℒ-1(1/∏p=1∞1+lλ/ps)(t),t≥0. Arguing as in [25], one yields that the function h(λ)=∏n=0∞(n2-λ)/(n2+λ),λ∈ℂ∖{±n2:n∈ℕ0};h(n2)=0,n∈ℕ, is analytic, and that there exists M>0 such that, for every γ∈(0,π(1/β-1/2)), there exist Mγ>0 and cγ>0 such that, for every λ∈Σγ,
(2.66)∥h(λb)(I-ã(λ)A)-1∥≤|λ|bmax((M+1|λ|β),Mγ(⌈2|λ|β+1⌉|λ|β+π23)exp(cγ|λ|βa)).
Furthermore, there exists an exponentially bounded continuous kernel k(t) such that k̃(λ)=k̃l(λ)h(λβ),λ∈ℂ,Reλ>0. By (2.66), it follows that A generates an exponentially bounded, analytic (a,k)-regularized resolvent of angle π(1/β-1/2). Furthermore, an application of Theorem 2.17 gives that Δ generates an exponentially bounded, analytic (a,k)-regularized resolvent of angle π(1/β-1/2). By Proposition 2.29, we have that Δ2 generates an exponentially bounded, analytic (a*a,k)-regularized resolvent of angle π(1/β-1/2). Arguing as in the case β∈[1/2,1), we have that, for every n∈ℕ, there exists an exponentially bounded kernel kn(t) such that the polyharmonic operator Δ2n generates an exponentially bounded, (a,kn)-regularized resolvent family of angle π/2. In the case β=2, it is known that A cannot be the generator of any exponentially bounded convoluted cosine function [15]; the case β∈(0,1/2) requires an additional analysis. Finally, it is worth noting that we can incorporate the above results in the study of the equation
(2.67)Dtβu(t,x)=(-Δ)2nu(t,x),x∈(0,π),t>0(n∈ℕ0),
where Dtβ denotes the Caputo fractional derivative [29].
The next theorem generalizes [6, Theorem  3.6, Corollary   3.8] (cf. also [31, Theorem  2.1] and [32, Theorem  3]).Theorem 2.32.
(i) AssumeC([0,∞))∋a satisfies (P1), (H5) holds, B∈L(E),Rang(B)⊆Rang(C) that and A is a subgenerator of an exponentially bounded (a,a)-regularized C-resolvent family (R(t))t≥0. Assume, further, that there exists ω≥0 such that, for every h≥0 and for every function f∈C([0,∞):E),
(Ma)
∫0hR(h-s)C-1Bf(s)ds∈D(A),(Mb)
∥A∫0hR(h-s)C-1Bf(s)ds∥≤eωtμB(h)∥f∥[0,h],t≥0, where ∥f∥[0,h]:=supt∈[0,h]∥f(t)∥,μB(t):[0,∞)→[0,∞) is continuous, nondecreasing and satisfies μB(0)=0,(Mc)
there exists an injective operatorC1∈L(E) such that  Rang(C1)⊆Rang(C)  and that C1A(I+B)⊆A(I+B)C1.Then A(I+B) is a subgenerator of an exponentially bounded (a,a)-regularized C1-resolvent family (S(t))t≥0 which satisfies the following integral equation
(2.68)S(t)x=R(t)C-1C1x+A∫0tR(t-s)C-1BS(s)xds,t≥0,x∈E.
(ii) LetA be a subgenerator of an exponentially bounded, once integrated C-cosine function and let ω,B, and C1 be as in (i).  Then A(I+B) is a subgenerator of an exponentially bounded, once integrated C1-cosine function.Remark 2.33.
(i) Assume thatA is a subgenerator of an exponentially bounded (a,a)-regularized C-resolvent family (R(t))t≥0 and that a Banach space (Z,|·|Z) satisfies the conditions (Za), (Zb), and (Zc) given in the formulation of [6, Definition  4.1]. (In particular, these conditions hold for [D(A)].) Then (Ma) and (Mb) are fulfilled if C-1B∈L(X,Z).
(ii) (References [32, 33]) Let B∈L(E), and let BC=CB.
(ii.1)
Assume thatBA is a subgenerator of a (local) (a,k)-regularized C-resolvent family, and (H5) holds for BA and C. Then AB is a subgenerator of an (a,k)-regularized C-resolvent family.(ii.2)
Assume thatAB is a subgenerator of a (local) (a,k)-regularized C-resolvent family and (H5) holds for AB and C. Then BA is a subgenerator of an (a,k)-regularized C-resolvent family, provided ρ(BA)≠∅.The proof of the next generalization of [15, Proposition  3] is provided for the sake of completeness.Theorem 2.34.
Assume thatτ∈(0,∞],Lloc1([0,τ))∋a1 is a kernel, Lloc1([0,τ))∋k  is a kernel, a(t)=(a1*a1)(t),t∈[0,τ), and k1(t)=(k*a1)(t),t∈[0,τ). Put 𝒜≡(0IA0),𝒞≡(C00C), and assume that (H5)  holds. Then A is a subgenerator of an (a,k)-regularized C-resolvent family (R(t))t∈[0,τ) if and only if  𝒜 is a subgenerator of an (a1,k1)-regularized 𝒞-resolvent family (S(t))t∈[0,τ).  If this is the case, then we have
(2.69)S(t)=((a1*R)(t)(a*R)(t)R(t)-k(t)C(a1*R)(t)),0≤t<τ,
and the integral generators of (R(t))t∈[0,τ) and (S(t))t∈[0,τ), denoted respectively by B and ℬ, satisfy ℬ=(0IB0).Proof.
It is immediately verified that(S(t))t∈[0,τ) is a nondegenerate, strongly continuous operator family in E×E which satisfies S(t)𝒜⊆𝒜S(t) and S(t)𝒞=𝒞S(t),0≤t<τ. Furthermore, the function k1(t) is a continuous kernel, S(0)=0=k1(0)𝒞,  and  𝒞𝒜⊆𝒜𝒞.  Let x∈D(A), and let y∈E. Then a simple computation involving (H5) shows that, for every t∈[0,τ),(2.70)∫0ta1(t-s)𝒜S(s)(xy)ds=∫0ta1(t-s)(R(s)x-k(s)Cx+(a1*R)(s)yA(a1*R)(s)x+A(a*R)(s)y)ds=((a1*R)(t)x-(a1*k)(t)Cx+(a1*a1*R)(t)yA(a*R)(t)x+(a1*(R-kC))(t)y)=((a1*R)(t)x-(a1*k)(t)Cx+(a1*a1*R)(t)yR(t)x-k(t)Cx+(a1*(R-kC))(t)y)=S(t)(xy)-k1(t)𝒞(xy).
Assume now that 𝒜 is a subgenerator of an (a1,k1)-regularized 𝒞-resolvent family (S(t))t∈[0,τ). Put S(t)=(S1(t)S2(t)S3(t)S4(t))t∈[0,τ), where Si(t)∈L(E),i∈{1,2,3,4}, and 0≤t<τ. A simple consequence of S(t)𝒞=𝒞S(t),t∈[0,τ) is Si(t)C=CSi(t),t∈[0,τ),i∈{1,2,3,4}. Since S(t)𝒜⊆𝒜S(t),t∈[0,τ), one gets
(2.71)S1(t)x+S2(t)y∈D(A),S1(t)y+S2(t)Ax=S3(t)x+S4(t)y,S3(t)y+S4(t)Ax=A(S1(t)x+S2(t)y),0≤t<τ,x∈D(A),y∈E.
Hence, S3(t)x=S2(t)Ax,x∈D(A), and S3(t)y=AS2(t)y,y∈E,0≤t<τ. This implies that, for every x∈D(A),S3(t)Ax=AS2(t)Ax=AS3(t)x,t∈[0,τ). Thereby, S3(t)A⊆AS3(t),t∈[0,τ), and (R(t)≡S3(t)+k(t)C)t∈[0,τ) is a strongly continuous operator family in E satisfying R(0)=k(0)C,R(t)C=CR(t) and R(t)A⊆AR(t),0≤t<τ. Since, for every λ∈ℂ,λ∈ρ𝒞(𝒜) if and only if  λ2∈ρC(A) [25], we have that (H5) holds for 𝒜 and 𝒞. Since, for every x∈E and y∈E,(2.72)𝒜((a1*S1)(t)(a1*S2)(t)(a1*S3)(t)(a1*S4)(t))(xy)=(S1(t)S2(t)S3(t)S4(t))(xy)-k1(t)𝒞(xy),
one gets (a1*S3)(t)x=S1(t)x-k1(t)Cx,(a1*S4)(t)x=S2(t)x,A(a1*S3)(t)x=S3(t)x and A(a1*S2)(t)x=S4(t)x-k1(t)Cx,0≤t<τ. Hence, A(a*R)(t)x=A(a*(S3+kC))(t)x=A(a1*a1*(S3+kC))(t)x=A(a1*(S1-k1C+(a1*k)C))(t)x=S3(t)x=R(t)x-k(t)Cx,t∈[0,τ). This implies that (R(t))t∈[0,τ) is a nondegenerate operator family, and we finally get that (R(t))t∈[0,τ) is an (a,k)-regularized C-resolvent family with a subgenerator A. The remnant of the proof follows from a slight technical modification of the final part of the proof of [15, Proposition  3].Remark 2.35.
(i) Letτ=∞, and let k(t) and a1(t) be exponentially bounded. Then (R(t))t∈[0,τ) is exponentially bounded if and only if  (S(t))t∈[0,τ) is exponentially bounded.
(ii) Letj∈ℕ,α>0,a1(t)=tα-1/Γ(α) and k∈Cj((0,τ)), respectivley, k∈C∞((0,τ)), and let the mapping t↦R(t),t∈(0,τ) be j-times differentiable, respectivley infinitely differentiable. Then the mapping t↦S(t),t∈(0,τ) is also j-times differentiable, resp. infinitely differentiable. Furthermore, if k(t) is of class CL, resp. CL (ρ-hypoanalytic, 1≤ρ<∞) and (R(t))t∈[0,τ) is of class CL, resp. CL (ρ-hypoanalytic), then (S(t))t∈[0,τ) is also of class CL, resp. CL (ρ-hypoanalytic).
(iii) Leta1(t)=1/πt and k1(t)=tn-(1/2)/Γ(n-(1/2)),n∈ℕ. Then Theorem 2.34 enables one to discuss the maximal interval of existence of a local (a1,k1)-regularized C-resolvent family and to construct an example of a local (a1,k1)-regularized C-resolvent family (R(t))t∈[0,τ) which cannot be extended beyond the interval [0,τ); combining with [25, Examples  1,  3,  5] and [34, Theorem  3.1], it is possible to construct examples of infinitely differentiable, nonanalytic (a1,k1)-regularized C-resolvent families and examples of (pseudo)differential operators generating (a1,k1)-regularized C-resolvent families of class CL.
(iv) Assumea1(t)=1/πt,A is a subgenerator of a (local) K-convoluted C-semigroup (SK(t))t∈[0,τ),k(t)=∫0tK(s)ds,t∈[0,τ), and (H3) holds (see Theorem 2.2(iii). Let k1(t) possess the same meaning as in Theorem 2.34. Then, for every x∈D(A) and y∈E, the system of integral equations
(2.73)u∈C([0,τ):[D(A)]),v∈C([0,τ):E),u(t)=k1(t)Cx+∫0tv(s)dsπ(t-s),t∈[0,τ),v(t)=k1(t)Cy+∫0tAu(s)dsπ(t-s),t∈[0,τ),
has a unique solution.



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*Source: 858242-2009-08-17.xml*