# Perturbation Theory for Abstract Volterra Equations

Authors: Marko Kostić
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We consider additive perturbation theorems for subgenerators of (a, k)-regularized C-resolvent families. A major part of our research is devoted to the study of perturbation properties of abstract time-fractional equations, primarily from their importance in modeling of various physical phenomena. We illustrate the results with several examples.


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## Body

## 1. Introduction and Preliminaries

A recently introduced notion of an(a,k)-regularized C-resolvent family on a sequentially complete locally convex space E plays an important role in the theory of abstract Volterra equations. A lot of effort has been directed towards characterizing spectral properties of subgenerators of (a,k)-regularized C-resolvent families, smoothing and duality properties, a generalized variation of parameters formula and subordination principles. The aim of this paper is to present a comprehensive survey of results about perturbation properties of abstract Volterra equations.The paper is organized as follows. In the second section, we consider bounded perturbation theorems for subgenerators of(a,k)-regularized C-resolvent families. A new line of approach to bounded commuting perturbations of abstract time-fractional equations is developed in Theorem 5. Our analysis is inspired, on the one side, by the incompleteness of the study of bounded perturbations of integrated C-cosine functions and, on the other side, by the possibilities of extension of [1, Theorem  2.5.3] to fractional operator families. We consider an exponentially equicontinuous (gα,k)-regularized C-resolvent family (R(t))t≥0 with a subgenerator A (gα(t)≡tα-1/Γ(α), t>0, α>0), a function k1(t) satisfying certain properties and an A-bounded perturbation B such that BA⊆AB and BC=CB. In order to prove the existence of perturbed (gα,k1)-regularized C-resolvent family (RB(t))t≥0 with a subgenerator A+B, we employ the method that involves only direct computations and differs from those established in [2–12] in that we do not consider (RB(t))t≥0 as the unique solution of a corresponding integral equation. The main objective in Theorem 7 is to show that, under some additional conditions, the perturbed (gα,k1)-regularized C-resolvent family (RB(t))t≥0 inherits analytical properties from (R(t))t≥0. In case α=2 and B satisfies the aforementioned conditions, Corollary 8 produces significantly better results compared with [13, Theorem 10.1] and [5, Theorem 3.1]. This is important since Hieber [14] proved that the Laplacian Δ with maximal distributional domain generates an exponentially bounded α-times integrated cosine function on Lp(ℝn) (1≤p<∞, n∈ℕ) for any α≥(n-1)|(1/2)-(1/p)|. Notice also that Keyantuo and Warma proved in [15] a similar result for the Laplacian Δ on Lp([0,π]n), with Dirichlet or Neumann boundary conditions. In Corollary 11, we focus our attention to the case k(t)=ℒ-1(λ-αe-ϱλσ)(t), t≥0(α>1,ϱ>0,σ∈(0,1)), which is important in the theory of ultradistribution semigroups of Gevrey type. As a special case of Corollary 11, we obtain that the class of tempered ultradistribution sines of (p!s)-class ({p!s}-class) is stable under bounded commuting perturbations (s>1); cf. [16], [17, Definition 13, Remark 15], [1, Section 3.5], [18], and the final part of the third section for more details. It is worthwhile to mention here the following fact: in order for the proof of Theorem 5 to work, one has to assume that the considered (gα,k)-regularized C-resolvent family (R(t))t≥0 is exponentially equicontinuous. It seems to be really difficult to prove an analogue of Theorem 5 in the context of local (gα,k)-regularized C-resolvent families (cf. [3, 7, 8, 13] and [1, Section 2.5, Theorem 3.5.17] for further information in this direction), which implies, however, that it is not clear whether the class of ultradistribution sines of (p!s)-class ({p!s}-class) retains the property stated above. In Theorems 13 and 14, Remark 15, and Corollary 17, we continue the researches of Arendt and Kellermann [2], Lizama and Sánchez [9], and Rhandi [4]. The local Hölder continuity with exponent σ∈(0,1] is the property stable under perturbations considered in these assertions, as explained in Remark 16.The final part of the paper is devoted to the study of unbounded perturbation theorems. The main purpose of Theorems20 and 21 is to generalize perturbation results of Kaiser and Weis [19]. The loss of regularity appearing in Theorem 20 is slightly reduced in Theorem 21 by assuming that the underlying Banach space E has certain geometrical properties. As an application, we consider (gα,gr+1)-regularized resolvent families generated by higher order differential operators (0<α≤2,r≥0). Perturbations of subgenerators of analytic (a,k)-regularized C-resolvent families are also analyzed in Theorem 24, which might be surprising in the case C≠I. The above result is applied to abstract time-fractional equations considered in [20, 21] and to differential operators in the spaces of Hölder continuous functions (von Wahl [22]). Possible applications of Corollary 8 and Theorem 7 can be also made to coercive differential operators considered by Li et al.  [23, Section 4] and by the author [24]. In the remainder of the third section, we reconsider and slightly improve results of Arendt and Batty [25] and Desch et al. [26] on rank-1 perturbations. Before we collect the material needed later on, we would like to draw the attention to paper [27] of Xiao et al. for the analysis of time-dependent perturbations of abstract Volterra equations. The results obtained in [27] can be straightforwardly generalized to the class of (a,C)-regularized resolvent families, and it is not the intention in this paper to go into further details (cf. also [28–30] and the review paper [31] for time-dependent perturbations).Henceforth,E denotes a Hausdorff sequentially complete locally convex space, SCLCS for short, and the abbreviation ⊛ stands for the fundamental system of seminorms which defines the topology of E; if E is a Banach space, then ∥x∥ denotes the norm of an element x∈E. If F is a SCLCS, then we denote by L(E,F) the space of all continuous linear mappings from E into F;L(E):=L(E,E). We assume that A is a closed linear operator acting on E and that (with the exception of assertions concerning rank-1 perturbations) L(E)∋C is an injective operator with CA⊆AC; the convolution like mapping * is given by f*g(t):=∫0t‍f(t-s)g(s)ds, and the principal branch is always used to take the powers. Given f∈Lloc1([0,∞)) and n∈ℕ, f*,n(t) denotes the nth convolution power of f(t), and f*,0(t) denotes the Dirac δ-distribution. If s∈ℝ and β∈(0,π], then ⌈s⌉:=inf{k∈ℤ:k≥s} and Σβ:={z∈ℂ:z≠0,|arg(z)|<β}. The domain, range, and resolvent set of A are denoted by D(A),R(A), and ρ(A), respectively. If D(A) is not dense in E, then D(A)¯ is a closed subspace of E and therefore a SCLCS itself; the fundamental system of seminorms which defines the topology of D(A)¯ is (p|D(A)¯)p∈⊛. Recall that the C-resolvent set of A, in short ρC(A), is defined by ρC(A):={λ∈ℂ:λ-A is injective and (λ-A)-1C∈L(E)}.Fairly complete information on the general theory of well-posed abstract Volterra equations in Banach spaces can be obtained by consulting the monograph [10] of Prüss. The following notion is crucially important in the theory of ill-posed Volterra equations (cf. [32–35]).Definition 1.
(i) LetE be an SCLCS, let 0<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 case τ=∞,(R(t))t≥0  is said to be exponentially equicontinuous if there exists ω∈ℝ such that the family {e-ωtR(t):t≥0} is equicontinuous.
(ii) Letβ∈(0,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 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.
It is said that(R(t))t≥0 is an exponentially equicontinuous, analytic (a,k)-regularized C-resolvent family of angle β, if for every γ∈(0,β), there exists ωγ≥0 such that the family {e-ωγℜzR(z):z∈Σγ} is equicontinuous.
Since there is no risk for confusion, we will identifyR(·) and R(·).
(iii) An(a,k)-regularized C-resolvent family (R(t))t≥0 is said to be entire if, for every x∈E, the mapping t↦R(t)x,t≥0 can be analytically extended to the whole complex plane.
In the sequel of the paper, we will consider only nondegenerate(a,k)-regularized C-resolvent families. The set which consists of all subgenerators of (R(t))t∈[0,τ) need not be finite. In case k(t)=gr+1(t), where r≥0, it is also said that (R(t))t∈[0,τ) is an r-times integrated(a,C)-regularized resolvent family; 0-times integrated (a,C)-regularized resolvent family is also called an (a,C)-regularized resolvent family. Instructive examples of integrated (a,C)-regularized resolvent families, providing possible applications of Theorem 14 and Corollary 17, can be constructed following the analysis given in the proof of [36, Proposition 2.4]. If k(t)=∫0t‍K(s)ds,t∈[0,τ), where K∈Lloc1([0,τ)) and K≠0, then we obtain the unification concept for (local) K-convoluted C-semigroups and cosine functions [1]. We refer the reader to [23, 28, 32, 37, 38] for some applications of (gα,k)-regularized C-resolvent families in the study of the following abstract time-fractional equation with α>0:(1)Dtαu(t)=Au(t),t>0,u(k)(0)=Cxk,k=0,1,…,⌈α⌉-1,
where xk∈D(A),k=0,1,…,⌈α⌉-1  and Dtα denotes the Caputo fractional derivative of orderα ([28]). Henceforth, we assume that k(t) and k1(t) are scalar-valued continuous kernels.
The following conditions will be used frequently:(P1)
:k(t) is Laplace transformable, that is, it is locally integrable on [0,∞), and there exists β∈ℝ such 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},δ~(λ):=1  and denote by ℒ-1 the inverse Laplace transform.(P2)
:k(t) satisfies (P1) and k~(λ)≠0, ℜλ>β for some β≥abs(k).
For the sake of convenience, we recall the following result from [32, 33].Lemma 2.
Letk(t) and a(t) satisfy (P1) and let (R(t))t≥0 be a strongly continuous operator family such that there exists ω≥0 satisfying that the family {e-ωtR(t):t≥0} is equicontinuous. Put ω0:=max(ω,abs(a),abs(k)).(i)
AssumeA is a subgenerator of the global (a,k)-regularized C-resolvent family (R(t))t≥0 and
(2)R(t)x=k(t)Cx+A∫0ta(t-s)R(s)xds,t≥0,x∈E.
Then, for every λ∈ℂ with ℜλ>ω0 and k~(λ)≠0, the operator I-a~(λ)A is injective and R(C)⊆R(I-a~(λ)A). Furthermore,
(3)k~(λ)(I-a~(λ)A)-1Cx=∫0∞e-λtR(t)xdt,∫0∞e-λt(t),x∈E,ℜλ>ω0,k~(λ)≠0,{1a~(λ):ℜλ>ω0,k~(λ)a~(λ)≠0}⊆ρC(A).(ii)
Assume (3). Then A is a subgenerator of the global (a,k)-regularized C-resolvent family (R(t))t≥0 satisfying (2).LetA be a subgenerator of a locally equicontinuous (a,k)-regularized C-resolvent family (R(t))t∈[0,τ) satisfying the equality (2) for all t∈[0,τ) and x∈E. Given s∈[0,τ) and x∈E, set u(t):=R(t)R(s)x-R(s)R(t)x,t∈[0,τ). Then it is not difficult to prove that u∈C([0,τ):E) and A∫0t‍a(t-s)u(s)ds=u(t),t∈[0,τ). Using the proof of [35, Theorem 2.7] (cf. also [33, Theorem 2.5]), it follows that ∫0t‍k(t-s)Cu(s)ds=0,t∈[0,τ). Since k(t) is a kernel and C is injective, we obtain R(t)R(s)=R(s)R(t),t,s∈[0,τ), which remains true for perturbed resolvent families considered in the paper. Assuming additionally that (R(t))t≥0 is a global exponentially equicontinuous (a,k)-regularized C-resolvent family as well as that a(t) and k(t) satisfy (P1), one can define the integral generator A^ of (R(t))t≥0 by setting
(4)A^≔{∫0ta(t-s)R(s)yds,t≥0(x,y)∈E×E:R(t)x-k(t)Cx=∫0ta(t-s)R(s)yds,t≥0}.
In case that a(t) is a kernel, the definition of integral generator A^ of (R(t))t≥0 coincides with the corresponding one introduced in [33]. Notice that A^ is the maximal subgenerator of (R(t))t≥0 with respect to the set inclusion and that Lemma 2 implies A^=C-1AC.


## 2. Bounded Perturbation Theorems

Assumeα>0 and l∈ℕ. Set, for any E-valued function f(t) satisfying (P1), Fα,f(z):=∫0∞‍e-z1/αtf(t)dt, z>max(abs(f),0)α. Using induction and elementary operational properties of vector-valued Laplace transform, one can simply prove that there exist uniquely determined real numbers (cl0,l,α)1≤l0≤l, independent of E and f(t), such that
(5)dldzlFα,f(z)=∑l0=1lcl0,l,αz(l0/α)-l∫0∞e-z(1/α)ttl0f(t)dt,z>max(abs(f),0)α.
Furthermore, cl,l,α=(-1)l/αl,l≥1,c1,l,α=((-1)/α)((1/α)-1)⋯((1/α)-(l-1)),l≥2    and the following nonlinear recursive formula holds:
(6)cl0,l+1,α=(-1)αcl0-1,l,α+(l0α-l)cl0,l,α,l0=2,…,l.
The precise computation of coefficients (cl0,l,α) is a nontrivial problem.Lemma 3.
There existsζ≥1 such that
(7)∑l0=1ll0!|cl0,l,α|≤ζll!∀l∈ℕ.Proof.
Clearly,Lα:=supn∈ℕ0|(1/αn)|<∞. Applying (6), one gets
(8)∑l0=1l+1l0!|cl0,l+1,α|≤|1α(1α-1)⋯(1α-l)|+∑l0=2l[l0!α|cl0-1,l,α|+l(1α+1)l0!|cl0,l,α|]+(l+1)!αl+1≤Lα(1α+l)l!+lα∑l0=1l-1l0!|cl0,l,α|+l(1α+1)∑l0=2l‍l0!|cl0,l,α|+(l+1)!αl+1,l≥2.
The preceding inequality implies inductively that (7) holds provided ζ≥4+(4/α)+4Lα(1+(1/α)).Setζα:=inf{ζ≥1:∑l0=1l‍l0!|cl0,l,α|≤ζll!foralll∈ℕ} and ((1/α)+1)⋯((1/α)+(l-1)):=1 if l=1. Clearly, ζ1=1, ζα>1/α, α∈(0,1) and ∑l0=1l‍l0!|cl0,l,α|≤ζαll! for all l∈ℕ.The following lemma will be helpful in the analysis of growth order of perturbed integrated(gα,C)-regularized resolvent families.Lemma 4.
Letα>1. Then ζα=1 and
(9)∑l0=1l‍l0!|cl0,l,α|=1α(1α+1)⋯(1α+(l-1))≤l!α∀l∈ℕ.Proof.
Pluggingf(t)≡1 in (5), we obtain
(10)∑l0=1l‍l0!cl0,l,α=(-1)l1α(1α+1)⋯(1α+(l-1))∀l∈ℕ.
Since α>1, it follows inductively from (6) that (-1)lcl0,l,α>0, provided l≥1 and 1≤l0≤l. Combined with (10), the above implies (9) and ζα=1.Now we are in a position to state the following important result.Theorem 5.
Supposeα>0,k(t) and k1(t) satisfy (P1), A is a subgenerator of a (gα,k)-regularized C-resolvent family (R(t))t≥0 satisfying (2) with a(t)=gα(t),ω≥max(abs(k),0), the family {e-ωtR(t):t≥0} is equicontinuous and the following conditions hold(i)
B∈L(E),  there exists |B|⊛>0 such that p(Bx)≤|B|⊛p(x), x∈E, p∈⊛, BA⊆AB and BC=CB.There existM≥1, ω′≥0, ω′′≥0 and ω′′′≥max(ω+ω′,ω+ω′′,abs(k1)) such that
(11){λ∈ℂ:ℜλ>ω′′′,k1~(λ)≠0}⊆{λ∈ℂ:ℜλ>ω′′′,k~(λ)≠0}as well as(ii)
For everyi,l0,l∈ℕ with 1≤l≤i and 1≤l0≤l, there exists a function ki,l0,l(t) satisfying (P1) and
(12)ℒ(ki,l0,l(t))(λ)=cl0,l,αλl0-α(l-1)k1~(λ)(1zk~(z1/α))z=λα(i-l),providedℜλ>ω′′′ and k1~(λ)≠0.(iii)
For everyi∈ℕ0, there exists a function ik(t) satisfying (P1) and a constant ci∈ℂ so that
(13)ci+k~i(λ)=λαk1~(λ)(1zk~(z1/α))z=λα(i),ℜλ>ω′′′,k1~(λ)≠0.(iv)
(14)∑i=0∞‍|ci||B|⊛ii!<∞,∑i=0∞‍|B|⊛ii!∫0t‍|ki(s)|ds≤Meω′t,t≥0,(v)
(15)∑i=1∞∑l=1i∑l0=1l|B|⊛ii!(il)∫0t(t-s)l0|ki,l0,l(s)|ds≤Meω′t,t≥0,(vi)
(16)∑i=2∞‍∑l=2i‍∑l0=1l-1‍|B|⊛ii!l(il)∫0t(t-s)l0|ki-1,l0,l-1(s)|ds≤Meω′′t,t≥0.Then A+B is a subgenerator of an exponentially equicontinuous (gα,k1)-regularized C-resolvent family (RB(t))t≥0, which is given by the following formula:
(17)RB(t)x:=∑i=0∞‍(-B)ii![ciR(t)x+(k*R(·)xi)(t)]+∑i=1∞‍∑l=1i‍∑l0=1l(-B)ii!(il)(ki,l0,l*·l0R(·)x)(t),t≥0,x∈E.
Furthermore,(18)RB(t)x=k1(t)Cx+(A+B)∫0tgα(t-s)RB(s)xds,t≥0,x∈E,
and the family {e-(ω+ω′)tRB(t):t≥0} is equicontinuous.Proof.
By (iv)-(v), we obtain that the series in (17) converge uniformly on compact subsets of [0,∞) as well as that (RB(t))t≥0 is strongly continuous and that the family {e-(ω+ω′)tRB(t):t≥0} is equicontinuous. By (i) and Lemma 2, (z-A)-1CB=B(z-A)-1C,z∈ρC(A)    and, for every λ∈ℂ with ℜλ>ω and k~(λ)≠0:λαk~(λ)(λα-A)-1CBx=∫0∞‍e-λtR(t)Bxdt,x∈E and λαk~(λ)B(λα-A)-1Cx=∫0∞‍e-λtBR(t)xdt, x∈E. By the uniqueness theorem for Laplace transform, one gets R(t)B=BR(t), t≥0.  The closedness of A,R(t)A⊆AR(t),t≥0  and (iv)-(v) taken together imply RB(t)A⊆ARB(t), t≥0. Hence, RB(t)(A+B)⊆(A+B)RB(t), t≥0. By Lemma 2,
(19)zk~(z1/α)(z-A)-1Cx=∫0∞e-z1/αtR(t)xdt,x∈E,ℜ(z1/α)>ω,k~(z1/α)≠0.
Exploiting the closedness of A and the product rule, we easily infer from (19) that, for every x∈E, l∈ℕ and for every z∈ℂ with ℜ(z1/α)>ω and k~(z1/α)≠0:(20)Adldzl∫0∞e-z1/αtR(t)xdt=Adldzl[zk~(z1/α)(z-A)-1Cx]=dldzl[zk~(z1/α)A(z-A)-1Cx]=zdldzl[zk~(z1/α)(z-A)-1Cx]+ldl-1dzl-1[zk~(z1/α)(z-A)-1Cx]-dldzl[zk~(z1/α)Cx].
Fix, for the time being, x∈E and λ∈ℂ with ℜλ>ω′′′ and k1~(λ)≠0. Then (11) implies k~(λ)≠0. By (iv)-(v) and the dominated convergence theorem, it follows that the Laplace transform of power series appearing in (17) can be computed term by term. Using this fact as well as (5), (19), and (ii)-(iii), we obtain that
(21)ℒ(RB(t)x)(λ)=λαk1~(λ)∑i=0∞‍(-B)ii!×∑l=0i‍(il)(1zk~(z1/α))z=λα(i-l)×(dldzl[zk~(z1/α)(z-A)-1Cx])z=λα.
Our goal is to prove that
(22)(I-A+Bλα)ℒ(RB(t)x)(λ)=k1~(λ)Cx.
By the product rule, we get(23)∑i=1∞‍(-B)ii!∑l=1i‍(il)(1zk~(z1/α))z=λα(i-l)(dldzl[zk~(z1/α)Cx])z=λα=-λαk~(λ)∑i=1∞‍(-B)ii!(1zk~(z1/α))z=λα(i)Cx;
notice that the convergence of last series follows from the conditions (iii)-(iv). Taking into account (5), (ii), and (vi), one yields that
(24)1λαk1~(λ)∫0∞e-λt∑i=2∞‍∑l=2i‍∑l0=1l-1‍(-B)ii!l(il)×∫0t(t-s)l0R(t-s)×ki-1,l0,l-1(s)xdsdt=1λαk1~(λ)∑i=2∞‍∑l=2i‍∑l0=1l-1‍(-B)ii!l(il)×∫0∞e-λt∫0t(t-s)l0R(t-s)×ki-1,l0,l-1(s)xdsdt=∑i=2∞‍∑l=2i‍∑l0=1l-1‍(-B)ii!l(il)cl0,l-1,αλl0-α(l-1)×∫0∞e-λttl0R(t)xdt=∑i=2∞‍(-B)ii!×∑l=2i‍l(il)(1zk~(z1/α))z=λα(i-l)×(dl-1dzl-1[zk~(z1/α)(z-A)-1Cx])z=λα,
which implies that the series
(25)∑i=1∞‍(-B)ii!∑l=1i‍l(il)(1zk~(z1/α))z=λα(i-l)×(dl-1dzl-1[zk~(z1/α)(z-A)-1Cx])z=λα
is also convergent. Now we get from (20)-(21) and (23)-(24):
(26)(I-A+Bλα)ℒ(RB(t)x)(λ)=λα∑i=0∞‍(-B)ii!∑l=0i‍(il)(1zk~(z1/α))z=λα(i-l)×(dldzl[zk~(z1/α)(z-A)-1Cx])z=λα-∑i=1∞‍(-B)ii!∑l=1i‍(il)(1zk~(z1/α))z=λα(i-l)×(zdldzl[zk~(z1/α)(z-A)-1Cx]+ldl-1dzl-1[zk~(z1/α)(z-A)-1Cx]-dldzl[zk~(z1/α)]Cx)z=λα-λαk~(λ)∑i=0∞‍(-B)ii!(1zk~(z1/α))z=λα(i)A(λα-A)-1Cx+λα∑i=0∞‍(-B)i+1i!∑l=0i‍(il)(1zk~(z1/α))z=λα(i-l)×(dldzl[zk~(z1/α)(z-A)-1Cx])z=λα=λ2αk~(λ)∑i=0∞‍(-B)ii!(1zk~(z1/α))z=λα(i)(λα-A)-1Cx-∑i=1∞‍(-B)ii!∑l=1i‍(il)(1zk~(z1/α))z=λα(i-l)×(ldl-1dzl-1[zk~(z1/α)(z-A)-1Cx])z=λα-λαk~(λ)∑i=1∞‍(-B)ii!(1zk~(z1/α))z=λα(i)Cx-λαk~(λ)∑i=0∞‍(-B)ii!(1zk~(z1/α))z=λα(i)×(-Cx+λα(λα-A)-1Cx)+λα∑i=0∞‍(-B)i+1i!∑l=0i‍(il)(1zk~(z1/α))z=λα(i-l)×(dldzl[zk~(z1/α)(z-A)-1Cx])z=λα=Cx-∑i=1∞‍(-B)ii!×∑l=1i‍(il)(1zk~(z1/α))z=λα(i-l)×(ldl-1dzl-1[zk~(z1/α)(z-A)-1Cx])z=λα+λα∑i=0∞‍(-B)i+1i!∑l=0i‍(il)(1zk~(z1/α))z=λα(i-l)×(dldzl[zk~(z1/α)(z-A)-1Cx])z=λα=Cx,
because the sum of coefficients of (-B)i (i≥1) in the last two series equals 0; this follows from an elementary calculus involving only the product rule. Assume now x∈D(A),ℜλ>ω′′′,k1~(λ)≠0 and (I-((A+B)/λα))x=0. By (22) and RB(t)(A+B)⊆(A+B)RB(t), t≥0, we obtain that
(27)k1~(λ)Cx=(I-A+Bλα)ℒ(RB(t)x)(λ)=ℒ(RB(t)(I-A+Bλα)x)(λ)=0,
which implies Cx=x=0. Thus, {λα:ℜλ>ω′′′,k1~(λ)≠0}⊆ρC(A+B) and
(28)k1~(λ)(I-A+Bλα)-1Cx=∫0∞e-λtRB(t)xdt,x∈Eℜλ>ω′′′,k1~(λ)≠0.
The proof of theorem completes an application of Lemma2.Remark 6.
(i) By [33, Proposition 2.4(i)], we get that (RB(t))t≥0 is a unique (gα,k1)-regularized C-resolvent family with the properties stated in the formulation of Theorem 5.
(ii) The following comment is also applicable to Theorem7 given below. Assume k(t)=k1(t),t≥0,n∈ℕ and the conditions (iv)-(vi) of Theorem 5 hold with |B|⊛i/i! replaced by |B|⊛i/nii! therein. Writing A+B as A+∑i=1n‍B/n and applying Theorem 5 successively n times, we obtain that A+B is a subgenerator of a global (gα,k)-regularized C-resolvent family (RB(t))t≥0 satisfying (18). Furthermore, the family {e-(ω+nω′)tRB(t):t≥0} is equicontinuous.
(iii) It is not clear whether there exist functionsk(t) and k1(t) such that the conditions (ii)–(vi) of Theorem 5 are fulfilled in the case α∈(0,1).Theorem 7.
Consider the situation of Theorem5 with (R(t))t≥0 being an exponentially equicontinuous, analytic (gα,k)-regularized C-resolvent family of angle β∈(0,π/2]. Assume that, for every γ∈(0,β), there exists ωγ≥0 such that the set {e-ωγℜzR(z):z∈Σγ} is equicontinuous. Assume, additionally, that there exists ε>0 such that, for every γ∈(0,β), there exist ωγ,1≥max(sup{abs(ik):i≥1},ωγ) and ωγ,2≥max(sup{abs(ki,l0,l):1≤l≤i,1≤l0≤l},ωγ+ε) with the following properties.(i)
For everyi∈ℕ0, the function λ↦ki~(λ),λ>ωγ,1  can be analytically extended to the sector ωγ,1+Σ(π/2)+γ and the following holds:
(29)∑i=0∞‍|B|⊛ii!supλ∈ωγ,1+Σ(π/2)+γ|ki~(λ)|<∞.(ii)
For everyi,l0,l∈ℕ with 1≤l≤i and 1≤l0≤l, the function λ↦ℒ(ki,l0,l(t))(λ), λ>ωγ,2 can be analytically extended to the sector ωγ,2+Σ(π/2)+γ and the following holds:
(30)∑i=1∞‍∑l=1i‍∑l0=1l‍|B|⊛ii!(il)l0!2πl0(εcosγ)l0×supλ∈ωγ,2+Σ(π/2)+γ|ℒ(ki,l0,l(t))(λ)|<∞.Then (RB(t))t≥0 is an exponentially equicontinuous, analytic (gα,k1)-regularized C-resolvent family of angle β.Proof.
Letp∈⊛,x∈E,γ∈(0,β)    and ε∈(0,(1/3)min(γ,(π/2)-γ)). Then Stirling’s formula implies that there exists κ≥1 such that
(31)|z|l0eωγℜz≤(ℜz)l0(cosγ)l0eωγℜz≤e-l0(cosγ)l0l0l0εl0e(ωγ+ε)ℜz≤κl0!2πl0(εcosγ)l0e(ωγ+ε)ℜz≤κl0!2πl0(εcosγ)l0eωγ,2ℜz
for all z∈Σγ and l0∈ℕ. By [33, Theorem 3.4(i)] and the proof of implication (i) ⇒ (ii) of [39, Theorem 2.6.1], we obtain that the mapping λ↦ℒ(ik*R(·)x)(λ),λ>ωγ,1, respectively,  λ↦ℒ(ki,l0,l*·l0R(·)x)(λ),λ>ωγ,2   can be analytically extended to the sector ωγ,1+Σ(π/2)+γ, respectively, ωγ,2+Σ(π/2)+γ, as well as that there exist cp>0 and qp∈⊛, independent of x, such that
(32)supλ∈ωγ,1+Σ(π/2)+γ-εp((λ-ωγ,1)ki~(λ)ℒ(R(t)x)(λ))≤cpqp(x)sinεsupλ∈ωγ,1+Σ(π/2)+γ-ε|ki~(λ)|
and that, for every i,l0,l∈ℕ with 1≤l≤i and 1≤l0≤l,
(33)supλ∈ωγ,2+Σ(π/2)+γ-εp((λ-ωγ,2)ℒ(ki,l0,l(t))(λ)ℒ(tl0R(t)x)(λ))≤κcpqp(x)sinεl0!2πl0(εcosγ)l0×supλ∈ωγ,2+Σ(π/2)+γ-ε|ℒ(ki,l0,l(t))(λ)|.
Using (32)-(33), [33, Theorem 3.4(i)] and the proof of implication (ii) ⇒ (i) of [39, Theorem 2.6.1], it follows that the functions t↦(ik*R(·)x)(t),t>0 and t↦(ki,l0,l*·l0R(·)x)(t),t>0 can be analytically extended to the sector Σγ and that the following estimates hold:
(34)p((k*R(·)xi)(z))≤cpqp(x)sinεsupλ∈ωγ,1+Σ(π/2)+γ-ε|ki~(λ)|(e1+ωγ,1ℜz+eωγ,1ℜzπsinε),z∈Σγ-3ε,p((ki,l0,l*·l0R(·)x)(z))≤κcpqp(x)sinεl0!2πl0(εcosγ)l0×supλ∈ωγ,2+Σ(π/2)+γ-ε|ℒ(ki,l0,l(t))(λ)|×(e1+ωγ,2ℜz+eωγ,2ℜzπsinε),z∈Σγ-3ε.
Since Vitali’s theorem holds in our framework (cf. e.g. [33, Lemma 3.3]), we easily infer from (29)-(30), (34), and the arbitrariness of γ and ε that the mapping t↦RB(t)x,t>0 can be analytically extended to the sector Σβ by the formula (17). Thanks to the proof of Theorem 5, the series appearing in (17) converge uniformly on compact subsets of [0,∞), which implies limt→0+∑i=0∞‍((-B)i/i!)ciR(t)x=∑i=0∞‍((-B)i/i!)ciR(0),limt→0+∑i=0∞‍((-B)i/i!)(ik*R(·)x)(t)=0, and limt→0+∑i=1∞‍∑l=1i‍∑l0=1l‍((-B)i/i!)(il)(ki,l0,l*·l0R(·)x)(t)=0. Furthermore, the functions z↦f1(z):=∑i=0∞‍((-B)i/i!)ciR(z)x,z∈Σβ,z↦f2(z):=∑i=0∞‍((-B)i/i!)(ik*R(·)x)(z),z∈Σβ, and z↦f3(z):=∑i=1∞‍∑l=1i‍∑l0=1l‍((-B)i/i!)(il)(ki,l0,l*·l0R(·)x)(z),z∈Σβ are analytic, and the set {e-(ωγ+ωγ,1+ωγ,2)ℜzfj(z):1≤j≤3,z∈Σγ-3ε} is bounded. An application of [33, Theorem 3.4(ii)] gives that the mapping z↦RB(z)x,z∈Σβ∪{0} is continuous on any closed subsector of Σβ∪{0}, which completes the proof of theorem.It would take too long to go into details concerning stability of certain differential properties ([40, 41]) under bounded commuting perturbations described in Theorem 5.Letα>0, let β>0, and let the Mittag-Leffler functionEα,β(z) be defined by Eα,β(z):=∑n=0∞‍zn/Γ(αn+β),z∈ℂ. Set Eα(z):=Eα,1(z),z∈ℂ. Then it is well known (cf. [28, 42–44]) that Eα′(z)=Eα,α(z)/α,   z∈ℂ and that, for every α>1, there exist bα≥1 and cα≥1 such that
(35)Eα,α(t)≤bαt(1-α)/αexp(t1/α),t>0,Eα(t)≤cαexp(t1/α),t≥0.
It is noteworthy that the assumptions of Theorems 5 and 7 hold provided α>1 and k(t)=k1(t)=gr+1(t), where r≥0. In this case,  c0=1,k0(t)=0,ci=0,i≥1,
(36)ik(t)=(r+1α-1)⋯(r+1α-i)gαi(t),t≥0,i≥1
and, for every i,l0,l∈ℕ with 1≤l≤i and 1≤l0≤l,
(37)ki,l0,l(t)=cl0,l,α(r+1α-1)⋯(r+1α-(i-l))gαi-l0(t),t>0.
In order to verify (iv)–(vi), notice that there exists a constant cr,α≥1 such that |(((r+1)/α)-1)⋯(((r+1)/α)-i)|≤cr,αi! for all i∈ℕ. Then we obtain from (35) and Lemmas 3–4 that
(38)∑i=1∞‍|B|⊛ii!|(r+1α-1)⋯(r+1α-i)|∫0t‍gαi(s)ds≤cr,α∑i=1∞‍(tα|B|⊛)iΓ(αi+1)=cr,α(Eα(tα|B|⊛)-1)≤cr,αcαet|B|⊛1/α,t≥0,(39)∑i=1∞∑l=1i∑l0=1l|B|⊛ii!(il)∫0t‍(t-s)l0|ki,l0,l(s)|ds≤∑i=1∞‍∑l=1i‍∑l0=1l‍|B|⊛ii!(il)|cl0,l,α|×|(r+1α-1)⋯(r+1α-(i-l))|l0!gαi+1(t)≤cr,αα∑i=1∞‍i(tα|B|⊛)iΓ(αi+1)=cr,ααtα|B|⊛∑i=1∞‍i(tα|B|⊛)i-1Γ(αi+1)=cr,ααtα|B|⊛(∑i=1∞‍ziΓ(αi+1))z=tα|B|⊛′=cr,αα2tα|B|⊛Eα,α(tα|B|⊛)≤cr,αbαα2tα|B|⊛(tα|B|⊛)(1-α)/αet|B|⊛1/α≤cr,αbαα2t|B|⊛1/αet|B|⊛1/α,t≥0,
proving the conditions (iv)-(v) and
(40)∑i=2∞‍∑l=2i‍∑l0=1l-1‍|B|⊛ii!l(il)∫0t(t-s)l0|ki-1,l0,l-1(s)|ds≤cr,α∑i=2∞‍∑l=2i‍∑l0=1l-1‍|B|⊛i(l-1)!l0!|cl0,l-1,α|gα(i-1)+1(t)≤cr,αα|B|⊛∑i=2∞‍(i-1)(tα|B|⊛)i-1Γ(α(i-1)+1)=cr,αα|B|⊛∑i=1∞‍i(tα|B|⊛)iΓ(αi+1)≤cr,αbαα2t|B|⊛(1+α)/αet|B|⊛1/α,t≥0,
proving the condition (vi). Assume now, with the notation used in the formulation of Theorem 7 that γ∈(0,β),ωγ,1≥ωγ,ωγ,1>0,(ωγ,1cosγ)α>|B|⊛,ε=1/cosγ,ωγ,2≥ωγ+ε, and (1+ωγ,2cosγ)α-1>|B|⊛. Then
(41)∑i=1∞‍|B|⊛ii!supλ∈ωγ,1+Σ(π/2)+γ|ki~(λ)|≤cr,α∑i=0∞‍|B|⊛isupλ∈ωγ,1+Σ(π/2)+γ1|λ|αi≤cr,α∑i=0∞‍|B|⊛i(ωγ,1cosγ)αi<∞,(42)∑i=1∞‍∑l=1i‍∑l0=1l‍|B|⊛ii!(il)l0!2πl0(εcosγ)l0×supλ∈ωγ,2+Σ(π/2)+γ|ℒ(ki,l0,l(t))(λ)|≤cr,α∑i=1∞‍∑l=1i‍∑l0=1l‍|B|⊛il!l0!|cl0,l,α|supλ∈ωγ,2+Σ(π/2)+γ1|λ|αi-l0≤cr,αα∑i=1∞‍∑l=1i‍∑l0=1l‍|B|⊛i(1+ωγ,2cosγ)αi-l0≤cr,αα∑i=1∞‍i2|B|⊛i(1+ωγ,2cosγ)(α-1)i<∞,
proving the conditions (29)-(30).Corollary 8.
Supposeα>1,ω≥0,r≥0 and A is a subgenerator of a global r-times integrated (gα,C)-regularized resolvent family (R(t))t≥0 satisfying (2) with a(t)=gα(t) and k(t)=gr+1(t). Let the family {e-ωtR(t):t≥0} be equicontinuous and let B∈L(E) satisfy the condition (i) quoted in the formulation of Theorem 5. Then A+B is a subgenerator of a global r-times integrated (gα,C)-regularized resolvent family (RB(t))t≥0 satisfying (18) with k1(t)=k(t). Furthermore, the family {(1+t)-1exp(-(ω+|B|⊛1/α)t)RB(t):t≥0} is equicontinuous, and (RB(t))t≥0 is an exponentially equicontinuous, analytic r-times integrated (gα,C)-regularized resolvent family of angle β∈(0,π/2] provided that (R(t))t≥0 is.Remark 9.
It is worthwhile to mention (cf. [1, Theorem 2.5.6]) that Corollary 8 remains true, with a different upper bound for the growth order of (RB(t))t≥0, in the case α=1. Using [33, Lemma 3.3] and the proof of cited theorem, it follows that (RB(t))t≥0 is entire provided that α∈ℕ and that (R(t))t≥0 is entire.Example 10.
Corollary8 is a proper extension of [45, Lemma 4.7] provided α=2 and B=zI (z∈ℂ), which can be applied in the analysis of the problem
(43)utt+2βut-Δu+2∑i=1n‍αiuxi+μu=0
in Lp([0,π]n), with Dirichlet boundary conditions; here we assume n∈ℕ,1≤p<∞ and β,αi,μ∈ℂ (see e.g., [46, pages 144-145] and [15, Theorem 4.2]). It is clear that Corollary 8 can be applied to (r-coercive) differential operators generating integrated cosine functions ([2, 14, 15, 47–50]) or exponentially equicontinuous (gα,C)-regularized resolvent families ([23, 24]); in what follows, we will apply Corollary 8 to abstract differential operators generating C-regularized cosine functions. Let E be one of the spaces Lp(ℝn) (1≤p≤∞),C0(ℝn),Cb(ℝn), BUC(ℝn), let 0≤l≤n and let ℱ-1 denote the inverse Fourier transform. Put ℕ0l:={α∈ℕ0n:αl+1=⋯=αn=0} and, for every l=0,1,…,n, El:={f∈E:f(α)∈E for all α∈ℕ0l}. Then the family of seminorms (pα(f):=||f(α)||E,f∈El;α∈ℕ0l) induces a Fréchet topology on El. Let Tl possess the same meaning as in [51] and let m∈ℕ,aα∈ℂ,0≤|α|≤m. Consider the operator P(D)f:=∑|α|≤m‍aαf(α) with its maximal distributional domain. Set P(x):=∑|α|≤m‍aαi|α|xα,x∈ℝn,ht,β(x)≔(1+|x|2)-β/2∑j=0∞‍(t2jP(x)j/(2j)!),x∈ℝn,t≥0,β≥0,Ω(ω):={λ2:ℜλ>ω}, if ω>0, and Ω(ω):=ℂ∖(-∞,ω2], if ω≤0. Assume r∈[0,m],ω∈ℝ, and the following condition:(♭)
P(x)∉Ω(ω),x∈ℝn and, in the case r∈(0,m], there exist σ>0 and σ′>0 such that ℜP(x)≤-σ|x|r+σ′,x∈ℝn.
Then, for everyl=0,1,…,n, there exists M≥1 such that, for every β>(m-(r/2))(n/4),P(D) generates an exponentially equicontinuous Tl(〈(1+|x|2)-β〉)-regularized cosine function (Cβ(t))t≥0 in El satisfying Cβ(t)f=ℱ-1ht,β*f,t≥0,f∈El and pα(Cβ(t)f)≤Mpα(f)gn/2(t),t≥0,f∈El,α∈ℕ0l, with gn/2(t) being the function defined on [52, page 40]; cf. [33, 51, 52] for full details. If 1<p<∞ and E=Lp(ℝn), then the previous result can be slightly refined by allowing that β takes the value (1/2)(m-r/2)n|1/p-1/2|. Given φ∈L1(ℝn), define the bounded linear operator B on El by (Bf)(x):=∫ℝn‍φ(x-t)f(t)dt,f∈El,x∈ℝn. Then BP(D)⊆P(D)B,Tl(〈(1+|x|2)-β〉)B=BTl(〈(1+|x|2)-β〉) and pα(Bf)≤||φ||L1(ℝn)pα(f),f∈E,α∈ℕ0l. Applying Corollary 8, we get that P(D)+B generates an exponentially equicontinuous Tl(〈(1+|x|2)-β〉)-regularized cosine function (Cβ,B(t))t≥0 in El.
Assume nowα>1,ϱ>0,σ∈(0,1) and k(t)=k1(t)=ℒ-1(λ-αe-ϱλσ)(t), t≥0. By the consideration given in [1, Remark 2.5.4(iii)], it follows that, for every l∈ℕ, there exist real numbers (pm,l,α,ϱ,σ)1≤m≤l such that p1,l,α,ϱ,σ=ϱ(σ/α)((σ/α)-1)⋯((σ/α)-(l-1)),pl,l,α,ϱ,σ=(ϱ(σ/α))l, and that the following holds:
(44)dldzl(1zk~(z1/α))=dldzleϱzσ/α=eϱzσ/α∑m=1l‍pm,l,α,ϱ,σzm(σ/α)-lz>0,pm,l+1,α,ϱ,σ=ϱσαpm-1,l,α,ϱ,σ+(mσα-l)pm,l,α,ϱ,σ,2≤m≤l.
This implies c0=1,k0(t)=0,ci=0,i≥1,
(45)ik(t)=∑m=1i‍pm,i,α,ϱ,σgαi-mσ(t),t>0,i≥1,ki,l0,l(t)=cl0,l,α∑m=1i-l‍pm,i-l,α,ϱ,σgαi-l0-mσ(t),t>0,1≤l<i,1≤l0≤l,(46)ki,l0,i(t)=cl0,i,αgαi-l0(t),t>0,1≤l0≤i.
By means of (44) and the proof of Lemma 3, we obtain the existence of a constant ζα,ϱ,σ≥1 such that
(47)∑m=1l‍m!|pm,l,α,ϱ,σ|≤ζα,ϱ,σll!∀l∈ℕ.
In what follows, we assume that ζα,ϱ,σ≥1 is minimal with respect to (47); notice that ζα,ϱ,σ>ϱσ/α and that it is not clear whether Lemma 4 can be reconsidered in the newly arisen situation. Then
(48)∑i=1∞‍∑l0=1i‍|B|⊛ii!∫0t(t-s)l0|ki,l0,i(s)|ds=∑i=1∞‍∑l0=1i‍|B|⊛ii!|cl0,i,α|l0!∫0t(t-s)l0l0!sαi-l0-1Γ(αi-l0)ds≤∑i=1∞‍(tα|B|⊛)iΓ(αi+1)≤Eα(tα|B|⊛)≤cαet|B|⊛1/α,t≥0.
Since Γ(·) is increasing in (ξ,∞), where ξ~1.4616…, we obtain that
(49)Γ(αi+1)Γ(αi-mσ+1)m!≤Γ(αi+1)Γ(αi-m+1)m!=αi(αi-1)⋯(αi-m+1)m!≤(⌈αi⌉m),
provided i≥2 and 1≤m≤i-1. Combining this with (35), Lemmas 3 and 4 and (47), we get
(50)∑i=2∞‍∑l=1i-1‍∑l0=1l‍|B|⊛ii!(il)∫0t(t-s)l0|ki,l0,l(s)|ds≤∑i=2∞‍∑l=1i-1‍∑l0=1l‍|B|⊛ii!(il)l0!|cl0,l,α|×∑m=1i-l‍m!|pm,i-l,α,ϱ,σ|tαi-mσΓ(αi-mσ+1)m!≤1α∑i=2∞‍∑l=1i-1‍|B|⊛iΓ(αi+1)ζα,ϱ,σi-l∑m=1i-l‍tαi-mσΓ(αi+1)Γ(αi-mσ+1)m!≤1α∑i=2∞‍∑l=1i-1‍|B|⊛iΓ(αi+1)ζα,ϱ,σi-l∑m=1i-l‍tαi-mσ(⌈αi⌉m)≤1α(1+t-σ)∑i=2∞‍∑l=1i-1‍|B|⊛iΓ(αi+1)ζα,ϱ,σi-l(t+t1-σ)αi≤1αζα,ϱ,σ(1+t-σ)∑i=1∞‍i(|B|⊛(t+t1-σ)αζα,ϱ,σ)i+1Γ(αi+1)≤1α(1+t-σ)|B|⊛(t+t1-σ)α×(∑i=1∞‍iziΓ(αi+1))z=|B|⊛(t+t1-σ)αζα,ϱ,σ(51)≤bαα2(1+t-σ)|B|⊛(1+α)/α×(t+t1-σ)1+αζα,ϱ,σ1/αe|B|⊛1/α(t+t1-σ)ζα,ϱ,σ1/α,t≥1.
Noticing that tαi-mσ≤1,t∈[0,1),i≥2,1≤m≤i-1, we obtain from (50) that there exists ξα,ϱ,σ≥1 such that
(52)∑i=2∞‍∑l=1i-1‍∑l0=1l‍|B|⊛ii!(il)∫0t(t-s)l0|ki,l0,l(s)|ds≤ξα,ϱ,σ,t∈[0,1).
By (48)–(52), (v) holds for any ω′>(|B|⊛ζα,ϱ,σ)1/α. In almost the same way, one can prove that (iv) and (vi) hold for any ω′>(|B|⊛ζα,ϱ,σ)1/α. Assume now that (R(t))t≥0 is an exponentially equicontinuous, analytic (gα,k)-regularized C-resolvent family of angle β∈(0,π/2],γ∈(0,β), ωγ,1≥ωγ, ωγ,1>0, (ωγ,1cosγ)α-σ>|B|⊛ζα,ϱ,σ, ε=1/cosγ, ωγ,2≥ωγ+ε and (1+ωγ,2cosγ)α-1>|B|⊛ζα,ϱ,σ. Then
(53)∑i=0∞‍|B|⊛ii!supλ∈ωγ,1+Σ(π/2)+γ|k~i(λ)|≤∑i=0∞‍|B|⊛ii!∑m=1i‍m!|pm,i,α,ϱ,σ|(ωγ,1cosγ)αi-mσ≤μσ,γ∑i=0∞‍|B|⊛iζα,ϱ,σi(ωγ,1cosγ)(α-σ)i<∞
for an appropriate constant μσ,γ≥1,
(54)∑i=1∞‍∑l0=1i‍|B|⊛ii!l0!2πl0(εcosγ)l0supλ∈ωγ,2+Σ(π/2)+γ|ℒ(ki,l0,i(t))(λ)|≤∑i=1∞‍i|B|⊛i(1+ωγ,2)(α-1)i<∞,
and, for every ε>0 with (1+ωγ,2cosγ)α-1>|B|⊛(ζα,ϱ,σ+ε), there exists με>0 such that
(55)∑i=2∞‍∑l=1i-1‍∑l0=1l‍|B|⊛ii!(il)l0!2πl0(εcosγ)l0×supλ∈ωγ,2+Σ(π/2)+γ|ℒ(ki,l0,l(t))(λ)|≤∑i=2∞‍∑l=1i-1‍∑l0=1l‍|B|⊛i(i-l)!l!l0!|cl0,l,α|∑m=1i-l‍|pm,i-l,α,ϱ,σ|(1+ωγ,2cosγ)αi-l0-mσ≤∑i=2∞‍∑l=1i-1‍∑l0=1l‍|B|⊛i(i-l)!l!l0!|cl0,l,α|∑m=1i-l‍|pm,i-l,α,ϱ,σ|(1+ωγ,2cosγ)(α-1)i≤1α∑i=2∞‍∑l=1i-1‍|B|⊛iζα,ϱ,σi-l(1+ωγ,2cosγ)(α-1)i≤1α∑i=2∞‍(i-1)|B|⊛i(ζα,ϱ,σ+ε)i(1+ωγ,2cosγ)(α-1)i<∞,
proving the conditions (29)-(30).Corollary 11.
Letα>1,ω≥0,ϱ>0,σ∈(0,1),k(t)=ℒ-1(λ-αe-ϱλσ)(t),t≥0, and let A be a subgenerator of a global (gα,k)-regularized C-resolvent family (R(t))t≥0 satisfying (2) with a(t)=gα(t). Let B∈L(E) satisfy the condition (i) quoted in the formulation of Theorem 5. Then A+B is a subgenerator of a global (gα,k)-regularized C-resolvent family (RB(t))t≥0 satisfying (18) with k1(t)=k(t). Furthermore, for every ε>0, the family {exp(-(ω+(|B|⊛ζα,ϱ,σ)1/α+ε)t)RB(t):t≥0} is equicontinuous, and (RB(t))t≥0 is an exponentially equicontinuous, analytic (gα,k)-regularized C-resolvent family of angle β∈(0,π/2] provided that (R(t))t≥0 is.Example 12.
Lets>1,
(56)E:={f∈C∞[0,1];∥f∥=supp≥0∥f(p)∥∞p!s<∞},(57)A:=-dds,D(A):={f∈E;f′∈E,f(0)=0}.
Then ρ(A)=ℂ,A generates a tempered ultradistribution semigroup of (p!s)-class, and A cannot be the generator of a distribution semigroup since A is not stationary dense (see e.g., [53, Example 1.6] and [41]). If f∈E,t∈[0,1] and λ∈ℂ, set fλ1(t):=∫0t‍e-λ(t-s)f(s)ds and fλ2(t):=∫0t‍eλ(t-s)f(s)ds. Then fλ1(·),fλ2(·)∈E,λ∈ℂ, and there exist b>0 and M≥1, independent of f(·), such that
(58)∥fλ1(·)∥≤M||f||eb|λ|1/s,ℜλ≥0,f∈E.
It is clear that ||fλ2(·)||L∞[0,1]≤e|λ|||f||,ℜλ≥0 and ||(d/dt)fλ2(·)||L∞[0,1]≤(|λ|e|λ|+1)||f||,ℜλ≥0. Proceeding by induction, we obtain that, for every n≥2,t∈[0,1] and λ∈ℂ with ℜλ≥0:(59)dndtnfλ2(t)=dn-1dtn-1f(t)+∑k=1n-1‍λkdn-1-kdtn-1-kf(t)+λnfλ2(t).
On the other hand, [54, Proposition 4.5] implies that there exists c>0 such that ∑p=0∞‍tp/p!s=O(exp(ct1/s)),t≥0. Combined with (59) and the logarithmic convexity, the last estimate yields
(60)1n!s∥dndtnfλ2(·)∥L∞[0,1]≤||f||+||f||ec|λ|1/s+|λ|nn!se|λ|||f||≤(1+ec|λ|1/s+ec|λ|1/se|λ|)||f||,ℜλ≥0,λ≠0.
In view of (60) we get that, for every η>1, there exists Mη≥1, independent of f(·), such that
(61)∥fλ2(·)∥≤Mη∥f∥eη|λ|,ℜλ≥0,f∈E.
Consider now the complex polynomial P(z)=∑j=0n‍ajzj, z∈ℂ, an≠0, n≥2. Set, for every λ∈ℂ,Pλ(·):=P(·)-λ and consider the operator P(A) defined by
(62)D(P(A))≔D(An),P(A)f≔∑j=0n‍ajAjf,f∈D(P(A)).
Clearly, P(A) is not stationary dense. Let r>0 and d>0 be such that P(z)≠0,|z|≥r and P′(z)≠0,|z|≥d. Let z1,λ,…,zn,λ denote the zeros of the polynomial z↦Pλ(z),z∈ℂ and let 0<m:=min|z|≥d+1|P′(z)|. Then an old result of Walsh [55] says that |zj,λ|≤r+|an|-1/n|λ|1/n,1≤j≤n,λ∈ℂ. Furthermore, it is checked at once that there exists a sufficiently large λ0>0 such that zj,λ is a simple zero of Pλ(z) and that |zj,λ|≥d+1, provided |λ|≥λ0 and 1≤j≤n. Therefore, for every λ∈ℂ with |λ|≥λ0 and for every i,j∈{1,…,n} with i≠j, the following holds:
(63)d+1≤|zj,λ|≤r+|an|-1/n|λ|1/n,|P′(zj,λ)|≥m,zi,λ≠zj,λ.
It is straightforward to verify that
(64)ρ(p(A))=ℂ,R(λ:p(A))=(-1)n+1an-1R(z1,λ:A)⋯R(zn,λ:A),λ∈ℂ.
Assume now |λ|≥λ0. Then de L'Hospital's rule implies
(65)an∏1≤i≤ni≠j(zi,λ-zj,λ)=(-1)n+1P′(zj,λ),1≤j≤n.
Using the resolvent equation, (58), (61)–(63), and (65), one can rewrite and evaluate the right-hand side of equality appearing in (64) as follows:
(66)∥(-1)n+1an-1R(z1,λ:A)⋯R(zn,λ:A)∥=∥(-1)n+1an-1∑j=1n‍R(zj,λ:A)∏1≤i≤ni≠j(zi,λ-zj,λ)∥=∥∑j=1n‍R(zj,λ:A)P′(zj,λ)∥≤1m∑j=1n‍∥R(zj,λ:A)∥.
By (64) and (66) we finally get that, for every η>1,
(67)∥R(λ:p(A))∥=O(eb|an|-1/n|λ|1/ns+eη|an|-1/n|λ|1/n),λ∈ℂ.
Since the preceding estimate holds for any λ∈ℂ, it is quite complicated to inscribe here all of its consequences (cf. [1, 16], [32, (2.35)–(2.37)], [56, 57]); for example, P(A) generates a tempered ultradistribution sine of (p!s)-class provided n≥2s, and P(A) generates an exponentially bounded, ℒ-1(e-ϱλ1/n)-convoluted group provided ϱ>|an|-1/n/cos(π/2n). Let us also mention that the consideration given in example following [41, Corollary 3.8] enables one to construct important examples of (pseudo-)differential operators generating ultradistribution sines, and that the estimate (67) can be derived, with insubstantial technical modifications, in the case of a general sequence (Mp) of positive numbers satisfying M0=1 and Mp+q≥MpMq(p,q≥0). In what follows, we will illustrate an application of Corollary 11. Suppose n>α≥1, δ∈(0,π/2], (π/2+δ)α/n<π/2, ϱ≥1/cos((π/2+δ)α/n) and k(t)=ℒ-1(λ-αe-ϱλα/n)(t), t≥0. By [32, Theorem 2.17] and (67), P(A) is the integral generator of an exponentially bounded, analytic (gα,k)-regularized resolvent family of angle δ (cf. also [41, Proposition 3.12]). Let φ∈E and Bf(t):=(φ*f)(t),t∈[0,1], f∈E. Then B∈L(E),BP(A)⊆P(A)B and, by Corollary 11, P(A)+B is the integral generator of an exponentially bounded, analytic (gα,k)-regularized resolvent family of angle δ.
The following extension of [9, Theorem 3.1], [32, Theorem 2.12] has been recently established in [33]; cf. also [39, Theorem 3.15.6], [4, Theorem 1.1].Theorem 13.
SupposeM>0,ω1≥ω≥0,A is a subgenerator of an (a,k)-regularized C-resolvent family (R(t))t≥0 such that p(R(t)x)≤Meωtp(x),x∈E,t≥0,p∈⊛ and z∈ℂ. Let B:D(A)¯→E be a linear operator such that BCx=CBx,x∈D(A) and that, for every p∈⊛, there exist cp>0 and q∈⊛ satisfying p(C-1Bx)≤cpq(x),x∈D(A)¯. Let (P1) hold for a(t),k(t),b(t) and let a~(λ)/k~(λ)=b~(λ)+z,ℜλ>ω1,k~(λ)≠0. Suppose μ>ω1,γ∈[0,1) and
(68)D(A)¯=E,(69)∫0∞e-μtp(C-1B∫0tb(t-s)R(s)xds+zC-1BR(t)x)dt≤γp(x),x∈D(A),p∈⊛
or
(70)(R(t))t≥0satisfies(2),D(A)¯≠Eand(69)holdsforanyx∈E,p∈⊛.
Then the operator A+B is a subgenerator of an (a,k)-regularized C-resolvent family (RB(t))t≥0 satisfying (18) with k1(t) and gα(t) replaced by k(t) and a(t) therein. Furthermore,
(71)p(RB(t)x)≤M1-γeμtp(x),x∈E,t≥0,p∈⊛,(72)RB(t)x=R(t)x+∫0tRB(t-r)(C-1B∫0rb(r-s)R(s)xds+zC-1BR(t)x∫0tRB(t-r))dr,t≥0,x∈D(A),
and (72) holds for any t≥0 and x∈E provided (70).In many cases, we do not have the existence of a functionb(t) and a complex number z such that a~(λ)/k~(λ)=b~(λ)+z,ℜλ>ω1,k~(λ)≠0. The following theorem is an attempt to fill this gap.Theorem 14.
SupposeM,M1>0,ω≥0,l∈ℕ and A is a subgenerator of an (a,k)-regularized C-resolvent family (R(t))t≥0 such that p(R(t)x)≤Meωtp(x),x∈E,t≥0,p∈⊛ and that (2) holds. Let a(t) and k(t) satisfy (P1) and let the following conditions hold:(i)
BCx=CBx,
x∈D(A),
(73)p(CAjC-1Bx)≤M1p(x),x∈D(A)¯,p∈⊛,0≤j≤l-1,p(AlC-1Bx)≤M1p(x),x∈D(A)¯,p∈⊛.(ii)
There exist a functionb(t) satisfying (P1) and a complex number z such that
(74)a~(λ)l+1k~(λ)=b~(λ)+z,ℜλ>max(ω,abs(a),abs(k)),k~(λ)≠0.(iii)
limλ→+∞∫0∞‍e-λt|a(t)|dt=0 and limλ→+∞∫0∞‍e-λt×|b(t)|dt=0.Then, for every x∈E,  there exists a unique solution of the integral equation
(75)RB(t)x=R(t)x+(S*RB)(t)x,t≥0;
furthermore, (RB(t))t≥0 is an (a,k)-regularized C-resolvent family with a subgenerator A+B, there exist μ≥max(ω,abs(a),abs(k)) and γ∈[0,1) such that (71) holds and that (18) holds with k1(t) and gα(t) replaced by k(t) and a(t) therein.Proof.
It is clear that∫0∞‍e-λt|a*,j(t)|dt≤(∫0∞‍e-λt|a(t)|dt)j, j∈ℕ, λ>abs(a). Define, for every x∈D(A)¯ and t≥0,
(76)S(t)x:=∑j=0l-1‍a*,j+1(t)CAjC-1Bx+∫0tb(t-s)R(s)AlC-1Bxds+zR(t)AlC-1Bx.
By [58, Theorem 1.7, page 3] it follows that R(t)x∈D(A)¯,   t≥0,x∈E. Using this fact and (i), we get that S(t)∈L(D(A)¯),   t≥0. Keeping in mind the condition (ii), it is not difficult to prove that, for every x∈D(A)¯,
(77)ℒ(S(t)x)(λ)=a~(λ)(I-a~(λ)A)-1CC-1Bx,x∈D(A)¯,ℜλ>ω1,k~(λ)≠0.
Using the conditions (i) and (iii), we obtain the existence of numbers μ>max(ω,abs(a),abs(k)) and γ∈[0,1) such that
(78)∫0∞e-μtp(S(t)x)dt≤γp(x),x∈D(A)¯,p∈⊛
and that (H1) holds, where(H1)
: For every strongly continuous functionf:[0,∞)→L(E,D(A)¯) such that p(f(t)x)≤Mp(x),x∈E,   t≥0,p∈⊛, the following inequality holds:
(79)∫0te-μ(t-s)p(S(t-s)f(s)x)ds≤γMp(x),x∈E,t≥0,p∈⊛.Now one can define inductively, for every t≥0, the sequence (Tn(t))n∈ℕ0 in L(E,D(A)¯) by T0(t):=R(t) and Tn+1(t)x:=∫0t‍S(t-s)Tn(s)xds,x∈E,n∈ℕ0; observe that, for every n∈ℕ0,   (Tn(t))t≥0 is strongly continuous and that the family {Tn(t):t≥0} is locally equicontinuous (with clear meaning). By (78), (H1), and the proof of [9, Theorem 3.1], it follows inductively that
(80)p(Tn(t)x)≤Mγneμtp(x),x∈E,t≥0,p∈⊛
and that, for every x∈E and t≥0, the sequence (RBn(t)x:=∑i=0n‍Ti(t)x)n is Cauchy in E and therefore convergent. Set RB(t)x:=limn→∞RBn(t)x,x∈E,t≥0. It is obvious that the mapping t↦RB(t)x,t≥0 is continuous for every fixed x∈E as well as that (71) and (75) hold. Therefore, it suffices to show that
(81)k~(λ)(I-a~(λ)(A+B))-1Cx=∫0∞e-λtRB(t)xdtx∈E,ℜλ>μ,k~(λ)≠0.
Towards this end, notice that (78) and (H1) together imply that I-S~(λ) is invertible for ℜλ>μ and (I-S~(λ))-1=∑n=0∞‍[S~(λ)]n,ℜλ>μ. Now we obtain from (75)
(82)RB~(λ)x=(I-S~(λ))-1k~(λ)(I-a~(λ)A)-1Cx,ℜλ>μ,x∈E,
which immediately implies with (77) the validity of (81) in case a~(λ)=0,ℜλ>μ and k~(λ)≠0. Assume now a~(λ)k~(λ)≠0 and ℜλ>μ. Then a straightforward computation involving the equality BCx=CBx,x∈D(A) as well as (77) and (82) shows that the operator I-a~(λ)(A+B) is injective and
(83)(I-S~(λ))-1k~(λ)(I-a~(λ)A)-1C(I-a~(λ)(A+B))x=k~(λ)Cx,x∈D(A).
The representation (I-S~(λ))-1=∑n=0∞‍[((1/a~(λ))-A)-1CC-1B]n implies
(84)(1a~(λ)-(A+B))(I-S~(λ))-1k~(λ)(1a~(λ)-A)-1Cx=k~(λ)Cx,x∈E,R(C)⊆R(I-a~(λ)(A+B)) and (81), finishing the proof of theorem.Remark 15.
Now we will explain how one can reformulate Theorem13 in case in which B is not necessarily bounded operator from D(A)¯ into E (cf. also [3, 7, 8] and the next section). Consider the situation of Theorem 13 with E being complete. Assume (69) and, instead of condition C-1B∈L(E):(♮)
C-1B:D(A)→E and, for every p∈⊛, there exist cp>0 and q∈⊛ such that p(C-1Bx)≤cp(q(x)+q(Ax)),x∈D(A).Denote, with a little abuse of notation, T0(t)=R(t),t≥0,S(t)x=C-1B(∫0t‍b(t-s)R(s)xds+zR(t)x),t≥0,x∈D(A) and T1(t)x=∫0t‍T0(t-s)S(s)xds,t≥0,x∈D(A). Then (♮) implies that the mapping t↦S(t)x,t≥0 is continuous for every x∈D(A) and p(T1(t)x)≤γMp(x),x∈D(A),t≥0,p∈⊛. By [59, Lemma 22.19] and the completeness of E, one can extend the operator T1(t) to the whole space E (t≥0). Proceeding inductively, one can define for each t≥0 the sequence (Tn(t)=∫0t‍Tn-1(t-s)S(s)ds)n∈ℕ0 in L(E) such that p(Tn(t)x)≤γnMp(x), x∈E, t≥0, p∈⊛. The preceding inequality implies that, for every x∈E, the sequence (RBn(t)x≡∑i=0n‍Ti(t)x)n is Cauchy in E and therefore convergent. Put RB(t)x=limn→∞RBn(t)x, x∈E, t≥0. As in the proof of Theorem 14, the mapping t↦RB(t)x, t≥0 is continuous for every fixed x∈E and (71)-(72) hold. Using the closedness of A and the condition (♮), we get ∫0∞‍e-λtS(t)xdt=C-1Ba~(λ)(I-a~(λ)A)-1Cx, x∈D(A), ℜλ>μ, k~(λ)≠0 and C-1Ba~(λ)(I-a~(λ)A)-1C∈L(E), ℜλ>max(ω1,μ), k~(λ)≠0. In view of (69), p(S~(λ)x)≤γp(x),x∈D(A),ℜλ>μ,p∈⊛; by the denseness of D(A) in E, the last estimate holds for all x∈E. Hence, the operator I-S~(λ) is invertible and (I-S~(λ))-1x=∑n=0∞‍[C-1Ba~(λ)(I-a~(λ)A)-1C]nx,x∈E,ℜλ>μ,k~(λ)≠0. Suppose, for the time being, ℜλ>μ and a~(λ)k~(λ)≠0. The closedness of the operator A+B can be proved as follows. Let a net (xτ)τ∈T in E satisfy xτ→x,τ→∞ and (I-a~(λ)(A+B))xτ→y,τ→∞. Then a simple computation shows that (I-a~(λ)(A+B))xτ=(I-S~(λ))(I-a~(λ)A)xτ, which implies (I-a~(λ)A)xτ=(I-S~(λ))-1(I-a~(λ)(A+B))xτ→(I-S~(λ))-1y,τ→∞. Since I-a~(λ)A is closed, we infer that x∈D(A),(I-a~(λ)A)x=(I-S~(λ))-1y and (I-a~(λ)(A+B))x=y. Therefore, the closedness of A+B follows from that of I-a~(λ)(A+B). Suppose now ℜλ>μ and k~(λ)≠0. Similarly as in the proof of Theorem 14, we get R~B(λ)x=R~(λ)(I-S~(λ))-1x, x∈E, the injectiveness of I-a~(λ)(A+B),R(C)⊆R(I-a~(λ)(A+B)) and k~(λ)(I-a~(λ)(A+B))-1Cx=R~B(λ)x,x∈E, which implies that the conclusions of Theorem 13 continue to hold. We left to the interested reader details concerning the possibilities of the extension of [8, Theorems 3.1 and 3.2] and results of [3, 7, 11, 12] to abstract Volterra equations in SCLCSs.Remark 16.
The local Hölder continuity with exponentσ∈(0,1] is an example of the property which is stable under perturbations described in Theorems 13–14 and Remark 15, as indicated below. Consider the situation of Theorem 13 in which D(A) is dense in E. Using the same notation as in Remark 15, one has p(S(t)x)≤cpMq(x)[∫0t‍|b(t-r)|eωrdr+|z|eωt],p∈⊛,t≥0,x∈D(A). Suppose now that, for every T>0 and p∈⊛, there exist cT,p>0 and hT,p∈⊛ such that
(85)p(R(t)x-R(s)x)≤cT,p(t-s)σhT,p(x),x∈E,0≤s<t≤T.
Let T>0 and p∈⊛ be fixed. Then, for every x∈D(A) and 0≤s<t≤T,
(86)p(S(t)x-S(s)x)≤cpq(∫0s‍b(r)(R(t-r)x-R(s-r)x)dr+∫ts‍b(r)R(t-r)xdr)+cp|z|q(R(t)x-R(s)x)≤cp[cT,q(t-s)σhT,q(x)(∫0T‍|b(r)|dr+|z|)+MeωTq(x)∫ts‍|b(r)|dr],
which implies by (72) that
(87)p((RB*S)(t)x-(RB*S)(s)x)≤MeμT1-γ(∫0s‍p(S(t-r)x-S(s-r)x)dr+∫st‍p(S(t-r)x)dr)≤cpMeμT1-γ(cT,q(t-s)σhT,q(x)∫0s‍∫0T‍|b(v)|dvdr+MeωTq(x)∫0s‍∫s-rt-r‍|b(v)|dvdr+|z|cT,qT(t-s)σhT,q(x)∫0s‍)+cpM2e2μTq(x)1-γ×(∫st‍∫0t-r‍|b(t-r-v)|dvdr+|z|(t-s)).
One can simply prove that there exists cT>0 such that, for 0≤s<t≤T,
(88)∫0s∫s-rt-r|b(v)|dvdr+∫st∫0t-r|b(t-r-v)|dvdr≤cT(t-s)σ,
which implies with (72), the previous computation and the denseness of A that there exists bT,p>0 such that, for every x∈E and 0≤s<t≤T:(89)p(RB(t)x-RB(s)x)≤bT,p(t-s)σmax(p(x),hT,p(x),q(x)).
The same estimate holds provided (70), while in the case of Remark 15 we obtain that, for every x∈D(A) and 0≤s<t≤T,
(90)p(RB(t)x-RB(s)x)(t-s)σ≤bT,pmax((Ax),q(x)+q(Ax))p(x)+p(Ax),hT,p(x)+hT,p(Ax),q(x)+q(Ax)).
Assuming additionally
(91)sup0≤s<t≤T1(t-s)σ∑j=0l-1‍(∫0t-s|a*,j+1(r)|dr+∫0s|a*,j+1(t-r)-a*,j+1(s-r)|dr)<∞,
then an estimate of the form (89) holds in the case of Theorem 14.
The following corollary is an immediate consequence of Theorems13–14 and Remark 15.Corollary 17.
SupposeM,M1>0,ω≥0,α>0,β≥0,A is a subgenerator of a (gα,gαβ+1)-regularized C-resolvent family (R(t))t≥0 satisfying p(R(t)x)≤Meωtp(x),x∈E,t≥0,p∈⊛ and (2) with a(t)=gα(t) and k(t)=gαβ+1(t). Assume exactly one of the following conditions:(i)
α-1-αβ≥0,BCx=CBx,x∈D(A),  and (a)  ∨  (b), where
(a)
p(C-1Bx)≤M1p(x),x∈D(A)¯,p∈⊛.(b)
E is complete, (69) and (♮).(ii)
α-1-αβ<0,BCx=CBx,x∈D(A),l=⌈(αβ+1-α)/α⌉  and (73) holds.Then there exist μ>ω and γ∈[0,1) such that A+B is a subgenerator of a (gα,gαβ+1)-regularized C-resolvent family (RB(t))t≥0 satisfying (71), and (18) with k1(t) replaced by gαβ+1(t) therein.Remark 18.
Let0<α<2 and let (R(t))t≥0 be an exponentially equicontinuous, analytic (gα,gαβ+1)-regularized C-resolvent family of angle δ∈(0,π/2]. Suppose additionally that, for every ζ∈(0,δ), there exist Mζ≥1 and ωζ≥0 such that p(R(z)x)≤Mζeωζℜzp(x),x∈E,z∈Σζ,p∈⊛. If (ii) or (i)(a) holds, then we obtain from Corollary 17 and the proofs of Kato's analyticity criteria [60, Theorems 4.3 and 4.6] that (RB(t))t≥0 is also an exponentially equicontinuous, analytic (gα,gαβ+1)-regularized C-resolvent family of angle δ; furthermore, for every ζ∈(0,δ), there exist Mζ′≥1 and ωζ′≥0 such that p(RB(z)x)≤Mζ′eωζ′ℜzp(x),x∈E,z∈Σζ,p∈⊛. If (i)(b) holds, then one has to assume additionally that there exist η>ω and γ∈[0,1) such that, for every ζ∈(-δ,δ),x∈D(A) and p∈⊛, the following holds:
(92)∫0∞e-μtp(C-1B∫0t‍b(t-s)R(seiζ)xds+zC-1BR(teiζ)x∫0t‍)dt≤γp(x).
The question whether perturbations considered in Theorems 13–14 retain analytical properties requires further analysis and will not be discussed in the context of this paper.Example 19 (cf. [28, Example 2.24]).
LetE:=l1,0<α<1 and l:=⌈(1-α)/α⌉. Define a closed densely defined linear operator Aα on E by D(Aα):={〈xn〉∈l1:∑n=1∞‍n|xn|<∞} and Aα〈xn〉:=〈eiα(π/2)nxn〉,〈xn〉∈D(Aα). Then Aα is the integral generator of a bounded (gα,1)-regularized resolvent family, Aα+I is not the integral generator of an exponentially bounded (gα,1)-regularized resolvent family, and σ(Aα)={eiα(π/2)n:n∈ℕ}. Suppose
(93)B∈L(E),R(B)⊆D(Al)={〈xn〉∈l1:∑n=1∞‍nl|xn|<∞}.
Then it follows from Corollary 17 that A+B is the integral generator of an exponentially bounded (gα,1)-regularized resolvent family.


## 3. Unbounded Perturbation Theorems

In the subsequent theorems, we transfer the assertions of [19, Theorems 3.1 and 3.3] and [1, Theorem 2.5.9, Corollary 2.5.10] to abstract Volterra equations.Theorem 20.
SupposeE is a Banach space, k(t) and a(t) satisfy (P1)-(P2) and A is the integral generator of an exponentially bounded (a,k)-regularized resolvent family (R(t))t≥0 satisfying (2) with C=I. Let M>0 and ω≥0 be such that ∥R(t)∥≤Meωt,t≥0 and let λ0>max(ω,abs(a),abs(k)) satisfy k~(λ)a~(λ)≠0,ℜλ≥λ0. Suppose that, for every ε>0, there exists Cε>0 such that
(94)1|k~(λ)|≤Cεeε|λ|,ℜλ≥λ0,|a~(λ)||a~(λ0+iℑλ)|≤Cεeε|λ|,ℜλ≥λ0.(i)
LetB be a linear operator, let D(A)⊆D(B) and let
(95)∥BR(1a~(λ):A)∥≤Mϱ|λ|-ϱ,ℜλ=λ0
for some ϱ>0 and Mϱ>0 (for ϱ=0 and some M0∈(0,1)). Then, for every ζ>1,A+B is the integral generator of an exponentially bounded, (a,k*0gζ)-regularized resolvent family (RB(t))t≥0 satisfying (18) with k1(t)=(k*0gζ)(t),C=I, and gα(t) replaced by a(t) therein.(ii)
LetB be a densely defined linear operator and let
(96)∥R(1a~(λ):A)Bx∥≤Mϱ|λ|-ϱ∥x∥,x∈D(B),ℜλ=λ0
for some ϱ>0 and Mϱ>0(for ϱ=0 and some M0∈(0,1)). Then there exists a closed extension D of the operator A+B such that, for every ζ>1, D is the integral generator of an exponentially bounded, (a,k*0gζ)-regularized resolvent family (RB(t))t≥0 satisfying (18) with k1(t)=(k*0gζ)(t),C=I, and gα(t) replaced by a(t) therein. Furthermore, if A and A* are densely defined, then D is the part of the operator (A*+B*)* in E.Proof.
By Lemma2, {1/a~(λ):ℜλ>λ0}⊆ρ(A) and
(97)∥R(1a~(λ):A)∥≤M|a~(λ)||k~(λ)|(ℜλ-ω),ℜλ>λ0.
Given z∈ℂ with ℜz>λ0, put λz:=λ0+iℑz. Then the prescribed assumptions combined with (97) imply
(98)∥BR(1a~(z):A)∥=∥BR(1a~(λz):A)(I+(1a~(λz)-1a~(z))R(1a~(z):A))∥≤∥BR(1a~(λz):A)∥[1+|1a~(λz)-1a~(z)|∥R(1a~(z):A)∥]≤Mϱλ0ϱ[1+|1a~(λz)-1a~(z)|M|a~(z)||k~(z)|(ℜz-ω)]≤Mϱλ0ϱ+MϱMλ0ϱ(λ0-ω)|k~(z)||a~(z)a~(λz)-1|.
Consider now the function h:{z∈ℂ:ℜz≥0}→L(E) defined by h(z):=zϱBR((1/a~(λ0+z)):A),ℜz≥0,z≠0,h(0):=0 if ϱ>0, and h(0):=BR((1/a~(λ0)):A) if ϱ=0. Then the function z↦h(z) is continuous for ℜz≥0 and analytic for ℜz>0. Furthermore, ∥h(it)∥≤Mϱ,t∈ℝ and, by (94)–(98), one has that, for every ε>0, there exists Cε′>0 such that ∥h(z)∥≤Cε′eε|z| for all z∈ℂ with ℜz≥0. By the Phragmén-Lindelöf type theorems (cf. for instance [39, Theorem 3.9.8]), we get that ∥h(z)∥≤Mϱ for all z∈ℂ with ℜz≥0. This, in turn, implies that there exists a>λ0 such that ∥BR(1/a~(λ):A)∥<1/2,ℜλ≥a if ϱ>0, and that ∥BR(1/a~(λ):A)∥<M0,ℜλ≥a if ϱ=0. Therefore, 1/a~(λ)∈ρ(A+B),ℜλ≥a and there exists cϱ>0 such that, for ℜλ≥a:(99)∥1a~(λ)R(1a~(λ):A+B)∥=∥1a~(λ)R(1a~(λ):A)(I-BR(1a~(λ):A))-1∥≤cρ|k~(λ)|.
The proof of (i) follows from [32, Theorem 2.7(i), Remark 2.3(v)]. Using [19, Lemma 3.2] and a similar argumentation, we obtain the validity of (ii).Recall that a Banach spaceE has Fourier type p∈[1,2] if and only if the Fourier transform extends to a bounded linear operator from Lp(ℝ:E) to Lq(ℝ:E), where 1/p+1/q=1. Each Banach space E has Fourier type 1, and E* has the same Fourier type as E. A space of the form Lp(Ω,μ) has Fourier type min(p,p/p-1), and there exist examples of nonreflexive Banach spaces which do have nontrivial Fourier type.Theorem 21.
LetE be a Banach space of Fourier type p∈(1,2].(i)
Let the assumptions of Theorem20(i) hold and let ζ>1/p. Assume that at least one of the following conditions holds:
(a)
A and A* are densely defined, there exist λ0′>λ0 and η>1 such that
(100)|k~(λ)|=O(|λ|ζ-η),ℜλ>λ0′,|a~(λ)|=O(|λ|ζ-η),ℜλ>λ0′.(b)
A is densely defined and E is reflexive.(c)
B(D(A2))⊆D(A) and BAx=ABx,x∈D(A2).ThenA+B is the integral generator of an exponentially bounded, (a,k*0gζ)-regularized resolvent family (RB(t))t≥0 satisfying (18) with k1(t)=(k*0gζ)(t),C=I, and gα(t) replaced by a(t) therein.(ii)
Let the assumptions of Theorem20(ii) hold and let ζ>1/p. Then there exists a closed extension D of the operator A+B such that D is the integral generator of an exponentially bounded, (a,k*0gζ)-regularized resolvent family (RB(t))t≥0 satisfying (18) with k1(t)=(k*0gζ)(t),C=I, and gα(t) replaced by a(t) therein. Furthermore, if A and A* are densely defined, then D is the part of the operator (A*+B*)* in E.Proof.
Assume that (c) holds. According to (100), R(1/a~(λ):A)(I-BR(1/a~(λ):A))-1=(I-BR(1/a~(λ):A))-1R(1/a~(λ):A),ℜλ≥a and (1/a~(λ))R(1/a~(λ):A+B)=(1/a~(λ))(I-BR(1/a~(λ):A))-1R(1/a~(λ):A),ℜλ≥a. Define
(101)RB(t)x≔12πi∫a-i∞a+i∞eλtλ-ζ[k~(λ)a~(λ)R(1a~(λ):A+B)x]dλ,x∈E,t≥0.
By the first part of the proof of [19, Theorem 3.3], A+B is the integral generator of an exponentially bounded, (a,k*0gζ)-regularized resolvent family (RB(t))t≥0 satisfying (18) with k1(t)=(k*0gζ)(t),C=I, and gα(t) replaced by a(t) therein. The property (18) holds in any particular case considered below and the assertion (ii) is also an immediate consequence of the proof of [19, Theorem 3.3]. Assume now that (b) holds. Then A* is densely defined and, by [33, Theorem 2.14(ii)], (R*(t))t≥0 is an exponentially bounded, (a,k)-regularized resolvent family with the integral generator A*. Let q be such that 1/p+1/q=1 and let J:E→E** denote the canonical embedding of E in its bidual E**. Since E* has Fourier type p and (1/a~(λ))R(1/a~(λ):A+B)*=(1/a~(λ))((I-BR((1/a~(λ)):A))-1)*R((1/a~(λ)):A)*,ℜλ≥a, it follows that there exists c1>0 such that, for every x*∈E* and r≥a,
(102)∫-∞∞∥k~(r+is)a~(r+is)R(1a~(r+is):A+B)*x*∥qds≤c1∥x*∥q.
Set, for every x*∈E* and t≥0:(103)RB,*(t)x*≔12πi∫a-i∞a+i∞eλtλ-ζ[k~(λ)a~(λ)R(1a~(λ):A+B)*x*]dλ.
Then (RB,*(t))t≥0⊆L(E*) is strongly continuous, exponentially bounded and
(104)k~(λ)a~(λ)R(1a~(λ):A+B)*x*=λζ∫0∞e-λtRB,*(t)x*dt,ℜλ>a,x*∈E*.
By Lemma2, (RB,*(t))t≥0 is an (a,k*0gζ)-regularized resolvent family with the integral generator (A+B)*. By [33, Theorem 2.14(iii)], it follows that (RB(t)≡J-1RB,*(t)*J)t≥0 is an (a,k*0gζ)-regularized resolvent family with the integral generator A+B=J-1((A+B)*)*J. We continue the proof by assuming that (a) holds. Using (99)-(100), we easily infer that the improper integral in (101) converges absolutely for x∈D(A) and that
(105)k~(λ)a~(λ)R(1a~(λ):A+B)x=λζ∫0∞e-λtRB(t)xdt,ℜλ>a,x∈D(A).
By (104)-(105) and the uniqueness theorem for Laplace transform, we get
(106)〈RB,*(t)x*,x〉=〈x*,RB(t)x〉,t≥0,x*∈E*,x∈D(A)
andRB,*(t)*Jx=JRB(t)x,t≥0,x∈D(A). Now one can simply prove that ((RB,*(t)*)|E)t≥0 is an exponentially bounded, (a,k*0gζ)-regularized resolvent family with the integral generator A+B.Remark 22.
(i) It is noteworthy that Kaiser and Weis analyzed in [61, Theorem 3.1] an analogue of Theorem 21 for operator semigroups in Hilbert spaces. The question whether the perturbed semigroup (RB(t))t≥0 is strongly continuous at t=0 was answered in the affirmative by Batty [62]; here we would like to note that it is not clear in which way one can transfer the assertion of [62, Theorem 1] to abstract Volterra equations.
(ii) To the author's knowledge, the denseness ofD(A*) in E* cannot be so simply dropped from the formulation of (a). The main problem is that we do not know whether the mapping t↦R(t)*x*, t≥0 is measurable provided x*∈E*∖D(A*)- (cf. [19, (5)-(6), page 221; l. 7-8, page 222] and [63, Section 3]). Notice also that the assertion (c), although practically irrelevant, may help one to better understand the proof of [19, Theorem 3.3].
(iii) Letα>0 and a(t)=gα(t). Then the assumptions of Theorems 20 and 21[(i)(b)-(c), (ii)] hold while the assumptions of Theorem 21(i)(a) hold provided ζ+α>1.
In the following nontrivial example, we will transfer the assertion of [19, Proposition 8.1] to abstract time-fractional equations.Example 23.
Let1<p<∞,1/p+1/q=1,k∈ℕ0,0<β≤2 and E:=Lp(ℝ). Define a closed linear operator Aβ,k on E by D(Aβ,k):=W4k+2,p(ℝ) and Aβ,kf:=ei(2-β)(π/2)f(4k+2),f∈D(Aβ,k). Put Bf(x):=V(x)f(l)(x),x∈ℝ with maximal domain D(B):={f∈E:V·f(l)∈E}; here V(x) is a potential and l∈ℕ0. Assume first that
(107)V∈Lp(ℝ),l≤1p((4k+2)p-1-(4k+2)(p-1)β).
Given ℜλ>0, denote by μj,λ(1≤j≤2k+1)(2k+1) solutions of the equation μj,λ4k+2=λβei(β(π/2)-π) with ℜμj,λ>0. Then D(A)⊆D(B),
(108)(R(λβ:Aβ,k)f)(x)=eiβ(π/2)4k+2∫-∞∞∑j=12k+1‍e-μj,λ|x-s|(-μj,λ)4k+1f(s)ds,
provided f∈E,x∈ℝ,ℜλ>0,
(109)(BR(λβ:Aβ,k)f)(x)=eiβ(π/2)4k+2V(x)×∑j=12k+1‍∫-∞x(e-μj,λ(x-s)(-μj,λ)4k-l+1f(s)ds-∫x∞eμj,λ(x-s)μj,λ4k-l+1f(s)ds),f∈E,x∈ℝ,ℜλ>0,∥R(λβ:Aβ,k)∥≤(|λ|β(1-(1/(4k+2)))min(ℜμ1,λ,…,ℜμ2k+1,λ))-1,ℜλ>0,∥BR(λβ:Aβ,k)∥≤∥V∥p(×min((ℜμ1,λ)1/q,…,(ℜμ2k+1,λ)1/q))-1|λ|β(1-((l+1)/(4k+2)))×min((ℜμ1,λ)1/q,…,(ℜμ2k+1,λ)1/q))-1,
provided ℜλ>0. Furthermore, ℜμj,λ=|λ|β/(4k+2)cos(arg(μj,λ)),ℜλ>0,1≤j≤2k+1, and
(110)min{ℜμj,λ:1≤j≤2k+1}=|λ|β/(4k+2)×min(cos(arg(λ)β+(βπ)/(2)4k+2+(2k-1)π4k+2),-cos(arg(λ)β+(βπ)/24k+2+π2)),
provided ℜλ>0. The above implies that there exists a constant cβ,k>0 such that
(111)|λ|β/(4k+2)cos(arg(λ))/min(ℜμ1,λ,…,ℜμ2k+1,λ)≤cβ,k,ℜλ>0.
Keeping in mind (107)-(111), we obtain that
(112)∥R(λβ:Aβ,k)∥=O(|λ|1-β(ℜλ)-1),ℜλ>0(113)∥BR(λβ:Aβ,k)∥=O(∥V∥p(ℜλ)-β(1-((l+1)/(4k+2))+(1/(4k+2)q)))=O(∥V∥p(ℜλ)(-1)/q),
provided ℜλ>0. Denote by βk the infimum of all nonnegative real numbers r≥0 such that the operator Aβ,k generates an exponentially bounded (gβ,gr+1)-regularized resolvent family. The precise computation of integration rate βk falls out from the framework of this paper (cf. also the representation formula [28, Example 3.7, (3.15)] and notice that it is not clear whether Theorem 13 or Remark 15 can be applied in case β∈(1,2]). Clearly, (112) yields the imprecise estimate βk≤1; furthermore, βk=0 provided p=2 ([24]), and βk≤|(1/2)-(1/p)| provided β∈{1,2} [14, 50]. Set κp:=min(1/p,(p-1)/p). By Theorem 21, Aβ,k+B generates an exponentially bounded (gβ,gσβ,k,p+1)-regularized resolvent family for any σβ,k,p>βk+κp. By (112)-(113) and the proof of [19, Proposition 8.1], the above remains true provided (4k+2)p-1-((4k+2)(p-1)/β)≥0,l=0 and V∈Lp(ℝ)+L∞(ℝ); similarly, one can consider the operators Aβ,k1(k∈ℕ,0<β≤2) and Aβ,k2(k∈ℕ,0<β≤1) given by Aβ,k1f:=e-iβ(π/2)f(4k),f∈W4k,p(ℝ):=D(Aβ,k1) and Aβ,k2f:=e±i(π/2)(1-β)f(2k+1),f∈W2k+1,p(ℝ):=D(Aβ,k2).
Notice that Lizama and Prado have recently analyzed in [21] the qualitative properties of the abstract relaxation equation:
(114)u′(t)-ADtαu(t)+u(t)=f(t),α∈(0,1),t≥0,u(0)=0,
where E is a Banach space and f∈Lloc1([0,∞):E). By a (strong) solution of (114) we mean any function u∈C1([0,∞):E) such that (114) holds for a.e. t≥0. The following extension of [28, Theorem 2.25] (cf. also [10, page 65]) will be helpful in the study of perturbation properties of (114).Theorem 24.
Letk(t) and a(t) satisfy (P1). Suppose δ∈(0,π/2], ω≥max(0,abs(a),abs(k)), there exist analytic functions k^:ω+Σ(π/2)+δ→ℂ and a^:ω+Σ(π/2)+δ→ℂ such that k^(λ)=k~(λ), ℜλ>ω,a^(λ)=a~(λ),ℜλ>ω and k^(λ)a^(λ)≠0, λ∈ω+Σ(π/2)+δ. Let A be a subgenerator of an analytic (a,k)-regularized C-resolvent family (R(t))t≥0 of angle δ and let (2) hold. Suppose that, for every η∈(0,δ), there exists cη>0 such that
(115)p(e-ωℜzR(z)x)≤cηp(x),x∈E,p∈⊛,z∈Ση
as well as b,c≥0,B is a linear operator satisfying D(C-1AC)⊆D(B),BCx=CBx,x∈D(C-1AC) and
(116)p(C-1Bx)≤bp(C-1ACx)+cp(x),x∈D(C-1AC),p∈⊛.
Assume that at least one of the following conditions holds:(i)
A is densely defined, the numbers b and c are sufficiently small, there exists |C|⊛>0 such that p(Cx)≤|C|⊛p(x),x∈E,p∈⊛ and, for every η∈(0,δ), there exists ωη≥ω such that |k^(λ)-1|=O(|λ|),λ∈ωη+Σ(π/2)+η and |a^(λ)/k^(λ)|=O(|λ|),λ∈ωη+Σ(π/2)+η.(ii)
A is densely defined, the number b is sufficiently small, there exists |C|⊛>0 such that p(Cx)≤|C|⊛p(x),x∈E,p∈⊛ and, for every η∈(0,δ), there exists ωη≥ω such that |k^(λ)-1|=O(|λ|),λ∈ωη+Σ(π/2)+η and a^(λ)/(λk^(λ))→0,|λ|→∞,λ∈ωη+Σ(π/2)+η.(iii)
A is densely defined, the number c is sufficiently small, b=0 and, for every η∈(0,δ), there exists ωη≥ω such that |a^(λ)/k^(λ)|=O(|λ|),λ∈ωη+Σ(π/2)+η.(iv)
b=0 and, for every η∈(0,δ), there exists ωη≥ω such that a^(λ)/(λk^(λ))→0,|λ|→∞,λ∈ωη+Σ(π/2)+η.Then C-1(C-1AC+B)C is a subgenerator of an exponentially equicontinuous, analytic (a,k)-regularized C-resolvent family (RB(t))t≥0 of angle δ, which satisfies RB(z)[C-1(C-1AC+B)C]⊆[C-1(C-1AC+B)C]RB(z),z∈Σδ and the following condition:
(117)∀η∈(0,δ)∃ωη′>0∃Mη>0∀p∈⊛:p(RB(z)x)≤Mηeωη′ℜzp(x),x∈E,z∈Ση.
Furthermore, in cases (iii) and (iv), the above remains true with the operator C-1(C-1AC+B)C replaced by C-1AC+B.Proof.
First of all, notice that the closedness of the operatorC-1AC+B in cases (iii) or (iv) trivially follows and that it is not clear how one can prove that the operator C-1AC+B is closed in cases (i) or (ii). We will only prove the assertion provided that (i) holds and remark the minor modifications in case that (iv) holds. Let η∈(0,δ) and σ∈(0,1). Clearly, A⊆C-1AC, C[C-1AC]⊆[C-1AC]C, C[C-1AC+B]⊆[C-1AC+B]C, C[C-1(C-1AC+B)C]⊆[C-1(C-1AC+B)C]C and C-1AC+B⊆C-1(C-1AC+B)C. Invoking (115), [33, Theorem 3.6] and the proof of [39, Theorem 2.6.1], we obtain that
(118)limλ→+∞λk^(λ)a^(λ)(1a^(λ)-C-1AC)-1Cx=k(0)Cx,x∈E
and that there existsNη>0 such that {(1/a^(λ)):λ∈ω+Σ(π/2)+η}⊆ρC(C-1AC) and
(119)supλ∈ω+Σ(π/2)+ηp((λ-ω)k^(λ)a^(λ)(1a^(λ)-C-1AC)-1Cx)≤Nηp(x),x∈E.
By (116) and (119), we infer that, for every λ∈ωη+Σ(π/2)+η,x∈E and p∈⊛:(120)p(C-1B(1a^(λ)-C-1AC)-1Cx)≤b|C|⊛p(x)+bNηcηp(x)|λ-ω||k~(λ)|+ccηNηp(x)|λ-ω||a^(λ)||k^(λ)|,
which implies by the given assumption the existence of a number ωη′>ωη such that p(C-1B((1/a^(λ))-C-1AC)-1Cx)≤σp(x),x∈E,λ∈ωη′+Σ(π/2)+η,p∈⊛, provided that the numbers b and c are sufficiently small; if (iv) holds, then
(121)limλ→+∞C-1B(1a^(λ)-C-1AC)-1Cx=0,x∈E.
Using the same argument as in the proof of Theorem 14, it follows that, for every λ∈ωη′+Σ(π/2)+η,R(C)⊆R((1/a^(λ))-(C-1AC+B))⊆R((1/a^(λ))-C-1(C-1AC+B)C) as well as that the operators (1/a^(λ))-(C-1AC+B) and (1/a^(λ))-C-1(C-1AC+B)C are injective. Moreover, for any λ∈ωη′+Σ(π/2)+η:(122)(1a^(λ)-(C-1AC+B))-1C=(1a^(λ)-C-1(C-1AC+B)C)-1C=(1a^(λ)-C-1AC)-1×C(I-C-1B(1a^(λ)-C-1AC)-1C)-1.
Now we will prove that the operator C-1(C-1AC+B)C is closed. Let (xτ)τ∈T be a net in E satisfying xτ→x, τ→∞ and C-1(C-1AC+B)Cxτ→y, τ→∞. Then ((1/a~(λ))-(C-1AC+B))-1CC-1(C-1AC+B)Cxτ→((1/a~(λ))-(C-1AC+B))-1Cy, τ→∞, that is, -Cxτ+(1/a~(λ))((1/a~(λ))-(C-1AC+B))-1Cxτ→((1/a~(λ))-(C-1AC+B))-1Cy, τ→∞, which simply implies Cx∈D(C-1AC+B) and (C-1AC+B)Cx=Cy. Therefore, x∈D(C-1(C-1AC+B)C), C-1(C-1AC+B)Cx=y and C-1(C-1AC+B)C is closed, as required. Notice that, for every x∈E, the analyticity of mapping
(123)λ↦(1a^(λ)-C-1(C-1AC+B)C)-1Cx=(1a^(λ)-C-1AC)-1C×∑n=0∞‍[C-1B(1a^(λ)-C-1AC)-1C]nx,λ∈ωη′+Σ(π/2)+η
follows from  [33, Lemma 3.3] and the fact that an E-valued mapping is analytic if and only if it is weakly analytic. By [33, Theorem 3.7], C-1(C-1AC+B)C is a subgenerator of an exponentially equicontinuous, analytic (a,k)-regularized C-resolvent family (RB(t))t≥0 of angle η and (117) holds; assuming (iv), we get from (119)
(124)p((λ-ωη′)k^(λ)a^(λ)(1a^(λ)-(C-1AC+B))-1Cx)=p((λ-ωη′)(λ-ω)(λ-ω)k^(λ)a^(λ)(1a^(λ)-C-1AC)-1C×(I-C-1B(1a^(λ)-C-1AC)-1C)-1x)≤(1+1cosη)Nη1-σp(x),x∈E,λ∈ωη′+Σ(π/2)+η.
In combination with (118) and (121), the above implies
(125)limλ→+∞λk^(λ)a^(λ)(1a^(λ)-(C-1AC+B))-1Cx=limλ→+∞λk^(λ)a^(λ)(1a^(λ)-C-1AC)-1Cx+limλ→+∞λk^(λ)a^(λ)(1a^(λ)-C-1AC)-1C×C-1B(1a^(λ)-C-1AC)-1×C(I-C-1B(1a^(λ)-C-1AC)-1C)-1x=k(0)Cx+0=k(0)Cx,x∈E,
and the proof follows again from an application of [33, Theorem 3.7].Remark 25.
Using the proof of [33, Theorem 3.7], we get that there exists ω0>0 such that, for every x∈E and for every λ∈ℂ with ℜλ>ω0:(126)k^(λ)(I-a^(λ)(C-1AC+B))-1Cx=k^(λ)(I-a^(λ)C-1(C-1AC+B)C)-1Cx=∫0∞e-λtRB(t)xdt.
By Lemma 2, we obtain that (18) holds with A+B,   k1(t) and gα(t) replaced, respectively, by C-1(C-1AC+B)C,k(t) and a(t) therein; clearly, the above assertion remains true with the operator C-1(C-1AC+B)C replaced by C-1AC+B, provided that (iii) or (iv) holds. Taking the Laplace transform, (126) simply implies that C-1(C-1AC+B)C is, in fact, the integral generator of (RB(t))t≥0.Example 26.
Letu(t) be a solution of (114). Set aα(t):=ℒ-1(λα/(λ+1))(t), t≥0, kα(t):=e-t, t≥0, and v(t)=u(t)+(1*u)(t), t≥0. Then u(t)=v(t)-(e-t*v)(t),  t≥0 and v(t)=A(aα*v)(t)+(1*f)(t), t≥0, which implies that the notion of an (aα,k)-regularized C-resolvent family is important in the study of (114). In [21], the authors mainly use the following conditions: k(t)=kα(t), C=I and A is the generator of a bounded analytic C0-semigroup. Set δ:=min(π/2,πα/(2(1-α))) and assume, more generally, that for every η∈(0,((π/2)+δ)(1-α)), there exists ωη>0 such that the family
(127){(1+|λ|)1-r(λ-A)-1C:λ∈ωη+Ση}
is equicontinuous (r≥0) and that the mapping
(128)λ↦(λ-A)-1Cx,λ∈ωη+Σηiscontinuousforeveryfixedx∈E.
Notice that (127)-(128) hold provided that A is a subgenerator of an exponentially equicontinuous r-times integrated C-semigroup (Rr(t))t≥0; furthermore, if
(129)∃M≥1∃ω≥0:p(Rr(t)x)≤Meωtp(x),x∈E,p∈⊛,
then, for every η∈(0,π/2) and ωη>ω, there exists Mη>0 such that
(130)p((λ-A)-1Cx)≤Mη(1+|λ|)r-1p(x),x∈E,λ∈ωη+Ση,p∈⊛.
We refer the reader to [58, Chapter 1] for examples of differential operators generating exponentially equicontinuous, r-times integrated C-semigroups satisfying (129). Assume, further, that there exist ω>max(0,abs(k)) and an analytic function k^:ω+Σ(π/2)+δ→ℂ such that k^(λ)=k~(λ),ℜλ>ω,k^(λ)≠0, λ∈ω+Σ(π/2)+δ and |k^(λ)|=O(|λ|-1), λ∈ω+Σ(π/2)+δ. Let γ∈(0,δ) and let ((π/2)+γ)(1-α)<η<π/2. Then there exists a sufficiently large ωγ′>ω such that (λ+1)/λα=λ1-α+λ-α∈ωη+Ση for all λ∈ωγ′+Σ(π/2)+γ, which implies with (127)-(128) and [33, Proposition 2.16(iii)] that the mapping λ↦(k^(λ)/a^(λ))((1/a^(λ))-A)-1Cx,λ∈ωγ′+Σ(π/2)+γ is analytic (x∈E) and that, for every σ≥r(1-α), the family {(λ-ωγ′)(k^(λ)/|λ|σ)(1/a^(λ))((1/a^(λ))-A)-1C:λ∈ωγ′+Σ(π/2)+γ} is equicontinuous (if (129) holds, then there exists Nγ>0 such that p((λ-ωγ′)(k^(λ)/|λ|σ)(1/a^(λ))((1/a^(λ))-A)-1Cx)≤Nγp(x),x∈E,p∈⊛,λ∈Ση). Using [33, Theorem 3.7] and the arbitrariness of η, we get that A is a subgenerator of an exponentially equicontinuous, analytic (aα,k*gζ)-regularized C-resolvent family (R(t))t≥0 of angle δ, where
(131)ζ={r(1-α),ifD(A)¯=E>r(1-α),ifD(A)¯≠E
and g0(t) stands for the Dirac distribution (if (129) holds, then for every η∈(0,δ) there exist ωη>0 and Lη>0 such that p(R(z)x)≤Lηeωηℜzp(x),x∈E,p∈⊛). This is a significant improvement of [21, Theorem 3.1]. In what follows, we will provide the basic information on the C-well-posedness of (114). Given β∈(0,1) and T>0, set
(132)Cβ0([0,T]:E)≔{f∈C([0,T]:E):f(0)=0,|f|β,T,p<∞,∀p∈⊛},
where
(133)|f|β,T,p:=sup0≤s<t≤Tp(f(t)-f(s))(t-s)β.
Let A be densely defined, let r=0 and let β∈(0,1) be such that C-1(1*f|[0,T])∈Cβ0([0,T]:E) for all T>0. Then ζ=0, and the proof of [10, Theorem 2.4] combined with the Cauchy integral formula (cf. also [33, Section 1, Theorem 3.4(i)]) indicates that the function
(134)v(t)=R(t)C-1(1*f)(t)+∫0tR′(t-s)(C-1(1*f)(s)-C-1(1*f)(t))ds,t≥0
satisfies A(aα*v)(t)=v(t)-(1*f)(t),t≥0 and that, for every T>0, one has v|[0,T]∈Cβ0([0,T]:E); in the above formula, we assume that (R(t))t≥0 is the exponentially equicontinuous, analytic (aα,C)-regularized resolvent family of angle δ. It is obvious that the function t↦u(t)=v(t)-(e-t*v)(t),t≥0 is a unique function satisfying (114) in integrated form
(135)u(t)-A(g1-α*u)(t)+(1*u)(t)=(1*f)(t),t≥0,u(0)=0
and that u|[0,T]∈Cβ0([0,T]:E) for all T>0. If x∈E, x≠0 and C-1(1*(f(·)-x)|[0,T])∈Cβ0([0,T]:E) for all T>0, then we obtain similarly the unique solution u(t) of the problem
(136)u(t)-x-A(g1-α*(u(·)-x))(t)+(1*u)(t)=(1*f)(t),t≥0,u(0)=x;
furthermore, u|[0,T]∈Cβ0([0,T]:E) for all T>0. Since aα∉BVloc([0,∞)), the above-described method does not work in the case r>0 (cf. [35, Corollary 2.11] and [33, Theorem 2.6(i)]).
We are turning back to the case in whichA is not necessarily densely defined. Let C-1f,AC-1f∈Lloc1([0,∞):E) and let (Rα(t))t≥0 denote the (aα,kα*gζ)-regularized C-resolvent family with a subgenerator A. By the proofs of [21, Theorem 3.5, Corollary 3.6], it follows that, for every x∈R(C), there exists a unique solution of the problem
(137)u(t)-A(g(1-α)*u)(t)+(1*u)(t)=(1*f*gζ)(t)+gζ+1(t)x,t≥0,
given by t↦u(t)=Rα(t)C-1x+∫0t‍Rα(t-s)C-1f(s)ds,t≥0. Only after assuming some additional conditions, one can differentiate the formulae (135)–(137), obtaining in such a way (114) or its slight modification. Now we are interested in the perturbation properties of (114). Assume r∈[0,1] and A is a subgenerator of an exponentially equicontinuous, r-times integrated C-semigroup satisfying (129). Let B be a linear operator such that D(A)⊆D(B),BCx=CBx,x∈D(A) and let b,c≥0 satisfy p(C-1Bx)≤bp(Ax)+cp(x),x∈D(A),p∈⊛. By Remark 25 and the proof of Theorem 24, we have the following(i)
Ifr=ζ=0,b is sufficiently small and |C|⊛>0 satisfies p(Cx)≤|C|⊛p(x),x∈E,p∈⊛, then C-1(A+B)C is the integral generator of an exponentially equicontinuous, analytic (aα,k)-regularized C-resolvent family (RB(t))t≥0 of angle δ (cf. [64, Chapter III] and [65, Chapter 7] for corresponding examples).(ii)
Ifb=0,c is sufficiently small, r=1, and ζ=1-α, then A+B, respectively, C-1(A+B)C, is a subgenerator, respectively the integral generator, of an exponentially equicontinuous, analytic (aα,k*gζ)-regularized C-resolvent family (RB(t))t≥0 of angle δ.(iii)
Ifb=0,0≤r<1, and ζ≥r(1-α), then A+B, respectively C-1(A+B)C, is a subgenerator, respectively the integral generator, of an exponentially equicontinuous, analytic (aα,k*gζ)-regularized C-resolvent family (RB(t))t≥0 of angle δ.
We continue this example by observing that Karczewska and Lizama [20] have recently analyzed the following stochastic fractional oscillation equation:
(138)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. [20, Theorems 3.1 and 3.2]) is essentially applied in the study of deterministic counterpart of (138) in integrated form
(139)u(t)+∫0tg2-α(t-s)Au(s)ds+∫0t(t-s)u(s)ds=∫0t(t-s)f(s)ds,t>0,
where f∈Lloc1([0,∞):E). Equation (139) models an oscillation process with fractional damping term and after differentiation becomes, in some sense,
(140)u′′(t)+ADtαu(t)+u(t)=f(t),t≥0.
Without any essential changes, one can consider the C-well-posedness and perturbation properties of (139).Example 27.
(See [22, 66, 67]). Let α∈(0,1),m∈ℕ, let Ω be a bounded domain in ℝn with boundary of class C4m, and let E:=Cα(Ω¯). Consider the operator A:D(A)⊆Cα(Ω¯)→Cα(Ω¯) given by
(141)Au(x):=∑|β|≤2m‍aβ(x)Dβu(x)∀x∈Ω¯
with domain D(A):={u∈C2m+α(Ω¯):Dβu|∂Ω=0forall|β|≤m-1}. Here β∈ℕ0n,|β|=∑i=1n‍βj,Dβ=∏i=1n‍((1/i)(∂/∂xi))βi, and aβ:Ω¯→ℂ satisfy the following conditions:(i)
aβ(x)∈ℝ for all x∈Ω¯ and |β|=2m,(ii)
aβ∈Cα(Ω¯) for all |β|≤2m, and(iii)
there existsM>0 such that
(142)M-1|ξ|2m≤∑|β|=2m‍aβ(x)ξβ≤M|ξ|2m∀ξ∈ℝn,x∈Ω¯.Then there exists a sufficiently large σ>0 such that the operator -Aσ≡-(A+σ) satisfies Σω∪{0}⊆ρ(-Aσ) with some ω∈((π/2),π) and
(143)∥R(λ:-Aσ)∥=O(|λ|(α/2m)-1),λ∈Σω.
Notice that A is not densely defined since D(A)⊆{u∈Cα(Ω¯):u|∂Ω=0}. Let ς∈[1,(2ω/π)) and τ∈((α/2m),1). By (143) and [32, Theorem 2.17], we get that -Aσ is the integral generator of an exponentially bounded, analytic (gς,gςτ+1)-regularized resolvent family of angle δ=(ω/ς)-(π/2)∈(0,(π/2)). Assume now that B:D(B)⊆Cα(Ω¯)→Cα(Ω¯) is a linear operator satisfying D(A)⊆D(B) and ||Bu||≤c||u||,u∈D(A) for some c>0. Applying Theorem 24(iv), we obtain that the operator -(A+B) is the integral generator of an exponentially bounded, analytic (gς,gςτ+1)-regularized resolvent family of angle δ. Suppose, for example, m=n=1 and Ω=(0,1). Let φ,ψ∈L1[0,1] and let the operator B:Cα[0,1]→Cα[0,1] be defined by
(144)Bu(x)≔∫0x(u(x-s)-u(0))φ(s)ds+∫0x(u(1-x+s)-u(1))ψ(s)ds,x∈[0,1].
Then B satisfies the conditions stated above since B∈L(Cα[0,1]) and ||Bu||≤(||φ||L1[0,1]+||ψ||L1[0,1])||u||,u∈Cα[0,1]. Finally, it could be interesting to construct an example in which there does not exist B^∈L(Cα[0,1]) such that B^x=Bx for all x∈D(A).
In the remaining part, which is mainly motivated by reading of the paper [25] by Arendt and Batty, we assume that E is a Banach space. We consider rank-1 perturbations of ultradistribution semigroups and sines whose generators possess polynomially bounded resolvent; our intention is also to prove generalizations of [25, Theorem 4.3] and [26, Theorem 1.3] for abstract time-fractional equations.
Givena∈E, b*∈E* and C∈L([D(A)],E), we consider the rank-1 perturbation B∈L([D(A)],E) of A given by
(145)Bx:=b*(Cx)a,x∈D(A).
We also denote this operator B by ab*C. Denote Bδ(a,b*):={(x,y*)∈E×E*:||x-a||≤δ,||x*-b*||≤δ}(a∈E,b^*∈E^*,δ>0).
For the sake of convenience to the reader, we will repeat the assertion of [25, Theorem 1.3].Lemma 28.
LetA be a closed linear operator on E, let C∈L([D(A)],E) and let ε>0. Assume that Ωn⊆ρ(A) and supλ∈Ωn||CR(λ:A)x||<∞ for all x in a dense subset En of E and all n∈ℕ. Let gn:Ωn→(0,∞)(n∈ℕ). Assume that for each (a,b*)∈Bε(0,0) there exists n∈ℕ such that Ωn⊆ρ(A+ab*C) and ||R(λ:A+ab*C)||≤gn(λ),λ∈Ωn. Then there exists m∈ℕ such that supλ∈Ωm||CR(λ:A)||<∞.Henceforth, we assume that(Mp) is a sequence of positive real numbers such that M0=1 and that the following conditions are fulfilled:
(M.1)Mp2≤Mp+1Mp-1,p∈ℕ,(M.2)Mp≤AHpsup0≤i≤pMiMp-i,p∈ℕ,forsomeA,H>1,(M.3′)∑p=1∞‍Mp-1Mp<∞.
Let s>1. Then the Gevrey sequences (p!s), (pps) and (Γ(1+ps)) satisfy the above conditions. The associated function of (Mp) is defined by M(t):=supp∈ℕln(tp/Mp),t>0 and M(0):=0. Recall [54], the function t↦M(t),t≥0 is increasing, limt→∞M(t)=∞ and limt→∞(M(t)/t)=0.Following [1, 16], a closed linear operator A is said to be the generator of an ultradistribution sine of (Mp)-class if and only if the operator 𝒜:=(0IA0) generates an ultradistribution semigroup of (Mp)-class (cf. [16, 18, 68, 69] for the notion). The following well-known lemma (cf. [69, Theorem 1.5], [16, Theorem 9] and [1, Chapter 3]) will be helpful in our further work.Lemma 29.
(i) LetA be a closed densely defined operator on E. Then A generates an ultradistribution semigroup of (Mp)-class if and only if there exist l≥1,α>0 and β∈ℝ such that
(146)Λl,α,β:={λ∈ℂ:ℜλ≥αM(l|ℑλ|)+β}⊆ρ(A),(147)∥R(λ:A)∥=O(exp(M(l|λ|))),λ∈Λl,α,β.
(ii) LetA be a closed densely defined operator on E. Then A generates an ultradistribution sine of (Mp)-class if and only if there exist l≥1,α>0 and β∈ℝ such that
(148){λ2:λ∈Λl,α,β}⊆ρ(A),(149)∥R(λ2:A)∥=O(exp(M(l|λ|))),λ∈Λl,α,β.Theorem 30.
Letl≥1,α>0,β∈ℝ,k∈ℕ and C>0. Let A be a closed densely defined operator on E.(i)
Assume (148) and
(150)∥R(λ2:A)∥≤C(1+|λ|)k,λ∈Λl,α,β.
Let ε>0 and z∈ℂ be such that for each (a,b*)∈Bε(0,0) the operator A+ab*(z-A) generates an ultradistribution sine of (Mp)-class. Then A must be bounded.(ii)
Assume (146) and
(151)∥R(λ:A)∥≤C(1+|λ|)k,λ∈Λl,α,β.
Let ε>0 and z∈ℂ be such that for each (a,b*)∈Bε(0,0) the operator A+ab*(z-A) generates an ultradistribution semigroup of (Mp)-class. Then A generates an analytic C0-semigroup.Proof.
We will only prove the first part of the theorem. PutΩn:={λ∈ℂ:ℜλ≥nM(n|ℑλ|)+n}. Then Ωn⊆Λl,α,β for all n≥max(l,α,|β|). By the generalized resolvent equation, it follows that for each x∈Yn≡D(A⌈k/2⌉+2), the set {||R(λ:A)x||:λ∈Ωn} is bounded. The prescribed assumption combined with Lemma 29(ii) implies that for each (a,b*)∈Bε(0,0) there exist n∈ℕ and a function gn:Ωn→(0,∞) such that Ωn2:={λ2:λ∈Ωn}⊆ρ(A+ab*(z-A)) and ||R(λ2:A+ab*(z-A))||≤gn(λ),λ∈Ωn. By Lemma 28, we obtain m∈ℕ such that supλ∈Ωm2||λR(λ:A)||<∞. Let ξ+iη=λ∈∂(Ωm2). Assume |η1|≤|η| and μ=ξ+η1. Then ξ=(mM(m|t|)+m)2-t2 and η=2t(mM(m|t|)+m) for some t∈ℝ. Since limt→∞(M(t)/t)=0, we easily infer that there exist t0>0 and L≥1 such that, for any |t|≥t0:(152)||(λ-μ)R(λ:A)||≤2|η|||R(λ:A)||≤2L|η||λ|≤4L|t|(mM(m|t|)+m)(((mM(m|t|)+m)2-t2)2+4t2(mM(m|t|)+m)2)1/2≤12,
which implies that R(μ:A) exists and ||R(μ:A)||≤2||R(λ:A)||≤2L/|λ|≤2L. Therefore, there exists ω0>0 such that ||R(·:A)|| is polynomially bounded on {λ∈ℂ:ℜλ≤-ω0}∖{λ2:λ∈Λl,α,β}. The set {λ∈ℂ:ℜλ≥-ω0}∖{λ2:λ∈Λl,α,β} is compact, which completes the proof by [25, Lemma 2.3].Remark 31.
(i) It is worth noting that Theorem30(ii) is an extension of [25, Theorem3.1], and that Theorem 30(i) is an extension of [25, Theorem 2.2] provided k>0 in the formulation of this result. Consider now the situation of [25, Theorem 2.2] with A being the generator of an exponentially bounded α-times integrated cosine function (α≥0). Then there exists ωA>0 such that supλ>ωA||λ(2-α)/2R(λ:A)||<∞. Let ω>ωA. Then, for every γ∈ℝ, one can define the fractional power Aγ:=(ω-A)γ (cf. [1, Section 1.4]). Assuming 0≤α<2 and 0<γ≤(2-α)/2, we obtain from [1, Theorem 1.4.10(iii),(x)] that D(A)⊆D(Aγ) and Aγ∈L([D(A)],E), which implies that one can define the rank-1 perturbation Bγ:=A+ab*Aγ of A; notice that the case γ=1 has been already considered in Theorem 30. Obviously, AγR(λ:A)x=Aγ-1R(λ:A)(ω-A)x for all x∈D(A) and λ∈ρ(A). By the proof of [25, Theorem 2.2], one gets that there exists n∈ℕ such that {λ2:ℜλ≥n}⊆ρ(A) and supℜλ≥n||AγR(λ2:A)||<∞. Unfortunately, it is not clear whether the above conclusions together with [25, Lemma 2.4] (cf. also [70, Lemma 2.3]) imply that supℜλ≥n||λ2γR(λ2:A)||<∞, unless α=0. Notice also that the assumption γ=1 must be imposed in the case α≥2.
(ii) In the formulation of Theorem30(ii) and Theorem 30(i), respectively, we do not assume that the operator A+ab*(z-A) has polynomially bounded resolvent on the square of Λl,α,β, respectively, on Λl,α,β. Furthermore, we may assume that the operator A+ab*(z-A) has a slightly different spectral properties (cf. [25, Remark 2.5] and the formulation of Theorem 32 below).
(iii) Givenε∈(0,1) and Cε>0, set
(153)Ωε:={λ∈ℂ:ℜλ≥ε|λ|+Cε}.
The proof of Theorem 30(i) and Theorem 30(ii) respectively, does not work any longer if, for every ε>0, the estimate (150), respectively (151), holds with Λl,α,β replaced by Ωε. Therefore, it is not clear whether Theorem 3.11 can be reformulated in case of certain classes of hyperfunction semigroups and sines [1, 71].
Recall [32], a (local) (a,k)-regularized C-resolvent family (R(t))t∈[0,τ) having A as a subgenerator is of classCL if and only if the following holds(i)
the mappingt↦R(t),t∈(0,τ) is infinitely differentiable (in the uniform operator topology), and(ii)
for every compact setK⊆(0,τ) there exists hK>0 such that
(154)supt∈K,p∈ℕ0∥hKp(dp/dtp)R(t)Mp∥<∞;(R(t))t∈[0,τ) is said to be ρ-hypoanalytic,1≤ρ<∞, if (R(t))t∈[0,τ) is of class CL with Mp=p!ρ.
By [72, Theorem 5.5] and [32, Theorem 2.23], a C0-semigroup (T(t))t≥0 is ρ-hypoanalytic for some ρ≥1 if (T(t))t≥0 is in the Crandall-Pazy class of semigroups. Recall that (T(t))t≥0 is in the Crandall-Pazy class [72] if and only if there exist γ∈(0,1], b>0, k>0 and c∈ℝ such that
(155)Eγ,b,c≔{λ∈ℂ:ℜλ≥c-b|ℑλ|γ}⊆ρ(A),||R(λ:A)||≤c,λ∈Eγ,b,c.
Keeping in mind (155), the subsequent theorem can be viewed as a generalization of [25, Theorem 4.3]. Observe that the operator (ω-A)γ(γ∈ℝ) is defined for a sufficiently large ω>0, provided that A generates an exponentially bounded (gα,gβ)-regularized resolvent family.Theorem 32.
Suppose0<α<2,(α-1)/α<γ<1,z∈ℂ,β≥0 and a densely defined operator A generates an exponentially bounded (gα,gβ)-regularized resolvent family (R(t))t≥0.(i)
Assume thatb=0 and for each (a,b)∈Bε(0,0) there exists a kernel ka,b*(t) satisfying (P1)-(P2) so that the operator A+ab*(ω-A)γ generates an exponentially bounded (gα,ka,b*)-regularized resolvent family. Then (R(t))t≥0 is (1/(αγ+1-α))-hypoanalytic.(ii)
Assume thatb>0 and for each (a,b)∈Bε(0,0) there exists a kernel ka,b*(t) satisfying (P1)-(P2) so that the operator A+ab*(z-A) generates an exponentially bounded (gα,ka,b*)-regularized resolvent family. Then A generates an exponentially bounded, analytic (gα,1)-regularized resolvent family.Proof.
Givenω≥0, set Φα,ω:={λα:ℜλ≥ω}. Making use of [25, Lemma 2.4], [32, Theorem 2.7], and Lemma 28, we get that there exist n∈ℕ and m>0 such that Ψn,α:=Φα,n∩{z∈ℂ:|z-λ|≤m|λ|γforsomeλ∈∂(Φα,n)}⊆ρ(A) and that ||R(λ:A)||=O(|λ|-γ),λ∈Ψn,α. Let ε∈(0,1) and let Kε>0 be such that
(156)αKε(1+Kε)α-1≤m.
Put ρ:=1/(αγ+1-α). Notice that ρ≥1 since α>0 and (α-1)/α<γ<1. With the help of (156) and the Darboux inequality, we obtain that for each η∈ℝ:(157)|(n-Kε|η|1/ρ+iη)α-(n+iη)α|≤αKε|η|1/ρsupν∈[n+iη,n+iη-Kε|η|1/ρ]|ν|α-1≤αKε|η|1/ρ(|n+iη|+Kε|η|1/ρ)α-1≤αKε(1+Kε)α-1|η|1/ρ|n+iη|α-1≤αKε(1+Kε)α-1|n+iη|α-1+(1/ρ)≤m|n+iη|αγ,
which implies that {λ∈ℂ:ℜλ≥-Kε|ℑλ|1/ρ+n}⊆Ψn,α. The proof of (i) is completed by an application of [32, Theorem 2.23]. Suppose now that the assumptions of (ii) hold. Then ω-A need not be sectorial, in general. We obtain similarly the existence of an integer n∈ℕ and a number c∈(0,1) such that Υn,α:=Φα,n∩{z∈ℂ:|z-λ|≤m|λ|forsomeλ∈∂(Φα,n)}⊆ρ(A) and that ||R(λ:A)||≤c|λ|-1,λ∈Υn,α. Then it readily follows from [32, Theorem 2.17] that A generates an exponentially bounded, analytic (gα,1)-regularized resolvent family of angle (arcsinc)/α.Now we will transfer the assertion of [26, Theorem 1.3] to abstract time-fractional equations. For a∈E and b*∈E*, define Aa,b* by D(Aa,b*):={x∈E:x+〈b*,x〉a∈D(A)} and Aa,b*x:=A(x+〈b*,x〉a),x∈D(Aa,b*).We need the following auxiliary lemma (cf. the proofs and formulations of [26, Lemmas 2.1 and 2.2]).Lemma 33.
Letω≥0,α∈(0,2),a∈E,b*∈E* and z∈ℂω, where ℂω:={z∈ℂ:ℜz>ω}.(i)
Thenzα is an eigenvalue of both, Aa,b* and A+ab*A, with Aa,b*(AR(zα:A)a)=zαAR(zα:A)a and (A+ab*A)(R(zα:A)a)=zαR(zα:A)a.(ii)
Let〈b*,AR(zα:A)a〉=1 and 〈b*,AR(zα:A)a〉≠0. Then for each ε>0 there exists δ>0 such that for all (a1,b1*)∈Bδ(a,b*) there exists some z1∈B(z,ε) such that 〈b1*,AR(z1α:A)a1〉=1 and 〈b1*,AR(z1α:A)a1〉≠0.The following fractional analogue of [26, Lemma 2.3] will be essentially utilized in the proof of Theorem 35 stated below.Lemma 34.
Supposeα∈(0,2), n∈ℕ, ω≥0, ε>0 and A is the generator of an exponentially bounded, nonanalytic (gα,1)-regularized resolvent family (Sα(t))t≥0 satisfying ||Sα(t)||≤Meωt,t≥0 for some M≥1. Let r>ω,k=⌈1/α⌉,a∈D(Ak),b*∈D((A*)k) and zj∈ℂω be such that 〈b*,AR(zjα:A)a〉=1 and 〈b*,AR(zjα:A)a〉≠0(1≤j≤n). Then there exist (a1,b1*)∈Bε(a,b*)∩(D(Ak)×D((A*)k)) and 1z,…,n+1z∈ℂω such that ℜn+1z=r,|jz-zj|<ε(1≤j≤n), 〈b1*,AR(n+1zα:A)a1〉=1 and 〈b1*,AR(n+1zα:A)a1〉≠0.Proof.
We will only outline the main details of the proof. First of all, notice that||R(zα:A*)||=||R(zα:A)||≤M/((ℜz-ω)|z|α-1),ℜz>ω. By [1, Lemma 1.4.10], we get that A* is stationary dense with n(A*)≤1, which implies that D((A*)k)¯=D(A*)¯ in the strong topology of E*. Let the numbers δ1>0,…,δn>0 be given by Lemma 33(i) and let δ:=min(1,ε,δ1,…,δn). By the generalized resolvent equation (see e.g., the proof of [25, Theorem 2.2]) and the fact that *:L(E)→L(E*) is an isometrically isomorphism, we obtain that for each (a,b*)∈D(Ak)×D((A*)k), the following supremum
(158)supℜz=rmax(1,||A*R(zα:A*)b*||,||AR(zα:A)a||,|〈b*,AR(zα:A)a〉|)≔K
is finite. The nonanalyticity of (Sα(t))t≥0 yields that supℜz=r||AR(zα:A)||=∞. By the denseness of D(Ak) in E, we get the existence of an element u∈D(Ak) and a complex number n+1z:=z such that ℜz=r, ||AR(zα:A)u||>(10K/δ2), AR(zα:A)2u≠0 and ||u||<1. Now one can proceed as in the proof of [26, Lemma 2.3] so as to obtain v*∈D(A*) such that |〈v*,AR(zα:A)u〉|>(10K/δ2), ||v*||<1 and 〈b*+(δ/2)v*,AR(zα:A)2u〉≠0. Since D((A*)k) is dense in D(A*) with respect to the strong topology of E*, we may assume that v*∈D((A*)k). Copying the final part of the proof of the aforementioned lemma, with AR(z:A) and AR(z:A)2 replaced by AR(zα:A) and AR(zα:A)2 there, we obtain that there exists (a1,b1*)∈Bε(a,b*)∩(D(Ak)×D(A*)k) with required properties (cf. [26, page 474,l.1-4]).Ifα∈(0,2) and A is the generator of an exponentially bounded (gα,1)-regularized resolvent family (Sα(t))t≥0 satisfying the properties stated above, then one can simply prove that for each r>ω there exist z∈ℂω with ℜz=r and (a,b*)∈D(Ak)×D((A*)k) such that 〈b*,AR(zα:A)a〉=1 and 〈b*,AR(zα:A)2a〉≠0. Using induction, Lemma 34 and the proof of [26, Theorem 1.3], we obtain the following theorem.Theorem 35.
Supposeα∈(0,2),n∈ℕ,ω≥0,ε>0 and A is the generator of an exponentially bounded, nonanalytic (gα,1)-regularized resolvent family (Sα(t))t≥0 satisfying ||Sα(t)||≤Meωt, t≥0 for some M≥1. Let Ij⊆(ωα,∞) be an open interval (1≤j<∞). Then there exist a∈E and b*∈E* such that the operators Aa,b* and A+ab*A have a sequence of eigenvalues zj with ℜzj∈Ij  for all j=1,2,….We close the paper with the observation that perturbation theorems for q-exponentially equicontinuous(a,k)-regularized C-resolvent families have been recently analyzed in [73].



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