Perturbation Theory for Abstract Volterra Equations

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.

Dedicated to the memory of Slobodan Novaković

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 (𝑔_𝛼,𝑘)-regularized 𝐶-resolvent family (𝑅(𝑡))_𝑡≥0 with a subgenerator 𝐴 (𝑔_𝛼(𝑡)≡𝑡^𝛼−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 (𝑔_𝛼,𝑘₁)-regularized 𝐶-resolvent family (𝑅_𝐵(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 (𝑅_𝐵(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 (𝑔_𝛼,𝑘₁)-regularized 𝐶-resolvent family (𝑅_𝐵(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 𝐿^𝑝(ℝ^𝑛) (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 𝐿^𝑝([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 (𝑔_𝛼,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 (𝑔_𝛼,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 Theorems 20 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 (𝑔_𝛼,𝑔_𝑟+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)∶=∫^𝑡₀𝑓(𝑡−𝑠)𝑔(𝑠)𝑑𝑠, and the principal branch is always used to take the powers. Given 𝑓∈𝐿¹_loc([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 Σ_𝛽∶={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 𝜌_𝐶(𝐴), is defined by 𝜌_𝐶(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) Let 𝐸 be an SCLCS, let 0<𝜏≤∞,𝑘∈𝐶([0,𝜏)),𝑘≠0, and let 𝑎∈𝐿¹_loc([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+∫^𝑡₀𝑎(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) Let 𝛽∈(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∶Σ_𝛽→𝐿(𝐸) satisfying that, for every 𝑥∈𝐸, the mapping 𝑧↦𝐑(𝑧)𝑥,𝑧∈Σ_𝛽 is analytic as well as that(a)𝐑(𝑡)=𝑅(𝑡),𝑡>0 and(b)lim_{𝑧→0,𝑧∈Σ𝛾}𝐑(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 𝜔_𝛾≥0 such that the family {𝑒^{−𝜔𝛾ℜ𝑧}𝐑(𝑧)∶𝑧∈Σ_𝛾} is equicontinuous.
Since there is no risk for confusion, we will identify 𝑅(⋅) 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)=𝑔_𝑟+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)=∫^𝑡₀𝐾(𝑠)𝑑𝑠,𝑡∈[0,𝜏), where 𝐾∈𝐿¹_loc([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 (𝑔_𝛼,𝑘)-regularized 𝐶-resolvent families in the study of the following abstract time-fractional equation with 𝛼>0∶𝐃^𝛼_𝑡𝑢(𝑡)(𝑡)𝑢=𝐴𝑢,𝑡>0,_(𝑘)(0)=𝐶𝑥_𝑘,𝑘=0,1,…,⌈𝛼⌉−1, (1) where 𝑥_𝑘∈𝐷(𝐴),𝑘=0,1,…,⌈𝛼⌉−1 and 𝐃^𝛼_𝑡 denotes the Caputo fractional derivative of order 𝛼 ([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)∶=lim_𝑏→∞∫^𝑏₀𝑒^−𝜆𝑡𝑘(𝑡)𝑑𝑡∶=∫^∞₀𝑒^−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. Let 𝑘(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)Assume 𝐴 is a subgenerator of the global (𝑎,𝑘)-regularized 𝐶-resolvent family (𝑅(𝑡))_𝑡≥0 and 𝑅(𝑡)𝑥=𝑘(𝑡)𝐶𝑥+𝐴^𝑡₀𝑎(𝑡−𝑠)𝑅(𝑠)𝑥𝑑𝑠,𝑡≥0,𝑥∈𝐸. (2) 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, 𝑘(𝜆)(𝐼−𝑎(𝜆)𝐴)⁻¹𝐶𝑥=^∞₀𝑒^−𝜆𝑡𝑅(𝑡)𝑥𝑑𝑡,𝑥∈𝐸,ℜ𝜆>𝜔₀,𝑘(𝜆)1≠0,𝑎(𝜆)∶ℜ𝜆>𝜔₀,𝑘(𝜆)𝑎(𝜆)≠0⊆𝜌_𝐶(𝐴). (3)(ii)Assume (3). Then 𝐴 is a subgenerator of the global (𝑎,𝑘)-regularized 𝐶-resolvent family (𝑅(𝑡))_𝑡≥0 satisfying (2).

Let 𝐴 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∫^𝑡₀𝑎(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 ∫^𝑡₀𝑘(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 (𝑡)(𝑡)=𝐴∶=𝑥,𝑦∈𝐸×𝐸∶𝑅𝑥−𝑘𝐶𝑥_𝑡0𝑎(𝑡−𝑠)𝑅(𝑠)𝑦𝑑𝑠,𝑡≥0. (4) 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

Assume 𝛼>0 and 𝑙∈ℕ. Set, for any 𝐸-valued function 𝑓(𝑡) satisfying (P1), 𝐹_𝛼,𝑓(𝑧)∶=∫^∞₀𝑒^{−𝑧1/𝛼𝑡}𝑓(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 (𝑐_{𝑙0,𝑙,𝛼})_{1≤𝑙0≤𝑙}, independent of 𝐸 and 𝑓(𝑡), such that 𝑑^𝑙𝑑𝑧^𝑙𝐹_𝛼,𝑓(𝑧)=^𝑙_𝑙0=1𝑐_{𝑙0,𝑙,𝛼}𝑧^{(𝑙0/𝛼)−𝑙∞}₀𝑒^{−𝑧(1/𝛼)𝑡}𝑡^𝑙0𝑓(𝑡)𝑓𝑑𝑡,𝑧>maxabs,0^𝛼. (5) Furthermore, 𝑐_{𝑙,𝑙,𝛼}=(−1)^𝑙/𝛼^𝑙,𝑙≥1,𝑐_1,𝑙,𝛼=((−1)/𝛼)((1/𝛼)−1)⋯((1/𝛼)−(𝑙−1)),𝑙≥2    and the following nonlinear recursive formula holds: 𝑐_{𝑙0,𝑙+1,𝛼}=(−1)𝛼𝑐_{𝑙0−1,𝑙,𝛼}+𝑙₀𝛼−𝑙𝑐_{𝑙0,𝑙,𝛼},𝑙₀=2,…,𝑙. (6) The precise computation of coefficients (𝑐_{𝑙0,𝑙,𝛼}) is a nontrivial problem.

Lemma 3. There exists 𝜁≥1 such that ^𝑙_𝑙0=1𝑙₀!𝑐_{𝑙0,𝑙,𝛼}≤𝜁^𝑙𝑙!∀𝑙∈ℕ. (7)

Proof. Clearly, 𝐿_𝛼∶=sup_𝑛∈ℕ0|_𝑛^1/𝛼|<∞. Applying (6), one gets ^𝑙+1_𝑙0=1𝑙₀!𝑐_{𝑙0,𝑙+1,𝛼}≤1𝛼1𝛼⋯1−1𝛼+−𝑙_𝑙𝑙0=2𝑙₀!𝛼𝑐_{𝑙0−1,𝑙,𝛼}1+𝑙𝛼+1𝑙₀!𝑐_{𝑙0,𝑙,𝛼}+(𝑙+1)!𝛼^𝑙+1≤𝐿_𝛼1𝛼𝑙+𝑙𝑙!+𝛼^𝑙−1_𝑙0=1𝑙₀!𝑐_{𝑙0,𝑙,𝛼}1+𝑙𝛼+1^𝑙_𝑙0=2𝑙₀!𝑐_{𝑙0,𝑙,𝛼}+(𝑙+1)!𝛼^𝑙+1,𝑙≥2. (8) The preceding inequality implies inductively that (7) holds provided 𝜁≥4+(4/𝛼)+4𝐿_𝛼(1+(1/𝛼)).

Set 𝜁_𝛼∶=inf{𝜁≥1∶∑^𝑙_𝑙0=1𝑙₀!|𝑐_{𝑙0,𝑙,𝛼}|≤𝜁^𝑙𝑙!forall𝑙∈ℕ} and ((1/𝛼)+1)⋯((1/𝛼)+(𝑙−1))∶=1 if 𝑙=1. Clearly, 𝜁₁=1, 𝜁_𝛼>1/𝛼, 𝛼∈(0,1) and ∑^𝑙_𝑙0=1𝑙₀!|𝑐_{𝑙0,𝑙,𝛼}|≤𝜁^𝑙_𝛼𝑙! for all 𝑙∈ℕ.

The following lemma will be helpful in the analysis of growth order of perturbed integrated (𝑔_𝛼,𝐶)-regularized resolvent families.

Lemma 4. Let 𝛼>1. Then 𝜁_𝛼=1 and ^𝑙_𝑙0=1𝑙₀!𝑐_{𝑙0,𝑙,𝛼}=1𝛼1𝛼⋯1+1𝛼+(𝑙−1)≤𝑙!𝛼∀𝑙∈ℕ. (9)

Proof. Plugging 𝑓(𝑡)≡1 in (5), we obtain ^𝑙_𝑙0=1𝑙₀!𝑐_{𝑙0,𝑙,𝛼}=(−1)^𝑙1𝛼1𝛼⋯1+1𝛼+(𝑙−1)∀𝑙∈ℕ. (10) Since 𝛼>1, it follows inductively from (6) that (−1)^𝑙𝑐_{𝑙0,𝑙,𝛼}>0, provided 𝑙≥1 and 1≤𝑙₀≤𝑙. Combined with (10), the above implies (9) and 𝜁_𝛼=1.

Now we are in a position to state the following important result.

Theorem 5. Suppose 𝛼>0,𝑘(𝑡) and 𝑘₁(𝑡) satisfy (P1), 𝐴 is a subgenerator of a (𝑔_𝛼,𝑘)-regularized 𝐶-resolvent family (𝑅(𝑡))_𝑡≥0 satisfying (2) with 𝑎(𝑡)=𝑔_𝛼(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 exist 𝑀≥1, 𝜔^′≥0, 𝜔^′′≥0 and 𝜔^′′′≥max(𝜔+𝜔^′,𝜔+𝜔^′′,abs(𝑘₁)) such that 𝜆∈ℂ∶ℜ𝜆>𝜔^′′′,𝑘₁(𝜆)⊆≠0𝜆∈ℂ∶ℜ𝜆>𝜔^′′′,𝑘(𝜆)≠0 (11) as well as(ii) For every 𝑖,𝑙₀,𝑙∈ℕ with 1≤𝑙≤𝑖 and 1≤𝑙₀≤𝑙, there exists a function 𝑘_{𝑖,𝑙0,𝑙}(𝑡) satisfying (P1) and ℒ𝑘_{𝑖,𝑙0,𝑙}(𝑡)(𝜆)=𝑐_{𝑙0,𝑙,𝛼}𝜆^{𝑙0−𝛼(𝑙−1)}𝑘₁(𝜆)1𝑧𝑘𝑧^{1/𝛼(𝑖−𝑙)}_𝑧=𝜆^𝛼, (12) provided ℜ𝜆>𝜔^′′′ and 𝑘₁(𝜆)≠0.(iii) For every 𝑖∈ℕ₀, there exists a function _𝑖𝑘(𝑡) satisfying (P1) and a constant 𝑐_𝑖∈ℂ so that 𝑐_𝑖+_𝑖𝑘(𝜆)=𝜆^𝛼𝑘₁(𝜆1)𝑧𝑘𝑧^1/𝛼(𝑖)_𝑧=𝜆^𝛼,ℜ𝜆>𝜔^′′′,𝑘₁(𝜆)≠0. (13)(iv)^∞_𝑖=0𝑐_𝑖|𝐵|^𝑖_⊛𝑖!<∞,^∞_𝑖=0|𝐵|^𝑖_⊛𝑖!^𝑡₀^𝑖𝑘(𝑠)𝑑𝑠≤𝑀𝑒^𝜔′𝑡,𝑡≥0, (14)(v)^∞𝑖_𝑖=1^𝑙_{𝑙=1𝑙0=1}|𝐵|^𝑖_⊛𝑖𝑙𝑖!^𝑡₀(𝑡−𝑠)^𝑙0𝑘_{𝑖,𝑙0,𝑙}(𝑠)𝑑𝑠≤𝑀𝑒^𝜔′𝑡,𝑡≥0, (15)(vi)^∞𝑖_𝑖=2𝑙=2^𝑙−1_𝑙0=1|𝐵|^𝑖_⊛𝑙𝑖𝑙𝑖!^𝑡₀(𝑡−𝑠)^𝑙0𝑘_{𝑖−1,𝑙0,𝑙−1}(𝑠)𝑑𝑠≤𝑀𝑒^{𝜔′′𝑡},𝑡≥0. (16)Then 𝐴+𝐵 is a subgenerator of an exponentially equicontinuous (𝑔_𝛼,𝑘₁)-regularized 𝐶-resolvent family (𝑅_𝐵(𝑡))_𝑡≥0, which is given by the following formula: 𝑅_𝐵(𝑡)𝑥∶=^∞_𝑖=0(−𝐵)^𝑖𝑐𝑖!^𝑖𝑅(𝑡)𝑥+_𝑖(⋅)𝑥(𝑡+𝑘∗𝑅)_{∞𝑖𝑖=1𝑙𝑙=1𝑙0=1}(−𝐵)^𝑖𝑖𝑙𝑘𝑖!^{𝑖,𝑙0,𝑙}∗⋅^𝑙0𝑅(⋅)𝑥(𝑡),𝑡≥0,𝑥∈𝐸. (17)
Furthermore, 𝑅_𝐵(𝑡)𝑥=𝑘₁(𝑡)𝐶𝑥+(𝐴+𝐵)^𝑡₀𝑔_𝛼(𝑡−𝑠)𝑅_𝐵(𝑠)𝑥𝑑𝑠,𝑡≥0,𝑥∈𝐸, (18) and the family {𝑒^{−(𝜔+𝜔′)𝑡}𝑅_𝐵(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 (𝑅_𝐵(𝑡))_𝑡≥0 is strongly continuous and that the family {𝑒^{−(𝜔+𝜔′)𝑡}𝑅_𝐵(𝑡)∶𝑡≥0} is equicontinuous. By (i) and Lemma 2, (𝑧−𝐴)⁻¹𝐶𝐵=𝐵(𝑧−𝐴)⁻¹𝐶,𝑧∈𝜌_𝐶(𝐴)    and, for every 𝜆∈ℂ with ℜ𝜆>𝜔 and 𝑘(𝜆)≠0∶ 𝜆^𝛼𝑘(𝜆)(𝜆^𝛼−𝐴)⁻¹𝐶𝐵𝑥=∫^∞₀𝑒^−𝜆𝑡𝑅(𝑡)𝐵𝑥𝑑𝑡,𝑥∈𝐸 and 𝜆^𝛼𝑘(𝜆)𝐵(𝜆^𝛼−𝐴)⁻¹𝐶𝑥=∫^∞₀𝑒^−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 𝑅_𝐵(𝑡)𝐴⊆𝐴𝑅_𝐵(𝑡), 𝑡≥0. Hence, 𝑅_𝐵(𝑡)(𝐴+𝐵)⊆(𝐴+𝐵)𝑅_𝐵(𝑡), 𝑡≥0. By Lemma 2, 𝑧𝑘𝑧^1/𝛼(𝑧−𝐴)⁻¹𝐶𝑥=^∞₀𝑒^{−𝑧1/𝛼𝑡}𝑅(𝑡)𝑧𝑥𝑑𝑡,𝑥∈𝐸,ℜ^1/𝛼>𝜔,𝑘𝑧^1/𝛼≠0. (19) 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(𝑧^1/𝛼)>𝜔 and 𝑘(𝑧^1/𝛼)≠0∶𝐴𝑑^𝑙𝑑𝑧^𝑙∞₀𝑒^{−𝑧1/𝛼𝑡}𝑅(𝑡)𝑑𝑥𝑑𝑡=𝐴^𝑙𝑑𝑧^𝑙𝑧𝑘𝑧^1/𝛼(𝑧−𝐴)⁻¹=𝑑𝐶𝑥_𝑙𝑑𝑧^𝑙𝑧𝑘𝑧^1/𝛼𝐴(𝑧−𝐴)⁻¹𝑑𝐶𝑥=𝑧^𝑙𝑑𝑧^𝑙𝑧𝑘𝑧^1/𝛼(𝑧−𝐴)⁻¹𝑑𝐶𝑥+𝑙^𝑙−1𝑑𝑧^𝑙−1𝑧𝑘𝑧^1/𝛼(𝑧−𝐴)⁻¹−𝑑𝐶𝑥_𝑙𝑑𝑧^𝑙𝑧𝑘𝑧^1/𝛼𝐶𝑥. (20) 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 ℒ𝑅_𝐵(𝑡)𝑥(𝜆)=𝜆^𝛼𝑘₁(𝜆)^∞_𝑖=0(−𝐵)^𝑖×𝑖!_𝑖𝑙=0𝑖𝑙1𝑧𝑘𝑧^{1/𝛼(𝑖−𝑙)}_𝑧=𝜆^𝛼×𝑑^𝑙𝑑𝑧^𝑙𝑧𝑘𝑧^1/𝛼(𝑧−𝐴)⁻¹𝐶𝑥_{𝑧=𝜆𝛼}. (21) Our goal is to prove that 𝐼−𝐴+𝐵𝜆^𝛼ℒ𝑅_𝐵(𝑡)𝑥(𝜆)=𝑘₁(𝜆)𝐶𝑥. (22)
By the product rule, we get ^∞_𝑖=1(−𝐵)^𝑖𝑖!^𝑖_𝑙=1𝑖𝑙1𝑧𝑘𝑧^{1/𝛼(𝑖−𝑙)}_𝑧=𝜆^𝛼𝑑^𝑙𝑑𝑧^𝑙𝑧𝑘𝑧^1/𝛼𝐶𝑥_{𝑧=𝜆𝛼}=−𝜆^𝛼𝑘(𝜆)^∞_𝑖=1(−𝐵)^𝑖1𝑖!𝑧𝑘𝑧^1/𝛼(𝑖)_𝑧=𝜆^𝛼𝐶𝑥; (23) notice that the convergence of last series follows from the conditions (iii)-(iv). Taking into account (5), (ii), and (vi), one yields that 1𝜆^𝛼𝑘₁(𝜆)^∞0𝑒^{∞−𝜆𝑡𝑖}_𝑖=2𝑙=2^𝑙−1_𝑙0=1(−𝐵)^𝑖𝑙𝑖𝑙×𝑖!_𝑡0(𝑡−𝑠)^𝑙0𝑅(𝑡−𝑠)×𝑘_{𝑖−1,𝑙0,𝑙−1}(𝑠)=1𝑥𝑑𝑠𝑑𝑡𝜆^𝛼𝑘₁(𝜆)^∞𝑖_𝑖=2𝑙=2^𝑙−1_𝑙0=1(−𝐵)^𝑖𝑙𝑖𝑙×𝑖!_∞0𝑒^{−𝜆𝑡𝑡}₀(𝑡−𝑠)^𝑙0𝑅(𝑡−𝑠)×𝑘_{𝑖−1,𝑙0,𝑙−1}(𝑠)=𝑥𝑑𝑠𝑑𝑡_{∞𝑖𝑖=2𝑙=2𝑙−1𝑙0=1}(−𝐵)^𝑖𝑙𝑖𝑙𝑐𝑖!^{𝑙0,𝑙−1,𝛼}𝜆^{𝑙0−𝛼(𝑙−1)}×^∞₀𝑒^−𝜆𝑡𝑡^𝑙0𝑅(𝑡)=𝑥𝑑𝑡_∞𝑖=2(−𝐵)^𝑖×𝑖!_𝑖𝑙=2𝑙𝑖𝑙1𝑧𝑘𝑧^{1/𝛼(𝑖−𝑙)}_𝑧=𝜆^𝛼×𝑑^𝑙−1𝑑𝑧^𝑙−1𝑧𝑘𝑧^1/𝛼(𝑧−𝐴)⁻¹𝐶𝑥_{𝑧=𝜆𝛼}, (24) which implies that the series ^∞_𝑖=1(−𝐵)^𝑖𝑖!^𝑖_𝑙=1𝑙𝑖𝑙1𝑧𝑘𝑧^{1/𝛼(𝑖−𝑙)}_𝑧=𝜆^𝛼×𝑑^𝑙−1𝑑𝑧^𝑙−1𝑧𝑘𝑧^1/𝛼(𝑧−𝐴)⁻¹𝐶𝑥_{𝑧=𝜆𝛼} (25) is also convergent. Now we get from (20)-(21) and (23)-(24): 𝐼−𝐴+𝐵𝜆^𝛼ℒ𝑅_𝐵(𝑡)𝑥(𝜆)=𝜆^𝛼∞_𝑖=0(−𝐵)^𝑖𝑖!^𝑖_𝑙=0𝑖𝑙1𝑧𝑘𝑧^{1/𝛼(𝑖−𝑙)}_𝑧=𝜆^𝛼×𝑑^𝑙𝑑𝑧^𝑙𝑧𝑘𝑧^1/𝛼(𝑧−𝐴)⁻¹𝐶𝑥_{𝑧=𝜆𝛼}−^∞_𝑖=1(−𝐵)^𝑖𝑖!^𝑖_𝑙=1𝑖𝑙1𝑧𝑘𝑧^{1/𝛼(𝑖−𝑙)}_𝑧=𝜆^𝛼×𝑧𝑑^𝑙𝑑𝑧^𝑙𝑧𝑘𝑧^1/𝛼(𝑧−𝐴)⁻¹𝑑𝐶𝑥+𝑙^𝑙−1𝑑𝑧^𝑙−1𝑧𝑘𝑧^1/𝛼(𝑧−𝐴)⁻¹−𝑑𝐶𝑥_𝑙𝑑𝑧^𝑙𝑧𝑘𝑧^1/𝛼𝐶𝑥_{𝑧=𝜆𝛼}−𝜆^𝛼𝑘(𝜆)^∞_𝑖=0(−𝐵)^𝑖1𝑖!𝑧𝑘𝑧^1/𝛼(𝑖)_𝑧=𝜆^𝛼𝐴𝜆^𝛼−𝐴⁻¹𝐶𝑥+𝜆^𝛼∞_𝑖=0(−𝐵)^𝑖+1𝑖!^𝑖_𝑙=0𝑖𝑙1𝑧𝑘𝑧^{1/𝛼(𝑖−𝑙)}_𝑧=𝜆^𝛼×𝑑^𝑙𝑑𝑧^𝑙𝑧𝑘𝑧^1/𝛼(𝑧−𝐴)⁻¹𝐶𝑥_{𝑧=𝜆𝛼}=𝜆^2𝛼𝑘(𝜆)^∞_𝑖=0(−𝐵)^𝑖1𝑖!𝑧𝑘𝑧^1/𝛼(𝑖)_𝑧=𝜆^𝛼𝜆^𝛼−𝐴⁻¹−𝐶𝑥_∞𝑖=1(−𝐵)^𝑖𝑖!^𝑖_𝑙=1𝑖𝑙1𝑧𝑘𝑧^{1/𝛼(𝑖−𝑙)}_𝑧=𝜆^𝛼×𝑙𝑑^𝑙−1𝑑𝑧^𝑙−1𝑧𝑘𝑧^1/𝛼(𝑧−𝐴)⁻¹𝐶𝑥_{𝑧=𝜆𝛼}−𝜆^𝛼𝑘(𝜆)^∞_𝑖=1(−𝐵)^𝑖1𝑖!𝑧𝑘𝑧^1/𝛼(𝑖)_𝑧=𝜆^𝛼𝐶𝑥−𝜆^𝛼𝑘(𝜆)^∞_𝑖=0(−𝐵)^𝑖1𝑖!𝑧𝑘𝑧^1/𝛼(𝑖)_𝑧=𝜆^𝛼×−𝐶𝑥+𝜆^𝛼𝜆^𝛼−𝐴⁻¹𝐶𝑥+𝜆^𝛼∞_𝑖=0(−𝐵)^𝑖+1𝑖!^𝑖_𝑙=0𝑖𝑙1𝑧𝑘𝑧^{1/𝛼(𝑖−𝑙)}_𝑧=𝜆^𝛼×𝑑^𝑙𝑑𝑧^𝑙𝑧𝑘𝑧^1/𝛼(𝑧−𝐴)⁻¹𝐶𝑥_{𝑧=𝜆𝛼}=𝐶𝑥−^∞_𝑖=1(−𝐵)^𝑖×𝑖!_𝑖𝑙=1𝑖𝑙1𝑧𝑘𝑧^{1/𝛼(𝑖−𝑙)}_𝑧=𝜆^𝛼×𝑙𝑑^𝑙−1𝑑𝑧^𝑙−1𝑧𝑘𝑧^1/𝛼(𝑧−𝐴)⁻¹𝐶𝑥_{𝑧=𝜆𝛼}+𝜆^𝛼∞_𝑖=0(−𝐵)^𝑖+1𝑖!^𝑖_𝑙=0𝑖𝑙1𝑧𝑘𝑧^{1/𝛼(𝑖−𝑙)}_𝑧=𝜆^𝛼×𝑑^𝑙𝑑𝑧^𝑙𝑧𝑘𝑧^1/𝛼(𝑧−𝐴)⁻¹𝐶𝑥_{𝑧=𝜆𝛼}=𝐶𝑥, (26) 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)/𝜆^𝛼))𝑥=0. By (22) and 𝑅_𝐵(𝑡)(𝐴+𝐵)⊆(𝐴+𝐵)𝑅_𝐵(𝑡), 𝑡≥0, we obtain that 𝑘₁(𝜆)𝐶𝑥=𝐼−𝐴+𝐵𝜆^𝛼ℒ𝑅_𝐵(𝑡)𝑥𝜆)𝑅(=ℒ_𝐵(𝑡)𝐼−𝐴+𝐵𝜆^𝛼𝑥(𝜆)=0, (27) which implies 𝐶𝑥=𝑥=0. Thus, {𝜆^𝛼∶ℜ𝜆>𝜔^′′′,𝑘₁(𝜆)≠0}⊆𝜌_𝐶(𝐴+𝐵) and 𝑘₁(𝜆)𝐼−𝐴+𝐵𝜆^𝛼−1𝐶𝑥=^∞₀𝑒^−𝜆𝑡𝑅_𝐵(𝑡)𝑥𝑑𝑡,𝑥∈𝐸ℜ𝜆>𝜔^′′′,𝑘₁(𝜆)≠0. (28)
The proof of theorem completes an application of Lemma 2.

Remark 6. (i) By [33, Proposition 2.4(i)], we get that (𝑅_𝐵(𝑡))_𝑡≥0 is a unique (𝑔_𝛼,𝑘₁)-regularized 𝐶-resolvent family with the properties stated in the formulation of Theorem 5.
(ii) The following comment is also applicable to Theorem 7 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_⊛/𝑛^𝑖𝑖! therein. Writing 𝐴+𝐵 as 𝐴+∑^𝑛_𝑖=1𝐵/𝑛 and applying Theorem 5 successively 𝑛 times, we obtain that 𝐴+𝐵 is a subgenerator of a global (𝑔_𝛼,𝑘)-regularized 𝐶-resolvent family (𝑅_𝐵(𝑡))_𝑡≥0 satisfying (18). Furthermore, the family {𝑒^{−(𝜔+𝑛𝜔′)𝑡}𝑅_𝐵(𝑡)∶𝑡≥0} is equicontinuous.
(iii) It is not clear whether there exist functions 𝑘(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 Theorem 5 with (𝑅(𝑡))_𝑡≥0 being an exponentially equicontinuous, analytic (𝑔_𝛼,𝑘)-regularized 𝐶-resolvent family of angle 𝛽∈(0,𝜋/2]. Assume that, for every 𝛾∈(0,𝛽), there exists 𝜔_𝛾≥0 such that the set {𝑒^{−𝜔𝛾ℜ𝑧}𝑅(𝑧)∶𝑧∈Σ_𝛾} is equicontinuous. Assume, additionally, that there exists 𝜀>0 such that, for every 𝛾∈(0,𝛽), there exist 𝜔_𝛾,1≥max(sup{abs(_𝑖𝑘)∶𝑖≥1},𝜔_𝛾) and 𝜔_𝛾,2≥max(sup{abs(𝑘_{𝑖,𝑙0,𝑙})∶1≤𝑙≤𝑖,1≤𝑙₀≤𝑙},𝜔_𝛾+𝜀) with the following properties.(i)For every i∈ℕ₀, the function 𝜆↦_𝑖𝑘(𝜆),𝜆>𝜔_𝛾,1 can be analytically extended to the sector 𝜔_𝛾,1+Σ_{(𝜋/2)+𝛾} and the following holds: ^∞_𝑖=0|𝐵|^𝑖_⊛𝑖!sup_{𝜆∈𝜔𝛾,1+Σ(𝜋/2)+𝛾𝑖}𝑘(𝜆)<∞. (29)(ii)For every 𝑖,𝑙₀,𝑙∈ℕ with 1≤𝑙≤𝑖 and 1≤𝑙₀≤𝑙, the function 𝜆↦ℒ(𝑘_{𝑖,𝑙0,𝑙}(𝑡))(𝜆), 𝜆>𝜔_𝛾,2 can be analytically extended to the sector 𝜔_𝛾,2+Σ_{(𝜋/2)+𝛾} and the following holds: ^∞𝑖_𝑖=1^𝑙_{𝑙=1𝑙0=1}|𝐵|^𝑖_⊛𝑖𝑙𝑙𝑖!⁰!2𝜋𝑙₀𝜀cos𝛾^𝑙0×sup_{𝜆∈𝜔𝛾,2+Σ(𝜋/2)+𝛾}ℒ𝑘_{𝑖,𝑙0,𝑙}(𝑡)(𝜆)<∞. (30)Then (𝑅_𝐵(𝑡))_𝑡≥0 is an exponentially equicontinuous, analytic (𝑔_𝛼,𝑘₁)-regularized 𝐶-resolvent family of angle 𝛽.

Proof. Let 𝑝∈⊛,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 |𝑧|^𝑙0𝑒^{𝜔𝛾ℜ𝑧}≤(ℜ𝑧)^𝑙0cos𝛾^𝑙0𝑒^{𝜔𝛾ℜ𝑧}≤𝑒^−𝑙0cos𝛾^𝑙0𝑙^𝑙0₀𝜀^𝑙0𝑒^{(𝜔𝛾+𝜀)ℜ𝑧}𝑙≤𝜅⁰!2𝜋𝑙₀𝜀cos𝛾^𝑙0𝑒^{(𝜔𝛾+𝜀)ℜ𝑧}𝑙≤𝜅⁰!2𝜋𝑙₀𝜀cos𝛾^𝑙0𝑒^{𝜔𝛾,2ℜ𝑧} (31) for all 𝑧∈Σ_𝛾 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>𝜔_𝛾,1, respectively, 𝜆↦ℒ(𝑘_{𝑖,𝑙0,𝑙}∗⋅^𝑙0𝑅(⋅)𝑥)(𝜆),𝜆>𝜔_𝛾,2   can be analytically extended to the sector 𝜔_𝛾,1+Σ_{(𝜋/2)+𝛾}, respectively, 𝜔_𝛾,2+Σ_{(𝜋/2)+𝛾}, as well as that there exist 𝑐_𝑝>0 and 𝑞_𝑝∈⊛, independent of 𝑥, such that sup_{𝜆∈𝜔𝛾,1+Σ(𝜋/2)+𝛾−𝜀}𝑝𝜆−𝜔_𝛾,1𝑖𝑘(𝜆)ℒ(𝑅(𝑡)𝑥𝜆≤𝑐)()_𝑝𝑞_𝑝(𝑥)sin𝜀sup_{𝜆∈𝜔𝛾,1+Σ(𝜋/2)+𝛾−𝜀𝑖}𝑘(𝜆) (32) and that, for every 𝑖,𝑙₀,𝑙∈ℕ with 1≤𝑙≤𝑖 and 1≤𝑙₀≤𝑙, sup_{𝜆∈𝜔𝛾,2+Σ(𝜋/2)+𝛾−𝜀}𝑝𝜆−𝜔_𝛾,2ℒ𝑘_{𝑖,𝑙0,𝑙}(𝑡)(𝜆)ℒ𝑡^𝑙0𝑅(𝑡)𝑥(𝜆)𝑐≤𝜅^𝑝𝑞_𝑝(𝑥)𝑙sin𝜀⁰!2𝜋𝑙₀𝜀cos𝛾^𝑙0×sup_{𝜆∈𝜔𝛾,2+Σ(𝜋/2)+𝛾−𝜀}ℒ𝑘_{𝑖,𝑙0,𝑙}(𝑡)(𝜆). (33) 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↦(𝑘_{𝑖,𝑙0,𝑙}∗⋅^𝑙0𝑅(⋅)𝑥)(𝑡),𝑡>0 can be analytically extended to the sector Σ_𝛾 and that the following estimates hold: 𝑝_𝑖(⋅)𝑥(𝑧)≤𝑐𝑘∗𝑅_𝑝𝑞_𝑝(𝑥)sin𝜀sup_{𝜆∈𝜔𝛾,1+Σ(𝜋/2)+𝛾−𝜀𝑖}𝑘(𝜆)𝑒^{1+𝜔𝛾,1ℜ𝑧}+𝑒^{𝜔𝛾,1ℜ𝑧},𝜋sin𝜀𝑧∈Σ_𝛾−3𝜀,𝑝𝑘_{𝑖,𝑙0,𝑙}∗⋅^𝑙0𝑅(⋅)𝑥(𝑧)𝑐≤𝜅^𝑝𝑞_𝑝(𝑥)𝑙sin𝜀⁰!2𝜋𝑙₀𝜀cos𝛾^𝑙0×sup_{𝜆∈𝜔𝛾,2+Σ(𝜋/2)+𝛾−𝜀}ℒ𝑘_{𝑖,𝑙0,𝑙}(𝑡)(𝜆×𝑒_{1+𝜔𝛾,2ℜ𝑧})+𝑒^{𝜔𝛾,2ℜ𝑧}𝜋sin𝜀,𝑧∈Σ_𝛾−3𝜀. (34) 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↦𝑅_𝐵(𝑡)𝑥,𝑡>0 can be analytically extended to the sector Σ_𝛽 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 lim_𝑡→0+∑^∞_𝑖=0((−𝐵)^𝑖/𝑖!)𝑐_𝑖𝑅(𝑡)𝑥=∑^∞_𝑖=0((−𝐵)^𝑖/𝑖!)𝑐_𝑖𝑅(0),lim_𝑡→0+∑^∞_𝑖=0((−𝐵)^𝑖/𝑖!)(_𝑖𝑘∗𝑅(⋅)𝑥)(𝑡)=0, and lim_𝑡→0+∑^∞_𝑖=1∑^𝑖_𝑙=1∑^𝑙_𝑙0=1((−𝐵)^𝑖/𝑖!)^𝑖_𝑙(𝑘_{𝑖,𝑙0,𝑙}∗⋅^𝑙0𝑅(⋅)𝑥)(𝑡)=0. Furthermore, the functions 𝑧↦𝑓₁(𝑧)∶=∑^∞_𝑖=0((−𝐵)^𝑖/𝑖!)𝑐_𝑖𝑅(𝑧)𝑥,𝑧∈Σ_𝛽,𝑧↦𝑓₂(𝑧)∶=∑^∞_𝑖=0((−𝐵)^𝑖/𝑖!)(_𝑖𝑘∗𝑅(⋅)𝑥)(𝑧),𝑧∈Σ_𝛽, and 𝑧↦𝑓₃(𝑧)∶=∑^∞_𝑖=1∑^𝑖_𝑙=1∑^𝑙_𝑙0=1((−𝐵)^𝑖/𝑖!)^𝑖_𝑙(𝑘_{𝑖,𝑙0,𝑙}∗⋅^𝑙0𝑅(⋅)𝑥)(𝑧),𝑧∈Σ_𝛽 are analytic, and the set {𝑒^{−(𝜔𝛾+𝜔𝛾,1+𝜔𝛾,2)ℜ𝑧}𝑓_𝑗(𝑧)∶1≤𝑗≤3,𝑧∈Σ_𝛾−3𝜀} is bounded. An application of [33, Theorem 3.4(ii)] gives that the mapping 𝑧↦𝑅_𝐵(𝑧)𝑥,𝑧∈Σ_𝛽∪{0} is continuous on any closed subsector of Σ_𝛽∪{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.

Let 𝛼>0, let 𝛽>0, and let the Mittag-Leffler function 𝐸_𝛼,𝛽(𝑧) be defined by 𝐸_𝛼,𝛽(𝑧)∶=∑^∞_𝑛=0𝑧^𝑛/Γ(𝛼𝑛+𝛽),𝑧∈ℂ. Set 𝐸_𝛼(𝑧)∶=𝐸_𝛼,1(𝑧),𝑧∈ℂ. Then it is well known (cf. [28, 42–44]) that 𝐸^′_𝛼(𝑧)=𝐸_𝛼,𝛼(𝑧)/𝛼,   𝑧∈ℂ and that, for every 𝛼>1, there exist 𝑏_𝛼≥1 and 𝑐_𝛼≥1 such that 𝐸_𝛼,𝛼(𝑡)≤𝑏_𝛼𝑡^{(1−𝛼)/𝛼}exp𝑡^1/𝛼𝐸,𝑡>0,_𝛼(𝑡)≤𝑐_𝛼exp𝑡^1/𝛼,𝑡≥0. (35) It is noteworthy that the assumptions of Theorems 5 and 7 hold provided 𝛼>1 and 𝑘(𝑡)=𝑘₁(𝑡)=𝑔_𝑟+1(𝑡), where 𝑟≥0. In this case, 𝑐₀=1,𝑘₀(𝑡)=0,𝑐_𝑖=0,𝑖≥1, _𝑖𝑘(𝑡)=𝑟+1𝛼−1⋯𝑟+1𝛼−𝑖𝑔_𝛼𝑖(𝑡),𝑡≥0,𝑖≥1 (36) and, for every 𝑖,𝑙₀,𝑙∈ℕ with 1≤𝑙≤𝑖 and 1≤𝑙₀≤𝑙, 𝑘_{𝑖,𝑙0,𝑙}(𝑡)=𝑐_{𝑙0,𝑙,𝛼}𝑟+1𝛼−1⋯𝑟+1𝛼−(𝑖−𝑙)𝑔_{𝛼𝑖−𝑙0}(𝑡),𝑡>0. (37) In order to verify (iv)–(vi), notice that there exists a constant 𝑐_𝑟,𝛼≥1 such that |(((𝑟+1)/𝛼)−1)⋯(((𝑟+1)/𝛼)−𝑖)|≤𝑐_𝑟,𝛼𝑖! for all 𝑖∈ℕ. Then we obtain from (35) and Lemmas 3–4 that ^∞_𝑖=1|𝐵|^𝑖_⊛𝑖!𝑟+1𝛼−1⋯𝑟+1𝛼−𝑖^𝑡₀𝑔_𝛼𝑖(𝑠)𝑑𝑠≤𝑐^∞_{𝑟,𝛼𝑖=1}𝑡^𝛼|𝐵|_⊛^𝑖Γ(𝛼𝑖+1)=𝑐_𝑟,𝛼𝐸_𝛼𝑡^𝛼|𝐵|_⊛−1≤𝑐_𝑟,𝛼𝑐_𝛼𝑒^{𝑡|𝐵|⊛1/𝛼},𝑡≥0,_{∞𝑖𝑖=1𝑙𝑙=1𝑙0=1}|𝐵|^𝑖_⊛𝑖𝑙𝑖!^𝑡₀(𝑡−𝑠)^𝑙0𝑘_{𝑖,𝑙0,𝑙}(𝑠≤)𝑑𝑠_{∞𝑖𝑖=1𝑙𝑙=1𝑙0=1}|𝐵|^𝑖_⊛𝑖𝑙𝑐𝑖!^{𝑙0,𝑙,𝛼}×𝑟+1𝛼−1⋯𝑟+1𝛼−(𝑖−𝑙)𝑙₀!𝑔_𝛼𝑖+1(𝑡)≤𝑐_𝑟,𝛼𝛼^∞_𝑖=1𝑖𝑡^𝛼|𝐵|_⊛^𝑖Γ()=𝑐𝛼𝑖+1𝛼^𝑟,𝛼𝑡^𝛼|𝐵|_⊛^∞_𝑖=1𝑖𝑡^𝛼|𝐵|_⊛^𝑖−1Γ()=𝑐𝛼𝑖+1_𝑟,𝛼𝛼𝑡^𝛼|𝐵|_⊛^∞_𝑖=1𝑧^𝑖Γ(𝛼𝑖+1)^′_{𝑧=𝑡𝛼|𝐵|⊛}=𝑐_𝑟,𝛼𝛼²𝑡^𝛼|𝐵|_⊛𝐸_𝛼,𝛼𝑡^𝛼|𝐵|_⊛≤𝑐_𝑟,𝛼𝑏_𝛼𝛼²𝑡^𝛼|𝐵|_⊛𝑡^𝛼|𝐵|_⊛^{(1−𝛼)/𝛼}𝑒^{𝑡|𝐵|⊛1/𝛼}≤𝑐_𝑟,𝛼𝑏_𝛼𝛼²𝑡|𝐵|_⊛^1/𝛼𝑒^{𝑡|𝐵|⊛1/𝛼},𝑡≥0, (38)(39) proving the conditions (iv)-(v) and ^∞𝑖_𝑖=2𝑙=2^𝑙−1_𝑙0=1|𝐵|^𝑖_⊛𝑙𝑖𝑙𝑖!^𝑡₀(𝑡−𝑠)^𝑙0𝑘_{𝑖−1,𝑙0,𝑙−1}(𝑠)𝑑𝑠≤𝑐^∞_𝑟,𝛼^𝑖_𝑖=2𝑙=2^𝑙−1_𝑙0=1|𝐵|^𝑖_⊛()!𝑙𝑙−1⁰!𝑐_{𝑙0,𝑙−1,𝛼}𝑔_{𝛼(𝑖−1)+1}(𝑡)≤𝑐_𝑟,𝛼𝛼|𝐵|_⊛^∞_𝑖=2()𝑡𝑖−1^𝛼|𝐵|_⊛^𝑖−1Γ(𝛼())=𝑐𝑖−1+1_𝑟,𝛼𝛼|𝐵|_⊛^∞_𝑖=1𝑖𝑡^𝛼|𝐵|_⊛^𝑖Γ()≤𝑐𝛼𝑖+1_𝑟,𝛼𝑏_𝛼𝛼²𝑡|𝐵|_⊛^{(1+𝛼)/𝛼}𝑒^{𝑡|𝐵|⊛1/𝛼},𝑡≥0, (40) proving the condition (vi). Assume now, with the notation used in the formulation of Theorem 7 that 𝛾∈(0,𝛽),𝜔_𝛾,1≥𝜔_𝛾,𝜔_𝛾,1>0,(𝜔_𝛾,1cos𝛾)^𝛼>|B|_⊛,𝜀=1/cos𝛾,𝜔_𝛾,2≥𝜔_𝛾+𝜀, and (1+𝜔_𝛾,2cos𝛾)^𝛼−1>|B|_⊛. Then ^∞_𝑖=1|𝐵|^𝑖_⊛𝑖!sup_{𝜆∈𝜔𝛾,1+Σ(𝜋/2)+𝛾𝑖}𝑘(𝜆)≤𝑐^∞_{𝑟,𝛼𝑖=0}|𝐵|^𝑖_⊛sup_{𝜆∈𝜔𝛾,1+Σ(𝜋/2)+𝛾}1|𝜆|^𝛼𝑖≤𝑐^∞_{𝑟,𝛼𝑖=0}|𝐵|^𝑖_⊛𝜔_𝛾,1cos𝛾^𝛼𝑖<∞,_{∞𝑖𝑖=1𝑙𝑙=1𝑙0=1}|𝐵|^𝑖_⊛𝑖𝑙𝑙𝑖!⁰!2𝜋𝑙₀𝜀cos𝛾^𝑙0×sup_{𝜆∈𝜔𝛾,2+Σ(𝜋/2)+𝛾}ℒ𝑘_{𝑖,𝑙0,𝑙}(𝑡)(𝜆)≤𝑐^∞_𝑟,𝛼^𝑖_𝑖=1^𝑙_{𝑙=1𝑙0=1}|𝐵|^𝑖_⊛𝑙𝑙!⁰!𝑐_{𝑙0,𝑙,𝛼}sup_{𝜆∈𝜔𝛾,2+Σ(𝜋/2)+𝛾}1|𝜆|^{𝛼𝑖−𝑙0}≤𝑐_𝑟,𝛼𝛼^∞𝑖_𝑖=1^𝑙_{𝑙=1𝑙0=1}|𝐵|^𝑖_⊛1+𝜔_𝛾,2cos𝛾^{𝛼𝑖−𝑙0}≤𝑐_𝑟,𝛼𝛼^∞_𝑖=1𝑖²|𝐵|^𝑖_⊛1+𝜔_𝛾,2cos𝛾^{(𝛼−1)𝑖}<∞, (41)(42) proving the conditions (29)-(30).

Corollary 8. Suppose 𝛼>1,𝜔≥0,𝑟≥0 and 𝐴 is a subgenerator of a global 𝑟-times integrated (𝑔_𝛼,𝐶)-regularized resolvent family (𝑅(𝑡))_𝑡≥0 satisfying (2) with 𝑎(𝑡)=𝑔_𝛼(𝑡) and 𝑘(𝑡)=𝑔_𝑟+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 (𝑔_𝛼,𝐶)-regularized resolvent family (𝑅_𝐵(𝑡))_𝑡≥0 satisfying (18) with 𝑘₁(𝑡)=𝑘(𝑡). Furthermore, the family {(1+𝑡)⁻¹exp(−(𝜔+|𝐵|_⊛^1/𝛼)𝑡)𝑅_𝐵(𝑡)∶𝑡≥0} is equicontinuous, and (𝑅_𝐵(𝑡))_𝑡≥0 is an exponentially equicontinuous, analytic 𝑟-times integrated (𝑔_𝛼,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 (𝑅_𝐵(𝑡))_𝑡≥0, in the case 𝛼=1. Using [33, Lemma 3.3] and the proof of cited theorem, it follows that (𝑅_𝐵(𝑡))_𝑡≥0 is entire provided that 𝛼∈ℕ and that (𝑅(𝑡))_𝑡≥0 is entire.

Example 10. Corollary 8 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 𝑢_𝑡𝑡+2𝛽𝑢_𝑡−Δ𝑢+2^𝑛_𝑖=1𝛼_𝑖𝑢_𝑥𝑖+𝜇𝑢=0 (43) in 𝐿^𝑝([0,𝜋]^𝑛), with Dirichlet boundary conditions; here we assume 𝑛∈ℕ,1≤𝑝<∞ and 𝛽,𝛼_i,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 (𝑔_𝛼,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 𝐿^𝑝(ℝ^𝑛) (1≤𝑝≤∞),𝐶₀(ℝ^𝑛),𝐶_𝑏(ℝ^𝑛), BUC(ℝ^𝑛), let 0≤𝑙≤𝑛 and let ℱ⁻¹ denote the inverse Fourier transform. Put ℕ^𝑙₀∶={𝛼∈ℕ^𝑛₀∶𝛼_𝑙+1=⋯=𝛼_𝑛=0} and, for every 𝑙=0,1,…,𝑛, 𝐸_𝑙∶={𝑓∈𝐸∶𝑓^(𝛼)∈𝐸 for all 𝛼∈ℕ^𝑙₀}. Then the family of seminorms (𝑝_𝛼(𝑓)∶=||𝑓^(𝛼)||_𝐸,𝑓∈𝐸_𝑙;𝛼∈ℕ^𝑙₀) induces a Fréchet topology on 𝐸_𝑙. Let 𝐓_𝐥 possess the same meaning as in [51] and let 𝑚∈ℕ,𝑎_𝛼∈ℂ,0≤|𝛼|≤𝑚. Consider the operator 𝑃(𝐷)𝑓∶=∑_{|𝛼|≤𝑚}𝑎_𝛼𝑓^(𝛼) with its maximal distributional domain. Set 𝑃(𝑥)∶=∑_{|𝛼|≤𝑚}𝑎_𝛼𝑖^|𝛼|𝑥^𝛼,𝑥∈ℝ^𝑛,ℎ_𝑡,𝛽(𝑥)∶= (1+|𝑥|²)^−𝛽/2∑^∞_𝑗=0(𝑡^2𝑗𝑃(𝑥)^𝑗/(2𝑗)!),𝑥∈ℝ^𝑛,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∈ℝ^𝑛 and, in the case 𝑟∈(0,𝑚], there exist 𝜎>0 and 𝜎^′>0 such that ℜ𝑃(𝑥)≤−𝜎|𝑥|^𝑟+𝜎^′,𝑥∈ℝ^𝑛.
Then, for every 𝑙=0,1,…,𝑛, there exists 𝑀≥1 such that, for every 𝛽>(𝑚−(𝑟/2))(𝑛/4),𝑃(𝐷) generates an exponentially equicontinuous 𝐓_𝐥(⟨(1+|𝑥|²)^−𝛽⟩)-regularized cosine function (𝐶_𝛽(𝑡))_𝑡≥0 in 𝐸_𝑙 satisfying 𝐶_𝛽(𝑡)𝑓=ℱ⁻¹ℎ_𝑡,𝛽∗𝑓,𝑡≥0,𝑓∈𝐸_𝑙 and 𝑝_𝛼(𝐶_𝛽(𝑡)𝑓)≤𝑀𝑝_𝛼(𝑓)𝑔_𝑛/2(𝑡),𝑡≥0,𝑓∈𝐸_𝑙,𝛼∈ℕ^𝑙₀, with 𝑔_𝑛/2(𝑡) being the function defined on [52, page 40]; cf. [33, 51, 52] for full details. If 1<𝑝<∞ and 𝐸=𝐿^𝑝(ℝ^𝑛), 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(ℝ^𝑛), define the bounded linear operator 𝐵 on 𝐸_𝑙 by (𝐵𝑓)(𝑥)∶=∫_ℝ𝑛𝜑(𝑥−𝑡)𝑓(𝑡)𝑑𝑡,𝑓∈𝐸_𝑙,𝑥∈ℝ^𝑛. Then 𝐵𝑃(𝐷)⊆𝑃(𝐷)𝐵,𝐓_𝐥(⟨(1+|𝑥|²)^−𝛽⟩)𝐵=𝐵𝐓_𝐥(⟨(1+|𝑥|²)^−𝛽⟩) and 𝑝_𝛼(𝐵𝑓)≤||𝜑||_𝐿¹_(ℝ^𝑛₎𝑝_𝛼(𝑓),𝑓∈𝐸,𝛼∈ℕ^𝑙₀. Applying Corollary 8, we get that 𝑃(𝐷)+𝐵 generates an exponentially equicontinuous 𝐓_𝐥(⟨(1+|𝑥|²)^−𝛽⟩)-regularized cosine function (𝐶_𝛽,𝐵(𝑡))_𝑡≥0 in 𝐸_𝑙.
Assume now 𝛼>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 (𝑝_{𝑚,𝑙,𝛼,𝜚,𝜎})_{1≤𝑚≤𝑙} such that 𝑝_{1,𝑙,𝛼,𝜚,𝜎}=𝜚(𝜎/𝛼)((𝜎/𝛼)−1)⋯((𝜎/𝛼)−(𝑙−1)),𝑝_{𝑙,𝑙,𝛼,𝜚,𝜎}=(𝜚(𝜎/𝛼))^𝑙, and that the following holds: 𝑑^𝑙𝑑𝑧^𝑙1𝑧𝑘𝑧^1/𝛼=𝑑^𝑙𝑑𝑧^𝑙𝑒^{𝜚𝑧𝜎/𝛼}=𝑒^{𝜚𝑧𝜎/𝛼𝑙}_𝑚=1𝑝_{𝑚,𝑙,𝛼,𝜚,𝜎}𝑧^{𝑚(𝜎/𝛼)−𝑙}𝑝𝑧>0,_{𝑚,𝑙+1,𝛼,𝜚,𝜎}𝜎=𝜚𝛼𝑝_{𝑚−1,𝑙,𝛼,𝜚,𝜎}+𝑚𝜎𝛼−𝑙𝑝_{𝑚,𝑙,𝛼,𝜚,𝜎},2≤𝑚≤𝑙. (44) This implies 𝑐₀=1,𝑘₀(𝑡)=0,𝑐_𝑖=0,𝑖≥1, _𝑖𝑘(𝑡)=^𝑖_𝑚=1𝑝_{𝑚,𝑖,𝛼,𝜚,𝜎}𝑔_{𝛼𝑖−𝑚𝜎}(𝑡)𝑘,𝑡>0,𝑖≥1,_{𝑖,𝑙0,𝑙}(𝑡)=𝑐_{𝑙0,𝑙,𝛼}^𝑖−𝑙_𝑚=1𝑝_{𝑚,𝑖−𝑙,𝛼,𝜚,𝜎}𝑔_{𝛼𝑖−𝑙0−𝑚𝜎}(𝑡),𝑡>0,1≤𝑙<𝑖,1≤𝑙₀≤𝑙,𝑘_{𝑖,𝑙0,𝑖}(𝑡)=𝑐_{𝑙0,𝑖,𝛼}𝑔_{𝛼𝑖−𝑙0}(𝑡),𝑡>0,1≤𝑙₀≤𝑖. (45)(46) By means of (44) and the proof of Lemma 3, we obtain the existence of a constant 𝜁_{𝛼,𝜚,𝜎}≥1 such that 𝑚!𝑝^𝑙_{𝑚=1𝑚,𝑙,𝛼,𝜚,𝜎}≤𝜁^𝑙_{𝛼,𝜚,𝜎}𝑙!∀𝑙∈ℕ. (47) In what follows, we assume that 𝜁_{𝛼,𝜚,𝜎}≥1 is minimal with respect to (47); notice that 𝜁_{𝛼,𝜚,𝜎}>𝜚𝜎/𝛼 and that it is not clear whether Lemma 4 can be reconsidered in the newly arisen situation. Then ^∞𝑖_{𝑖=1𝑙0=1}|𝐵|^𝑖_⊛𝑖!^𝑡₀(𝑡−𝑠)^𝑙0𝑘_{𝑖,𝑙0,𝑖}(𝑠=)𝑑𝑠_{∞𝑖𝑖=1𝑙0=1}|𝐵|^𝑖_⊛𝑐𝑖!^{𝑙0,𝑖,𝛼}𝑙₀!^𝑡₀(𝑡−𝑠)^𝑙0𝑙₀!𝑠^{𝛼𝑖−𝑙0−1}Γ𝛼𝑖−𝑙₀≤𝑑𝑠_∞𝑖=1𝑡^𝛼|𝐵|_⊛^𝑖Γ(𝛼𝑖+1)≤𝐸_𝛼𝑡^𝛼|𝐵|_⊛≤𝑐_𝛼𝑒^{𝑡|𝐵|⊛1/𝛼},𝑡≥0. (48) Since Γ(⋅) is increasing in (𝜉,∞), where 𝜉∼1.4616…, we obtain that Γ(𝛼𝑖+1)Γ()≤Γ()𝛼𝑖−𝑚𝜎+1𝑚!𝛼𝑖+1Γ()=()⋯()𝛼𝑖−𝑚+1𝑚!𝛼𝑖𝛼𝑖−1𝛼𝑖−𝑚+1≤⌈⌉𝑚,𝑚!𝛼𝑖 (49) provided 𝑖≥2 and 1≤𝑚≤𝑖−1. Combining this with (35), Lemmas 3 and 4 and (47), we get ^∞_𝑖=2^𝑖−1𝑙_{𝑙=1𝑙0=1}|𝐵|^𝑖_⊛𝑖𝑙𝑖!^𝑡₀(𝑡−𝑠)^𝑙0𝑘_{𝑖,𝑙0,𝑙}(𝑠≤)𝑑𝑠_{∞𝑖=2𝑖−1𝑙𝑙=1𝑙0=1}|𝐵|^𝑖_⊛𝑖𝑙𝑙𝑖!⁰!𝑐_{𝑙0,𝑙,𝛼}×^𝑖−𝑙_𝑚=1𝑚!𝑝_{𝑚,𝑖−𝑙,𝛼,𝜚,𝜎}𝑡^{𝛼𝑖−𝑚𝜎}Γ()≤1𝛼𝑖−𝑚𝜎+1𝑚!𝛼^∞_𝑖=2^𝑖−1_𝑙=1|𝐵|^𝑖_⊛Γ()𝜁𝛼𝑖+1^{𝑖−𝑙𝛼,𝜚,𝜎𝑖−𝑙}_𝑚=1𝑡^{𝛼𝑖−𝑚𝜎}Γ(𝛼𝑖+1)Γ()≤1𝛼𝑖−𝑚𝜎+1𝑚!𝛼^∞_𝑖=2^𝑖−1_𝑙=1|𝐵|^𝑖_⊛Γ()𝜁𝛼𝑖+1^{𝑖−𝑙𝛼,𝜚,𝜎𝑖−𝑙}_𝑚=1𝑡^{𝛼𝑖−𝑚𝜎}⌈⌉𝑚≤1𝛼𝑖𝛼1+𝑡^−𝜎∞_𝑖=2^𝑖−1_𝑙=1|𝐵|^𝑖_⊛Γ()𝜁𝛼𝑖+1^{𝑖−𝑙𝛼,𝜚,𝜎}𝑡+𝑡^{1−𝜎𝛼𝑖}≤1𝛼𝜁_{𝛼,𝜚,𝜎}1+𝑡^−𝜎∞_𝑖=1𝑖|𝐵|_⊛𝑡+𝑡^1−𝜎𝛼𝜁_{𝛼,𝜚,𝜎}^𝑖+1Γ()≤1𝛼𝑖+1𝛼1+𝑡^−𝜎|𝐵|_⊛𝑡+𝑡^1−𝜎𝛼×^∞_𝑖=1𝑖𝑧^𝑖Γ(𝛼𝑖+1)_{𝑧=|𝐵|⊛(𝑡+𝑡1−𝜎)𝛼𝜁𝛼,𝜚,𝜎}≤𝑏_𝛼𝛼²1+𝑡^−𝜎|𝐵|_⊛^{(1+𝛼)/𝛼}×𝑡+𝑡^{1−𝜎1+𝛼}𝜁^1/𝛼_{𝛼,𝜚,𝜎}𝑒^{|𝐵|⊛1/𝛼(𝑡+𝑡1−𝜎)𝜁1/𝛼𝛼,𝜚,𝜎},𝑡≥1. (50)(51) Noticing that 𝑡^{𝛼𝑖−𝑚𝜎}≤1,𝑡∈[0,1),𝑖≥2,1≤𝑚≤𝑖−1, we obtain from (50) that there exists 𝜉_{𝛼,𝜚,𝜎}≥1 such that ^∞_𝑖=2^𝑖−1𝑙_{𝑙=1𝑙0=1}|𝐵|^𝑖_⊛𝑖𝑙𝑖!^𝑡₀(𝑡−𝑠)^𝑙0𝑘_{𝑖,𝑙0,𝑙}(𝑠)𝑑𝑠≤𝜉_{𝛼,𝜚,𝜎},𝑡∈[0,1). (52) By (48)–(52), (v) holds for any 𝜔^′>(|𝐵|_⊛𝜁_{𝛼,𝜚,𝜎})^1/𝛼. In almost the same way, one can prove that (iv) and (vi) hold for any 𝜔^′>(|𝐵|_⊛𝜁_{𝛼,𝜚,𝜎})^1/𝛼. Assume now that (𝑅(𝑡))_𝑡≥0 is an exponentially equicontinuous, analytic (𝑔_𝛼,𝑘)-regularized 𝐶-resolvent family of angle 𝛽∈(0,𝜋/2],𝛾∈(0,𝛽), 𝜔_𝛾,1≥𝜔_𝛾, 𝜔_𝛾,1>0, (𝜔_𝛾,1cos𝛾)^𝛼−𝜎>|𝐵|_⊛𝜁_{𝛼,𝜚,𝜎}, 𝜀=1/cos𝛾, 𝜔_𝛾,2≥𝜔_𝛾+𝜀 and (1+𝜔_𝛾,2cos𝛾)^𝛼−1>|𝐵|_⊛𝜁_{𝛼,𝜚,𝜎}. Then ^∞_𝑖=0|𝐵|^𝑖_⊛𝑖!sup_{𝜆∈𝜔𝛾,1+Σ(𝜋/2)+𝛾𝑖}𝑘(𝜆)≤^∞_𝑖=0|𝐵|^𝑖_⊛𝑖!^𝑖_𝑚=1𝑚!𝑝_{𝑚,𝑖,𝛼,𝜚,𝜎}𝜔_𝛾,1cos𝛾^{𝛼𝑖−𝑚𝜎}≤𝜇^∞_{𝜎,𝛾𝑖=0}|𝐵|^𝑖_⊛𝜁^𝑖_{𝛼,𝜚,𝜎}𝜔_𝛾,1cos𝛾^{(𝛼−𝜎)𝑖}<∞ (53) for an appropriate constant 𝜇_𝜎,𝛾≥1, ^∞𝑖_{𝑖=1𝑙0=1}|𝐵|^𝑖_⊛𝑙𝑖!⁰!2𝜋𝑙₀𝜀cos𝛾^𝑙0sup_{𝜆∈𝜔𝛾,2+Σ(𝜋/2)+𝛾}ℒ𝑘_{𝑖,𝑙0,𝑖}(𝑡𝜆≤)()_∞𝑖=1𝑖|𝐵|^𝑖_⊛1+𝜔_𝛾,2^{(𝛼−1)𝑖}<∞, (54) and, for every 𝜀>0 with (1+𝜔_𝛾,2cos𝛾)^𝛼−1>|𝐵|_⊛(𝜁_{𝛼,𝜚,𝜎}+𝜀), there exists 𝜇_𝜀>0 such that ^∞_𝑖=2^𝑖−1𝑙_{𝑙=1𝑙0=1}|𝐵|^𝑖_⊛𝑖𝑙𝑙𝑖!⁰!2𝜋𝑙₀𝜀cos𝛾^𝑙0×sup_{𝜆∈𝜔𝛾,2+Σ(𝜋/2)+𝛾}ℒ𝑘_{𝑖,𝑙0,𝑙}(𝑡)(𝜆≤_{∞𝑖=2𝑖−1𝑙𝑙=1𝑙0=1})|𝐵|^𝑖_⊛()𝑙𝑖−𝑙!𝑙!⁰!𝑐_{𝑙0,𝑙,𝛼}^𝑖−𝑙_𝑚=1𝑝_{𝑚,𝑖−𝑙,𝛼,𝜚,𝜎}1+𝜔_𝛾,2cos𝛾^{𝛼𝑖−𝑙0−𝑚𝜎}≤^∞_𝑖=2^𝑖−1𝑙_{𝑙=1𝑙0=1}|𝐵|^𝑖_⊛()𝑙𝑖−𝑙!𝑙!⁰!𝑐_{𝑙0,𝑙,𝛼}^𝑖−𝑙_𝑚=1𝑝_{𝑚,𝑖−𝑙,𝛼,𝜚,𝜎}1+𝜔_𝛾,2cos𝛾^{(𝛼−1)𝑖}≤1𝛼^∞_𝑖=2^𝑖−1_𝑙=1|𝐵|^𝑖_⊛𝜁^𝑖−𝑙_{𝛼,𝜚,𝜎}1+𝜔_𝛾,2cos𝛾^{(𝛼−1)𝑖}≤1𝛼^∞_𝑖=2()|𝐵|𝑖−1^𝑖⊛𝜁_{𝛼,𝜚,𝜎}+𝜀^𝑖1+𝜔_𝛾,2cos𝛾^{(𝛼−1)𝑖}<∞, (55) proving the conditions (29)-(30).

Corollary 11. Let 𝛼>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 (𝑔_𝛼,𝑘)-regularized 𝐶-resolvent family (𝑅(𝑡))_𝑡≥0 satisfying (2) with 𝑎(𝑡)=𝑔_𝛼(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 (𝑔_𝛼,𝑘)-regularized 𝐶-resolvent family (𝑅_𝐵(𝑡))_𝑡≥0 satisfying (18) with 𝑘₁(𝑡)=𝑘(𝑡). Furthermore, for every 𝜀>0, the family {exp(−(𝜔+(|𝐵|_⊛𝜁_{𝛼,𝜚,𝜎})^1/𝛼+𝜀)𝑡)𝑅_𝐵(𝑡)∶𝑡≥0} is equicontinuous, and (𝑅_𝐵(𝑡))_𝑡≥0 is an exponentially equicontinuous, analytic (𝑔_𝛼,𝑘)-regularized 𝐶-resolvent family of angle 𝛽∈(0,𝜋/2] provided that (𝑅(𝑡))_𝑡≥0 is.

Example 12. Let 𝑠>1, 𝐸∶=𝑓∈𝐶^∞[0,1];𝑓=sup_𝑝≥0𝑓^(𝑝)_∞𝑝!^𝑠<∞,𝐴∶=−𝑑(𝐴)𝑑𝑠,𝐷∶=𝑓∈𝐸;𝑓^′∈𝐸,𝑓(0)=0. (56)(57) 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 𝑓¹_𝜆(𝑡)∶=∫^𝑡₀𝑒^{−𝜆(𝑡−𝑠)}𝑓(𝑠)𝑑𝑠 and 𝑓²_𝜆(𝑡)∶=∫^𝑡₀𝑒^{𝜆(𝑡−𝑠)}𝑓(𝑠)𝑑𝑠. Then 𝑓¹_𝜆(⋅),𝑓²_𝜆(⋅)∈𝐸,𝜆∈ℂ, and there exist 𝑏>0 and 𝑀≥1, independent of 𝑓(⋅), such that 𝑓¹_𝜆(⋅)≤𝑀𝑓𝑒^{𝑏|𝜆|1/𝑠},ℜ𝜆≥0,𝑓∈𝐸. (58) It is clear that ||𝑓²_𝜆(⋅)||_𝐿^∞_[0,1]≤𝑒^|𝜆|||𝑓||,ℜ𝜆≥0 and ||(𝑑/𝑑𝑡)𝑓²_𝜆(⋅)||_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∶𝑑^𝑛𝑑𝑡^𝑛𝑓²_𝜆(𝑡)=𝑑^𝑛−1𝑑𝑡^𝑛−1𝑓(𝑡)+^𝑛−1_𝑘=1𝜆^𝑘𝑑^{𝑛−1−𝑘}𝑑𝑡^{𝑛−1−𝑘}𝑓(𝑡)+𝜆^𝑛𝑓²_𝜆(𝑡). (59) On the other hand, [54, Proposition 4.5] implies that there exists 𝑐>0 such that ∑^∞_𝑝=0𝑡^𝑝/𝑝!^𝑠=𝑂(exp(𝑐𝑡^1/𝑠)),𝑡≥0. Combined with (59) and the logarithmic convexity, the last estimate yields 1𝑛!^𝑠𝑑^𝑛𝑑𝑡^𝑛𝑓²_𝜆(⋅)_𝐿∞[0,1]≤𝑓+𝑓𝑒^{𝑐|𝜆|1/𝑠}+|𝜆|^𝑛𝑛!^𝑠𝑒^|𝜆|𝑓≤1+𝑒^{𝑐|𝜆|1/𝑠}+𝑒^{𝑐|𝜆|1/𝑠}𝑒^|𝜆|𝑓,ℜ𝜆≥0,𝜆≠0. (60)
In view of (60) we get that, for every 𝜂>1, there exists 𝑀_𝜂≥1, independent of 𝑓(⋅), such that 𝑓²_𝜆(⋅)≤𝑀_𝜂𝑓𝑒^𝜂|𝜆|,ℜ𝜆≥0,𝑓∈𝐸. (61) Consider now the complex polynomial 𝑃(𝑧)=∑^𝑛_𝑗=0𝑎_𝑗𝑧^𝑗, 𝑧∈ℂ, 𝑎_𝑛≠0, 𝑛≥2. Set, for every 𝜆∈ℂ,𝑃_𝜆(⋅)∶=𝑃(⋅)−𝜆 and consider the operator 𝑃(𝐴) defined by 𝐷(𝑃(𝐴))∶=𝐷𝐴^𝑛,𝑃(𝐴)𝑓∶=^𝑛_𝑗=0𝑎_𝑗𝐴^𝑗𝑓,𝑓∈𝐷(𝑃(𝐴)). (62) 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 𝑧_1,𝜆,…,𝑧_𝑛,𝜆 denote the zeros of the polynomial 𝑧↦𝑃_𝜆(𝑧),𝑧∈ℂ and let 0<𝑚∶=min_{|𝑧|≥𝑑+1}|𝑃^′(𝑧)|. Then an old result of Walsh [55] says that |𝑧_𝑗,𝜆|≤𝑟+|𝑎_𝑛|^−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 𝑧_𝑗,𝜆 is a simple zero of 𝑃_𝜆(𝑧) and that |𝑧_𝑗,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: 𝑑+1≤𝑧_𝑗,𝜆≤𝑟+𝑎_𝑛^−1/𝑛|𝜆|^1/𝑛,𝑃^′𝑧_𝑗,𝜆≥𝑚,𝑧_𝑖,𝜆≠𝑧_𝑗,𝜆. (63) It is straightforward to verify that 𝜌𝑝(𝐴)𝑅(𝐴)=()=ℂ,𝜆∶𝑝−1^𝑛+1𝑎_𝑛⁻¹𝑅𝑧_1,𝜆∶𝐴⋯𝑅𝑧_𝑛,𝜆∶𝐴,𝜆∈ℂ. (64) Assume now |𝜆|≥𝜆₀. Then de L'Hospital's rule implies 𝑎_{𝑛1≤𝑖≤𝑛𝑖≠𝑗}𝑧_𝑖,𝜆−𝑧_𝑗,𝜆=(−1)^𝑛+1𝑃^′𝑧_𝑗,𝜆,1≤𝑗≤𝑛. (65) 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: (−1)^𝑛+1𝑎_𝑛⁻¹𝑅𝑧_1,𝜆∶𝐴⋯𝑅𝑧_𝑛,𝜆=()∶𝐴−1^𝑛+1𝑎_𝑛^𝑛−1_𝑗=1𝑅𝑧_𝑗,𝜆∶𝐴∏_{1≤𝑖≤𝑛𝑖≠𝑗}𝑧_𝑖,𝜆−𝑧_𝑗,𝜆=^𝑛_𝑗=1𝑅𝑧_𝑗,𝜆∶𝐴𝑃^′𝑧_𝑗,𝜆≤1𝑚^𝑛_𝑗=1𝑅𝑧_𝑗,𝜆∶𝐴. (66) By (64) and (66) we finally get that, for every 𝜂>1, 𝑅𝜆∶𝑝(𝐴)=𝑂𝑒^{𝑏|𝑎𝑛|−1/𝑛|𝜆|1/𝑛𝑠}+𝑒^{𝜂|𝑎𝑛|−1/𝑛|𝜆|1/𝑛},𝜆∈ℂ. (67) 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>|𝑎_𝑛|^−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 (𝑀_𝑝) of positive numbers satisfying 𝑀₀=1 and 𝑀_𝑝+𝑞≥𝑀_𝑝𝑀_𝑞(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 (𝑔_𝛼,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 (𝑔_𝛼,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. Suppose 𝑀>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)≤𝑀𝑒^𝜔𝑡𝑝(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 𝑐_𝑝>0 and 𝑞∈⊛ satisfying 𝑝(𝐶⁻¹𝐵𝑥)≤𝑐_𝑝𝑞(𝑥),𝑥∈𝐷(𝐴). Let (P1) hold for 𝑎(𝑡),𝑘(𝑡),𝑏(𝑡) and let 𝑎(𝜆)/𝑘(𝜆)=𝑏̃(𝜆)+𝑧,ℜ𝜆>𝜔₁,𝑘(𝜆)≠0. Suppose 𝜇>𝜔₁,𝛾∈[0,1) and 𝐷(𝐴)=𝐸,_∞0𝑒^−𝜇𝑡𝑝𝐶⁻¹𝐵^𝑡₀𝑏(𝑡−𝑠)𝑅(𝑠)𝑥𝑑𝑠+𝑧𝐶⁻¹(𝑡)𝑥(𝑥)(𝐴)𝐵𝑅𝑑𝑡≤𝛾𝑝,𝑥∈𝐷,𝑝∈⊛ (68)(69) or (𝑅(𝑡))_𝑡≥0𝑠𝑎𝑡𝑖𝑠𝑓𝑖𝑒𝑠(2),𝐷(𝐴)≠𝐸𝑎𝑛𝑑(69)ℎ𝑜𝑙𝑑𝑠𝑓𝑜𝑟𝑎𝑛𝑦𝑥∈𝐸,𝑝∈⊛. (70) Then the operator 𝐴+𝐵 is a subgenerator of an (𝑎,𝑘)-regularized 𝐶-resolvent family (𝑅_𝐵(𝑡))_𝑡≥0 satisfying (18) with 𝑘₁(𝑡) and 𝑔_𝛼(𝑡) replaced by 𝑘(𝑡) and 𝑎(𝑡) therein. Furthermore, 𝑝𝑅_𝐵(𝑡)𝑥≤𝑀𝑒1−𝛾^𝜇𝑡𝑝(𝑥),𝑥∈𝐸,𝑡≥0,𝑝∈⊛,𝑅_𝐵(𝑡)𝑥=𝑅(𝑡)𝑥+^𝑡₀𝑅_𝐵(𝑡−𝑟)𝐶⁻¹𝐵^𝑟₀𝑏(𝑟−𝑠)𝑅(𝑠)𝑥𝑑𝑠+𝑧𝐶⁻¹(𝑡)𝑥(𝐴),𝐵𝑅𝑑𝑟,𝑡≥0,𝑥∈𝐷 (71)(72) and (72) holds for any 𝑡≥0 and 𝑥∈𝐸 provided (70).

In many cases, we do not have the existence of a function 𝑏(𝑡) and a complex number 𝑧 such that 𝑎(𝜆)/𝑘(𝜆)=𝑏̃(𝜆)+𝑧,ℜ𝜆>𝜔₁,𝑘(𝜆)≠0. The following theorem is an attempt to fill this gap.

Theorem 14. Suppose 𝑀,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)≤𝑀𝑒^𝜔𝑡𝑝(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), 𝑝𝐶𝐴^𝑗𝐶⁻¹𝐵𝑥≤𝑀₁𝑝(𝑥),𝑥∈𝐷(𝐴)𝑝𝐴,𝑝∈⊛,0≤𝑗≤𝑙−1,_𝑙𝐶⁻¹𝐵𝑥≤𝑀₁𝑝(𝑥),𝑥∈𝐷(𝐴),𝑝∈⊛. (73)(ii)There exist a function 𝑏(𝑡) satisfying (P1) and a complex number 𝑧 such that 𝑎(𝜆)^𝑙+1𝑘(𝜆)=𝑏̃(𝜆)((𝑎)(𝑘,𝑘(𝜆)+𝑧,ℜ𝜆>max𝜔,abs,abs))≠0. (74)(iii)lim_𝜆→+∞∫^∞₀𝑒^−𝜆𝑡|𝑎(𝑡)|𝑑𝑡=0 and lim_𝜆→+∞∫^∞₀𝑒^−𝜆𝑡×|𝑏(𝑡)|𝑑𝑡=0.Then, for every 𝑥∈𝐸, there exists a unique solution of the integral equation 𝑅_𝐵(𝑡)𝑥=𝑅(𝑡)𝑥+𝑆∗𝑅_𝐵(𝑡)𝑥,𝑡≥0; (75) furthermore, (𝑅_𝐵(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 𝑔_𝛼(𝑡) replaced by 𝑘(𝑡) and 𝑎(𝑡) therein.

Proof. It is clear that ∫^∞₀𝑒^−𝜆𝑡|𝑎^∗,𝑗(𝑡)|𝑑𝑡≤(∫^∞₀𝑒^−𝜆𝑡|𝑎(𝑡)|𝑑𝑡)^𝑗, 𝑗∈ℕ, 𝜆>abs(𝑎). Define, for every 𝑥∈𝐷(𝐴) and 𝑡≥0, 𝑆(𝑡)𝑥∶=^𝑙−1_𝑗=0𝑎^∗,𝑗+1(𝑡)𝐶𝐴^𝑗𝐶⁻¹+𝐵𝑥_𝑡0𝑏(𝑡−𝑠)𝑅(𝑠)𝐴^𝑙𝐶⁻¹𝐵𝑥𝑑𝑠+𝑧𝑅(𝑡)𝐴^𝑙𝐶⁻¹𝐵𝑥. (76) 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), ℒ(𝑆(𝑡)𝑥)(𝜆)=𝑎(𝜆)(𝐼−𝑎(𝜆)𝐴)⁻¹𝐶𝐶⁻¹𝐵𝑥,𝑥∈𝐷(𝐴),ℜ𝜆>𝜔₁,𝑘(𝜆)≠0. (77) 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 ^∞₀𝑒^−𝜇𝑡𝑝(𝑆(𝑡)𝑥)𝑑𝑡≤𝛾𝑝(𝑥),𝑥∈𝐷(𝐴),𝑝∈⊛ (78) and that (H1) holds, where(H1): For every strongly continuous function 𝑓∶[0,∞)→𝐿(𝐸,𝐷(𝐴)) such that 𝑝(𝑓(𝑡)𝑥)≤𝑀𝑝(𝑥), 𝑥∈𝐸,   𝑡≥0, 𝑝∈⊛, the following inequality holds: ^𝑡₀𝑒^{−𝜇(𝑡−𝑠)}𝑝𝑆(𝑡−𝑠)𝑓(𝑠)𝑥𝑑𝑠≤𝛾𝑀𝑝(𝑥),𝑥∈𝐸,𝑡≥0,𝑝∈⊛. (79)Now one can define inductively, for every 𝑡≥0, the sequence (𝑇_𝑛(𝑡))_𝑛∈ℕ0 in 𝐿(𝐸,𝐷(𝐴)) by 𝑇₀(𝑡)∶=𝑅(𝑡) and 𝑇_𝑛+1(𝑡)𝑥∶=∫^𝑡₀𝑆(𝑡−𝑠)𝑇_𝑛(𝑠)𝑥𝑑𝑠,𝑥∈𝐸,𝑛∈ℕ₀; observe that, for every 𝑛∈ℕ₀,   (𝑇_𝑛(𝑡))_𝑡≥0 is strongly continuous and that the family {𝑇_𝑛(𝑡)∶𝑡≥0} is locally equicontinuous (with clear meaning). By (78), (H1), and the proof of [9, Theorem 3.1], it follows inductively that 𝑝𝑇_𝑛(𝑡)𝑥≤𝑀𝛾^𝑛𝑒^𝜇𝑡𝑝(𝑥),𝑥∈𝐸,𝑡≥0,𝑝∈⊛ (80) and that, for every 𝑥∈𝐸 and 𝑡≥0, the sequence (𝑅^𝑛_𝐵(𝑡)𝑥∶=∑^𝑛_𝑖=0𝑇_𝑖(𝑡)𝑥)_𝑛 is Cauchy in 𝐸 and therefore convergent. Set 𝑅_𝐵(𝑡)𝑥∶=lim_𝑛→∞𝑅^𝑛_𝐵(𝑡)𝑥,𝑥∈𝐸,𝑡≥0. It is obvious that the mapping 𝑡↦𝑅_𝐵(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 𝑘(𝜆)(𝐼−𝑎(𝜆)(𝐴+𝐵))⁻¹𝐶𝑥=^∞₀𝑒^−𝜆𝑡𝑅_𝐵(𝑡)𝑘(𝜆)𝑥𝑑𝑡𝑥∈𝐸,ℜ𝜆>𝜇,≠0. (81) 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) 𝑅_𝐵(𝜆)𝑥=𝐼−𝑆(𝜆)⁻¹𝑘(𝜆)(𝐼−𝑎(𝜆)𝐴)⁻¹𝐶𝑥,ℜ𝜆>𝜇,𝑥∈𝐸, (82) 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 𝐼−𝑆(𝜆)⁻¹𝑘(𝜆)(𝐼−𝑎(𝜆)𝐴)⁻¹𝐶(𝑎(𝜆𝑥=𝑘(𝜆)(𝐴).𝐼−)(𝐴+𝐵))𝐶𝑥,𝑥∈𝐷 (83) The representation (𝐼−𝑆(𝜆))⁻¹=∑^∞_𝑛=0[((1/𝑎(𝜆))−𝐴)⁻¹𝐶𝐶⁻¹𝐵]^𝑛 implies 1𝑎(𝜆)−(𝐴+𝐵)𝐼−𝑆(𝜆)⁻¹𝑘(𝜆)1𝑎(𝜆)−𝐴⁻¹=𝑘(𝜆)𝐶𝑥𝐶𝑥,𝑥∈𝐸, (84)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 Theorem 13 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 𝑐_𝑝>0 and 𝑞∈⊛ such that 𝑝(𝐶⁻¹𝐵𝑥)≤𝑐_𝑝(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(∫^𝑡₀𝑏(𝑡−𝑠)𝑅(𝑠)𝑥𝑑𝑠+𝑧𝑅(𝑡)𝑥),𝑡≥0,𝑥∈𝐷(𝐴) and 𝑇₁(𝑡)𝑥=∫^𝑡₀𝑇₀(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 (𝑇_𝑛(𝑡)=∫^𝑡₀𝑇_𝑛−1(𝑡−𝑠)𝑆(𝑠)𝑑𝑠)_𝑛∈ℕ0 in 𝐿(𝐸) such that 𝑝(𝑇_𝑛(𝑡)𝑥)≤𝛾^𝑛𝑀𝑝(𝑥), 𝑥∈𝐸, 𝑡≥0, 𝑝∈⊛. The preceding inequality implies that, for every 𝑥∈𝐸, the sequence (𝑅^𝑛_𝐵(𝑡)𝑥≡∑^𝑛_𝑖=0𝑇_𝑖(𝑡)𝑥)_𝑛 is Cauchy in 𝐸 and therefore convergent. Put 𝑅_𝐵(𝑡)𝑥=lim_𝑛→∞𝑅^𝑛_𝐵(𝑡)𝑥, 𝑥∈𝐸, 𝑡≥0. As in the proof of Theorem 14, the mapping 𝑡↦𝑅_𝐵(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 ∫^∞₀𝑒^−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 (𝑥_𝜏)_𝜏∈𝑇 in 𝐸 satisfy 𝑥_𝜏→𝑥,𝜏→∞ and (𝐼−𝑎(𝜆)(𝐴+𝐵))𝑥_𝜏→𝑦,𝜏→∞. Then a simple computation shows that (𝐼−𝑎(𝜆)(𝐴+𝐵))𝑥_𝜏=(𝐼−𝑆(𝜆))(𝐼−𝑎(𝜆)𝐴)𝑥_𝜏, which implies (𝐼−𝑎(𝜆)𝐴)𝑥_𝜏=(𝐼−𝑆(𝜆))⁻¹(𝐼−𝑎(𝜆)(𝐴+𝐵))𝑥_𝜏→(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 exponent 𝜎∈(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)≤𝑐_𝑝𝑀𝑞(𝑥)[∫^𝑡₀|𝑏(𝑡−𝑟)|𝑒^𝜔𝑟𝑑𝑟+|𝑧|𝑒^𝜔𝑡],𝑝∈⊛,𝑡≥0,𝑥∈𝐷(𝐴). Suppose now that, for every 𝑇>0 and 𝑝∈⊛, there exist 𝑐_𝑇,𝑝>0 and ℎ_𝑇,𝑝∈⊛ such that 𝑝(𝑅(𝑡)𝑥−𝑅(𝑠)𝑥)≤𝑐_𝑇,𝑝(𝑡−𝑠)^𝜎ℎ_𝑇,𝑝(𝑥),𝑥∈𝐸,0≤𝑠<𝑡≤𝑇. (85) Let 𝑇>0 and 𝑝∈⊛ be fixed. Then, for every 𝑥∈𝐷(𝐴) and 0≤𝑠<𝑡≤𝑇, 𝑝(𝑆(𝑡)𝑥−𝑆(𝑠)𝑥)≤𝑐_𝑝𝑞^𝑠₀𝑏(𝑟𝑅()()𝑥)+)(𝑡−𝑟𝑥−𝑅𝑠−𝑟𝑑𝑟_𝑠𝑡𝑏(𝑟)𝑅(𝑡−𝑟)𝑥𝑑𝑟+𝑐_𝑝|𝑧|𝑞(𝑅(𝑡)𝑥−𝑅(𝑠)𝑥)≤𝑐_𝑝𝑐_𝑇,𝑞(𝑡−𝑠)^𝜎ℎ_𝑇,𝑞(𝑥)^𝑇₀|𝑏(𝑟)|𝑑𝑟+|𝑧|+𝑀𝑒^𝜔𝑇𝑞(𝑥)^𝑠_𝑡|𝑏(𝑟)|𝑑𝑟, (86) which implies by (72) that 𝑝𝑅_𝐵∗𝑆(𝑡)𝑥−𝑅_𝐵(𝑠)𝑥≤∗𝑆𝑀𝑒^𝜇𝑇1−𝛾^𝑠₀𝑝(𝑆()()𝑥)+𝑡−𝑟𝑥−𝑆s−𝑟𝑑𝑟_𝑡𝑠𝑝(𝑆()𝑥)≤𝑐𝑡−𝑟𝑑𝑟_𝑝𝑀𝑒^𝜇𝑇𝑐1−𝛾^𝑇,𝑞(𝑡−𝑠)^𝜎ℎ_𝑇,𝑞(𝑥)^𝑠₀^𝑇₀|𝑏(𝑣)|𝑑𝑣𝑑𝑟+𝑀𝑒^𝜔𝑇𝑞(𝑥)^𝑠₀^𝑡−𝑟_𝑠−𝑟|𝑏(𝑣+|𝑧|𝑐)|𝑑𝑣𝑑𝑟_𝑇,𝑞𝑇(𝑡−𝑠)^𝜎ℎ_𝑇,𝑞(𝑥)+𝑐_𝑝𝑀²𝑒^2𝜇𝑇𝑞(𝑥)×1−𝛾_{𝑡𝑠0𝑡−𝑟}|𝑏(𝑡−𝑟−𝑣)|𝑑𝑣𝑑𝑟+|𝑧|(𝑡−𝑠). (87) One can simply prove that there exists 𝑐_𝑇>0 such that, for 0≤𝑠<𝑡≤𝑇, ^𝑠₀^𝑡−𝑟_𝑠−𝑟|𝑏(𝑣)|𝑑𝑣𝑑𝑟+^𝑡_𝑠0^𝑡−𝑟|𝑏(𝑡−𝑟−𝑣)|𝑑𝑣𝑑𝑟≤𝑐_𝑇(𝑡−𝑠)^𝜎, (88) which implies with (72), the previous computation and the denseness of 𝐴 that there exists 𝑏_𝑇,𝑝>0 such that, for every 𝑥∈𝐸 and 0≤𝑠<𝑡≤𝑇∶𝑝𝑅_𝐵(𝑡)𝑥−𝑅_𝐵(𝑠)𝑥≤𝑏_𝑇,𝑝(𝑡−𝑠)^𝜎max𝑝(𝑥),ℎ_𝑇,𝑝(𝑥),𝑞(𝑥). (89) 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, 𝑝𝑅_𝐵(𝑡)𝑥−𝑅_𝐵(𝑠)𝑥(𝑡−𝑠)^𝜎≤𝑏_𝑇,𝑝max𝑝(𝑥)+𝑝(𝐴𝑥),ℎ_𝑇,𝑝(𝑥)+ℎ_𝑇,𝑝(𝐴𝑥),𝑞(𝑥)+𝑞(𝐴𝑥). (90) Assuming additionally sup_{0≤𝑠<𝑡≤𝑇}1(𝑡−𝑠)^𝜎𝑙−1_𝑗=00^𝑡−𝑠𝑎^∗,𝑗+1(𝑟+)𝑑𝑟_𝑠0𝑎^∗,𝑗+1(𝑡−𝑟)−𝑎^∗,𝑗+1(𝑠−𝑟)𝑑𝑟<∞, (91) then an estimate of the form (89) holds in the case of Theorem 14.
The following corollary is an immediate consequence of Theorems 13–14 and Remark 15.

Corollary 17. Suppose 𝑀,𝑀₁>0,𝜔≥0,𝛼>0,𝛽≥0, 𝐴 is a subgenerator of a (𝑔_𝛼,𝑔_𝛼𝛽+1)-regularized 𝐶-resolvent family (𝑅(𝑡))_𝑡≥0 satisfying 𝑝(𝑅(𝑡)𝑥)≤𝑀𝑒^𝜔𝑡𝑝(𝑥),𝑥∈𝐸,𝑡≥0,𝑝∈⊛ and (2) with 𝑎(𝑡)=𝑔_𝛼(𝑡) and 𝑘(𝑡)=𝑔_𝛼𝛽+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 (𝑔_𝛼,𝑔_𝛼𝛽+1)-regularized 𝐶-resolvent family (𝑅_𝐵(𝑡))_𝑡≥0 satisfying (71), and (18) with 𝑘₁(𝑡) replaced by 𝑔_𝛼𝛽+1(𝑡) therein.

Remark 18. Let 0<𝛼<2 and let (𝑅(𝑡))_𝑡≥0 be an exponentially equicontinuous, analytic (𝑔_𝛼,𝑔_𝛼𝛽+1)-regularized 𝐶-resolvent family of angle 𝛿∈(0,𝜋/2]. Suppose additionally that, for every 𝜁∈(0,𝛿), there exist 𝑀_𝜁≥1 and 𝜔_𝜁≥0 such that 𝑝(𝑅(𝑧)𝑥)≤𝑀_𝜁𝑒^{𝜔𝜁ℜ𝑧}𝑝(𝑥),𝑥∈𝐸,𝑧∈Σ_𝜁,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 (𝑅_𝐵(𝑡))_𝑡≥0 is also an exponentially equicontinuous, analytic (𝑔_𝛼,𝑔_𝛼𝛽+1)-regularized 𝐶-resolvent family of angle 𝛿; furthermore, for every 𝜁∈(0,𝛿), there exist 𝑀^′_𝜁≥1 and 𝜔^′_𝜁≥0 such that 𝑝(𝑅_𝐵(𝑧)𝑥)≤𝑀^′_𝜁𝑒^{𝜔′𝜁ℜ𝑧}𝑝(𝑥),𝑥∈𝐸,𝑧∈Σ_𝜁,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: ^∞₀𝑒^−𝜇𝑡𝑝𝐶⁻¹𝐵^𝑡₀𝑏(𝑡−𝑠)𝑅𝑠𝑒^𝑖𝜁𝑥𝑑𝑠+𝑧𝐶⁻¹𝐵𝑅𝑡𝑒^𝑖𝜁𝑥𝑑𝑡≤𝛾𝑝(𝑥). (92) 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]). Let 𝐸∶=𝑙¹,0<𝛼<1 and 𝑙∶=⌈(1−𝛼)/𝛼⌉. Define a closed densely defined linear operator 𝐴_𝛼 on 𝐸 by 𝐷(𝐴_𝛼)∶={⟨𝑥_𝑛⟩∈𝑙¹∶∑^∞_𝑛=1𝑛|𝑥_𝑛|<∞} and 𝐴_𝛼⟨𝑥_𝑛⟩∶=⟨𝑒^{𝑖𝛼(𝜋/2)}𝑛𝑥_𝑛⟩,⟨𝑥_𝑛⟩∈𝐷(𝐴_𝛼). Then 𝐴_𝛼 is the integral generator of a bounded (𝑔_𝛼,1)-regularized resolvent family, 𝐴_𝛼+𝐼 is not the integral generator of an exponentially bounded (𝑔_𝛼,1)-regularized resolvent family, and 𝜎(𝐴_𝛼)={𝑒^{𝑖𝛼(𝜋/2)}𝑛∶𝑛∈ℕ}. Suppose 𝐵∈𝐿(𝐸),𝑅(𝐵)⊆𝐷𝐴^𝑙=𝑥_𝑛∈𝑙¹∶^∞_𝑛=1𝑛^𝑙𝑥_𝑛<∞. (93) Then it follows from Corollary 17 that 𝐴+𝐵 is the integral generator of an exponentially bounded (𝑔_𝛼,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. Suppose 𝐸 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)‖≤𝑀𝑒^𝜔𝑡,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 𝐶_𝜀>0 such that 1𝑘(𝜆)≤𝐶_𝜀𝑒^𝜀|𝜆|,ℜ𝜆≥𝜆₀,|𝑎(𝜆)|𝑎𝜆₀+𝑖ℑ𝜆≤𝐶_𝜀𝑒^𝜀|𝜆|,ℜ𝜆≥𝜆₀. (94)(i)Let 𝐵 be a linear operator, let 𝐷(𝐴)⊆𝐷(𝐵) and let 1𝐵𝑅𝑎(𝜆)∶𝐴≤𝑀_𝜚|𝜆|^−𝜚,ℜ𝜆=𝜆₀ (95) for some 𝜚>0 and 𝑀_𝜚>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∗₀𝑔_𝜁)-regularized resolvent family (𝑅_𝐵(𝑡))_𝑡≥0 satisfying (18) with 𝑘₁(𝑡)=(𝑘∗₀𝑔_𝜁)(𝑡),𝐶=𝐼, and 𝑔_𝛼(𝑡) replaced by 𝑎(𝑡) therein.(ii)Let 𝐵 be a densely defined linear operator and let 𝑅1𝑎(𝜆)∶𝐴𝐵𝑥≤𝑀_𝜚|𝜆|^−𝜚‖𝑥‖,𝑥∈𝐷(𝐵),ℜ𝜆=𝜆₀ (96) for some 𝜚>0 and 𝑀_𝜚>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∗₀𝑔_𝜁)-regularized resolvent family (𝑅_𝐵(𝑡))_𝑡≥0 satisfying (18) with 𝑘₁(𝑡)=(𝑘∗₀𝑔_𝜁)(𝑡),𝐶=𝐼, and 𝑔_𝛼(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 Lemma 2, {1/𝑎(𝜆)∶ℜ𝜆>𝜆₀}⊆𝜌(𝐴) and 𝑅1𝑎(𝜆)≤𝑀|𝑎(𝜆∶𝐴)|𝑘(𝜆)(ℜ𝜆−𝜔),ℜ𝜆>𝜆₀. (97) Given 𝑧∈ℂ with ℜ𝑧>𝜆₀, put 𝜆_𝑧∶=𝜆₀+𝑖ℑ𝑧. Then the prescribed assumptions combined with (97) imply 1𝐵𝑅𝑎(𝑧)=1∶𝐴𝐵𝑅𝑎𝜆_𝑧1∶𝐴𝐼+𝑎𝜆_𝑧−1𝑎(𝑧)𝑅1𝑎(𝑧)≤1∶𝐴𝐵𝑅𝑎𝜆_𝑧1∶𝐴1+𝑎𝜆_𝑧−1𝑎(𝑧)𝑅1𝑎(𝑧)≤𝑀∶𝐴_𝜚𝜆^𝜚₀11+𝑎𝜆_𝑧−1𝑎(𝑧)𝑀|𝑎(𝑧)|𝑘(𝑧)≤𝑀)(ℜ𝑧−𝜔_𝜚𝜆^𝜚₀+𝑀_𝜚𝑀𝜆^𝜚₀𝜆₀𝑘(𝑧𝑎(𝑧)−𝜔)𝑎𝜆_𝑧−1. (98) Consider now the function ℎ∶{𝑧∈ℂ∶ℜ𝑧≥0}→𝐿(𝐸) defined by ℎ(𝑧)∶=𝑧^𝜚𝐵𝑅((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)‖≤𝑀_𝜚,𝑡∈ℝ and, by (94)–(98), one has that, for every 𝜀>0, there exists 𝐶^′_𝜀>0 such that ‖ℎ(𝑧)‖≤𝐶^′_𝜀𝑒^𝜀|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)‖≤𝑀_𝜚 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 𝑐_𝜚>0 such that, for ℜ𝜆≥𝑎∶1𝑎(𝜆)𝑅1𝑎(𝜆)=1∶𝐴+𝐵𝑎(𝜆)𝑅1𝑎(𝜆)1∶𝐴𝐼−𝐵𝑅𝑎(𝜆)∶𝐴⁻¹≤𝑐_𝜌𝑘(𝜆.) (99) 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 space 𝐸 has Fourier type 𝑝∈[1,2] if and only if the Fourier transform extends to a bounded linear operator from 𝐿^𝑝(ℝ∶𝐸) to 𝐿^𝑞(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 𝐿^𝑝(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. Let 𝐸 be a Banach space of Fourier type 𝑝∈(1,2].(i) Let the assumptions of Theorem 20(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 𝜆^′₀>𝜆₀ and 𝜂>1 such that 𝑘(𝜆)=𝑂|𝜆|^𝜁−𝜂,ℜ𝜆>𝜆^′₀,|𝑎(𝜆)|=𝑂|𝜆|^𝜁−𝜂,ℜ𝜆>𝜆^′₀. (100)(b)𝐴 is densely defined and 𝐸 is reflexive.(c)𝐵(𝐷(𝐴²))⊆𝐷(𝐴) and 𝐵𝐴𝑥=𝐴𝐵𝑥,𝑥∈𝐷(𝐴²). Then 𝐴+𝐵 is the integral generator of an exponentially bounded, (𝑎,𝑘∗₀𝑔_𝜁)-regularized resolvent family (𝑅_𝐵(𝑡))_𝑡≥0 satisfying (18) with 𝑘₁(𝑡)=(𝑘∗₀𝑔_𝜁)(𝑡),𝐶=𝐼, and 𝑔_𝛼(𝑡) replaced by 𝑎(𝑡) therein.(ii) Let the assumptions of Theorem 20(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∗₀𝑔_𝜁)-regularized resolvent family (𝑅_𝐵(𝑡))_𝑡≥0 satisfying (18) with 𝑘₁(𝑡)=(𝑘∗₀𝑔_𝜁)(𝑡),𝐶=𝐼, and 𝑔_𝛼(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 𝑅_𝐵(𝑡)𝑥1∶=2𝜋𝑖^𝑎+𝑖∞_{𝑎−𝑖∞}𝑒^𝜆𝑡𝜆^−𝜁𝑘(𝜆)𝑎(𝜆)𝑅1𝑎(𝜆)∶𝐴+𝐵𝑥𝑑𝜆,𝑥∈𝐸,𝑡≥0. (101) By the first part of the proof of [19, Theorem 3.3], 𝐴+𝐵 is the integral generator of an exponentially bounded, (𝑎,𝑘∗₀𝑔_𝜁)-regularized resolvent family (𝑅_𝐵(𝑡))_𝑡≥0 satisfying (18) with 𝑘₁(𝑡)=(𝑘∗₀𝑔_𝜁)(𝑡),𝐶=𝐼, and 𝑔_𝛼(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, ^∞_−∞𝑘(𝑟+𝑖𝑠)𝑎()𝑅1𝑟+𝑖𝑠𝑎(𝑟+𝑖𝑠)∶𝐴+𝐵^∗𝑥^∗𝑞𝑑𝑠≤𝑐₁𝑥^∗𝑞. (102) Set, for every 𝑥^∗∈𝐸^∗ and 𝑡≥0∶𝑅_𝐵,∗(𝑡)𝑥^∗1∶=2𝜋𝑖^𝑎+𝑖∞_{𝑎−𝑖∞}𝑒^𝜆𝑡𝜆^−𝜁𝑘(𝜆)𝑎(𝜆)𝑅1𝑎(𝜆)∶𝐴+𝐵^∗𝑥^∗𝑑𝜆. (103) Then (𝑅_𝐵,∗(𝑡))_𝑡≥0⊆𝐿(𝐸^∗) is strongly continuous, exponentially bounded and 𝑘(𝜆)𝑎(𝜆)𝑅1𝑎(𝜆)∶𝐴+𝐵^∗𝑥^∗=𝜆^𝜁∞₀𝑒^−𝜆𝑡𝑅_𝐵,∗(𝑡)𝑥^∗𝑑𝑡,ℜ𝜆>𝑎,𝑥^∗∈𝐸^∗. (104)
By Lemma 2, (𝑅_𝐵,∗(𝑡))_𝑡≥0 is an (𝑎,𝑘∗₀𝑔_𝜁)-regularized resolvent family with the integral generator (𝐴+𝐵)^∗. By [33, Theorem 2.14(iii)], it follows that (𝑅_𝐵(𝑡)≡𝐽⁻¹𝑅_𝐵,∗(𝑡)^∗𝐽)_𝑡≥0 is an (𝑎,𝑘∗₀𝑔_𝜁)-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 𝑘(𝜆)𝑎(𝜆)𝑅1𝑎(𝜆)∶𝐴+𝐵𝑥=𝜆^𝜁∞₀𝑒^−𝜆𝑡𝑅_𝐵(𝑡)(𝐴).𝑥𝑑𝑡,ℜ𝜆>𝑎,𝑥∈𝐷 (105) By (104)-(105) and the uniqueness theorem for Laplace transform, we get 𝑅_𝐵,∗(𝑡)𝑥^∗,𝑥=𝑥^∗,𝑅_𝐵(𝑡)𝑥,𝑡≥0,𝑥^∗∈𝐸^∗,𝑥∈𝐷(𝐴) (106)
and 𝑅_𝐵,∗(𝑡)^∗𝐽_𝑥=𝐽_{𝑅𝐵(𝑡)𝑥},𝑡≥0,𝑥∈𝐷(𝐴). Now one can simply prove that ((𝑅_𝐵,∗(𝑡)^∗)_|𝐸)_𝑡≥0 is an exponentially bounded, (𝑎,𝑘∗₀𝑔_𝜁)-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 (𝑅_𝐵(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 of 𝐷(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) Let 𝛼>0 and 𝑎(𝑡)=𝑔_𝛼(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. Let 1<𝑝<∞,1/𝑝+1/𝑞=1,𝑘∈ℕ₀,0<𝛽≤2 and 𝐸∶=𝐿^𝑝(ℝ). Define a closed linear operator 𝐴_𝛽,𝑘 on 𝐸 by 𝐷(𝐴_𝛽,𝑘)∶=𝑊^4𝑘+2,𝑝(ℝ) and 𝐴_𝛽,𝑘𝑓∶=𝑒^{𝑖(2−𝛽)(𝜋/2)}𝑓^(4𝑘+2),𝑓∈𝐷(𝐴_𝛽,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 𝑉∈𝐿^𝑝(ℝ)1,𝑙≤𝑝()()4𝑘+2𝑝−1−4𝑘+2𝑝−1𝛽. (107) Given ℜ𝜆>0, denote by 𝜇_𝑗,𝜆 (1≤𝑗≤2𝑘+1)(2𝑘+1) solutions of the equation 𝜇^4𝑘+2_𝑗,𝜆=𝜆^𝛽𝑒^{𝑖(𝛽(𝜋/2)−𝜋)} with ℜ𝜇_𝑗,𝜆>0. Then 𝐷(𝐴)⊆𝐷(𝐵), 𝑅𝜆^𝛽∶𝐴_𝛽,𝑘𝑓(𝑥)=𝑒^{𝑖𝛽(𝜋/2)}4𝑘+2^∞_−∞^2𝑘+1_𝑗=1𝑒^{−𝜇𝑗,𝜆|𝑥−𝑠|}−𝜇_𝑗,𝜆^4𝑘+1𝑓(𝑠)𝑑𝑠, (108) provided 𝑓∈𝐸,𝑥∈ℝ,ℜ𝜆>0, 𝐵𝑅𝜆^𝛽∶𝐴_𝛽,𝑘𝑓(𝑥)=𝑒^{𝑖𝛽(𝜋/2)}𝑉(𝑥)×4𝑘+2_{2𝑘+1𝑗=1𝑥−∞}𝑒^{−𝜇𝑗,𝜆(𝑥−𝑠)}−𝜇_𝑗,𝜆^{4𝑘−𝑙+1}𝑓(𝑠)−𝑑𝑠_∞𝑥𝑒^{𝜇𝑗,𝜆(𝑥−𝑠)}𝜇^{4𝑘−𝑙+1}_𝑗,𝜆𝑓(𝑠),𝑅𝜆𝑑𝑠𝑓∈𝐸,𝑥∈ℝ,ℜ𝜆>0,_𝛽∶𝐴_𝛽,𝑘≤|𝜆|^{𝛽(1−(1/(4𝑘+2)))}minℜ𝜇_1,𝜆,…,ℜ𝜇_2𝑘+1,𝜆⁻¹,𝜆ℜ𝜆>0,𝐵𝑅^𝛽∶𝐴_𝛽,𝑘≤‖𝑉‖_𝑝|𝜆|^{𝛽(1−((𝑙+1)/(4𝑘+2)))}×minℜ𝜇_1,𝜆^1/𝑞,…,ℜ𝜇_2𝑘+1,𝜆^1/𝑞−1, (109) provided ℜ𝜆>0. Furthermore, ℜ𝜇_𝑗,𝜆=|𝜆|^{𝛽/(4𝑘+2)}cos(arg(𝜇_𝑗,𝜆)),ℜ𝜆>0,1≤𝑗≤2𝑘+1, and minℜ𝜇_𝑗,𝜆=|𝜆|∶1≤𝑗≤2𝑘+1_{𝛽/(4𝑘+2)}(𝜆)/(2)×mincosarg𝛽+𝛽𝜋+()𝜋4𝑘+22𝑘−1,(𝜆)4𝑘+2−cosarg𝛽+𝛽𝜋/2+𝜋4𝑘+22, (110) provided ℜ𝜆>0. The above implies that there exists a constant 𝑐_𝛽,𝑘>0 such that |𝜆|^{𝛽/(4𝑘+2)}cosarg(𝜆)/minℜ𝜇_1,𝜆,…,ℜ𝜇_2𝑘+1,𝜆≤𝑐_𝛽,𝑘,ℜ𝜆>0. (111) Keeping in mind (107)-(111), we obtain that 𝑅𝜆^𝛽∶𝐴_𝛽,𝑘=𝑂|𝜆|^1−𝛽(ℜ𝜆)⁻¹,ℜ𝜆>0𝐵𝑅𝜆^𝛽∶𝐴_𝛽,𝑘=𝑂‖𝑉‖_𝑝(ℜ𝜆)^{−𝛽(1−((𝑙+1)/(4𝑘+2))+(1/(4𝑘+2)𝑞))}=𝑂‖𝑉‖_𝑝(ℜ𝜆)^(−1)/𝑞, (112)(113) provided ℜ𝜆>0. Denote by 𝛽_𝑘 the infimum of all nonnegative real numbers 𝑟≥0 such that the operator 𝐴_𝛽,𝑘 generates an exponentially bounded (𝑔_𝛽,𝑔_𝑟+1)-regularized resolvent family. The precise computation of integration rate 𝛽_𝑘 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 𝛽_𝑘≤1; furthermore, 𝛽_𝑘=0 provided 𝑝=2 ([24]), and 𝛽_𝑘≤|(1/2)−(1/𝑝)| provided 𝛽∈{1,2} [14, 50]. Set 𝜅_𝑝∶=min(1/𝑝,(𝑝−1)/𝑝). By Theorem 21, 𝐴_𝛽,𝑘+𝐵 generates an exponentially bounded (𝑔_𝛽,𝑔_{𝜎𝛽,𝑘,𝑝+1})-regularized resolvent family for any 𝜎_{𝛽,𝑘,𝑝}>𝛽_𝑘+𝜅_𝑝. 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∈𝐿^𝑝(ℝ)+𝐿^∞(ℝ); similarly, one can consider the operators 𝐴¹_𝛽,𝑘(𝑘∈ℕ,0<𝛽≤2) and 𝐴²_𝛽,𝑘(𝑘∈ℕ,0<𝛽≤1) given by 𝐴¹_𝛽,𝑘𝑓∶=𝑒^{−𝑖𝛽(𝜋/2)}𝑓^(4𝑘),𝑓∈𝑊^4𝑘,𝑝(ℝ)∶=𝐷(𝐴¹_𝛽,𝑘) and 𝐴²_𝛽,𝑘𝑓∶=𝑒^{±𝑖(𝜋/2)(1−𝛽)}𝑓^(2𝑘+1),𝑓∈𝑊^2𝑘+1,𝑝(ℝ)∶=𝐷(𝐴²_𝛽,𝑘).
Notice that Lizama and Prado have recently analyzed in [21] the qualitative properties of the abstract relaxation equation: 𝑢^′(𝑡)−𝐴𝐃^𝛼_𝑡𝑢(𝑡)(𝑡)(𝑡),()(0)+𝑢=𝑓𝛼∈0,1,𝑡≥0,𝑢=0, (114) where 𝐸 is a Banach space and 𝑓∈𝐿¹_loc([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. Let 𝑘(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 𝑐_𝜂>0 such that 𝑝𝑒^{−𝜔ℜ𝑧}𝑅(𝑧)𝑥≤𝑐_𝜂𝑝(𝑥),𝑥∈𝐸,𝑝∈⊛,𝑧∈Σ_𝜂 (115) as well as 𝑏,𝑐≥0,𝐵 is a linear operator satisfying 𝐷(𝐶⁻¹𝐴𝐶)⊆𝐷(𝐵),𝐵𝐶𝑥=𝐶𝐵𝑥,𝑥∈𝐷(𝐶⁻¹𝐴𝐶) and 𝑝𝐶⁻¹𝐵𝑥≤𝑏𝑝𝐶⁻¹(𝑥),𝐶𝐴𝐶𝑥+𝑐𝑝𝑥∈𝐷⁻¹𝐴𝐶,𝑝∈⊛. (116)
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 𝜔_𝜂≥𝜔 such that |𝑘(𝜆)⁻¹|=𝑂(|𝜆|),𝜆∈𝜔_𝜂+Σ_{(𝜋/2)+𝜂} and |𝑎(𝜆)/𝑘(𝜆)|=𝑂(|𝜆|),𝜆∈𝜔_𝜂+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 𝜔_𝜂≥𝜔 such that |𝑘(𝜆)⁻¹|=𝑂(|𝜆|),𝜆∈𝜔_𝜂+Σ_{(𝜋/2)+𝜂} and 𝑎(𝜆)/(𝜆𝑘(𝜆))→0,|𝜆|→∞,𝜆∈𝜔_𝜂+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 𝜔_𝜂≥𝜔 such that |𝑎(𝜆)/𝑘(𝜆)|=𝑂(|𝜆|),𝜆∈𝜔_𝜂+Σ_{(𝜋/2)+𝜂}.(iv)𝑏=0 and, for every 𝜂∈(0,𝛿), there exists 𝜔_𝜂≥𝜔 such that 𝑎(𝜆)/(𝜆𝑘(𝜆))→0,|𝜆|→∞,𝜆∈𝜔_𝜂+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 (𝑅_𝐵(𝑡))_𝑡≥0 of angle 𝛿, which satisfies 𝑅_𝐵(𝑧)[𝐶⁻¹(𝐶⁻¹𝐴𝐶+𝐵)𝐶]⊆[𝐶⁻¹(𝐶⁻¹𝐴𝐶+𝐵)𝐶]𝑅_𝐵(𝑧),𝑧∈Σ_𝛿 and the following condition: ∀𝜂∈(0,𝛿)∃𝜔^′_𝜂>0∃𝑀_𝜂>0∀𝑝∈⊛∶𝑝𝑅_𝐵(𝑧)𝑥≤𝑀_𝜂𝑒^{𝜔′𝜂ℜ𝑧}𝑝(𝑥),𝑥∈𝐸,𝑧∈Σ_𝜂. (117) 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 operator 𝐶⁻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 lim_𝜆→+∞𝜆𝑘(𝜆)𝑎(𝜆)1𝑎(𝜆)−𝐶⁻¹𝐴𝐶⁻¹𝐶𝑥=𝑘(0)𝐶𝑥,𝑥∈𝐸 (118)
and that there exists 𝑁_𝜂>0 such that {(1/𝑎(𝜆))∶𝜆∈𝜔+Σ_{(𝜋/2)+𝜂}}⊆𝜌_𝐶(𝐶⁻¹𝐴𝐶) and sup_{𝜆∈𝜔+Σ(𝜋/2)+𝜂}𝑝()𝑘(𝜆)𝜆−𝜔𝑎(𝜆)1𝑎(𝜆)−𝐶⁻¹𝐴𝐶⁻¹𝐶𝑥≤𝑁_𝜂𝑝(𝑥),𝑥∈𝐸. (119) By (116) and (119), we infer that, for every 𝜆∈𝜔_𝜂+Σ_{(𝜋/2)+𝜂},𝑥∈𝐸 and 𝑝∈⊛∶𝑝𝐶⁻¹𝐵1𝑎(𝜆)−𝐶⁻¹𝐴𝐶⁻¹|𝐶|𝐶𝑥≤𝑏_⊛𝑝(𝑥)𝑁+𝑏^𝜂𝑐_𝜂𝑝(𝑥)|𝜆−𝜔|𝑘(𝜆)+𝑐𝑐_𝜂𝑁_𝜂𝑝(𝑥)|||𝑎(𝜆𝜆−𝜔)|𝑘(𝜆,) (120) which implies by the given assumption the existence of a number 𝜔^′_𝜂>𝜔_𝜂 such that 𝑝(𝐶⁻¹𝐵((1/𝑎(𝜆))−𝐶⁻¹𝐴𝐶)⁻¹𝐶𝑥)≤𝜎𝑝(𝑥),𝑥∈𝐸,𝜆∈𝜔^′_𝜂+Σ_{(𝜋/2)+𝜂},𝑝∈⊛, provided that the numbers 𝑏 and 𝑐 are sufficiently small; if (iv) holds, then lim_𝜆→+∞𝐶⁻¹𝐵1𝑎(𝜆)−𝐶⁻¹𝐴𝐶⁻¹𝐶𝑥=0,𝑥∈𝐸. (121) Using the same argument as in the proof of Theorem 14, it follows that, for every 𝜆∈𝜔^′_𝜂+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∈𝜔^′_𝜂+Σ_{(𝜋/2)+𝜂}∶1𝑎(𝜆)−𝐶⁻¹𝐴𝐶+𝐵⁻¹𝐶=1𝑎(𝜆)−𝐶⁻¹𝐶⁻¹𝐴𝐶+𝐵𝐶⁻¹𝐶=1𝑎(𝜆)−𝐶⁻¹𝐴𝐶⁻¹×𝐶𝐼−𝐶⁻¹𝐵1𝑎(𝜆)−𝐶⁻¹𝐴𝐶⁻¹𝐶⁻¹. (122) Now we will prove that the operator 𝐶⁻¹(𝐶⁻¹𝐴𝐶+𝐵)𝐶 is closed. Let (𝑥_𝜏)_𝜏∈𝑇 be a net in 𝐸 satisfying 𝑥_𝜏→𝑥, 𝜏→∞ and 𝐶⁻¹(𝐶⁻¹𝐴𝐶+𝐵)𝐶𝑥_𝜏→𝑦, 𝜏→∞. Then ((1/𝑎(𝜆))− (𝐶⁻¹𝐴𝐶+𝐵))⁻¹𝐶𝐶⁻¹(𝐶⁻¹𝐴𝐶+𝐵)𝐶𝑥_𝜏→ ((1/𝑎(𝜆))−(𝐶⁻¹𝐴𝐶+𝐵))⁻¹𝐶𝑦, 𝜏→∞, that is, −𝐶𝑥_𝜏+(1/𝑎(𝜆))((1/𝑎(𝜆))− (𝐶⁻¹𝐴𝐶+𝐵))⁻¹𝐶𝑥_𝜏 →((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 1𝜆↦𝑎(𝜆)−𝐶⁻¹𝐶⁻¹𝐴𝐶+𝐵𝐶⁻¹=1𝐶𝑥𝑎(𝜆)−𝐶⁻¹𝐴𝐶⁻¹𝐶×^∞_𝑛=0𝐶⁻¹𝐵1𝑎(𝜆)−𝐶⁻¹𝐴𝐶⁻¹𝐶^𝑛𝑥,𝜆∈𝜔^′_𝜂+Σ_{(𝜋/2)+𝜂} (123) 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 (𝑅_𝐵(𝑡))_𝑡≥0 of angle 𝜂 and (117) holds; assuming (iv), we get from (119) 𝑝𝜆−𝜔^′_𝜂𝑘(𝜆)𝑎(𝜆)1𝑎(𝜆)−𝐶⁻¹𝐴𝐶+𝐵⁻¹𝐶𝑥=𝑝𝜆−𝜔^′_𝜂()()𝑘(𝜆)𝜆−𝜔𝜆−𝜔𝑎(𝜆)1𝑎(𝜆)−𝐶⁻¹𝐴𝐶⁻¹𝐶×𝐼−𝐶⁻¹𝐵1𝑎(𝜆)−𝐶⁻¹𝐴𝐶⁻¹𝐶⁻¹𝑥≤11+𝑁cos𝜂^𝜂𝑝(𝑥)1−𝜎,𝑥∈𝐸,𝜆∈𝜔^′_𝜂+Σ_{(𝜋/2)+𝜂}. (124)
In combination with (118) and (121), the above implies lim_𝜆→+∞𝜆𝑘(𝜆)𝑎(𝜆)1𝑎(𝜆)−𝐶⁻¹𝐴𝐶+𝐵⁻¹𝐶𝑥=lim_𝜆→+∞𝜆𝑘(𝜆)𝑎(𝜆)1𝑎(𝜆)−𝐶⁻¹𝐴𝐶⁻¹𝐶𝑥+lim_𝜆→+∞𝜆𝑘(𝜆)𝑎(𝜆)1𝑎(𝜆)−𝐶⁻¹𝐴𝐶⁻¹𝐶×𝐶⁻¹𝐵1𝑎(𝜆)−𝐶⁻¹𝐴𝐶⁻¹×𝐶𝐼−𝐶⁻¹𝐵1𝑎(𝜆)−𝐶⁻¹𝐴𝐶⁻¹𝐶⁻¹𝑥=𝑘(0)𝐶𝑥+0=𝑘(0)𝐶𝑥,𝑥∈𝐸, (125) 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∶𝑘(𝜆)𝐼−𝑎(𝜆)𝐶⁻¹𝐴𝐶+𝐵⁻¹=𝑘(𝜆)𝑎(𝜆)𝐶𝐶𝑥𝐼−⁻¹𝐶⁻¹𝐴𝐶+𝐵𝐶⁻¹=𝐶𝑥_∞0𝑒^−𝜆𝑡𝑅_𝐵(𝑡)𝑥𝑑𝑡. (126) By Lemma 2, we obtain that (18) holds with 𝐴+𝐵,   𝑘₁(𝑡) and 𝑔_𝛼(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 (𝑅_𝐵(𝑡))_𝑡≥0.

Example 26. Let 𝑢(𝑡) be a solution of (114). Set 𝑎_𝛼(𝑡)∶=ℒ⁻¹(𝜆^𝛼/(𝜆+1))(𝑡), 𝑡≥0, 𝑘_𝛼(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(𝑎_𝛼∗𝑣)(𝑡)+(1∗𝑓)(𝑡), 𝑡≥0, which implies that the notion of an (𝑎_𝛼,𝑘)-regularized 𝐶-resolvent family is important in the study of (114). In [21], the authors mainly use the following conditions: 𝑘(𝑡)=𝑘_𝛼(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 𝜔_𝜂>0 such that the family (1+|𝜆|)^1−𝑟(𝜆−𝐴)⁻¹𝐶∶𝜆∈𝜔_𝜂+Σ_𝜂 (127) is equicontinuous (𝑟≥0) and that the mapping 𝜆↦(𝜆−𝐴)⁻¹𝐶𝑥,𝜆∈𝜔_𝜂+Σ_𝜂iscontinuousforeveryfixed𝑥∈𝐸. (128) Notice that (127)-(128) hold provided that 𝐴 is a subgenerator of an exponentially equicontinuous 𝑟-times integrated 𝐶-semigroup (𝑅_𝑟(𝑡))_𝑡≥0; furthermore, if ∃𝑀≥1∃𝜔≥0∶𝑝𝑅_𝑟(𝑡)𝑥≤𝑀𝑒^𝜔𝑡𝑝(𝑥),𝑥∈𝐸,𝑝∈⊛, (129) then, for every 𝜂∈(0,𝜋/2) and 𝜔_𝜂>𝜔, there exists 𝑀_𝜂>0 such that 𝑝(𝜆−𝐴)⁻¹𝐶𝑥≤𝑀_𝜂(1+|𝜆|)^𝑟−1𝑝(𝑥),𝑥∈𝐸,𝜆∈𝜔_𝜂+Σ_𝜂,𝑝∈⊛. (130) 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 𝜔^′_𝛾>𝜔 such that (𝜆+1)/𝜆^𝛼=𝜆^1−𝛼+𝜆^−𝛼∈𝜔_𝜂+Σ_𝜂 for all 𝜆∈𝜔^′_𝛾+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∈𝜔^′_𝛾+Σ_{(𝜋/2)+𝛾} is analytic (𝑥∈𝐸) and that, for every 𝜎≥𝑟(1−𝛼), the family {(𝜆−𝜔^′_𝛾) (𝑘(𝜆)/|𝜆|^𝜎)(1/𝑎(𝜆))((1/𝑎(𝜆))−𝐴)⁻¹𝐶∶ 𝜆∈𝜔^′_𝛾+Σ_{(𝜋/2)+𝛾}} is equicontinuous (if (129) holds, then there exists 𝑁_𝛾>0 such that 𝑝((𝜆−𝜔^′_𝛾)(𝑘(𝜆)/|𝜆|^𝜎)(1/𝑎(𝜆))((1/𝑎(𝜆))−𝐴)⁻¹𝐶𝑥)≤𝑁_𝛾𝑝(𝑥),𝑥∈𝐸,𝑝∈⊛,𝜆∈Σ_𝜂). Using [33, Theorem 3.7] and the arbitrariness of 𝜂, we get that 𝐴 is a subgenerator of an exponentially equicontinuous, analytic (𝑎_𝛼,𝑘∗𝑔_𝜁)-regularized 𝐶-resolvent family (𝑅(𝑡))_𝑡≥0 of angle 𝛿, where 𝑟()𝜁=1−𝛼,if𝐷(𝐴)()=𝐸>𝑟1−𝛼,if𝐷(𝐴)≠𝐸 (131) and 𝑔₀(𝑡) stands for the Dirac distribution (if (129) holds, then for every 𝜂∈(0,𝛿) there exist 𝜔_𝜂>0 and 𝐿_𝜂>0 such that 𝑝(𝑅(𝑧)𝑥)≤𝐿_𝜂𝑒^{𝜔𝜂ℜ𝑧}𝑝(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 𝐶⁰_𝛽])])(0)𝑓([0,𝑇∶𝐸∶=𝑓∈𝐶([0,𝑇∶𝐸∶𝑓=0,_{𝛽,𝑇,𝑝}<∞,∀𝑝∈⊛, (132) where 𝑓_{𝛽,𝑇,𝑝}∶=sup_{0≤𝑠<𝑡≤𝑇}𝑝𝑓(𝑡)−𝑓(𝑠)(𝑡−𝑠)^𝛽. (133) Let 𝐴 be densely defined, let 𝑟=0 and let 𝛽∈(0,1) be such that 𝐶⁻¹(1∗𝑓_|[0,𝑇])∈𝐶⁰_𝛽([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 𝑣(𝑡)=𝑅(𝑡)𝐶⁻¹1∗𝑓(𝑡)+^𝑡₀𝑅^′(𝑡−𝑠)𝐶⁻¹1∗𝑓(𝑠)−𝐶⁻¹1∗𝑓(𝑡)𝑑𝑠,𝑡≥0 (134) satisfies 𝐴(𝑎_𝛼∗𝑣)(𝑡)=𝑣(𝑡)−(1∗𝑓)(𝑡),𝑡≥0 and that, for every 𝑇>0, one has 𝑣_|[0,𝑇]∈𝐶⁰_𝛽([0,𝑇]∶𝐸); in the above formula, we assume that (𝑅(𝑡))_𝑡≥0 is the exponentially equicontinuous, analytic (𝑎_𝛼,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 𝑢(𝑡)−𝐴𝑔_1−𝛼(𝑡)+(𝑡)=(𝑡),(0)∗𝑢1∗𝑢)(1∗𝑓𝑡≥0,𝑢=0 (135) and that 𝑢_|[0,𝑇]∈𝐶⁰_𝛽([0,𝑇]∶𝐸) for all 𝑇>0. If 𝑥∈𝐸, 𝑥≠0 and 𝐶⁻¹(1∗(𝑓(⋅)−𝑥)_|[0,𝑇])∈𝐶⁰_𝛽([0,𝑇]∶𝐸) for all 𝑇>0, then we obtain similarly the unique solution 𝑢(𝑡) of the problem 𝑢(𝑡)−𝑥−𝐴𝑔_1−𝛼∗(𝑢(⋅))(𝑡)+(𝑡)=𝑡)(0)−𝑥1∗𝑢)(1∗𝑓(,𝑡≥0,𝑢=𝑥; (136) furthermore, 𝑢_|[0,𝑇]∈𝐶⁰_𝛽([0,𝑇]∶𝐸) for all 𝑇>0. Since 𝑎_𝛼∉𝐵𝑉_loc([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 which 𝐴 is not necessarily densely defined. Let 𝐶⁻¹𝑓,𝐴𝐶⁻¹𝑓∈𝐿¹_loc([0,∞)∶𝐸) and let (𝑅_𝛼(𝑡))_𝑡≥0 denote the (𝑎_𝛼,𝑘_𝛼∗𝑔_𝜁)-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 𝑢(𝑡)−𝐴𝑔_(1−𝛼)(𝑡)+(𝑡)=∗𝑢1∗𝑢)(1∗𝑓∗𝑔_𝜁(𝑡)+𝑔_𝜁+1(𝑡)𝑥,𝑡≥0, (137) given by 𝑡↦𝑢(𝑡)=𝑅_𝛼(𝑡)𝐶⁻¹𝑥+∫^𝑡₀𝑅_𝛼(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)If 𝑟=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 (𝑎_𝛼,𝑘)-regularized 𝐶-resolvent family (𝑅_𝐵(𝑡))_𝑡≥0 of angle 𝛿 (cf. [64, Chapter III] and [65, Chapter 7] for corresponding examples).(ii)If 𝑏=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 (𝑎_𝛼,𝑘∗𝑔_𝜁)-regularized 𝐶-resolvent family (𝑅_𝐵(𝑡))_𝑡≥0 of angle 𝛿.(iii)If 𝑏=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 (𝑎_𝛼,𝑘∗𝑔_𝜁)-regularized 𝐶-resolvent family (𝑅_𝐵(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: 𝑢(𝑡)+^𝑡₀(𝑡−𝑠)𝐴𝐃^𝛼_𝑠𝑢(𝑠)+𝑢(𝑠)𝑑𝑠=𝑊(𝑡),𝑡>0, (138) 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 𝑢(𝑡)+^𝑡₀𝑔_2−𝛼(𝑡−𝑠)𝐴𝑢(𝑠)𝑑𝑠+^𝑡₀()𝑢(𝑠)=𝑡−𝑠𝑑𝑠_𝑡0(𝑡−𝑠)𝑓(𝑠)𝑑𝑠,𝑡>0, (139) where 𝑓∈𝐿¹_loc([0,∞)∶𝐸). Equation (139) models an oscillation process with fractional damping term and after differentiation becomes, in some sense, 𝑢^′′(𝑡)+𝐴𝐃^𝛼_𝑡𝑢(𝑡)+𝑢(𝑡)=𝑓(𝑡),𝑡≥0. (140) 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 ℝ^𝑛 with boundary of class 𝐶^4𝑚, and let 𝐸∶=𝐶^𝛼(Ω). Consider the operator 𝐴∶𝐷(𝐴)⊆𝐶^𝛼(Ω)→𝐶^𝛼(Ω) given by 𝐴𝑢(𝑥)∶=_𝛽≤2𝑚𝑎_𝛽(𝑥)𝐷^𝛽𝑢(𝑥)∀𝑥∈Ω (141) with domain 𝐷(𝐴)∶={𝑢∈𝐶^2𝑚+𝛼(Ω)∶𝐷^𝛽𝑢_|𝜕Ω=0forall|𝛽|≤𝑚−1}. Here 𝛽∈ℕ^𝑛₀,|𝛽|=∑^𝑛_𝑖=1𝛽_𝑗,𝐷^𝛽=∏^𝑛_𝑖=1((1/𝑖)(𝜕/𝜕𝑥_𝑖))^𝛽𝑖, and 𝑎_𝛽∶Ω→ℂ satisfy the following conditions:(i)𝑎_𝛽(𝑥)∈ℝ for all 𝑥∈Ω and |𝛽|=2𝑚,(ii)𝑎_𝛽∈𝐶^𝛼(Ω) for all |𝛽|≤2𝑚, and(iii)there exists 𝑀>0 such that 𝑀⁻¹𝜉^2𝑚≤_𝛽=2𝑚𝑎_𝛽(𝑥)𝜉^𝛽≤𝑀𝜉^2𝑚∀𝜉∈ℝ^𝑛,𝑥∈Ω. (142)Then there exists a sufficiently large 𝜎>0 such that the operator −𝐴_𝜎≡−(𝐴+𝜎) satisfies Σ_𝜔∪{0}⊆𝜌(−𝐴_𝜎) with some 𝜔∈((𝜋/2),𝜋) and 𝑅𝜆∶−𝐴_𝜎=𝑂|𝜆|^{(𝛼/2𝑚)−1},𝜆∈Σ_𝜔. (143) Notice that 𝐴 is not densely defined since 𝐷(𝐴)⊆{𝑢∈𝐶^𝛼(Ω)∶𝑢_|𝜕Ω=0}. Let 𝜍∈[1,(2𝜔/𝜋)) and 𝜏∈((𝛼/2𝑚),1). By (143) and [32, Theorem 2.17], we get that −𝐴_𝜎 is the integral generator of an exponentially bounded, analytic (𝑔_𝜍,𝑔_𝜍𝜏+1)-regularized resolvent family of angle 𝛿=(𝜔/𝜍)−(𝜋/2)∈(0,(𝜋/2)). Assume now that 𝐵∶𝐷(𝐵)⊆𝐶^𝛼(Ω)→𝐶^𝛼(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 (𝑔_𝜍,𝑔_𝜍𝜏+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∶𝐶^𝛼[0,1]→𝐶^𝛼[0,1] be defined by 𝐵𝑢(𝑥)∶=^𝑥₀(𝑢()(0𝜑(𝑠)+𝑥−𝑠−𝑢))𝑑𝑠_𝑥0(𝑢(1−𝑥+𝑠)−𝑢(1))𝜓(𝑠)𝑑𝑠,𝑥∈[0,1]. (144) Then 𝐵 satisfies the conditions stated above since 𝐵∈𝐿(𝐶^𝛼[0,1]) and ||𝐵𝑢||≤(||𝜑||_𝐿¹_[0,1]+||𝜓||_𝐿¹_[0,1])||𝑢||,𝑢∈𝐶^𝛼[0,1]. Finally, it could be interesting to construct an example in which there does not exist 𝐵∈𝐿(𝐶^𝛼[0,1]) such that 𝐵𝑥=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.
Given 𝑎∈𝐸, 𝑏^∗∈𝐸^∗ and 𝐶∈𝐿([𝐷(𝐴)],𝐸), we consider the rank-1 perturbation 𝐵∈𝐿([𝐷(𝐴)],𝐸) of 𝐴 given by 𝐵𝑥∶=𝑏^∗(𝐶𝑥)𝑎,𝑥∈𝐷(𝐴). (145) We also denote this operator 𝐵 by 𝑎𝑏^∗𝐶. Denote 𝐵_𝛿(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. Let 𝐴 be a closed linear operator on 𝐸, let 𝐶∈𝐿([𝐷(𝐴)],𝐸) and let 𝜀>0. Assume that Ω_𝑛⊆𝜌(𝐴) and sup_{𝜆∈Ω𝑛}||𝐶𝑅(𝜆∶𝐴)𝑥||<∞ for all 𝑥 in a dense subset 𝐸_𝑛 of 𝐸 and all 𝑛∈ℕ. Let 𝑔_𝑛∶Ω_𝑛→(0,∞)(𝑛∈ℕ). Assume that for each (𝑎,𝑏^∗)∈𝐵_𝜀(0,0) there exists 𝑛∈ℕ such that Ω_𝑛⊆𝜌(𝐴+𝑎𝑏^∗𝐶) and ||𝑅(𝜆∶𝐴+𝑎𝑏^∗𝐶)||≤𝑔_𝑛(𝜆),𝜆∈Ω_𝑛. Then there exists 𝑚∈ℕ such that sup_{𝜆∈Ω𝑚}||𝐶𝑅(𝜆∶𝐴)||<∞.

Henceforth, we assume that (𝑀_𝑝) is a sequence of positive real numbers such that 𝑀₀=1 and that the following conditions are fulfilled: 𝑀²_𝑝≤𝑀_𝑝+1𝑀_𝑝−1,𝑝∈ℕ,𝑀_𝑝≤𝐴𝐻^𝑝sup_{0≤𝑖≤𝑝}𝑀_𝑖𝑀_𝑝−𝑖,𝑝∈ℕ,forsome𝐴,𝐻>1,_∞𝑝=1𝑀_𝑝−1𝑀_𝑝<∞. (M.1)(M.2)M.3^′ Let 𝑠>1. Then the Gevrey sequences (𝑝!^𝑠), (𝑝^𝑝𝑠) and (Γ(1+𝑝𝑠)) satisfy the above conditions. The associated function of (𝑀_𝑝) is defined by 𝑀(𝑡)∶=sup_𝑝∈ℕln(𝑡^𝑝/𝑀_𝑝),𝑡>0 and 𝑀(0)∶=0. Recall [54], the function 𝑡↦𝑀(𝑡),𝑡≥0 is increasing, lim_𝑡→∞𝑀(𝑡)=∞ and lim_𝑡→∞(𝑀(𝑡)/𝑡)=0.

Following [1, 16], a closed linear operator 𝐴 is said to be the generator of an ultradistribution sine of (𝑀_𝑝)-class if and only if the operator 𝒜∶=^0𝐼_𝐴0 generates an ultradistribution semigroup of (𝑀_𝑝)-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) Let 𝐴 be a closed densely defined operator on 𝐸. Then 𝐴 generates an ultradistribution semigroup of (𝑀_𝑝)-class if and only if there exist 𝑙≥1,𝛼>0 and 𝛽∈ℝ such that Λ_{𝑙,𝛼,𝛽}∶=𝜆∈ℂ∶ℜ𝜆≥𝛼𝑀(𝑙|ℑ𝜆|)+𝛽⊆𝜌(𝐴),‖𝑅(𝜆∶𝐴)‖=𝑂exp(𝑀(𝑙|𝜆|)),𝜆∈Λ_{𝑙,𝛼,𝛽}. (146)(147)
(ii) Let 𝐴 be a closed densely defined operator on 𝐸. Then 𝐴 generates an ultradistribution sine of (𝑀_𝑝)-class if and only if there exist 𝑙≥1,𝛼>0 and 𝛽∈ℝ such that 𝜆²∶𝜆∈Λ_{𝑙,𝛼,𝛽}⊆𝜌(𝐴),𝑅𝜆²∶𝐴=𝑂exp(𝑀(𝑙|𝜆|)),𝜆∈Λ_{𝑙,𝛼,𝛽}. (148)(149)

Theorem 30. Let 𝑙≥1,𝛼>0,𝛽∈ℝ,𝑘∈ℕ and 𝐶>0. Let 𝐴 be a closed densely defined operator on 𝐸.(i)Assume (148) and 𝑅𝜆²∶𝐴≤𝐶(1+|𝜆|)^𝑘,𝜆∈Λ_{𝑙,𝛼,𝛽}. (150) Let 𝜀>0 and 𝑧∈ℂ be such that for each (𝑎,𝑏^∗)∈𝐵_𝜀(0,0) the operator 𝐴+𝑎𝑏^∗(𝑧−𝐴) generates an ultradistribution sine of (𝑀_𝑝)-class. Then 𝐴 must be bounded.(ii)Assume (146) and ‖𝑅(𝜆∶𝐴)‖≤𝐶(1+|𝜆|)^𝑘,𝜆∈Λ_{𝑙,𝛼,𝛽}. (151) Let 𝜀>0 and 𝑧∈ℂ be such that for each (𝑎,𝑏^∗)∈𝐵_𝜀(0,0) the operator 𝐴+𝑎𝑏^∗(𝑧−𝐴) generates an ultradistribution semigroup of (𝑀_𝑝)-class. Then 𝐴 generates an analytic 𝐶₀-semigroup.

Proof. We will only prove the first part of the theorem. Put Ω_𝑛∶={𝜆∈ℂ∶ℜ𝜆≥𝑛𝑀(𝑛|ℑ𝜆|)+𝑛}. Then Ω_𝑛⊆Λ_{𝑙,𝛼,𝛽} for all 𝑛≥max(𝑙,𝛼,|𝛽|). By the generalized resolvent equation, it follows that for each 𝑥∈𝑌_𝑛≡𝐷(𝐴^{⌈𝑘/2⌉+2}), the set {||𝑅(𝜆∶𝐴)𝑥||∶𝜆∈Ω_𝑛} is bounded. The prescribed assumption combined with Lemma 29(ii) implies that for each (𝑎,𝑏^∗)∈𝐵_𝜀(0,0) there exist 𝑛∈ℕ and a function 𝑔_𝑛∶Ω_𝑛→(0,∞) such that Ω²_𝑛∶={𝜆²∶𝜆∈Ω_𝑛}⊆𝜌(𝐴+𝑎𝑏^∗(𝑧−𝐴)) and ||𝑅(𝜆²∶𝐴+𝑎𝑏^∗(𝑧−𝐴))||≤𝑔_𝑛(𝜆),𝜆∈Ω_𝑛. By Lemma 28, we obtain 𝑚∈ℕ such that sup_𝜆∈Ω²_𝑚||𝜆𝑅(𝜆∶𝐴)||<∞. Let 𝜉+𝑖𝜂=𝜆∈𝜕(Ω²_𝑚). 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 lim_𝑡→∞(𝑀(𝑡)/𝑡)=0, we easily infer that there exist 𝑡₀>0 and 𝐿≥1 such that, for any |𝑡|≥𝑡₀∶𝑅(𝜂𝑅(≤𝜂𝜆−𝜇𝜆∶𝐴)≤2||𝜆∶𝐴)||2𝐿|𝜆|≤4𝐿|𝑡|(𝑚𝑀(𝑚|𝑡|)+𝑚)(𝑚𝑀(𝑚|𝑡|)+𝑚)²−𝑡²²+4𝑡²(𝑚𝑀(𝑚|𝑡|)+𝑚)^21/2≤12, (152) 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∈Λ_{𝑙,𝛼,𝛽}}. The set {𝜆∈ℂ∶ℜ𝜆≥−𝜔₀}⧵{𝜆²∶𝜆∈Λ_{𝑙,𝛼,𝛽}} is compact, which completes the proof by [25, Lemma 2.3].

Remark 31. (i) It is worth noting that Theorem 30(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 𝜔_𝐴>0 such that sup_{𝜆>𝜔𝐴}||𝜆^(2−𝛼)/2𝑅(𝜆∶𝐴)||<∞. Let 𝜔>𝜔_𝐴. Then, for every 𝛾∈ℝ, one can define the fractional power 𝐴_𝛾∶=(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(𝐴_𝛾) and 𝐴_𝛾∈𝐿([𝐷(𝐴)],𝐸), which implies that one can define the rank-1 perturbation 𝐵_𝛾∶=𝐴+𝑎𝑏^∗𝐴_𝛾 of 𝐴; notice that the case 𝛾=1 has been already considered in Theorem 30. Obviously, 𝐴_𝛾𝑅(𝜆∶𝐴)𝑥=𝐴_𝛾−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 sup_{ℜ𝜆≥𝑛}||𝐴_𝛾𝑅(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 sup_{ℜ𝜆≥𝑛}||𝜆^2𝛾𝑅(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 Theorem 30(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 Λ_{𝑙,𝛼,𝛽}, respectively, on Λ_{𝑙,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) Given 𝜀∈(0,1) and 𝐶_𝜀>0, set Ω_𝜀∶=𝜆∈ℂ∶ℜ𝜆≥𝜀|𝜆|+𝐶_𝜀. (153) 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 Λ_{𝑙,𝛼,𝛽} replaced by Ω_𝜀. 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 class 𝐶^𝐿 if and only if the following holds(i)the mapping 𝑡↦𝑅(𝑡),𝑡∈(0,𝜏) is infinitely differentiable (in the uniform operator topology), and(ii)for every compact set 𝐾⊆(0,𝜏) there exists ℎ_𝐾>0 such that sup_{𝑡∈𝐾,𝑝∈ℕ0}ℎ^𝑝_𝐾𝑑^𝑝/𝑑𝑡^𝑝𝑅(𝑡)𝑀_𝑝<∞; (154)(𝑅(𝑡))_{𝑡∈[0,𝜏)} is said to be 𝜌-hypoanalytic, 1≤𝜌<∞, if (𝑅(𝑡))_{𝑡∈[0,𝜏)} is of class 𝐶^𝐿 with 𝑀_𝑝=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 𝐸_{𝛾,𝑏,𝑐}∶=𝜆∈ℂ∶ℜ𝜆≥𝑐−𝑏|ℑ𝜆|^𝛾(𝐴),𝑅(⊆𝜌||𝜆∶𝐴)||≤𝑐,𝜆∈𝐸_{𝛾,𝑏,𝑐}. (155) 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 (𝑔_𝛼,𝑔_𝛽)-regularized resolvent family.

Theorem 32. Suppose 0<𝛼<2,(𝛼−1)/𝛼<𝛾<1,𝑧∈ℂ,𝛽≥0 and a densely defined operator 𝐴 generates an exponentially bounded (𝑔_𝛼,𝑔_𝛽)-regularized resolvent family (𝑅(𝑡))_𝑡≥0.(i)Assume that 𝑏=0 and for each (𝑎,𝑏)∈𝐵_𝜀(0,0) there exists a kernel 𝑘_𝑎,𝑏^∗(𝑡) satisfying (P1)-(P2) so that the operator 𝐴+𝑎𝑏^∗(𝜔−𝐴)^𝛾 generates an exponentially bounded (𝑔_𝛼,𝑘_𝑎,𝑏^∗)-regularized resolvent family. Then (𝑅(𝑡))_𝑡≥0 is (1/(𝛼𝛾+1−𝛼))-hypoanalytic.(ii)Assume that 𝑏>0 and for each (𝑎,𝑏)∈𝐵_𝜀(0,0) there exists a kernel 𝑘_𝑎,𝑏^∗(𝑡) satisfying (P1)-(P2) so that the operator 𝐴+𝑎𝑏^∗(𝑧−𝐴) generates an exponentially bounded (𝑔_𝛼,𝑘_𝑎,𝑏^∗)-regularized resolvent family. Then 𝐴 generates an exponentially bounded, analytic (𝑔_𝛼,1)-regularized resolvent family.

Proof. Given 𝜔≥0, set Φ_𝛼,𝜔∶={𝜆^𝛼∶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 Ψ_𝑛,𝛼∶=Φ_𝛼,𝑛∩{𝑧∈ℂ∶|𝑧−𝜆|≤𝑚|𝜆|^𝛾forsome𝜆∈𝜕(Φ_𝛼,𝑛)}⊆𝜌(𝐴) and that ||𝑅(𝜆∶𝐴)||=𝑂(|𝜆|^−𝛾),𝜆∈Ψ_𝑛,𝛼. Let 𝜀∈(0,1) and let 𝐾_𝜀>0 be such that 𝛼𝐾_𝜀1+𝐾_𝜀^𝛼−1≤𝑚. (156) 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∶𝑛−𝐾_𝜀𝜂^1/𝜌+𝑖𝜂^𝛼−𝑛+𝑖𝜂^𝛼≤𝛼𝐾_𝜀𝜂^1/𝜌sup_{𝜈∈𝑛+𝑖𝜂,𝑛+𝑖𝜂−𝐾𝜀𝜂1/𝜌}|𝜈|^𝛼−1≤𝛼𝐾_𝜀𝜂^1/𝜌𝑛+𝑖𝜂+𝐾_𝜀𝜂^{1/𝜌𝛼−1}≤𝛼𝐾_𝜀1+𝐾_𝜀^𝛼−1𝜂^1/𝜌𝑛+𝑖𝜂^𝛼−1≤𝛼𝐾_𝜀1+𝐾_𝜀^𝛼−1𝑛+𝑖𝜂^{𝛼−1+(1/𝜌)}≤𝑚𝑛+𝑖𝜂^𝛼𝛾, (157) which implies that {𝜆∈ℂ∶ℜ𝜆≥−𝐾_𝜀|ℑ𝜆|^1/𝜌+𝑛}⊆Ψ_𝑛,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 Υ_𝑛,𝛼∶=Φ_𝛼,𝑛∩{𝑧∈ℂ∶|𝑧−𝜆|≤𝑚|𝜆|forsome𝜆∈𝜕(Φ_𝛼,𝑛)}⊆𝜌(𝐴) and that ||𝑅(𝜆∶𝐴)||≤𝑐|𝜆|⁻¹,𝜆∈Υ_𝑛,𝛼. Then it readily follows from [32, Theorem 2.17] that 𝐴 generates an exponentially bounded, analytic (𝑔_𝛼,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 𝐴_𝑎,𝑏^∗ by 𝐷(𝐴_𝑎,𝑏^∗)∶={𝑥∈𝐸∶𝑥+⟨𝑏^∗,𝑥⟩𝑎∈𝐷(𝐴)} and 𝐴_𝑎,𝑏^∗𝑥∶=𝐴(𝑥+⟨𝑏^∗,𝑥⟩𝑎),𝑥∈𝐷(𝐴_𝑎,𝑏^∗).

We need the following auxiliary lemma (cf. the proofs and formulations of [26, Lemmas 2.1 and 2.2]).

Lemma 33. Let 𝜔≥0,𝛼∈(0,2),𝑎∈𝐸,𝑏^∗∈𝐸^∗ and 𝑧∈ℂ_𝜔, where ℂ_𝜔∶={𝑧∈ℂ∶ℜ𝑧>𝜔}.(i)Then 𝑧^𝛼 is an eigenvalue of both, 𝐴_𝑎,𝑏^∗ and 𝐴+𝑎𝑏^∗𝐴, with 𝐴_𝑎,𝑏^∗(𝐴𝑅(𝑧^𝛼∶𝐴)𝑎)=𝑧^𝛼𝐴𝑅(𝑧^𝛼∶𝐴)𝑎 and (𝐴+𝑎𝑏^∗𝐴)(𝑅(𝑧^𝛼∶𝐴)𝑎)=𝑧^𝛼𝑅(𝑧^𝛼∶𝐴)𝑎.(ii)Let ⟨𝑏^∗,𝐴𝑅(𝑧^𝛼∶𝐴)𝑎⟩=1 and ⟨𝑏^∗,𝐴𝑅(𝑧^𝛼∶𝐴)𝑎⟩≠0. Then for each 𝜀>0 there exists 𝛿>0 such that for all (𝑎₁,𝑏^∗₁)∈𝐵_𝛿(𝑎,𝑏^∗) there exists some 𝑧₁∈𝐵(𝑧,𝜀) such that ⟨𝑏^∗₁,𝐴𝑅(𝑧^𝛼₁∶𝐴)𝑎₁⟩=1 and ⟨𝑏^∗₁,𝐴𝑅(𝑧^𝛼₁∶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. Suppose 𝛼∈(0,2), 𝑛∈ℕ, 𝜔≥0, 𝜀>0 and 𝐴 is the generator of an exponentially bounded, nonanalytic (𝑔_𝛼,1)-regularized resolvent family (𝑆_𝛼(𝑡))_𝑡≥0 satisfying ||𝑆_𝛼(𝑡)||≤𝑀𝑒^𝜔𝑡,𝑡≥0 for some 𝑀≥1. Let 𝑟>𝜔,𝑘=⌈1/𝛼⌉,𝑎∈𝐷(𝐴^𝑘),𝑏^∗∈𝐷((𝐴^∗)^𝑘) and 𝑧_𝑗∈ℂ_𝜔 be such that ⟨𝑏^∗,𝐴𝑅(𝑧^𝛼_𝑗∶𝐴)𝑎⟩=1 and ⟨𝑏^∗,𝐴𝑅(𝑧^𝛼_𝑗∶𝐴)𝑎⟩≠0(1≤𝑗≤𝑛). Then there exist (𝑎₁,𝑏^∗₁)∈𝐵_𝜀(𝑎,𝑏^∗)∩(𝐷(𝐴^𝑘)×𝐷((𝐴^∗)^𝑘)) and ₁𝑧,…,_𝑛+1𝑧∈ℂ_𝜔 such that ℜ_𝑛+1𝑧=𝑟,|_𝑗𝑧−𝑧_𝑗|<𝜀(1≤𝑗≤𝑛), ⟨𝑏^∗₁,𝐴𝑅(_𝑛+1𝑧^𝛼∶𝐴)𝑎₁⟩=1 and ⟨𝑏^∗₁,𝐴𝑅(_𝑛+1𝑧^𝛼∶𝐴)𝑎₁⟩≠0.

Proof. We will only outline the main details of the proof. First of all, notice that ||𝑅(𝑧^𝛼∶𝐴^∗)||=||𝑅(𝑧^𝛼∶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,…,𝛿_𝑛>0 be given by Lemma 33(i) and let 𝛿∶=min(1,𝜀,𝛿₁,…,𝛿_𝑛). 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(𝐴^𝑘)×𝐷((𝐴^∗)^𝑘), the following supremum sup_ℜ𝑧=𝑟max1,𝐴^∗𝑅𝑧^𝛼∶𝐴^∗𝑏^∗,𝐴𝑅𝑧^𝛼𝑎,𝑏∶𝐴_∗,𝐴𝑅𝑧^𝛼∶𝐴𝑎∶=𝐾 (158) is finite. The nonanalyticity of (𝑆_𝛼(𝑡))_𝑡≥0 yields that sup_ℜ𝑧=𝑟||𝐴𝑅(𝑧^𝛼∶𝐴)||=∞. By the denseness of 𝐷(𝐴^𝑘) in 𝐸, we get the existence of an element 𝑢∈𝐷(𝐴^𝑘) and a complex number _𝑛+1𝑧∶=𝑧 such that ℜ𝑧=𝑟, ||𝐴𝑅(𝑧^𝛼∶𝐴)𝑢||>(10𝐾/𝛿²), 𝐴𝑅(𝑧^𝛼∶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(𝑧^𝛼∶𝐴)𝑢⟩|>(10𝐾/𝛿²), ||𝑣^∗||<1 and ⟨𝑏^∗+(𝛿/2)𝑣^∗,𝐴𝑅(𝑧^𝛼∶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(𝑧^𝛼∶𝐴) and 𝐴𝑅(𝑧^𝛼∶𝐴)² there, we obtain that there exists (𝑎₁,𝑏^∗₁)∈𝐵_𝜀(𝑎,𝑏^∗)∩(𝐷(𝐴^𝑘)×𝐷(𝐴^∗)^𝑘) with required properties (cf. [26, page 474,𝑙.1−4]).

If 𝛼∈(0,2) and 𝐴 is the generator of an exponentially bounded (𝑔_𝛼,1)-regularized resolvent family (𝑆_𝛼(𝑡))_𝑡≥0 satisfying the properties stated above, then one can simply prove that for each 𝑟>𝜔 there exist 𝑧∈ℂ_𝜔 with ℜ𝑧=𝑟 and (𝑎,𝑏^∗)∈𝐷(𝐴^𝑘)×𝐷((𝐴^∗)^𝑘) such that ⟨𝑏^∗,𝐴𝑅(𝑧^𝛼∶𝐴)𝑎⟩=1 and ⟨𝑏^∗,𝐴𝑅(𝑧^𝛼∶𝐴)²𝑎⟩≠0. Using induction, Lemma 34 and the proof of [26, Theorem 1.3], we obtain the following theorem.

Theorem 35. Suppose 𝛼∈(0,2),𝑛∈ℕ,𝜔≥0,𝜀>0 and 𝐴 is the generator of an exponentially bounded, nonanalytic (𝑔_𝛼,1)-regularized resolvent family (𝑆_𝛼(𝑡))_𝑡≥0 satisfying ||𝑆_𝛼(𝑡)||≤𝑀𝑒^𝜔𝑡, 𝑡≥0 for some 𝑀≥1. Let 𝐼_𝑗⊆(𝜔^𝛼,∞) be an open interval (1≤𝑗<∞). Then there exist 𝑎∈𝐸 and 𝑏^∗∈𝐸^∗ such that the operators 𝐴_𝑎,𝑏^∗ and 𝐴+𝑎𝑏^∗𝐴 have a sequence of eigenvalues 𝑧_𝑗 with ℜ𝑧_𝑗∈𝐼_𝑗 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].

Acknowledgment

The author is partially supported by Grant 144016 of Ministry of Science and Technological Development, Republic of Serbia.