Abstract and Applied Analysis

Volume 2013 (2013), Article ID 307684, 26 pages

http://dx.doi.org/10.1155/2013/307684

Research Article

Perturbation Theory for Abstract Volterra Equations

Marko Kostić

Faculty of Technical Sciences, University of Novi Sad, Trg Dositeja Obradovića 6, 21125 Novi Sad, Serbia

Received 6 August 2012; Accepted 14 November 2012

Academic Editor: Irena Lasiecka

Copyright © 2013 Marko Kostić. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 -regularized -resolvent family on a sequentially complete locally convex space 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 -regularized -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 -regularized -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 -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 -regularized -resolvent family with a subgenerator (, , ), a function satisfying certain properties and an -bounded perturbation such that and . In order to prove the existence of perturbed -regularized -resolvent family with a subgenerator , we employ the method that involves only direct computations and differs from those established in [2–12] in that we do not consider 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 -regularized -resolvent family inherits analytical properties from . In case and 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 with maximal distributional domain generates an exponentially bounded -times integrated cosine function on (, ) for any . Notice also that Keyantuo and Warma proved in [15] a similar result for the Laplacian on , with Dirichlet or Neumann boundary conditions. In Corollary 11, we focus our attention to the case , , 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 -class (-class) is stable under bounded commuting perturbations (); 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 -regularized -resolvent family is exponentially equicontinuous. It seems to be really difficult to prove an analogue of Theorem 5 in the context of local -regularized -resolvent families (cf. [3, 7, 8, 13] and [1, Section 2.5, Theorem ] for further information in this direction), which implies, however, that it is not clear whether the class of ultradistribution sines of -class (-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 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 has certain geometrical properties. As an application, we consider -regularized resolvent families generated by higher order differential operators (). Perturbations of subgenerators of analytic -regularized -resolvent families are also analyzed in Theorem 24, which might be surprising in the case . 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 -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, 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 ; if is a Banach space, then denotes the norm of an element . If is a SCLCS, then we denote by the space of all continuous linear mappings from into . We assume that is a closed linear operator acting on and that (with the exception of assertions concerning rank-1 perturbations) is an injective operator with ; the convolution like mapping is given by , and the principal branch is always used to take the powers. Given and , denotes the th convolution power of , and denotes the Dirac -distribution. If and , then and . The domain, range, and resolvent set of are denoted by , and , respectively. If is not dense in , then is a closed subspace of and therefore a SCLCS itself; the fundamental system of seminorms which defines the topology of is . Recall that the -resolvent set of , in short , is defined by is injective and .

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 be an SCLCS, let , and let . A strongly continuous operator family is called a (local, if ) -regularized -resolvent family having as a subgenerator if and only if the following holds:

(a), , and ;(b);(c).

is said to be nondegenerate if the condition implies , and is said to be locally equicontinuous if, for every , the family is equicontinuous. In case is said to be exponentially equicontinuous if there exists such that the family is equicontinuous.

(ii) Let and let be an -regularized -resolvent family. Then it is said that is an analytic -regularized -resolvent family of angle , if there exists a function satisfying that, for every , the mapping is analytic as well as that

(a) and(b) for all and .

It is said that is an exponentially equicontinuous, analytic -regularized -resolvent family of angle , if for every , there exists such that the family is equicontinuous.

Since there is no risk for confusion, we will identify and .

(iii) An -regularized -resolvent family is said to be entire if, for every , the mapping can be analytically extended to the whole complex plane.

In the sequel of the paper, we will consider only nondegenerate -regularized -resolvent families. The set which consists of all subgenerators of need not be finite. In case , where , it is also said that is an -times integrated -regularized resolvent family; -times integrated -regularized resolvent family is also called an -regularized resolvent family. Instructive examples of integrated -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 , where and , then we obtain the unification concept for (local) -convoluted -semigroups and cosine functions [1]. We refer the reader to [23, 28, 32, 37, 38] for some applications of -regularized -resolvent families in the study of the following abstract time-fractional equation with

where and denotes the Caputo fractional derivative of order ([28]). Henceforth, we assume that and are scalar-valued continuous kernels.

The following conditions will be used frequently:

(P1): is Laplace transformable, that is, it is locally integrable on , and there exists such that exists for all with . Put inf and denote by the inverse Laplace transform.(P2): satisfies (P1) and , for some .

For the sake of convenience, we recall the following result from [32, 33].

Lemma 2. Let and satisfy and let be a strongly continuous operator family such that there exists satisfying that the family is equicontinuous. Put .

(i)Assume is a subgenerator of the global -regularized -resolvent family and

Then, for every with and , the operator is injective and . Furthermore,

(ii)Assume (3). Then is a subgenerator of the global -regularized -resolvent family satisfying (2).

Let be a subgenerator of a locally equicontinuous -regularized -resolvent family satisfying the equality (2) for all and . Given and , set . Then it is not difficult to prove that and . Using the proof of [35, Theorem 2.7] (cf. also [33, Theorem 2.5]), it follows that . Since is a kernel and is injective, we obtain , which remains true for perturbed resolvent families considered in the paper. Assuming additionally that is a global exponentially equicontinuous -regularized -resolvent family as well as that and satisfy , one can define the integral generator of by setting

In case that is a kernel, the definition of integral generator of coincides with the corresponding one introduced in [33]. Notice that is the maximal subgenerator of with respect to the set inclusion and that Lemma 2 implies .

2. Bounded Perturbation Theorems

Assume and . Set, for any -valued function satisfying (P1), , . Using induction and elementary operational properties of vector-valued Laplace transform, one can simply prove that there exist uniquely determined real numbers , independent of and , such that

Furthermore,     and the following nonlinear recursive formula holds:

The precise computation of coefficients is a nontrivial problem.

Lemma 3. There exists such that

Proof. Clearly, . Applying (6), one gets

The preceding inequality implies inductively that (7) holds provided .

Set and if . Clearly, , , and for all .

The following lemma will be helpful in the analysis of growth order of perturbed integrated -regularized resolvent families.

Lemma 4. Let . Then and

Proof. Plugging in (5), we obtain

Since , it follows inductively from (6) that , provided and . Combined with (10), the above implies (9) and .

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

Theorem 5. Suppose and satisfy , is a subgenerator of a -regularized -resolvent family satisfying (2) with the family is equicontinuous and the following conditions hold

 (i), there exists such that , , , and . There exist , , and such that

 as well as(ii) For every with and , there exists a function satisfying (P1) and

 provided and .(iii) For every , there exists a function satisfying (P1) and a constant so that

(iv)

(v)

(vi)

Then is a subgenerator of an exponentially equicontinuous -regularized -resolvent family , which is given by the following formula:

Furthermore,

and the family is equicontinuous.

Proof. By (iv)-(v), we obtain that the series in (17) converge uniformly on compact subsets of as well as that is strongly continuous and that the family is equicontinuous. By (i) and Lemma 2,    and, for every with and and , . By the uniqueness theorem for Laplace transform, one gets , . The closedness of   and (iv)-(v) taken together imply , . Hence, , . By Lemma 2,

Exploiting the closedness of and the product rule, we easily infer from (19) that, for every , and for every with and

Fix, for the time being, and with and . Then (11) implies . 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

Our goal is to prove that

By the product rule, we get

notice that the convergence of last series follows from the conditions (iii)-(iv). Taking into account (5), (ii), and (vi), one yields that

which implies that the series

is also convergent. Now we get from (20)-(21) and (23)-(24):

because the sum of coefficients of () in the last two series equals ; this follows from an elementary calculus involving only the product rule. Assume now and . By (22) and , , we obtain that

which implies . Thus, and

The proof of theorem completes an application of Lemma 2.

Remark 6. (i) By [33, Proposition ], we get that is a unique -regularized -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 and the conditions (iv)-(vi) of Theorem 5 hold with replaced by therein. Writing as and applying Theorem 5 successively times, we obtain that is a subgenerator of a global -regularized -resolvent family satisfying (18). Furthermore, the family is equicontinuous.

(iii) It is not clear whether there exist functions and such that the conditions (ii)–(vi) of Theorem 5 are fulfilled in the case .

Theorem 7. Consider the situation of Theorem 5 with being an exponentially equicontinuous, analytic -regularized -resolvent family of angle . Assume that, for every , there exists such that the set is equicontinuous. Assume, additionally, that there exists such that, for every , there exist and with the following properties.

(i)For every , the function can be analytically extended to the sector and the following holds:

(ii)For every with and , the function , can be analytically extended to the sector and the following holds:

Then is an exponentially equicontinuous, analytic -regularized -resolvent family of angle .

Proof. Let    and . Then Stirling’s formula implies that there exists such that

for all and . By [33, Theorem ] and the proof of implication (i) (ii) of [39, Theorem ], we obtain that the mapping , respectively,    can be analytically extended to the sector , respectively, , as well as that there exist and , independent of , such that

and that, for every with and ,

Using (32)-(33), [33, Theorem ] and the proof of implication (ii) (i) of [39, Theorem ], it follows that the functions and can be analytically extended to the sector and that the following estimates hold:

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 and that the mapping can be analytically extended to the sector by the formula (17). Thanks to the proof of Theorem 5, the series appearing in (17) converge uniformly on compact subsets of , which implies , and . Furthermore, the functions , and are analytic, and the set is bounded. An application of [33, Theorem ] gives that the mapping is continuous on any closed subsector of , 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 , let , and let the Mittag-Leffler function be defined by . Set . Then it is well known (cf. [28, 42–44]) that ,    and that, for every , there exist and such that

It is noteworthy that the assumptions of Theorems 5 and 7 hold provided and , where . In this case, ,

and, for every with and ,

In order to verify (iv)–(vi), notice that there exists a constant such that for all . Then we obtain from (35) and Lemmas 3–4 that

proving the conditions (iv)-(v) and

proving the condition (vi). Assume now, with the notation used in the formulation of Theorem 7 that , and . Then

proving the conditions (29)-(30).

Corollary 8. Suppose and is a subgenerator of a global -times integrated -regularized resolvent family satisfying (2) with and . Let the family be equicontinuous and let satisfy the condition (i) quoted in the formulation of Theorem 5. Then is a subgenerator of a global -times integrated -regularized resolvent family satisfying (18) with . Furthermore, the family is equicontinuous, and is an exponentially equicontinuous, analytic -times integrated -regularized resolvent family of angle provided that is.

Remark 9. It is worthwhile to mention (cf. [1, Theorem ]) that Corollary 8 remains true, with a different upper bound for the growth order of , in the case . Using [33, Lemma 3.3] and the proof of cited theorem, it follows that is entire provided that and that is entire.

Example 10. Corollary 8 is a proper extension of [45, Lemma 4.7] provided and (), which can be applied in the analysis of the problem

in , with Dirichlet boundary conditions; here we assume and (see e.g., [46, pages 144-145] and [15, Theorem 4.2]). It is clear that Corollary 8 can be applied to (-coercive) differential operators generating integrated cosine functions ([2, 14, 15, 47–50]) or exponentially equicontinuous -regularized resolvent families ([23, 24]); in what follows, we will apply Corollary 8 to abstract differential operators generating -regularized cosine functions. Let be one of the spaces (, BUC, let and let denote the inverse Fourier transform. Put and, for every , for all . Then the family of seminorms induces a Fréchet topology on . Let possess the same meaning as in [51] and let . Consider the operator with its maximal distributional domain. Set , if , and , if . Assume , and the following condition:

() and, in the case , there exist and such that .

Then, for every , there exists such that, for every generates an exponentially equicontinuous -regularized cosine function in satisfying and , with being the function defined on [52, page 40]; cf. [33, 51, 52] for full details. If and , then the previous result can be slightly refined by allowing that takes the value . Given , define the bounded linear operator on by . Then and . Applying Corollary 8, we get that generates an exponentially equicontinuous -regularized cosine function in .

Assume now and , . By the consideration given in [1, Remark ], it follows that, for every , there exist real numbers such that , and that the following holds:

This implies ,

By means of (44) and the proof of Lemma 3, we obtain the existence of a constant such that

In what follows, we assume that is minimal with respect to (47); notice that and that it is not clear whether Lemma 4 can be reconsidered in the newly arisen situation. Then

Since is increasing in , where , we obtain that

provided and . Combining this with (35), Lemmas 3 and 4 and (47), we get

Noticing that , we obtain from (50) that there exists such that

By (48)–(52), (v) holds for any . In almost the same way, one can prove that (iv) and (vi) hold for any . Assume now that is an exponentially equicontinuous, analytic -regularized -resolvent family of angle , , , , , and . Then

for an appropriate constant ,

and, for every with , there exists such that

proving the conditions (29)-(30).

Corollary 11. Let , and let be a subgenerator of a global -regularized -resolvent family satisfying (2) with . Let satisfy the condition (i) quoted in the formulation of Theorem 5. Then is a subgenerator of a global -regularized -resolvent family satisfying (18) with . Furthermore, for every , the family is equicontinuous, and is an exponentially equicontinuous, analytic -regularized -resolvent family of angle provided that is.

Example 12. Let ,

Then generates a tempered ultradistribution semigroup of -class, and cannot be the generator of a distribution semigroup since is not stationary dense (see e.g., [53, Example 1.6] and [41]). If and , set and . Then , and there exist and , independent of , such that

It is clear that and . Proceeding by induction, we obtain that, for every and with

On the other hand, [54, Proposition 4.5] implies that there exists such that . Combined with (59) and the logarithmic convexity, the last estimate yields

In view of (60) we get that, for every , there exists , independent of , such that

Consider now the complex polynomial , , , . Set, for every and consider the operator defined by

Clearly, is not stationary dense. Let and be such that and . Let denote the zeros of the polynomial and let . Then an old result of Walsh [55] says that . Furthermore, it is checked at once that there exists a sufficiently large such that is a simple zero of and that , provided and . Therefore, for every with and for every with , the following holds:

It is straightforward to verify that

Assume now . Then de L'Hospital's rule implies

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:

By (64) and (66) we finally get that, for every ,

Since the preceding estimate holds for any , it is quite complicated to inscribe here all of its consequences (cf. [1, 16], [32, (2.35)–(2.37)], [56, 57]); for example, generates a tempered ultradistribution sine of -class provided , and generates an exponentially bounded, -convoluted group provided . 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 of positive numbers satisfying and . In what follows, we will illustrate an application of Corollary 11. Suppose , , , and , . By [32, Theorem 2.17] and (67), is the integral generator of an exponentially bounded, analytic -regularized resolvent family of angle (cf. also [41, Proposition 3.12]). Let and , . Then and, by Corollary 11, is the integral generator of an exponentially bounded, analytic -regularized resolvent family of angle .

The following extension of [9, Theorem 3.1], [32, Theorem 2.12] has been recently established in [33]; cf. also [39, Theorem ], [4, Theorem 1.1].

Theorem 13. Suppose is a subgenerator of an -regularized -resolvent family such that and . Let be a linear operator such that and that, for every , there exist and satisfying . Let (P1) hold for and let . Suppose and

or

Then the operator is a subgenerator of an -regularized -resolvent family satisfying (18) with and replaced by and therein. Furthermore,

and (72) holds for any and provided (70).

In many cases, we do not have the existence of a function and a complex number such that . The following theorem is an attempt to fill this gap.

Theorem 14. Suppose and is a subgenerator of an -regularized -resolvent family such that and that (2) holds. Let and satisfy (P1) and let the following conditions hold:

(i), ,

(ii)There exist a function satisfying (P1) and a complex number such that

(iii) and .

Then, for every , there exists a unique solution of the integral equation

furthermore, is an -regularized -resolvent family with a subgenerator , there exist and such that (71) holds and that (18) holds with and replaced by and therein.

Proof. It is clear that , , . Define, for every and ,

By [58, Theorem 1.7, page 3] it follows that ,    . Using this fact and (i), we get that ,   . Keeping in mind the condition (ii), it is not difficult to prove that, for every ,

Using the conditions (i) and (iii), we obtain the existence of numbers and such that

and that (H1) holds, where

(H1): For every strongly continuous function such that ,    , the following inequality holds:

Now one can define inductively, for every , the sequence in by and ; observe that, for every ,    is strongly continuous and that the family is locally equicontinuous (with clear meaning). By (78), (H1), and the proof of [9, Theorem 3.1], it follows inductively that

and that, for every and , the sequence is Cauchy in and therefore convergent. Set . It is obvious that the mapping is continuous for every fixed as well as that (71) and (75) hold. Therefore, it suffices to show that

Towards this end, notice that (78) and (H1) together imply that is invertible for and . Now we obtain from (75)

which immediately implies with (77) the validity of (81) in case and . Assume now and . Then a straightforward computation involving the equality as well as (77) and (82) shows that the operator is injective and

The representation implies

and (81), finishing the proof of theorem.

Remark 15. Now we will explain how one can reformulate Theorem 13 in case in which is not necessarily bounded operator from into (cf. also [3, 7, 8] and the next section). Consider the situation of Theorem 13 with being complete. Assume (69) and, instead of condition

(♮) and, for every , there exist and such that .

Denote, with a little abuse of notation, and . Then (♮) implies that the mapping is continuous for every and . By [59, Lemma 22.19] and the completeness of , one can extend the operator to the whole space (). Proceeding inductively, one can define for each the sequence in such that , , , . The preceding inequality implies that, for every , the sequence is Cauchy in and therefore convergent. Put , , . As in the proof of Theorem 14, the mapping , is continuous for every fixed and (71)-(72) hold. Using the closedness of and the condition (♮), we get , , , and , , . In view of (69), ; by the denseness of in , the last estimate holds for all . Hence, the operator is invertible and . Suppose, for the time being, and . The closedness of the operator can be proved as follows. Let a net in satisfy , and . Then a simple computation shows that , which implies . Since is closed, we infer that and . Therefore, the closedness of follows from that of . Suppose now and . Similarly as in the proof of Theorem 14, we get , , the injectiveness of and , 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 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 is dense in . Using the same notation as in Remark 15, one has . Suppose now that, for every and , there exist and such that

Let and be fixed. Then, for every and ,

which implies by (72) that

One can simply prove that there exists such that, for ,

which implies with (72), the previous computation and the denseness of that there exists such that, for every and

The same estimate holds provided (70), while in the case of Remark 15 we obtain that, for every and ,

Assuming additionally

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 is a subgenerator of a -regularized -resolvent family satisfying and (2) with and . Assume exactly one of the following conditions:

(i), and (a)    (b), where

(a). (b) is complete, (69) and (♮).

(ii) and (73) holds.

Then there exist and such that is a subgenerator of a -regularized -resolvent family satisfying (71), and (18) with replaced by therein.

Remark 18. Let and let be an exponentially equicontinuous, analytic -regularized -resolvent family of angle . Suppose additionally that, for every , there exist and such that . 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 is also an exponentially equicontinuous, analytic -regularized -resolvent family of angle ; furthermore, for every , there exist and such that . If (i)(b) holds, then one has to assume additionally that there exist and such that, for every and , the following holds:

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 ]). Let and . Define a closed densely defined linear operator on by and . Then is the integral generator of a bounded -regularized resolvent family, is not the integral generator of an exponentially bounded -regularized resolvent family, and . Suppose

Then it follows from Corollary 17 that is the integral generator of an exponentially bounded -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 , Corollary ] to abstract Volterra equations.

Theorem 20. Suppose is a Banach space, and satisfy (P1)-(P2) and is the integral generator of an exponentially bounded -regularized resolvent family satisfying (2) with . Let and be such that and let satisfy . Suppose that, for every , there exists such that

(i)Let be a linear operator, let and let

for some and (for and some ). Then, for every is the integral generator of an exponentially bounded, -regularized resolvent family satisfying (18) with , and replaced by therein.(ii)Let be a densely defined linear operator and let

for some and for and some . Then there exists a closed extension of the operator such that, for every , is the integral generator of an exponentially bounded, -regularized resolvent family satisfying (18) with , and replaced by therein. Furthermore, if and are densely defined, then is the part of the operator in .

Proof. By Lemma 2, and

Given with , put . Then the prescribed assumptions combined with (97) imply

Consider now the function defined by if , and if . Then the function is continuous for and analytic for . Furthermore, and, by (94)–(98), one has that, for every , there exists such that for all with . By the Phragmén-Lindelöf type theorems (cf. for instance [39, Theorem ]), we get that for all with . This, in turn, implies that there exists such that if , and that if . Therefore, and there exists such that, for

The proof of (i) follows from [32, Theorem , Remark ]. Using [19, Lemma 3.2] and a similar argumentation, we obtain the validity of (ii).

Recall that a Banach space has Fourier type if and only if the Fourier transform extends to a bounded linear operator from to , where . Each Banach space has Fourier type 1, and has the same Fourier type as . A space of the form has Fourier type , and there exist examples of nonreflexive Banach spaces which do have nontrivial Fourier type.

Theorem 21. Let be a Banach space of Fourier type .

(i) Let the assumptions of Theorem 20(i) hold and let . Assume that at least one of the following conditions holds:

(a) and are densely defined, there exist and such that

(b) is densely defined and is reflexive.(c) and .

 Then is the integral generator of an exponentially bounded, -regularized resolvent family satisfying (18) with , and replaced by therein.(ii) Let the assumptions of Theorem 20(ii) hold and let . Then there exists a closed extension of the operator such that is the integral generator of an exponentially bounded, -regularized resolvent family satisfying (18) with , and replaced by therein. Furthermore, if and are densely defined, then is the part of the operator in .

Proof. Assume that (c) holds. According to (100), and . Define

By the first part of the proof of [19, Theorem 3.3], is the integral generator of an exponentially bounded, -regularized resolvent family satisfying (18) with , and replaced by 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 is densely defined and, by [33, Theorem ], is an exponentially bounded, -regularized resolvent family with the integral generator . Let be such that and let denote the canonical embedding of in its bidual . Since has Fourier type and , it follows that there exists such that, for every and ,

Set, for every and

Then is strongly continuous, exponentially bounded and

By Lemma 2, is an -regularized resolvent family with the integral generator . By [33, Theorem ], it follows that is an -regularized resolvent family with the integral generator . We continue the proof by assuming that (a) holds. Using (99)-(100), we easily infer that the improper integral in (101) converges absolutely for and that

By (104)-(105) and the uniqueness theorem for Laplace transform, we get

and . Now one can simply prove that is an exponentially bounded, -regularized resolvent family with the integral generator .

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 is strongly continuous at 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 in cannot be so simply dropped from the formulation of (a). The main problem is that we do not know whether the mapping , is measurable provided (cf. [19, -, 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 and . Then the assumptions of Theorems 20 and 21[(i)(b)-(c), (ii)] hold while the assumptions of Theorem 21(i)(a) hold provided .

In the following nontrivial example, we will transfer the assertion of [19, Proposition 8.1] to abstract time-fractional equations.

Example 23. Let and . Define a closed linear operator on by and . Put with maximal domain ; here is a potential and . Assume first that

Given , denote by solutions of the equation with . Then ,

provided ,

provided . Furthermore, , and

provided . The above implies that there exists a constant such that

Keeping in mind (107)-(111), we obtain that

provided . Denote by the infimum of all nonnegative real numbers such that the operator generates an exponentially bounded -regularized resolvent family. The precise computation of integration rate falls out from the framework of this paper (cf. also the representation formula [28, Example 3.7, ] and notice that it is not clear whether Theorem 13 or Remark 15 can be applied in case ). Clearly, (112) yields the imprecise estimate ; furthermore, provided ([24]), and provided [14, 50]. Set . By Theorem 21, generates an exponentially bounded -regularized resolvent family for any . By (112)-(113) and the proof of [19, Proposition 8.1], the above remains true provided and ; similarly, one can consider the operators and given by and .

Notice that Lizama and Prado have recently analyzed in [21] the qualitative properties of the abstract relaxation equation:

where is a Banach space and . By a (strong) solution of (114) we mean any function such that (114) holds for a.e. . 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 and satisfy (P1). Suppose , , there exist analytic functions and such that , and , . Let be a subgenerator of an analytic -regularized -resolvent family of angle and let (2) hold. Suppose that, for every , there exists such that

as well as is a linear operator satisfying and

Assume that at least one of the following conditions holds:

(i) is densely defined, the numbers and are sufficiently small, there exists such that and, for every , there exists such that and .(ii) is densely defined, the number is sufficiently small, there exists such that and, for every , there exists such that and .(iii) is densely defined, the number is sufficiently small, and, for every , there exists such that .(iv) and, for every , there exists such that .

Then is a subgenerator of an exponentially equicontinuous, analytic -regularized -resolvent family of angle , which satisfies and the following condition:

Furthermore, in cases (iii) and (iv), the above remains true with the operator replaced by .

Proof. First of all, notice that the closedness of the operator in cases (iii) or (iv) trivially follows and that it is not clear how one can prove that the operator 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 and . Clearly, , , , and . Invoking (115), [33, Theorem 3.6] and the proof of [39, Theorem ], we obtain that

and that there exists such that and

By (116) and (119), we infer that, for every and

which implies by the given assumption the existence of a number such that , provided that the numbers and are sufficiently small; if (iv) holds, then

Using the same argument as in the proof of Theorem 14, it follows that, for every as well as that the operators and are injective. Moreover, for any

Now we will prove that the operator is closed. Let be a net in satisfying , and , . Then , , that is, , , which simply implies and . Therefore, , and is closed, as required. Notice that, for every , the analyticity of mapping

follows from [33, Lemma 3.3] and the fact that an -valued mapping is analytic if and only if it is weakly analytic. By [33, Theorem 3.7], is a subgenerator of an exponentially equicontinuous, analytic -regularized -resolvent family of angle and (117) holds; assuming (iv), we get from (119)

In combination with (118) and (121), the above implies

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 such that, for every and for every with

By Lemma 2, we obtain that (18) holds with ,    and replaced, respectively, by and therein; clearly, the above assertion remains true with the operator replaced by , provided that (iii) or (iv) holds. Taking the Laplace transform, (126) simply implies that is, in fact, the integral generator of .

Example 26. Let be a solution of (114). Set , , , , and , . Then ,   and , , which implies that the notion of an -regularized -resolvent family is important in the study of (114). In [21], the authors mainly use the following conditions: , and is the generator of a bounded analytic -semigroup. Set and assume, more generally, that for every , there exists such that the family

is equicontinuous () and that the mapping

Notice that (127)-(128) hold provided that is a subgenerator of an exponentially equicontinuous -times integrated -semigroup ; furthermore, if

then, for every and , there exists such that

We refer the reader to [58, Chapter 1] for examples of differential operators generating exponentially equicontinuous, -times integrated -semigroups satisfying (129). Assume, further, that there exist and an analytic function such that , and , . Let and let . Then there exists a sufficiently large such that for all , which implies with (127)-(128) and [33, Proposition ] that the mapping is analytic () and that, for every , the family is equicontinuous (if (129) holds, then there exists such that ). Using [33, Theorem 3.7] and the arbitrariness of , we get that is a subgenerator of an exponentially equicontinuous, analytic -regularized -resolvent family of angle , where

and stands for the Dirac distribution (if (129) holds, then for every there exist and such that ). This is a significant improvement of [21, Theorem 3.1]. In what follows, we will provide the basic information on the -well-posedness of (114). Given and , set

where

Let be densely defined, let and let be such that for all . Then , and the proof of [10, Theorem 2.4] combined with the Cauchy integral formula (cf. also [33, Section 1, Theorem ]) indicates that the function

satisfies and that, for every , one has ; in the above formula, we assume that is the exponentially equicontinuous, analytic -regularized resolvent family of angle . It is obvious that the function is a unique function satisfying (114) in integrated form

and that for all . If , and for all , then we obtain similarly the unique solution of the problem

furthermore, for all . Since , the above-described method does not work in the case (cf. [35, Corollary 2.11] and [33, Theorem ]).

We are turning back to the case in which is not necessarily densely defined. Let and let denote the -regularized -resolvent family with a subgenerator . By the proofs of [21, Theorem 3.5, Corollary 3.6], it follows that, for every , there exists a unique solution of the problem

given by . 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 and is a subgenerator of an exponentially equicontinuous, -times integrated -semigroup satisfying (129). Let be a linear operator such that and let satisfy . By Remark 25 and the proof of Theorem 24, we have the following

(i)If is sufficiently small and satisfies , then is the integral generator of an exponentially equicontinuous, analytic -regularized -resolvent family of angle (cf. [64, Chapter ] and [65, Chapter 7] for corresponding examples).(ii)If is sufficiently small, , and , then , respectively, , is a subgenerator, respectively the integral generator, of an exponentially equicontinuous, analytic -regularized -resolvent family of angle .(iii)If , and , then , respectively , is a subgenerator, respectively the integral generator, of an exponentially equicontinuous, analytic -regularized -resolvent family of angle .

We continue this example by observing that Karczewska and Lizama [20] have recently analyzed the following stochastic fractional oscillation equation:

where is the generator of a bounded analytic -semigroup on a Hilbert space and denotes an -valued Wiener process defined on a stochastic basis . The theory of -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

where . Equation (139) models an oscillation process with fractional damping term and after differentiation becomes, in some sense,

Without any essential changes, one can consider the -well-posedness and perturbation properties of (139).

Example 27. (See [22, 66, 67]). Let , let be a bounded domain in with boundary of class , and let . Consider the operator given by

with domain . Here , and satisfy the following conditions:

(i) for all and ,(ii) for all , and(iii)there exists such that

Then there exists a sufficiently large such that the operator satisfies with some and

Notice that is not densely defined since . Let and . By (143) and [32, Theorem 2.17], we get that is the integral generator of an exponentially bounded, analytic -regularized resolvent family of angle . Assume now that is a linear operator satisfying and for some . Applying Theorem 24(iv), we obtain that the operator is the integral generator of an exponentially bounded, analytic -regularized resolvent family of angle . Suppose, for example, and . Let and let the operator be defined by

Then satisfies the conditions stated above since and . Finally, it could be interesting to construct an example in which there does not exist such that for all .

In the remaining part, which is mainly motivated by reading of the paper [25] by Arendt and Batty, we assume that 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 , and , we consider the rank-1 perturbation of given by

We also denote this operator by . Denote .

For the sake of convenience to the reader, we will repeat the assertion of [25, Theorem 1.3].

Lemma 28. Let be a closed linear operator on , let and let . Assume that and for all in a dense subset of and all . Let . Assume that for each there exists such that and . Then there exists such that

Henceforth, we assume that is a sequence of positive real numbers such that and that the following conditions are fulfilled:

Let . Then the Gevrey sequences , and satisfy the above conditions. The associated function of is defined by and . Recall [54], the function is increasing, and .

Following [1, 16], a closed linear operator is said to be the generator of an ultradistribution sine of -class if and only if the operator generates an ultradistribution semigroup of -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 be a closed densely defined operator on . Then generates an ultradistribution semigroup of -class if and only if there exist and such that

(ii) Let be a closed densely defined operator on . Then generates an ultradistribution sine of -class if and only if there exist and such that

Theorem 30. Let and . Let be a closed densely defined operator on .

(i)Assume (148) and

Let and be such that for each the operator generates an ultradistribution sine of -class. Then must be bounded.(ii)Assume (146) and

Let and be such that for each the operator generates an ultradistribution semigroup of -class. Then generates an analytic -semigroup.

Proof. We will only prove the first part of the theorem. Put . Then for all . By the generalized resolvent equation, it follows that for each , the set is bounded. The prescribed assumption combined with Lemma 29(ii) implies that for each there exist and a function such that and . By Lemma 28, we obtain such that . Let . Assume and . Then and for some . Since , we easily infer that there exist and such that, for any

which implies that exists and . Therefore, there exists such that is polynomially bounded on . The set 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, Theorem], and that Theorem 30(i) is an extension of [25, Theorem 2.2] provided in the formulation of this result. Consider now the situation of [25, Theorem 2.2] with being the generator of an exponentially bounded -times integrated cosine function (). Then there exists such that . Let . Then, for every , one can define the fractional power (cf. [1, Section 1.4]). Assuming and , we obtain from [1, Theorem ] that and , which implies that one can define the rank-1 perturbation of ; notice that the case has been already considered in Theorem 30. Obviously, for all and . By the proof of [25, Theorem 2.2], one gets that there exists such that and . Unfortunately, it is not clear whether the above conclusions together with [25, Lemma 2.4] (cf. also [70, Lemma 2.3]) imply that , unless . Notice also that the assumption must be imposed in the case .

(ii) In the formulation of Theorem 30(ii) and Theorem 30(i), respectively, we do not assume that the operator has polynomially bounded resolvent on the square of , respectively, on . Furthermore, we may assume that the operator has a slightly different spectral properties (cf. [25, Remark 2.5] and the formulation of Theorem 32 below).

(iii) Given and , set

The proof of Theorem 30(i) and Theorem 30(ii) respectively, does not work any longer if, for every , the estimate (150), respectively (151), holds with replaced by . 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) -regularized -resolvent family having as a subgenerator is of class if and only if the following holds

(i)the mapping is infinitely differentiable (in the uniform operator topology), and(ii)for every compact set there exists such that

is said to be -hypoanalytic, , if is of class with .

By [72, Theorem 5.5] and [32, Theorem 2.23], a -semigroup is -hypoanalytic for some if is in the Crandall-Pazy class of semigroups. Recall that is in the Crandall-Pazy class [72] if and only if there exist , , and such that

Keeping in mind (155), the subsequent theorem can be viewed as a generalization of [25, Theorem 4.3]. Observe that the operator is defined for a sufficiently large , provided that generates an exponentially bounded -regularized resolvent family.

Theorem 32. Suppose and a densely defined operator generates an exponentially bounded -regularized resolvent family .

(i)Assume that and for each there exists a kernel satisfying (P1)-(P2) so that the operator generates an exponentially bounded -regularized resolvent family. Then is -hypoanalytic.(ii)Assume that and for each there exists a kernel satisfying (P1)-(P2) so that the operator generates an exponentially bounded -regularized resolvent family. Then generates an exponentially bounded, analytic -regularized resolvent family.

Proof. Given , set . Making use of [25, Lemma 2.4], [32, Theorem 2.7], and Lemma 28, we get that there exist and such that and that . Let and let be such that

Put . Notice that since and . With the help of (156) and the Darboux inequality, we obtain that for each

which implies that . The proof of (i) is completed by an application of [32, Theorem 2.23]. Suppose now that the assumptions of (ii) hold. Then need not be sectorial, in general. We obtain similarly the existence of an integer and a number such that and that . Then it readily follows from [32, Theorem 2.17] that generates an exponentially bounded, analytic -regularized resolvent family of angle .

Now we will transfer the assertion of [26, Theorem 1.3] to abstract time-fractional equations. For and , define by and .

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

Lemma 33. Let and , where .

(i)Then is an eigenvalue of both, and , with and .(ii)Let and . Then for each there exists such that for all there exists some such that and .

The following fractional analogue of [26, Lemma 2.3] will be essentially utilized in the proof of Theorem 35 stated below.

Lemma 34. Suppose , , , and is the generator of an exponentially bounded, nonanalytic -regularized resolvent family satisfying for some . Let and be such that and . Then there exist and such that , and .

Proof. We will only outline the main details of the proof. First of all, notice that . By [1, Lemma ], we get that is stationary dense with , which implies that in the strong topology of . Let the numbers be given by Lemma 33(i) and let . By the generalized resolvent equation (see e.g., the proof of [25, Theorem 2.2]) and the fact that is an isometrically isomorphism, we obtain that for each , the following supremum

is finite. The nonanalyticity of yields that . By the denseness of in , we get the existence of an element and a complex number such that , , and . Now one can proceed as in the proof of [26, Lemma 2.3] so as to obtain such that , and . Since is dense in with respect to the strong topology of , we may assume that . Copying the final part of the proof of the aforementioned lemma, with and replaced by and there, we obtain that there exists with required properties (cf. [26, page ]).

If and is the generator of an exponentially bounded -regularized resolvent family satisfying the properties stated above, then one can simply prove that for each there exist with and such that and . Using induction, Lemma 34 and the proof of [26, Theorem 1.3], we obtain the following theorem.

Theorem 35. Suppose and is the generator of an exponentially bounded, nonanalytic -regularized resolvent family satisfying , for some . Let be an open interval (). Then there exist and such that the operators and have a sequence of eigenvalues with for all .

We close the paper with the observation that perturbation theorems for q-exponentially equicontinuous -regularized -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.

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