Research Article | Open Access

# Perturbation Theory for Abstract Volterra Equations

**Academic Editor:**Irena Lasiecka

#### 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