#### Abstract

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

#### 1. Introduction and Preliminaries

In this review, we will report how a large number of known results concerning -regularized resolvents [1–6], -regularized resolvents [7], and (local) convoluted -semigroups and cosine functions [8, 9] can be formulated in the case of general -regularized -resolvent families.

The paper is organized as follows. In Theorem 2.2, Remark 2.3, and Theorems 2.5, 2.6, and 2.7, we analyze the properties of subgenerators of -regularized -resolvent families and slightly improve results from [1]. With a view to further study the problem describing heat conduction in materials with memory and the Rayleigh problem of viscoelasticity in type spaces, we prove in Theorem 2.8 several different forms of subordination principles [10]. The main objective in Theorems 2.9–2.12, 2.26, 2.28, and 2.32 is to continue the researches raised in [3] and [5, 6]. Our main contributions are Theorems 2.16–2.17, 2.20–2.25, 2.27, and 2.30 clarifying the basic regularity properties of -regularized -resolvent families and a fairly general form of the abstract Weierstrass formula.

It is noteworthy that the complete spectral characterization of subgenerators of -regularized -resolvent families exists only in the exponential case and that it is not clear, with exception of various types of local convoluted -semigroups and cosine functions [9, 11], in what way one can prove a satisfactory Hille-Yosida theorem for local -regularized -resolvent families.

Throughout this paper denotes a nontrivial complex Banach space, denotes the space of boundedlinear operators from into denotes the dual space of and denotes a closed linear operator acting on The range and the resolvent set of aredenoted by and respectively; denotes the Banach space equipped with the graph norm. From now on, we assume that is an injective operator which satisfies and employ the convolution like mapping which is given by Recall, the -resolvent set of denoted by is defined to be the set of all complex numbers satisfying that the operator is injective and that Let us recall that a linear subspace is called a core for if is dense in with respect to the graph norm. Henceforth we identify a closed linear operator with its graph given two closed linear operators and on the inclusion means If is a closed subspace of then denotes the part of in that is,

We mainly use the following conditions.

(H1):*A*is densely defined.(H2):(H3): and (H4):(H5):(H1) (H2) (H3).(P1): is Laplace transformable, that is, it is locally integrable on and there exists so that exists for all with Put absinf

Let us remind that a function is called a kernel, if for every the supposition implies due to the famous Titchmarsh's theorem [12], the condition supp? implies that is a kernel. Set

#### 2. -Regularized -Resolvent Families

We start with the following definition.

*Definition 2.1. *Let and let A strongly continuousoperator family is called a (local, if ) -regularized -resolventfamily having as a subgeneratorif and only ifthe following holds: (i) and (ii)(iii)In the case is said to be exponentially bounded if, additionally, there exist and such that is said to be nondegenerate if the condition implies

From now on, we consider only nondegenerate -regularized -resolvent families. Notice that is nondegenerate provided that or that (H5) holds for a subgenerator of

In the case where and denotes the Gamma function, it is also said that is an -times integrated -resolvent family; in such a way, we unify the notion of (local) -times integrated -semigroups () and cosine functions () [1, 13, 14]. Furthermore, in the case where and we obtain the unification concept for (local) -convoluted -semigroups and cosine functions [15]. In the case is said to be a (local) -regularized resolvent family with a subgenerator (cf. also [16] for the definition which does not include the condition (ii) of Definition 2.1).

Designate by the set which consists of all subgenerators of

Then the following holds.

(i) implies (ii)If and then
(iii)Assume, additionally, that is a kernel. Then one can define the integral generator of by setting
The integral generator of is a closed linear operator which satisfies and extends an arbitrary subgenerator of Furthermore, if

Recall that in the case of convoluted -semigroups and cosine functions, the set becomes a complete lattice under suitable algebraic operations and that induced partial ordering coincides with the usual set inclusion. In general, needs not to be finite [9].

Henceforth we assume that the scalar-valued kernels are continuous on and that in

Assume temporarily and put

Following the proof of [1, Lemma ?2.2], we have where the last two equalities follow on account of and Hence,

The closedness of implies that (2.3) holds for every and

Theorem 2.2 (see [1]). *(i) Let be a subgenerator of an -regularized -resolvent family and let (H5) hold. Then (2.3) holds for every and If then (2.3) holds for every and **(ii) Let be a subgenerator of an -regularized -resolvent family Then whenever (H4) holds.**(iii) Let and be two -regularized -resolvent families having as a subgenerator. Then and if (H4) holds.**(iv) Let be a subgenerator of an -regularized -resolvent family If is absolutely continuous and then is a subgenerator of an -regularized resolvent family on *

*Remark 2.3. *(i) Let be an -regularized -resolvent family with a subgenerator ?? and let Then

(ii) Let be an -regularized -resolvent family with a subgenerator Then, for every and is an -regularized -resolvent family with a subgenerator

(iii) Let be an -regularized -resolvent family with a subgenerator and let be a kernel. Then is a subgenerator of an -regularized -resolvent family

(iv) Let be an -regularized resolvent family having as a subgenerator. Then is an -regularized -resolvent family with a subgenerator

(v) Suppose is an -regularized -resolvent family with a subgenerator (H1) or (H3) holds, and is a kernel. Then the integral generator of satisfies Toward this end, let Then , and Since and in it follows that , and On the other hand, is a subgenerator of whenever is; this implies and proves the claim. If (H2) holds, then In what follows, we also assume that and that (H5) holds for and Proceeding as in the proof of [9, Proposition ?2.1.1.6], one gets what follows.

(v.1) and (v.2) and have the same eigenvalues.(v.3)The assumption implies (v.4)card if is a core for (v.5) and furthermore, the property (v.5) holds whenever and is a kernel.

We refer the reader to [1, page 283] for the definition of (weak) solutions of the problem

where , and to [1, page 285] for the notion of spaces and

Define a subset of (the use of symbol is clear from the context) by In the case when is densely defined, is a linear mapping from into

Lemma 2.4 (see [17]). *Let be a closed linear operator. Assume and for all Then and *

Define the mapping by Then is linear, bounded, and injective.

Keeping in mind Lemma 2.4 and the proofs of [1, Theorem ?2.7, Corollary ?2.9, Remark ?2.10, Corollary ?2.11, and Corollary ??2.13], we have the following.

Theorem 2.5. *(i) Suppose is a subgenerator of a (local) -regularized -resolvent family and (H5) holds. Then (2.4) has a unique solution if and only if **(ii) (cf. also [18]) Assume is a subgenerator of a (local) -times integrated -resolvent family and (H5) holds. Then (2.4) has a unique solution if and only if **(iii) Let the assumptions of the item (i) of this theorem hold, and let Then if and only if **(iv) Let (H5) hold. Assume that is a subgenerator of an -times integrated -regularized resolvent, and respectively is a subgenerator of an -regularized resolvent family. Assume, further, that and respectively Then (2.4) has a unique solution.**(v) Assume that (H5) holds, is a subgenerator of an -regularized -resolvent family, is absolutely continuous, and If then there exists a unique solution of (2.4).*

The proof of following theorem follows from a standard application of Laplace transform techniques.

Theorem 2.6. *Let and satisfy (P1), and let be a strongly continuous operator family satisfying for some and Put *(i)*Suppose is a subgenerator of the exponentially bounded -regularized -resolvent family , and (H5) holds. Then, for every with and the operator is injective,
?
and *(ii)*Assume that (2.5)-(2.6) hold. Then is a subgenerator of the exponentially bounded -regularized -resolvent family *

The preceding theorem enables one to establish the real and complex characterization of subgenerators of (locally Lipschitz continuous) exponentially bounded -regularized -resolvent families [1, 9, 12]:

Theorem 2.7. *(i) Let and satisfy (P1), and let Assume that, for every with and the operator is injective and that If there exists an analytic function with:*

(i.1)*, *(i.2)* for some and then, for every is a subgenerator of a norm continuous, exponentially bounded -regularized -resolvent family.**(ii) Suppose and satisfy (P1) and (H2) or (H3) holds, and is a subgenerator of an exponentially bounded -regularized -resolvent family which satisfies the next condition:**
Then there exists such that
**(ii) Suppose is a subgenerator of an -regularized -resolvent family satisfying and (H5) holds. Then (2.18)-(2.19) hold. *

Theorem 2.10. *(i) Suppose is a subgenerator of an -regularized -resolvent family satisfying and Then **(ii) Suppose is a subgenerator of an -regularized -resolvent family satisfying and (H5) holds. If and then and **(iii) Suppose is reflexive, is a subgenerator of an -regularized -resolvent family satisfying and (H5) holds. If and then *

Theorem 2.11 (cf. also [22]). *Suppose and is a subgenerator of an -times integrated -semigroup respectively, an -times integrated -cosine function which satisfies respectively, Then, for every such that **
then is a subgenerator of an exponentially bounded, analytic -regularized -resolvent family of angle *

*Example 2.18 (cf. also [9, Theorem ?2.1.4.7]). *Let and let Let be densely defined. Then is a subgenerator of an exponentially bounded, analytic -regularized -resolvent family of angle if and only if for every there exist and such that
the mapping is analytic (continuous).

Let be a sequence of positive real numbers such that and that

(M.1)(M.2)(M.3)

The Gevrey sequences and satisfy the above conditions, where Put by (M.1), is increasing, and implies The associated function of is defined by As is known, the function is increasing, absolutely continuous, and For consistency of terminology with [24], we also employ the sequence and set

We need the following family of kernels. Define, for every the next entire function of exponential type zero Then

and this implies that It is noteworthy that, for every and

This yields

Put now
Then, for every and is infinitely differentiable in

*Definition 2.19. *Let be a (local) -regularized -resolvent family having as a subgenerator, and let the mapping be infinitely differentiable (in the uniform operator topology). Then it is said that is of class resp. of class if and only if for every compact set there exists resp. for every compact set and for every :
is said to be *-hypoanalytic, * if is of class with

By the proof of the scalar-valued version of the Pringsheim theorem, it follows that the mapping is real analytic if and only if is -hypoanalytic with

The main objective in Theorems 2.20–2.24 is to enquire into the basic differential properties of -regularized -resolvent families.

Theorem 2.20 ([25]). *Suppose is a closed linear operator, and satisfy (P1), and there exists such that, for every we have that the operator is injective and that If, additionally, for every there exist and an open neighborhood of the region
**
and an analytic mapping such that and that then, for every is a subgenerator of a norm continuous, exponentially bounded -regularized -resolvent family satisfying that the mapping is infinitely differentiable.*

Theorem 2.21. *Suppose and satisfy (P1), (H5) hold and is a subgenerator of an -regularized -resolvent family satisfying for appropriate constants and If there exists such that, for every there exist and so that *(i)*there exist an open neighborhood of the region and the analytic mappings and such that and *(ii)*for every with the operator is injective and *(iii)*,*(iv)* and **then the mapping is infinitely differentiable for every fixed Furthermore, if is dense in then the mapping , is infinitely differentiable. *

*Proof. *Assume and put and The curves and are oriented so that increases along and Set, for a sufficiently large One can simply prove that Let Then (iv) implies
for all Since one gets that, for every and the sequence is convergent in and that the convergence is uniform on every compact subset of Put Then it is obvious that This implies that the mapping is -times differentiable and that On the other hand, it is clear that, for every , and
By (2.41), we get that, for every and
With (iv) and the residue theorem in view, it follows that, for every and
Put By Theorem 2.6(i), we get that
This implies that the function is bounded on some right half plane. Taking into account (2.41), we have that, for every and
By (2.43)-(2.45), The arbitrariness of implies that the mapping is infinitely differentiable, finishing the proof.

Using the argumentation given in [25], one can prove the following theorems.

Theorem 2.22. *Suppose and satisfy (P1), is a subgenerator of a (local) -regularized -resolvent family , and Denote, for every and a corresponding **
Assume that, for every there exist an open neighborhood of the region and analytic mappings and such that *(i)(ii)*for every the operator is injective and *(iii)(iv)* and **Then is of class *

Theorem 2.23. *Suppose and satisfy (P1), is a subgenerator of a (local) -regularized -resolvent family , and Denote, for every and a corresponding **
Assume that, for every there exist an open neighborhood of the region and analytic mappings and such that the conditions (i)–(iv) of Theorem 2.22 hold with resp. replaced by respectively, Then is of class *

Theorem 2.24. *Suppose and is a (local) -regularized -resolvent family with a subgenerator Set
**Then is an -regularized -resolvent family with a subgenerator Furthermore, if the mapping is -times differentiable, then the mapping is likewise -times differentiable. If this isthe case, then we have, for every and **
and we have the following. *(i)*If is of class resp. of class then is likewise of class resptivley, of class *(ii)*If is -hypoanalytic, then is likewise -hypoanalytic.*

Before going further, notice that we can slightly reformulate Theorem 2.16 and Theorems 2.21–2.23 in the case when the functions and possess the meromorphic extensions on the corresponding regions defined in formulation of mentioned theorems. Having in mind [9, Theorem ?2.1.1.11, Theorem ??2.1.1.14], we have the following interesting analogue of [25, Theorem ?2.8] which cannot be so easily interpreted in the case of general -regularized -resolvent families.

Theorem 2.25. *(i) Let be a subgenerator of a local -convoluted -cosine function () resp. is of class (), and let for an appropriate complex-valued function (Put since it makes no misunderstanding, we will also write and for and respectively, and denote by the restriction of this function to any subinterval of ) Let the mapping be infinitely differentiable (-times differentiable, ), respectively, and let be of class (). Then is a subgenerator of a local -convoluted -cosine function satisfying that the mapping is infinitely differentiable (-times differentiable), resp. is of class (). Furthermore, the suppositions and imply the following: if the mapping is -times differentiable, then the mapping is likewise -times differentiable.**(ii) Suppose ?? and is a subgenerator of a local -times integrated -cosine function Then is a subgenerator of a local -times integrated -cosine function and the following holds.*

(ii.1)*If the mapping is infinitely differentiable (-times differentiable, ), then the mapping is infinitely differentiable (-times differentiable; *