Abstract and Applied Analysis

Volume 2012 (2012), Article ID 131652, 41 pages

doi:10.1155/2012/131652

Research Article

On a Class of Abstract Time-Fractional Equations on Locally Convex Spaces

Marko Kostić,¹ Cheng-Gang Li,² and Miao Li²

¹Faculty of Technical Sciences, University of Novi Sad, Trg D. Obradovića 6, 21125 Novi Sad, Serbia

²Department of Mathematics, Sichuan University, Chengdu 610064, China

Received 18 June 2012; Revised 5 August 2012; Accepted 5 August 2012

Academic Editor: Dumitru Băleanu

Copyright © 2012 Marko Kostić et al. 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.

This paper is devoted to the study of abstract time-fractional equations of the following form: , , , , where , and are closed linear operators on a sequentially complete locally convex space , , is an -valued function, and denotes the Caputo fractional derivative of order (Bazhlekova (2001)). We introduce and systematically analyze various classes of -regularized ()-existence and uniqueness (propagation) families, continuing in such a way the researches raised in (de Laubenfels (1999, 1991), Kostić (Preprint), and Xiao and Liang (2003, 2002). The obtained results are illustrated with several examples.

1. Introduction and Preliminaries

A great number of abstract time-fractional equations appearing in engineering, mathematical physics, and chemistry can be modeled through the abstract Cauchy problem

For further information about the applications of fractional calculus, the interested reader may consult the monographs by Baleanu et al. [1], Klafter et al. (Eds.) [2], Kilbas et al. [3], Mainardi [4], Podlubny [5], and Samko et al. [6]; we also refer to the references [7–19].

The aim of this paper is to develop some operator theoretical methods for solving the abstract time-fractional equations of the form (1.1). We start by quoting some special cases. The study of qualitative properties of the abstract Basset-Boussinesq-Oseen equation:

describing the unsteady motion of a particle accelerating in a viscous fluid under the action of the gravity, has been initiated by Lizama and Prado in [17]. For further results concerning the -wellposedness of (1.2), [20, 21] are of importance. In [12], Karczewska and Lizama 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. [12, Theorems  3.1 and 3.2]) can be applied in the study of deterministic counterpart of (1.3) in integrated form:

where denotes the Gamma function and . Equation (1.4) generalizes the so-called Bagley-Torvik equation, which can be obtained by plugging in (1.4), and models an oscillation process with fractional damping term (cf. [21] for the analysis of -wellposedness and perturbation properties of (1.4)). After differentiation, (1.4) becomes, in some sense,

Notice also that the periodic solutions for the equation

where and are closed linear operators defined on a complex Banach space , , and denotes the Liouville-Grünwald fractional derivative of order , have been studied by Keyantuo and Lizama in [13]. Observe also that Diethelm analyzed in [22, Chapter 8] scalar-valued multiterm Caputo fractional differential equations. Consider, for illustration purposes, the following abstract time-fractional equation:

where , and is a certain complex constant. Applying the Laplace transform (see, e.g., [10, (1.23)]), we get:

Therefore,

By (24) and (26) in [19], it readily follows that:

where

is the generalized Mittag-Leffler function. Here () and . The formula (1.10) shows that it is quite complicated to apply Fourier multiplier theorems to the abstract time-fractional equations of the form (1.1); for some basic references in this direction, the reader may consult [16, 23]. Before going any further, we would also like to observe that Atanacković et al. considered in [8], among many other authors, the following fractional generalization of the telegraph equation:

where , and . In that paper, solutions to signalling and Cauchy problems in terms of a series and integral representation are given.

In the second section, we continue the analysis from our recent paper [15], where it has been assumed that for some complex constants (); here, and in the sequel of the second section, denotes the identity operator on . We introduce and clarify the basic structural properties of various types of -regularized -existence and uniqueness propagation families. This is probably the best concept for the investigation of integral solutions of the abstract time-fractional equation (1.1) with , . If there exists an index such that , then the vector-valued Laplace transform cannot be so easily applied (cf. Theorems 2.10–2.11), which implies, however, that there exist some limitations to the introduced classes of propagation families. The notion of a strong solution of (1.1) is introduced in Definition 2.1, and the notions of strong and mild solutions of inhomogeneous equations of the form (2.15) below are introduced in Definition 2.7. The generalized variation of parameters formula is proved in Theorem 2.8.

On the other hand, the notions of -existence families and -uniqueness families for the higher order abstract Cauchy problem () were introduced by Xiao and Liang in [24, Definition  2.1]. In the third section, we will introduce more general classes of (local) -regularized -existence families for (1.1), -regularized -uniqueness families for (1.1), and -regularized -resolvent families for (1.1). Our intention in this section is to transfer results of [24] to abstract time-fractional equations. In addition, various adjoint type theorems for -regularized -resolvent families are considered in Theorem 3.6.

Throughout this paper, we will always assume that is a Hausdorff sequentially complete locally convex space over the field of complex numbers, SCLCS for short, and that the abbreviation stands for the fundamental system of seminorms which defines the topology of ; in this place, we would like to mention in passing that the locally convex spaces are very important to describe a set of mixed states in quantum theory [2]. The completeness of , if needed, will be explicitly emphasized. By is denoted the space of all continuous linear mappings from into . Let be the family of bounded subsets of and let , , , . Then is a seminorm on and the system induces the Hausdorff locally convex topology on . Recall that is sequentially complete provided that is barreled. Henceforth is a closed linear operator acting on , is an injective operator, and the convolution like mapping is given by . The domain, resolvent set and range of are denoted by , and , respectively. Since it makes no misunderstanding, we will identify with its graph. Recall that the -resolvent set of , denoted by , is defined by

Suppose is a linear subspace of . Then the part of in , denoted by , is a linear operator defined by and .

Define (). Then the norm of a class is defined by (). The canonical mapping is continuous and the completion of under the norm is denoted by . Since no confusion seems likely, we will also denote the norms on and ( and ) by ; denotes the subspace of which consists of those bounded linear operators on such that, for every , there exists satisfying , . If and , then the operator , defined by , , belongs to . This operator is uniquely extensible to a bounded linear operator on , and the following holds: . The function , defined by , , is a continuous homomorphism of onto , and extends therefore, to a continuous linear homomorphism of onto . The reader may consult [25] for the basic facts about projective limits of Banach spaces (closed linear operators acting on Banach spaces) and their projective limits. Recall, a closed linear operator acting on is said to be compartmentalized (w.r.t. ) if, for every , is a function. Therefore, is a compartmentalized operator.

Given in advance, set and . The principal branch is always used to take the powers. Set , , , (, ) and the Dirac -distribution. If , then we define . We refer the reader to [26] and references cited there for the basic material concerning integration in sequentially complete locally convex spaces and vector-valued analytic functions.

Let , let , and let the Mittag-Leffler function be defined by , . In this place, we assume that if . Set, for short, , . The Wright function is defined by , , where denotes the inverse Laplace transform. For further information concerning Mittag-Leffler and Wright functions, we refer the reader to [10, Section  1.3].

The following definition has been recently introduced in [27].

Definition 1.1. Suppose , , , , and is a closed linear operator on .

(i)Then it is said that is a subgenerator of a (local, if ) -regularized -existence and uniqueness family if and only if the mapping , is continuous for every fixed and if the following conditions hold:

(a), , (b) is injective, (c)

(ii)Let be strongly continuous. Then it is said that is a subgenerator of a (local, if ) -regularized -existence family if and only if and (1.14) holds. (iii)Let be strongly continuous. Then it is said that is a subgenerator of a (local, if ) -regularized -uniqueness family if and only if , is injective and (1.15) holds.

It will be convenient to remind us of the following definitions from [14, 20, 26].

Definition 1.2. (i) 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 the case is said to be exponentially equicontinuous (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 ,   is said to be an exponentially equicontinuous, analytic -regularized -resolvent family, respectively, equicontinuous analytic -regularized -resolvent family of angle , if for every , there exists , respectively, , such that the set is equicontinuous. Since there is no risk for confusion, we will identify in the sequel and .

Definition 1.3. (i) Let and . Suppose that is a global -regularized -resolvent family having as a subgenerator. Then it is said that is a quasi-exponentially equicontinuous (-exponentially equicontinuous, for short) -regularized -resolvent family having as subgenerator if and only if, for every , there exist , and such that:

(ii) Let , and let be a subgenerator of an analytic -regularized -resolvent family of angle . Then it is said that is a -exponentially equicontinuous, analytic -regularized -resolvent family of angle , if for every and , there exist and such that

For a global -regularized -existence and uniqueness family having as subgenerator, it is said that is locally equicontinuous (exponentially equicontinuous, (-)exponentially equicontinuous, analytic, (-)exponentially analytic,…) if and only if both and are.

The reader may consult [26, Theorems  2.7 and 2.8] for the basic Hille-Yosida type theorems for exponentially equicontinuous -regularized -resolvent families. The characterizations of exponentially equicontinuous, analytic -regularized -resolvent families in terms of spectral properties of their subgenerators are given in [26, Theorems  3.6 and 3.7]. For further information concerning -exponentially equicontinuous -regularized -resolvent families, we refer the reader to [20, 25].

Henceforth, we assume that are scalar-valued kernels and that in . All considered operator families will be nondegenerate.

The following conditions will be used in the sequel:

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

2. The Main Structural Properties of -Regularized -Existence and Uniqueness Propagation Families

In this section, we will always assume that is a SCLCS, and are closed linear operators acting on , , and . Our intention is to clarify the most important results concerning the -wellposedness of (1.1). Set , , , and .

Definition 2.1. A function is called a (strong) solution of (1.1) if and only if for , and (1.1) holds. The abstract Cauchy problem (1.1) is said to be (strongly) -wellposed if:

(i)for every , there exists a unique solution of (1.1); (ii)for every and , there exist and such that, for every , the following holds:

In the case of abstract Cauchy problem (), the definition of -wellposedness introduced above is slightly different from the corresponding definition introduced by Xiao and Liang [28, Definition  5.2, page 116] in the Banach space setting (cf. also [28, Definition  1.2, page 46] for the case ). Recall that the notion of a strong -propagation family is important in the study of existence and uniqueness of strong solutions of the abstract Cauchy problem (); compare [28, Section  3.5, pages 115–130] for further information in this direction. Suppose now that , is a strong solution of (1.1), with and initial values . Convoluting both sides of (1.1) with , and making use of the equality [10, (1.21)], it readily follows that , satisfies the following:

In the sequel of this section, we will primarily consider various types of solutions of the integral equation (2.2).

Given in advance, set . Then it is clear that . Plugging , , , in (2.2), one gets:

where appears in the th place () starting from 0. Suppose now , and , . Denote , , . Convoluting formally both sides of (2.3) with , , one obtains that, for :

Motivated by the above analysis, we introduce the following definition.

Definition 2.2. Suppose , , , , , and are injective. A sequence of strongly continuous operator families in is called a (local, if ):

(i)-regularized -existence propagation family for (1.1) if and only if and the following holds:

for any .(ii) -regularized -uniqueness propagation family for (1.1) if and only if and

for any and . (iii)-regularized -resolvent propagation family for (1.1), in short -regularized -propagation family for (1.1), if is a -regularized -uniqueness propagation family for (1.1), and if for every , and , one has , and .

The above classes of propagation families can be defined by purely algebraic equations (cf. [11, 15, 27]). We will not go into further details about this topic here.

As indicated before, we will consider only nondegenerate -regularized -resolvent propagation families for (1.1). In case , where , it is also said that is a -times integrated -resolvent propagation family for (1.1); -times integrated -resolvent propagation family for (1.1) is simply called -resolvent propagation family for (1.1). For a -regularized -existence and uniqueness family , it is said that is locally equicontinuous (exponentially equi-continuous, (-)exponentially equicontinuous, analytic, (-)exponentially analytic,…) if and only if all single operator families are. The above terminological agreements and abbreviations can be simply understood for the classes of -regularized -existence propagation families and -regularized -uniqueness propagation families. The class of -regularized -existence and uniqueness propagation families for (1.1) can be also introduced (cf. Definitions 1.1 and 3.1 below).

In case that , where for , it is also said that the operator is a subgenerator of . Now we would like to notice the following: if is a subgenerator of a -regularized -resolvent propagation family for (1.1), then, in general, there do not exist , and such that is an -regularized -resolvent family with subgenerator ; the same observation holds for the classes of -regularized -existence propagation families and -regularized -uniqueness propagation families. Despite this fact, the structural results for -regularized -resolvent propagation families can be derived by using appropriate modifications of the proofs of corresponding results for -regularized -resolvent families. Furthermore, these results can be clarified for any single operator family of the tuple .

Let be a -regularized -resolvent propagation family with subgenerator . Then one can simply prove that the validity of condition (H5) implies the following functional equation:

for any . The set consisted of all subgenerators of , denoted by , need not to be finite. Notice that the supposition obviously implies . The integral generator of is defined as the set of all pairs such that, for every and , the following holds:

It is a linear operator on which extends any subgenerator and satisfies . We have the following.

(i), , provided , and . (ii)Let be locally equicontinuous. Then:

(a) is a closed linear operator. (b), if , , , . (c), if and (H5) holds. Furthermore, the condition (H5) can be replaced by (2.7).

(iii)Let . Then , , and . Assume that (2.7) holds for , and that (2.7) holds for replaced by . Then we have the following:

(a) and .(b) and have the same eigenvalues. (c).

Albeit the similar assertions can be considered in general case, we will omit the corresponding discussion even in the case that for .

Proposition 2.3. Let , and let be a locally equicontinuous -regularized -resolvent propagation family for (1.1). If (2.5) holds with , then the following holds:

(i) the equality

 holds provided and the following condition:

() any of the assumptions , , or , for some , implies , ;

(ii) the equality (2.9) holds provided , and the following condition:

() if , , for some , then , .

Proof. Let and be fixed. Define , . Using (2.5), it is not difficult to prove that

Let . Convoluting both sides of (2.10) with , we easily infer that , and , . Now the equality (2.9) follows from (). The proof is quite similar in the case .

Remark 2.4. The equations (1.1) with are much easier to deal with, since in this case, and for all . In general, (1.1) with cannot be reduced to an equivalent equation of the previously considered form.

Proposition 2.5. Suppose is a locally equicontinuous -regularized -resolvent propagation family for (1.1), , and . Then we have the following.

(i) If and () holds, then

 If, additionally,

 then (2.11) holds for all . (ii) The equality (2.11) holds provided , and (); assuming additionally (2.12), we have the validity of (2.11) for all .

Proof. We will only prove the second part of proposition. Let . Then the functional equation of () implies:

which yields after a tedious computation:

In view of (), the above equality shows that , . It can be simply verified that the condition (2.12) implies that (2.9) holds for all .

Proposition 2.6. Let be a locally equicontinuous -regularized -existence propagation family (-regularized -unique-ness propagation family, -regularized -resolvent propagation family) for (1.1), and let be a kernel. Then the tuple is a locally equicontinuous -regularized -existence propagation family (-regularized -uniqueness propagation family, -regularized -resolvent propagation family) for (1.1).

Suppose now is complete, (1.1) is -wellposed, is dense in and . Set , , , where and appears in the th place in the preceding expression. Since we have assumed that is complete, the operator () can be uniquely extended (cf. also (ii) of Definition 2.1) to a bounded linear operator on . It can be easily proved that is a locally equicontinuous -uniqueness propagation family for (1.1), and that the assumption , implies , . In case that , where for , one can apply the arguments given in the proof of [29, Proposition  1.1, page 32] in order to see that is a locally equicontinuous -resolvent propagation family for (1.1). Regrettably, it is not clear how one can prove in general case that , , .

The following definition also appears in [15].

Definition 2.7. Let and . Consider the following inhomogeneous equation:

A function is said to be

(i)a strong solution of (2.15) if and only if , and (2.15) holds for every ;(ii)a mild solution of (2.15) if and only if , , and

It is clear that every strong solution of (2.15) is also a mild solution of the same problem. The converse statement is not true, in general. One can similarly define the notion of a strong (mild) solution of the problem (2.2).

Let , and let . Then the following holds:

(a)if is a -existence propagation family for (1.1), then the function , , is a mild solution of (2.2) with for ; (b)if is a -uniqueness propagation family for (1.1), and , , , , , then the function , , is a strong solution of (2.2), provided for .

Theorem 2.8. Suppose is a locally equicontinuous -regularized -uniqueness propagation family for (1.1), (2.5) holds, and . Then the following holds:

(i) if , then any strong solution of (2.15) satisfies the equality:

 for any . Therefore, there is at most one strong (mild) solution for (2.15), provided that () holds, (ii)if , then any strong solution of (2.15) satisfies the equality:

 Therefore, there is at most one strong (mild) solution for (2.15), provided that and that () holds.

Proof. We will only prove the second part of theorem. Let . Taking into account (2.6), we get:

This implies the uniqueness of strong solutions to (2.15), provided that and that () holds. The uniqueness of mild solutions in the above case follows from the fact that, for every such a solution , there exists a sufficiently large such that the function is a strong solution of (2.15), with replaced by therein.

If is a (local) -regularized -resolvent propagation family for (1.1), then Theorem 2.8 shows that there exist certain relations between single operator families , and (cf. also [15] and [28, page 116]). It would take too long to analyze such relations in detail.

The subsequent theorems can be shown by modifying the arguments given in the proof of [30, Theorem  2.2.1].

Theorem 2.9. Suppose satisfies (P1), , is strongly continuous, and the family is equicontinuous, provided . Let be a closed linear operator on , let , and let be injective. Set , .

(i) Suppose , . Then is a global -regularized -existence propagation family for (1.1) if and only if the following conditions hold.

(a) The equality

 holds provided , , and . (b)The equality

 holds provided , , and .

(ii) Suppose , , . Then is a global -regularized -uniqueness propagation family for (1.1) if and only if, for every with , and for every , the following equality holds:

Theorem 2.10. Suppose satisfies (P1), , is strongly continuous, and the family is equicontinuous, provided . Let , , , , , and , . Assume, additionally, that the operator is injective for every with and for every with and , and that the operator is injective for every with and for every with and . Then is a global -regularized -resolvent propagation family for (1.1), and (2.5) holds, if and only if the equalities (2.20)-(2.21) are fulfilled.

Keeping in mind Theorem 2.10, one can simply clarify the most important Hille-Yosida type theorems for exponentially equicontinuous -regularized -resolvent propagation families (cf. also [15] and [26, Theorem  2.8] for further information in this direction). Notice also that the preceding theorem can be slightly reformulated for -regularized -existence and uniqueness resolvent propagation families.

The analytical properties of -regularized -resolvent propagation families are stated in the following two theorems whose proofs are omitted (cf. [14, Theorems  2.16-2.17] and [26, Lemma  3.3, Theorems  3.4, 3.6, and 3.7]).

Theorem 2.11. Suppose , is an analytic -regularized -resolvent propagation family for (1.1), satisfies (P1), (2.5) holds, and can be analytically continued to a function , where . Suppose , , , , , and , . Let the family

and let the set

be bounded provided and . Set

provided , and

provided . Suppose is an open connected subset of , and the set has a limit point in , for any . Then the operator is injective for every and ,

provided and , and

provided and . Suppose, additionally, that there exists such that . Then the family

provided and , respectively, the family

provided and , the mapping

defined for , is analytic, provided and , and the mapping

is analytic, provided and .

Theorem 2.12. Assume satisfies (P1), , and, for every with , the function can be analytically extended to a function satisfying that, for every , the set is bounded. Let , , , , , and , . Assume, additionally, that for each the set contains the set , and that , provided and , respectively, , provided and (cf. the formulation of preceding theorem). Suppose also that the operator is injective, provided and , and that the operator is injective, provided and . Let ( satisfy that, for every , the mapping , is analytic as well as that:

provided ,

provided ,

and, in the case ,

Then there exists an exponentially equicontinuous, analytic -regularized -resolvent propagation family for (1.1). Furthermore, the family is equicontinuous for all and , (2.5) holds, and , , .

In this paper, we will not consider differential properties of -regularized -resolvent (propagation) families. For more details, the interested reader may consult [30], and especially, [26, Theorems  3.18–3.20]. Notice also that the assertion of [26, Proposition  3.12] can be reformulated for -regularized -resolvent (propagation) families.

In the following theorem, which possesses several obvious consequences, we consider -exponentially equicontinuous -regularized -resolvent propagation families in complete locally convex spaces.

Theorem 2.13. (i) Suppose , is a -exponentially equicontinuous -regularized -resolvent propagation family for (1.1), , , and for every , there exist and such that

Then is a compartmentalized operator and, for every seminorm , is an exponentially bounded -regularized -resolvent propagation family for (1.1), in , with replaced by (). Furthermore,

and is a -exponentially equicontinuous, analytic -regularized -resolvent propagation family of angle , provided that is. Assume additionally that (2.5) holds. Then, for every , (2.5) holds with and replaced by and .

(ii) Suppose satisfies (P1), is complete, is a compartmentalized operator in , for some () and, for every , is a subgenerator (the integral generator, in fact) of an exponentially bounded -regularized -resolvent propagation family in satisfying (2.38), and (2.5) with and replaced, respectively, by and . Suppose, additionally, that and , provided . Then, for every , (2.37) holds () and is a subgenerator (the integral generator, in fact) of a -exponentially equicontinuous -regularized -resolvent propagation family satisfying (2.5). Furthermore, is a -exponentially equicontinuous, analytic -regularized -resolvent propagation family of angle provided that, for every , is a q-exponentially bounded, analytic -regularized -resolvent propagation family of angle .

Proof. The proof is almost completely similar to that of [20, Theorem  3.1], and we will only outline a few relevant facts needed for the proof of (i). Suppose and for some . Then (2.6) in combination with (2.37) implies that , , provided , and , , provided . In any case, , which implies , , and in particular . Since , we obtain and . Therefore, is a compartmentalized operator. It is clear that (2.38) holds and that the mapping , is continuous for any . This implies by the standard limit procedure that the mapping , is continuous for any . Now we will prove that, for every , the operator is closable for the topology of . In order to do that, suppose is a sequence in with and , in . Using the dominated convergence theorem, (2.6) and (2.37), we get that , for any . Taking the Laplace transform, one obtains . Since , we get that and that is closable, as claimed. Suppose . It is checked at once that , , , . The functional equation (2.6) for the operators , and can be trivially verified, which also holds for the functional equation (2.6) in case of its validity for the operators , , and . The remaining part of the proof can be obtained by copying the final part of the proof of [20, Theorem  3.1(i)].

Remark 2.14. In the second part of Theorem 2.13, we must restrict ourselves to the case in which for some (). As a matter of fact, it is not clear how one can prove that the operator is injective, provided , and , as well as that the operator is injective, provided , and . Then Theorem 2.10 is inapplicable, which implies that the argumentation used in the proof of [20, Theorem  3.1(ii)] does not work for the proof of fact that, for every and , is a projective family of operators.

3. -Regularized -Existence and Uniqueness Families for (1.1)

Throughout this section, we will always assume that and are sequentially complete locally convex spaces. By is denoted the space which consists of all bounded linear operators from into . The fundamental system of seminorms which defines the topology on , respectively, , is denoted by , respectively, . The symbol designates the identity operator on .

Let . A strongly continuous operator family is said to be locally equicontinuous if and only if, for every and for every , there exist and such that , , ; the notion of equicontinuity of is defined similarly. Notice that is automatically locally equicontinuous in case that the space is barreled.

Following Xiao and Liang [24], we introduce the following definition.

Definition 3.1. Suppose , , , and is injective.

(i)A strongly continuous operator family is said to be a (local, if ) -regularized -existence family for (1.1) if and only if, for every , the following holds: , for every with , for , and

for any .(ii)A strongly continuous operator family is said to be a (local, if ) -regularized -uniqueness family for (1.1) if and only if, for every and , the following holds:

(iii)A strongly continuous family is said to be a (local, if ) -regularized -existence and uniqueness family for (1.1) if and only if is a -regularized -existence family for (1.1), and is a -regularized -uniqueness family for (1.1). (iv)Suppose and . Then a strongly continuous operator family is said to be a (local, if ) -regularized -resolvent family for (1.1) if and only if is a -regularized -uniqueness family for (1.1), , for and , as well as , , and , for .

In case , where , it is also said that is a -times integrated -existence family for (1.1); -times integrated -existence family for (1.1) is also said to be a -existence family for (1.1). The notion of (exponential) analyticity of -existence families for (1.1) is taken in the sense of Definition 1.2(ii); the above terminological agreement can be simply understood for all other classes of uniqueness and resolvent families introduced in Definition 3.1.

Integrating both sides of (3.1) sufficiently many times, we easily infer that (cf. [24, Definition 2.1, page 151; and (2.8), page 153]):

for any , and . In this place, it is worth noting that the identity (3.3), with , , and (), has been used in [24] for the definition of a -existence family for (). It can be simply proved that this definition is equivalent with the corresponding one given by Definition 3.1.

Proposition 3.2. Let be a -regularized -existence and uniqueness family for (1.1), and let be locally equicontinuous. If , or , then , , .

Proof. Let be fixed. Using the local equicontinuity of , we easily infer that the mappings , and , are continuous and coincide. The prescribed assumptions also imply that, for every , and ,

Keeping in mind (3.2)-(3.3) and the foregoing arguments, we get that

This, in turn, implies the required equality , .

Definition 3.3. Suppose . Then we define , and

In the first part of subsequent theorem (cf. also [24, Remark  2.2, Example  2.5, Remark  2.6]), we will consider the most important case . The analysis is similar if for some .

Theorem 3.4. (i) Suppose is a -existence family for (1.1), , and for . Then the function

is a strong solution of the problem (2.2) on , where satisfy for .

(ii) Suppose is a locally equicontinuous -regularized -uniqueness family for (1.1), and . Then there exists at most one strong (mild) solution of (2.2) on , with , .

Proof. A straightforward computation involving (3.3) shows that

since

This implies that is a mild solution of (2.2) on . In order to complete the proof of (i), it suffices to show that and for all . Towards this end, notice that the partial integration implies that, for every ,

Therefore, and

Suppose, for the time being, . Then for . Moreover, the inequality holds provided and , and for and . Now it is not difficult to prove that

finishing the proof of (i). The second part of theorem can be proved as follows. Suppose is a strong solution of (2.2) on , with , . Using this fact and the equality

for any and , we easily infer that (for more general results, see [31, Proposition  2.4(i)], and [29, page 155]):

Therefore, , and , .

Before proceeding further, we would like to notice that the solution , given by (3.7), need not to be of class , in general. Using integration by parts, it is checked at once that (3.7) is an extension of the formula [24, (2.5); Theorem  2.4, page 152]. Notice, finally, that the proof of Theorem 3.4(ii) is much simpler than that of [24, Theorem  2.4(ii)].

The standard proof of following theorem is omitted (cf. also [24, Theorem  2.7, Remark  2.8, Theorem  2.9] and [28, Chapter 1]).

Theorem 3.5. Suppose satisfies (P1), , , , and is injective. Set , .

(i) (a) Let be a -regularized -existence family for (1.1), let the family be equicontinuous, and let the family be equicontinuous (). Then the following holds:

(b) Let the operator be injective for every with . Suppose, additionally, that there exist strongly continuous operator families and such that and are equicontinuous () as well as that

for every with and , and . Then there exists a -regularized -existence family for (1.1), denoted by . Furthermore, , , and , , , . (ii) Let the assumptions of (i) hold with . If , then one suppose additionally that, for every , there exists a strongly continuous operator family such that is equicontinuous as well as that

for every with , and . Let , and let for some (). Then, for every , there exist and such that the corresponding solution satisfies the following estimate:

(iii) Suppose is strongly continuous and the operator family is equicontinuous. Then is a -regularized -uniqueness family for (1.1) if and only if, for every , the following holds:

The Hausdorff locally convex topology on defines the system of seminorms on , where , , . Let us recall that is sequentially complete provided that is barreled. Following Wu and Zhang [32], we also define on the topology of uniform convergence on compacts of , denoted by ; more precisely, given a functional , the basis of open neighborhoods of with respect to is given by , where runs over all compacts of and . Then is locally convex, complete and the topology is finer than the topology induced by the calibration .

Now we focus our attention to the adjoint type theorems for (local) -regularized -resolvent families. The proof of following theorem follows from the arguments given in the proofs of [26, Theorems  2.14 and 2.15]; because of that, we will omit it.

Theorem 3.6. (i) Suppose is barreled, , is a -regularized -resolvent family for (1.1), and . Then is a -regularized -resolvent family for (1.1), with replaced by ().

(ii) Suppose is barreled, is a (local, global exponentially equicontinuous) -regularized -resolvent family for (1.1), and . Put . Then , is a (local, global exponentially equicontinuous) -regularized -resolvent family for (1.1), in .

(iii) Suppose is a locally equicontinuous -regularized -resolvent family for (1.1), and . Then is a locally equicontinuous -regularized -resolvent family for (1.1), in , with replaced by (). Furthermore, if is exponentially equicontinuous, then is also exponentially equicontinuous.

Notice here that a similar theorem can be proved for the class of -regularized -resolvent propagation families.

Let . Convoluting both sides of (1.1) with , we get that

In the subsequent theorem, whose proof follows from a slight modification of the proof of [24, Theorem  3.1(i)], we will analyze inhomogeneous Cauchy problem (3.21) in more detail.

Theorem 3.7. Suppose is a locally equicontinuous -existence family for (1.1), , and for . Let , let satisfy , , and let satisfy , . Then the function

is a mild solution of the problem (3.21) on , where satisfy for . If, additionally, and is locally equicontinuous, then the solution , given by (3.22), is a strong solution of (1.1) on .

Remark 3.8. Suppose that all conditions quoted in the first part of the above theorem hold, and the family is locally equicontinuous. We assume, instead of condition , that there exists a locally equicontinuous -uniqueness family for (1.1) on , as well as that there exist functions such that , , (cf. also the formulation of [24, Theorem  3.1(ii)]). Using the functional equation for , one can simply prove that, for every , the function

is a mild solution of the problem

By the uniqueness of solutions, we have that the following holds:

provided , and . Fix . Then the above equality implies that, for every with , one has:

provided , and . For such an index , we conclude from (3.26) that the mapping , is continuous. Observe now that the condition

which holds in the case of abstract Cauchy problem (), shows that the mapping , is continuous as well as that the mapping , is continuous. Hence, the validity of condition (3.27) implies that the function , given by (3.22), is a strong solution of (1.1) on .

4. Subordination Principles

The proof of following theorem can be derived by using Theorem 3.5 and the argumentation given in [10, Section 3].

Theorem 4.1. Suppose , is injective and .

(i) Let , and let the assumptions of Theorem 3.5(i)-(b) hold. Put

 Define, for every and , by replacing in (4.1) with . Suppose that there exist a number and a continuous kernel satisfying (P1) and , . Then there exists an exponentially bounded -regularized -existence family for (1.1), with replaced by therein (). Furthermore, the family is equicontinuous. (ii) Let , let the assumptions of Theorem 3.5(ii) hold, and let . Define, for every and , by replacing in (4.1) with . Then, for every , the family is equicontinuous,

 for every with , and . Let (defined in the obvious way), and let for some (). Then, for every , there exist and such that the corresponding solution satisfies the following estimate:

(iii) Suppose is a -regularized -uniqueness family for (1.1), and the family is equicontinuous. Define, for every , by replacing in (4.1) with . Suppose that there exist a number and a continuous kernel satisfying (P1) and , . Then there exists a -regularized -uniqueness family for (1.1), with replaced by therein (). Furthermore, the family is equicontinuous.

Remark 4.2. (i) Consider the situation of Theorem 4.1(iii). Then we have the obvious equality . If and (this inequality holds provided ), then .

(ii) Let be a kernel, and let be a (local) -regularized -uniqueness family for (1.1). Then is a -regularized -uniqueness family for (1.1).

(iii) Concerning the analytical properties of -regularized -existence families in Theorem 4.1(i), the following facts should be stated.

(a)The mapping , admits an extension to and, for every , the mapping , is analytic. (b)Let , and let be equicontinuous. Then is an exponentially equicontinuous, analytic -regularized -existence family of angle , and for every , there exist and such that

(c) is an exponentially equicontinuous, analytic -regularized -existence family of angle .

The similar statements hold for the -regularized -uniqueness family in Theorem 4.1(iii).

The results on -regularized -existence and uniqueness families can be applied in the study of following abstract Volterra equation:

where , , , and are closed linear operators on . As in Definition 2.7, by a mild solution, respectively, strong solution, of (4.5), we mean any function such that , and that

respectively, any function such that , and that (4.5) holds.

We need the following definition.

Definition 4.3. Suppose , , , and is injective.

(i)A strongly continuous operator family is said to be a (local, if ) -regularized -existence family for (4.5) if and only if

(ii)A strongly continuous operator family is said to be a (local, if ) -regularized -uniqueness family for (4.5) if and only if

Notice also that one can introduce the classes of -regularized -existence and uniqueness families as well as -regularized -resolvent families for (4.5); compare Definition 3.1. The full analysis of -regularized -existence and uniqueness families for (4.5) falls out from the framework of this paper.

The following facts are clear.

(i)Suppose is a -regularized -existence family for (4.5). Then, for every , the function , , is a mild solution of (4.5) with , . (ii)Let be a locally equicontinuous -regularized -uniqueness family for (4.5). Then there exists at most one mild (strong) solution of (4.5).

The proof of following subordination principle is standard and therefore omitted (cf. the proofs of [29, Theorem  4.1, page 101] and [24, Theorem  2.7]).

Theorem 4.4. (i) Suppose there is an exponentially equicontinuous -regularized -existence family for (1.1). Let be completely positive, let , and satisfy (P1), and let be such that, for every with and , the following holds:

Assume, additionally, that there exist a number and a function satisfying (P1) so that, for every with and , one has:

Then there exists an exponentially equicontinuous -regularized -existence family for (4.5).

(ii) Suppose there is an exponentially equicontinuous -regularized -uniqueness family for (1.1). Let be completely positive, let , and satisfy (P1), and let be such that, for every with and , the following holds:

Then there exists an exponentially equicontinuous -regularized -uniqueness family for (4.5).

It is not difficult to reformulate Theorem 4.4 for the class of strong -propagation families (cf. also Example 5.3 below).

Although our analysis tends to be exhaustive, we cannot cover, in this limited space, many interested subjects. For example, the characterizations of some special classes of -exponentially equicontinuous -regularized -existence and uniqueness families in complete locally convex spaces. We also leave to the interested reader the problem of clarifying the Trotter-Kato type theorems for introduced classes.

5. Examples and Applications

We start this section with the following example.

Example 5.1. Suppose () and, for every with , one has and . Let for .

(i)

(a) Suppose , , , and is a subgenerator of an exponentially equicontinuous -regularized -resolvent family which satisfies the following equality:

 Put and . By [26, Theorem  2.7], we have that, for every sufficiently small , there exists such that and the family is equicontinuous. Notice also that

 Due to the choice of , we have that, for every sufficiently small , there exists such that, for every , one has:

 Therefore, we have the following: if the operator is densely defined, then the above inequality in combination with Theorem 2.12 indicates that is a subgenerator of an exponentially equicontinuous, analytic -times integrated -resolvent propagation family for (1.1), with being the angle of analyticity; if the operator is not densely defined, then the above conclusion continues to hold with replaced by any number . (a′) Suppose , , , is a subgenerator of an exponentially equicontinuous, analytic -regularized -resolvent family of angle , and (5.1) holds. Put and . If the operator is densely defined, then it follows from [26, Theorem  3.6] and the above analysis that the operator is the integral generator of an exponentially equicontinuous, analytic -times integrated -resolvent propagation family for (1.1), with being the angle of analyticity; if the operator is not densely defined, then the above conclusion continues to hold with replaced by any number . Now we will apply this result to the following fractional analogue of the telegraph equation:

 where , and . Let be one of the spaces (, , , and . Put and recall that the space () is defined by . The totality of seminorms induces a Fréchet topology on . Let possess the same meaning as in [33], and let act with its maximal distributional domain. Suppose first and . Then the operator is the integral generator of an exponentially equicontinuous, analytic -semigroup of angle , which implies that is the integral generator of an exponentially equicontinuous, analytic -regularized resolvent propagation family , if , respectively, if , of angle ; the established conclusion also holds in the Fréchet nuclear space which consists of those smooth functions on with period along each coordinate axis [26]. In this place, we would like to observe that it is not clear whether the angle of analyticity of constructed -regularized resolvent propagation families, in the case that , can be improved by allowing that takes the value . Suppose now or . Then, for every , the operator is the integral generator of an exponentially equicontinuous, analytic -times integrated -regularized resolvent propagation family , if , respectively, if , of angle . (b) Suppose , , , , , , and is a subgenerator of an exponentially equicontinuous -regularized -resolvent family which satisfies the following equality:

 Let be defined as in (a). Then it is checked at once that and . Put , , where and . It is clear that, for every , there exists a sufficiently large such that, for every ,

 Arguing as in (a), we reveal that is a subgenerator of an exponentially equicontinuous, analytic -regularized -resolvent propagation family for (1.1), with being the angle of analyticity. (b′) Suppose , , , (), , , is a subgenerator of an exponentially equicontinuous, analytic -regularized -resolvent family of angle , and (5.5) holds. Assume, additionally, that . Define as in (a)’, and , , where and . Then one can simply verify that and . Making use of [26, Theorem  3.6] and the foregoing arguments, we obtain that the operator is the integral generator of an exponentially equicontinuous, analytic -regularized -resolvent propagation family for (1.1), with being the angle of analyticity. Before proceeding further, we would like to recommend for the reader [14, 20, 21, 26, 30, 34] for some examples of (nondensely defined, in general) differential operators generating various types of -regularized -resolvent families.

(ii) Suppose is complete, , , and is the densely defined generator of a -exponentially equicontinuous -regularized -resolvent family which satisfies that, for every , there exist and such that , , . By [20, Theorem  3.1], we infer that is a compartmentalized operator and that, for every , the operator is the integral generator of an exponentially bounded -regularized -resolvent family in . Then the first part of this example shows that is the integral generator of an exponentially bounded, analytic -resolvent propagation family, with being the angle of analyticity. By Theorem 2.13(ii), we obtain that is the integral generator of a -exponentially equicontinuous, analytic -resolvent propagation family for (1.1), and that the corresponding angle of analyticity is . It can be simply shown that, for every and , there exist and such that , , . In the continuation, we will also present some other applications of -regularized -resolvent families in the analysis of some special cases of (1.1); as already mentioned, this theory is inapplicable if some of initial values is a non-zero element of . Consider the abstract Basset-Boussinesq-Oseen equation (1.2) and assume that is complete. Set , , , and . Suppose is the integral generator of a q-exponentially equicontinuous -regularized -resolvent family satisfying (2.37); cf. [20, 25] for important examples of differential operators generating q-exponentially equicontinuous -regularized -resolvent families. Then it has been proved in [20] that is the integral generator of a -exponentially equicontinuous, analytic -regularized resolvent family of angle . Notice, finally, that the choice of function instead of has some advantages.

Example 5.2. Suppose , , is measurable, , , , () and , , with maximal domain. Assume , , and , . Denote by the associated function of the sequence [30] and put , , , . Clearly, there exists a constant such that , . Hereafter we assume that the following condition holds:

(H) for every , there exist , and such that and

Notice that the above condition holds provided , , and , (cf. [31]), and that the validity of condition (H) does not imply, in general, the essential boundedness of the function . We will prove that is the integral generator of a global (not exponentially bounded, in general) -regularized -resolvent propagation family for (1.1). Clearly, it suffices to show that, for every , is the integral generator of a local -regularized -resolvent propagation family for (1.1) on . Suppose that is given in advance, and that , and satisfy (H), for this . Let denote the upwards oriented boundary of ultralogarithmic region . Put, for every , and ,

if , and

if . It is clear that, for every , , , and that is strongly continuous. Furthermore, the Cauchy theorem implies that , . Now we will prove that the identity (2.6) holds provided and . Let . Then a straightforward computation involving Cauchy theorem shows that (2.6) holds, with replaced by therein, if and only if:

Using [28, Lemma  5.5, page 23] and the Cauchy theorem, the above equality is equivalent with:

which is true because the integrands appearing on both sides of this equality are equal identically. One can similarly prove that the identity (2.6) holds provided and , so that , defined in the obvious way, is a -regularized -resolvent propagation family for (1.1), with subgenerator . Notice that the condition (H) implies for all , which has as a further consequence that , provided and , and that , provided and . The equality (2.5) holds for , the integral generator of , defined similarly as in the second section, coincides with the operator , which is the unique subgenerator of . Notice that, for every compact set , there exists such that

and that one can similarly consider the generation of local -regularized -resolvent propagation families which oblige a modification of the property stated above with . Now we would like to give an example of -regularized -resolvent propagation family for (1.1) in which for some . Assume , , , , and , . Define , and as before (). Then the established conclusions continue to hold since, for every , there exist , and such that (H) holds as well as that:

Notice, finally, that it is not so difficult to construct examples of local -regularized -resolvent propagation families which cannot be extended beyond its maximal interval of existence.

Example 5.3. Suppose , , , , , , , , and . Put and with maximal distributional domain. Now we will focus our attention to the following fractional analogue of damped Klein-Gordon equation:

The case has been analyzed in [24, Example  4.1], showing that there exists an exponentially bounded -uniqueness family for (5.14) and that, for every , there exists an exponentially bounded -existence family for (5.14) with . It is worth noting that the estimates obtained in cited example enables one to simply verify that the conditions of Theorem 4.1(i)-(ii) hold with and , and that the conditions of Theorem 4.1(iii) hold with and . This implies that there exists an exponentially bounded -regularized -uniqueness family for (5.14) with , , and that there exists an exponentially bounded -existence family for (5.14) with , . Applying Theorem 3.7, we obtain that, for every and , there exists a unique mild solution of the corresponding problem (3.21) as well as that there exist and such that the following estimate holds for each :

It is checked at once that the solution is analytically extensible to the sector , provided that . Suppose now , , provided , respectively, , provided , and . Then there exists a strong -propagation family for the problem (5.14) with (cf. [28, Example  5.8, page 130]). Using [10, (1.23), page 12; Theorems  3.1–3.3, pages 40–42] and [28, Proposition  5.3(iii), page 116], it readily follows that, for every and , the function , , given by

is a unique strong solution of the corresponding integral equation (3.21) with and ; obviously, this solution is analytically extensible to the sector . Notice also that one can similarly consider (cf. [24, Example  4.2] for more details) the results concerning the existence and uniqueness of mild solutions of the following time-fractional equation:

and that Theorem 4.4 can be applied in the analysis of the following integral equation:

for certain kernels and . We leave details to the interested reader.

Consider now the following slight modification of (5.14):

Suppose now that (for further information concerning the case , [21, 23] may be of some importance). Although the equality does not hold in general, we would like to point out that the existence and uniqueness of mild solutions to the homogeneous counterpart of (5.20) cannot be so easily proved for initial values belonging to the Sobolev space , for some . In order to better explain this, we will introduce the new function by . Then (5.20) can be rewritten in the following equivalent matricial form:

where ; see, for example, [35, 36]. The characteristic values of associated polynomial matrix  := [] are , , which implies that the condition of Petrovskii for systems of abstract time-fractional equations, that is, , is not satisfied [36]. Notice, finally, that (1.1) cannot be converted to an equivalent matrix form, except for some very special values of .

Before proceeding further, we would like to observe that several examples of -times integrated -existence and uniqueness families, acting on products of possibly different Banach spaces (), can be constructed following the consideration given in [37, Section  7].

Example 5.4. Let ,

Then , and for every , , [21]. Consider now the complex non-zero polynomials , , (), and define, for every and , the operator by and , . Our intention is to analyze the smoothing properties of solutions of the equation (3.21) with , , , , and a suitable chosen function . In order to do that, set , (, ), and after that, . Then it is not difficult to prove (cf. [21, Example  2.10]) that, for every , and that

where denote the zeroes of and , . Suppose now that the following condition holds:

(H) there exist , and such that, for every , one has: , , .  It is well known from the elementary courses of numerical analysis [38] that the condition: (H₁) there exist , and such that, for every , one has:

implies (H). The validity of last condition can be simply verified in many concrete situations, and it seems that slightly better estimates can be obtained only in the case of very special equations of the form (1.1). We would also like to point out that the condition (H) need not to be satisfied, in general. Using (5.23), the inequality (), as well as the continuity of mappings , and , , for , we obtain the existence of a positive polynomial such that

In what follows, we will use the following family of kernels. Define, for every , the entire function by , , where . Then it is clear that , , . Hence, , , , where , and . It is also worth noting that, for every , and , we have , and

Put now , , . Then, for every , , and is infinitely differentiable for . By Theorem 3.5(i)-(b) and (iii), we easily infer from (5.25) that there exists such that, for every , there exists an exponentially bounded -regularized -resolvent family for (1.1), with . Furthermore, the mapping , is infinitely differentiable in the uniform operator topology of and, for every compact set and for every , there exists such that

One can similarly construct examples of exponentially bounded, analytic -regularized -resolvent families.

Acknowledgments

The first named author is partially supported by Grant 144016 of Ministry of Science and Technological Development, Republic of Serbia. The second and third authors are supported by the NSFC of China (Grant no. 10971146).

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