Advanced Theoretical and Applied Studies of Fractional Differential EquationsView this Special Issue
Research Article | Open Access
On a Class of Abstract Time-Fractional Equations on Locally Convex Spaces
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. , Klafter et al. (Eds.) , Kilbas et al. , Mainardi , Podlubny , and Samko et al. ; 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 . For further results concerning the -wellposedness of (1.2), [20, 21] are of importance. In , 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.  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 . 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 , 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 , 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 , 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  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 . 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  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  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 .
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.
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 .
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