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Journal of Function Spaces
Volume 2015 (2015), Article ID 158145, 9 pages
http://dx.doi.org/10.1155/2015/158145
Research Article

A Subordination Principle on Wright Functions and Regularized Resolvent Families

Departamento de Matemáticas, Instituto Universitario de Matemáticas y Aplicaciones, Universidad de Zaragoza, 50009 Zaragoza, Spain

Received 11 December 2014; Accepted 20 February 2015

Academic Editor: Gestur Ólafsson

Copyright © 2015 Luciano Abadias and Pedro J. Miana. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We obtain a vector-valued subordination principle for -regularized resolvent families which unified and improves various previous results in the literature. As a consequence, we establish new relations between solutions of different fractional Cauchy problems. To do that, we consider scaled Wright functions which are related to Mittag-Leffler functions, the fractional calculus, and stable Lévy processes. We study some interesting properties of these functions such as subordination (in the sense of Bochner), convolution properties, and their Laplace transforms. Finally we present some examples where we apply these results.

1. Introduction

A function is a Bernstein function if is of class , for all andThe celebrated Bochner subordination theorem characterizes Bernstein functions: given , a Bernstein function, there exists a unique convolution semigroup of subprobability measures on such that Conversely, given a convolution semigroup of subprobability measures on , then there exists a unique Bernstein function such that (2) holds true; see, for example, [1, Theorem 5.2]. The original subordination principle for stochastic processes in connection with diffusion equations and semigroups was introduced in [2]. In [3, Chapters 4.3, and 4.4], a detailed study of stochastic processes, their transition semigroups, generators, and subordination results are developed.

Now let be a densely defined closed linear operator on a Banach space which generates a -contraction semigroup . Then the solution of the first order abstract Cauchy problem is given by for . Now, suppose that is a vaguely continuous convolution semigroup of subprobability measures on with the corresponding Bernstein function . Then the Bochner integral defines again a -contraction semigroup on [1, Proposition 12.1]. Then the semigroup is called subordinate (in line with Bochner) to the semigroup with respect to the Bernstein function . In particular, given and (where are the stable Lévy processes, see (32)), then is an analytic semigroup generated by , the fractional powers of the generator according to Balakrishnan. For more details, see [4, Chapter IX].

Other subordination formulae allow defining new families of operators from some previous ones by integration. Let be the generator of a cosine function on a Banach space (see definition in [5, Section 3.14]). Then generates a holomorphic -semigroup of angle , given by [5, Theorem 3.14.17]. Remember that the solution of the second order Cauchy problem is for [5, Section 3.14].

In [6], a two-kernel dependent family of strong continuous operators defined in a Banach space is introduced. This family allows us to consider in a unified treatment the notions of, among others, -semigroups of operators, cosine families, -times integrated semigroups, resolvent families, and -generalized solutions. Let and . The family is a -regularized resolvent family generated by if the following conditions are fulfilled: is strongly continuous for and for all ; , that is, , for and ; andsee [6, Definition 2.1]. In the case , we obtain the resolvent families which are treated in detail in [7]; for , this family of operators is a -semigroup, and we also retrieve cosine functions for and . Some spectral properties for -regularized resolvent families are proved in [8].

Subordination theorems for -regularized resolvents have been considered in some different works. In [7, Section I.4], the use of the theory of Bernstein functions, completely positive functions, and the Laplace transform allows showing some subordination results for -regularized resolvents [7, Theorem 4.1, Corollaries 4.4 and 4.5]. A nice subordination theorem for -times integrated semigroup is proved in [6, Theorem 3.7]. In the case of -regularized functions, this subordination theorem is improved in [9, Theorem 3.1] and [10, Theorem 3.1] and an integral representation, similar to formula (6), is also proved involving Wright functions. In [11, Theorem 3.1], using holomorphic functional calculus, the authors prove a subordination result for -regularized resolvent families generated by fractional powers of closed operators, which extends both [4, Chapter IX, Section 11, Theorem 2] and [9, Theorem 3.1]. Finally, in [12, Theorem 2.8], a subordination principle for -regularized resolvents, inspired in the original proof of [7, Theorem 4.1], is shown. In all these results, note that the subordination integral formula is only present in [9, Theorem 3.1] and [11, Theorem 3.1].

The main aim of this paper is to obtain subordination integral formulae to -regularized resolvents (Theorem 12). To achieve this, we present a detailed presentation of Wright and Mittag-Leffler functions in Section 2, which includes some basic results and known connections of these functions and fractional differential equations.

In Section 3, we introduce a new family of biparameter special functions in two variables defined by scaling Wright functions (Definition 2). This family of functions plays a fundamental role in the subordination principle for -regularized resolvent families; see formula (51). Moreover, these functions satisfy a nice subordination formula, Theorem 8, which extends some known results for Wright -function and stable Lévy processes; see Remark 9. In fact the algebraic nature (for convolution products) of these functions is shown in Propositions 5 and 7.

In Section 4, we consider -regularized resolvents in abstract Banach spaces. We prove the main subordination principle, Theorem 12, and some consequences in Remark 13 and Corollary 14.

Finally, in Section 5 we present some comments, concrete examples, and applications to fractional Cauchy problems which illustrate the main results of this paper.

Notation. Let , , and be the Lebesgue Banach algebra of integrable function on with the usual convolution product The usual Laplace transform of a function , , is defined by for . Let , and we denote , , and for .

The set of continuous functions on such that is denoted as , and the set of holomorphic functions on such that is denoted as . We denote by an abstract Banach space, the set of linear and bounded operators on the Banach space , and the set of functions of compact support and infinitely differentiable on into .

2. Mittag-Leffler and Wright Functions

In this section we present definitions and basic properties of Mittag-Leffler and Wright functions. The algebraic structure of those functions has been partially considered in [13] and formulae (15) and (16) seem to be new.

The Mittag-Leffler functions are defined by We write . The Mittag-Leffler functions satisfy the following fractional differential problems: for , under certain initial conditions, where and denote the Caputo and Riemann-Liouville fractional derivatives of order , respectively; see Section 5 and [14, 15]. Their Laplace transform is For more details see [10, Section 1.3].

Recently, the next algebraic property has been proved for and ; see [13, Theorem 1]. In fact, a similar identity holds for generalized Mittag-Leffler function with , , and for and . The proof of this result is a straightforward consequence of [16, Theorem 5]. In the case for , the algebraic property is which is a direct consequence of Theorems 2.1 and 2.2 of [17].

The Wright function that we denote by was introduced and investigated by . Maitland Wright in a series of notes starting from 1933 in the framework of the theory of partitions; see [18]. This entire function is defined by the series representation, convergent in the whole complex plane:

The equivalence between the above series and the following integral representations of is easily proven by using the Hankel formula for the Gamma function: where denotes the Hankel path defined as a contour that begins at   , encircles the branch cut that lies along the negative real axis, and ends up at ; for more details see [14, Appendix F]. It is clear that

In addition, as discussed below, the following special cases are of considerable interest: interrelated through

The Wright -function in two variables is defined by This function has been studied, for example, in [14, page 257] and [19, Section 6]; a subordination formula for time fractional diffusion process is given in [19, Formula (6.3)] and [14, Formula (F.55)]: for , the following subordination formula holds true;This subordination formula had previously appeared in [10, Formula (3.28)].

The deep connection between fractional differential equations (in space and in time) and Wright-type functions has been studied in detail in [14, 19, 20].

It is known that that is, is the Laplace transform of in the whole complex plane; see [14, Formula (F.25)]. Then, observe that for where both functions are related to the solutions of the fractional differential problems mentioned above.

Nice connections between Mittag-Leffler functions and Wright functions are obtained by the Laplace transform; see formula (24) and [14, Formula (F.22)]. In the next proposition, we present some interesting properties of Wright functions. The next result extends the study which was done in [10, Chapter 1, page 14] for the case with .

Proposition 1. Let and . Then the following properties hold:(i); .(ii),  for .

Proof. (i) Using the definition of , we havewhere we have applied the Fubini theorem and the Laplace transform of .
(ii) The positivity of follows from (24), the complete monotonicity of for , , and (see [14, Appendix E, Formula (E.32)]), and the Post-Widder inversion formula; see [10, Lemma 1.6].

3. Scaled Wright Functions

In this section, we introduce two-parameter Wright functions in Definition 2, which we call scaled Wright functions. This class of functions includes the Wright -function introduced in [19, Formula (6.2)] and also considered in [14, Formula (F.51)] and stable Lévy processes. They satisfy important properties (Theorem 3 and Proposition 5) and a subordination principle (Theorem 8) and play a crucial role in this paper.

Definition 2. For and , we define the function in two variables by

Note that, using the change of variable , we get the integral representation

The function is considered in the literature in some particular cases.(i)For , where is the Wright -function in two variables studied in [14, page 257] and is considered in [10, Theorem 3.1]; for , see [14, Appendix F, Formula (F.16)].(ii)For ,is the stable Lévy process of order ; see Section 1, [2], and [4, Chapter IX], in particular

In the next theorem, we join some properties which are satisfied by functions .

Theorem 3. Let and ; one has the following. (i),  for .(ii),  for .(iii),  for , .(iv)  for , .(v),  for .(vi),  for .

Proof. (i) It is clear by Definition 2 and Proposition 1. (ii) It is easy to see that is of exponential growth in . Let , (where and are involved in the definition of Hankel path ). Then, by Cauchy theorem of residue, (iii) Using (24), the result is direct by a change of variable. (iv) We combine parts (ii) and (iii) to obtain the equality. (v) It is clear using Laplace transform and (i). (vi) It is clear by a change of variable and applying Proposition 1(i).

We combine Theorem 3(ii) and (iii) and formula (13) to get the following corollary.

Corollary 4. For ,  , one denotes by the function given byThen , andfor .

Note that

The following key lemma includes the particular case and proved in [21, Lemma 1].

Proposition 5. For , , and , the following identity holds: for .

Proof. Note that is a Laplace transformable function and locally integrable in two variables. We apply the Laplace transform in variable to get in the right side with , where we have applied Theorem 3(ii) and [15, Chapter II, Formula (5.11)].
In the left side, we also apply the Laplace transform in the variable to get that with , where we have used Theorem 3(ii).

Remark 6. For in Proposition 5, we obtain the equality proved in Theorem 3(vi).

Finally, we show an algebraic identity which satisfies functions .

Proposition 7. Take , , and . Then (i) = /;(ii)/.

Proof. Note that , for ,  , and ; see Theorem 3(iii). Then we apply the Laplace transform in the variable in both equalities to get identities (15) and (16). The injectivity of Laplace transform allows us to finish the proof.

To finish this section, we prove a subordination formula for functions which expands some well-known results.

Theorem 8. For , , and , the following identity holds:

Proof. To show this theorem, we apply the Laplace transform in both variables , the so-called double Laplace transform, Fubini theorem, Theorem 3(ii) and (iii), and finally formula (13) to get that for . Due to Theorem 3(iv) and the uniqueness of the double Laplace transform (see, e.g., [22, page 346]), we conclude the equality.

Remark 9. In the case that , we obtain formula (23). For and , we get the following subordination formula for stable Lévy processes: for , . Finally, for and , we obtain that where we use equality (31).

4. Subordination Principle for Regularized Resolvent Families

In the following we consider that the operator is a densely defined closed linear operator on a Banach space . Let . A family is a -regularized resolvent family generated by if the following conditions are satisfied.(a) is strongly continuous for and for all .(b); that is, for and .(c)The integral equation holds for and .This family of operators was formerly introduced for general kernels in [6, Definition 2.1]. The above definition is also considered for and in [23], for in [17], and for and in [9, Definition 2.1].

The reason why we do not consider the value of at in condition (a) (compare with [6, Definition 2.1, Condition (R1)]) is that the function has a singularity at if .

Let be a strongly continuous operator family such that , for any . The operator family is said to be Laplace-transformable if there exists such that the Laplace transform of , converges for ; see, for example, [5, Definition 3.1.4]. If generates a -regularized resolvent family such that for and is Laplace transformable of parameter , we write . We denote . The case of -regularized resolvent families exponentially bounded, for , is considered in [11, Definition 2.5] and [10, Definition 2.4]; in this case, .

The next theorem characterizes the Laplace transform of -regularized family and extends [23, Theorem 3.11] and the proof is similar to the proof of [6, Proposition 3.1].

Theorem 10. Let . Then if and only if and there exists a strongly continuous function , locally integrable, for all , and Laplace transformable such that for all . Furthermore, the family is the -regularized resolvent family generated by .

Example 11. For and , the family defined by is a -regularized resolvent family generated by on the Banach space . First note that the family is well defined since for ; see Theorem 3(vi). In addition, by Theorem 3(ii), we have that The strong continuity at the origin is a simple check, and the reader can follow the steps on the proof of Theorem 12; in particular, the stable Lévy processes are convolution -regularized resolvent families. Similarly, for , Mittag-Leffler functions are -regularized resolvent families generated by on the Banach space (or ); see formula (13). These two families of functions are canonical examples of -regularized resolvent families.

The next theorem is the main one of this paper.

Theorem 12. Let ,  , and . If generates a -regularized resolvent family , then generates the following -regularized resolvent family: for and . Moreover the following equality holds: for .

Proof. First we show that is Laplace transformable of parameter . Note that using Theorem 3(ii) we get that for , and .
The family is strongly continuous on and now we prove the strong continuity at the origin. Let , and then where we have used Theorem 3(vi). We apply the dominated convergence theorem to the above term and Proposition 1(i) to conclude thatsince is a -regularized resolvent family. Finally, by Theorem 10, we obtain that the family is a -regularized resolvent family generated by . The proof of equality (52) is a straightforward consequence of Theorem 3(v).

Remark 13. Note that the above subordination theorem extends some subordination results which have been considered in this paper and previous papers. (i)Now, we consider the family of functions which is a -regularized resolvent family for and (see Example 11). Then we apply Theorem 12 for and , and we obtain the formula which is shown in Theorem 8.(ii)If and in Theorem 12, we retrieve the subordination principle for -regularized resolvent families given in [9, Theorem 3.1] and [10, Theorem 3.1] for and .(iii)By [12, Theorem 2.8(i)], given a -regularized resolvent family generated by , we obtain a -regularized resolvent family generated by , where and . This is a particular case of Theorem 12 for and .

Corollary 14. Let and . If is a -regularized resolvent family generated by , then generates the following -regularized resolvent family:

In particular, we have the following remarkable particular cases.(i)If generates a -semigroup , that is, a -regularized resolvent family, satisfying , , then generates the following -regularized resolvent family: (ii)Let and be a -regularized resolvent family generated by . Then generates the following -semigroup: In the case that is the generator of an exponentially bounded sine function ,   (see details in [5, Section 3.15]), that is, is a -regularized resolvent family, then generates the following -semigroup: with exponential bound , which is proved in [24, Theorem 5.2], and see also [25, Proposition 2.5 and Theorem 2.6].

5. Applications, Examples, and Final Comments

In this last section we present some applications of our results to fractional Cauchy problems and classical semigroups in Lebesgue spaces. Now we recall some basic definitions on fractional Cauchy problems. Let ; we call Riemann-Liouville fractional integral of of order , to the function given by and the Riemann-Liouville fractional derivative of of order is given by and with . Also, we consider the Caputo fractional derivative of of order ,with ; see, for example [14, 15] and [11, Section 4]. Note that, in the above definitions, the function can be taken in a larger space than where the definitions make sense.

5.1. Fractional Powers in Fractional Cauchy Problems

The next results extend [11, Theorem 4.9(a), (c)]: for , the solutions of Caputo fractional problems are -regularized resolvent families (see [10, Definition 2.3]) which may be obtained by integration from -regularized resolvent families by (52). Note that these are solutions of Riemann-Liouville fractional problems; see [17, Theorem 1.1].

Theorem 15. Let and be the generator of a uniformly bounded -semigroup .(i)The fractional Cauchy problem is well-posed and its unique solution is given by (ii)The fractional Cauchy problem is well-posed and its unique solution is given by

Proof. (i) By Corollary 14(i), the operator generates a -regularized resolvent family which provides the solution of fractional Cauchy problem (64); see [17, Theorem 1.1]. (ii) By [4, Chapter IX], the operator generates a -semigroup given by (5). By part (i), Fubini theorem, and the definition of in Corollary 4, we conclude the proof.

5.2. Convolution Semigroups on

Let be a uniformly bounded convolution semigroup in generated by ; that is, Two well-known examples of convolution semigroups are Gaussian and Poisson semigroups, and , whose generators are the Laplacian and , respectively; see, for example, [4, Chapter IX]. Then, by Theorem 15(i), the solution of the Riemann-Liouville fractional diffusion problem of order is given by In addition, to the particular case of the Laplacian, we obtain the solution of the Caputo fractional diffusion problem of order , (considered in [11, Example 4.13]) which is given by , for .

5.3. Multiplication Families on

Let be a multiplication semigroup in generated by ; that is,Some examples are , treated in [26]. Then, by Theorem 15(i), the solution of the Riemann-Liouville fractional diffusion problem of order is given by where we have applied Theorem 3(iii). Even more, by Theorem 15(ii), the solution of the Riemann-Liouville fractional diffusion problem of order , , is given by where we have applied Theorem 3(ii) and (iii).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors thank Professor Carlos Lizama for some comments and nice ideas that have contributed to improve the final version of this paper. The authors also thank the referee for several comments and, in particular, [8]. Authors have been partially supported by Project MTM2013-42105-P, DGI-FEDER, of the MCYTS, Project E-64, D. G. Aragón, and Project UZCUD2014-CIE-09, Universidad de Zaragoza.

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