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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 683021, 14 pages
The Maximal Subspace for Generation of -Regularized Families
1Facultad de Ciencias Básicas, Universidad Tecnológica de Bolívar, Cartagena, Colombia
2Departamento de Matemática y Ciencia de la Computación, Facultad de Ciencia, Universidad de Santiago de Chile, Casilla 307, Correo 2, Santiago, Chile
Received 25 May 2012; Revised 7 September 2012; Accepted 12 September 2012
Academic Editor: Patricia J. Y. Wong
Copyright © 2012 Edgardo Alvarez-Pardo and Carlos Lizama. 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.
Let be a linear operator in a Banach space . We define a subspace of and a norm such that the part of in such subspace generates an (, )-regularized resolvent family. This space is maximal-unique in a suitable sense and nontrivial, under certain conditions on the kernels and .
Inspired by the well-known Hille-Yosida theorem, Kantorovitz defined in 1988 a linear subspace and a norm such that the restriction of to this subspace generates a strongly continuous semigroup of contractions (see ). This so-called Hille-Yosida space is maximal unique in a suitable sense. The same problem has been considered in the context of strongly continuous operator families of contractions by Cioranescu in . In this case, the generation theorem of Sova and Fattorini was fundamental for her work. Later, Lizama in  used the generation theorem for resolvent families due to Da Prato and Iannelli (see ) as basis to generalize the results of Kantorovitz and Cioranescu to the context of resolvent families of bounded and linear operators. In this paper, also some applications to Volterra equations were given.
It is remarkable that resolvent families do not include -times integrated semigroups, -times integrated cosine functions, -convoluted semigroups, -convoluted cosine families, and integrated Volterra equations. For a historical account of these classes of operators, see (, page 234). Actually, these types of families are -regularized. The concept of -regularized resolvent families was introduced in . The systematic treatment based on techniques of Laplace transforms was developed in several papers (see, e.g., [7–13]). The theory of -regularized families has been developed in many directions and we refer to the recent monograph of Kostić  for further information. In this context, the problem to find maximal subspaces for generation of -regularized families remained open in case . In this paper, we are able to close this gap generalizing, in particular, [1, 2, 6].
In this work, we will use the generation theorem for -regularized resolvent family (see ) to show that there exists a linear subspace in and a norm majorizing the given norm, such that is a Banach space, and the part of in generates a -regularized resolvent family of contractions in . Moreover, the space is a maximal-unique in a sense to be defined below. Concerning the non-triviality of , we prove that it contains the eigenvectors corresponding to non-positive eigenvalues of . We close this paper with illustrative examples concerning the cases and in some region and .
This paper is organized as follows. In the first section, we recall the definition as well as basic results about -regularized families.
In Section 2, we show the existence of the maximal subspace such that the part of in this subspace generates an -regularized family. We prove that such subspace is a Banach space with the norm defined below. The maximality is also proved and we show how this is used to obtain a relation with the Hille-Yosida space corresponding to the semigroup case.
In Section 3, we present some applications of the theory developed in the preceding section. Here we show the particular cases of generation corresponding to resolvent families, cosine operator families, semigroups, -times semigroups and -times cosine operator families. After that, we give concrete conditions on a given operator to obtain the non-triviality of the maximal spaces and hence the well posedness on these spaces, for the abstract Cauchy problems of first and second order.
In this section, we recall some useful results in the literature about -regularized resolvent families. Let us fix some notations. From now on, we take to be a Banach space with the norm . We denote by the Banach algebra of all bounded linear operators on endowed with the operator norm, which again is denoted by . The identity operator on is denoted by , and denotes the interval . For a closed operator , we denote by , , the spectrum, the point spectrum, and resolvent of , respectively.
Definition 2.1. Let , , and be given. Assume that is a linear operator with domain . A strongly continuous family is called -regularized family on having as a generator if the following hold:(a); (b) and for all and ; (c) for all and .
In the case where , this definition corresponds to the resolvent family for the Volterra equation of convolution type in . Moreover, if, in addition, then this family is a -semigroup on or if is a cosine family on .
We note that the study of -regularized families is associated to a wide class of linear evolution equation, including, for example, fractional abstract differential equations (see ).
Definition 2.2. We say that is of type if there exist constants and such that for all .
We will require the following theorem on generation of -regularized families (see ).
Theorem 2.3. Let be a closed and densely defined operator on a Banach space . Then is an -regularized family of type if and only if the following hold: (a) and for all ;(b) satisfies the estimates
In the case where , Theorem 2.3 is well known. In fact, if then it is just the Hille-Yosida theorem; if , then it is the generation theorem due essentially to Da Prato and Iannelli in . In the case where and , it is the generation theorem for -times integrated semigroups ; if and is arbitrary, it corresponds to the generation theorem for integrated solutions of Volterra equations due to Arendt and Kellerman .
In order to give applications to our results we recall the following concepts of fractional calculus. The Mittag-Leffler function (see, e.g., [18–20]) is defined as follows: where is a Hankel path, that is, a contour which starts and ends at and encircles the disc counter clockwise. The function is an entire function which provides a generalization of several usual functions. For a recent review, we refer to the monograph .
An interesting property related with the Laplace transform of the Mittag-Leffler function is the following (cf. , () page 267):
Remark 2.4 (see ). If and , then where as with . This implies that for each , there is a constant such that
3. The Maximal Subspace
In this section, is a Banach space with norm . Let be a linear operator and and , . Assume that and for all . Observe that we are implicitly assuming that the inequality (2.2) holds with . Let for , and define where for the product is defined as . It is clear that is a norm on .
Proposition 3.1. is a Banach space.
Proof. Let be a Cauchy sequence. We observe that if . Then is a Cauchy sequence on . Let .
First, we show that . Indeed, let , be fixed. Then
Second, we prove that converges to . Let . There exists such that for . Since in the norm of , we also have where is the constant given in (2.2). Hence for every , , we have by inequality (2.2)
Taking supremum over all , , we obtain
Therefore the sequence converges to in the norm . Consequently is a Banach space.
Definition 3.2. Let be defined by , where
This operator is sometimes called the part of in .
We denote where the closure is taken in the norm .
Lemma 3.3. With the preceding definitions and hypothesis, we have(a) is a closed linear operator on ,(b) is invertible on for each , (c) for each and , where .
Proof. Let be fixed. Since , then
Note that is closed because is a Banach space and is closed. So the first part is done. Now, let be fixed. Since , then . Moreover, therefore . On the other hand, from identities above, we have thus . Hence , and we conclude that Now if then, in particular, , and therefore, This proves the second assertion. In particular, and hence
Finally, let , , be fixed. We have where are arbitrary for , and . This proves the third part of the lemma.
Lemma 3.4. Let be defined by , where Then is a closed operator such that and (a) is invertible on for each , (b) for each and where .
Proof. We observe that . Then the result is a direct consequence of (, Lemma 3.10.2).
As a consequence, we obtain the main result of this section on the existence of -regularized families.
Theorem 3.5. Let be a linear operator defined in a Banach space and and , . Assume that for all . Then there exist a linear subspace and a norm such that is a Banach space and generates an -regularized family of contractions in .
Concerning the non-triviality of , we will prove that it contains the eigenvectors corresponding to nonpositive eigenvalues of .
Let be fixed. Let be the unique solution to the scalar equation
Thus, provided the kernels and are Laplace transformable, we have
Proposition 3.6. Let be an eigenvector of corresponding to the eigenvalue . Then .
Proof. Let be an eigenvector of corresponding to the eigenvalue such that the map is bounded. Let and be fixed. Then This implies that and, consequently, .
The following result shows us that the spaces are maximal-unique in a certain sense.
Theorem 3.7. Under the same hypothesis of Theorem 3.5, if is a Banach space such that , and the operator with generates an -regularized family of contractions in , then , and .
Proof. Suppose that , are as in the statement of theorem. Since is the Laplace transform of the -regularized family , we have that for , , and We conclude that for , , that is, . It follows that Hence Finally, this implies that and .
The next result treats the “maximal property”. In order to obtain the analogous result to the resolvent families case, we need more information about the function .
Theorem 3.8. Let be a linear operator defined in a Banach space and , with absolutely convergent Laplace transform for . Assume that for all . Suppose that . Then generates a strongly continuous semigroup of contractions on .
Proof. By Hille-Yosida theorem it is sufficient to have and for all , for some real .
In order to show this, we take in the second part of Lemma 3.4 and obtain Let . Then (3.23) gives Since as , and by the initial value theorem, we obtain for all sufficiently large. Since , we get for sufficiently large, which concludes the proof.
Remark 3.9. From the maximal uniqueness of and under the same hypothesis of preceding theorem we obtain that , and . Note that, in particular, which include [3, Remark 2.10].
Taking we obtain the main result in [3, Theorem 2.5].
Corollary 4.1 (see , Theorem 2.5). Let be a linear operator defined in a Banach space and , with absolutely convergent Laplace transform for . Assume that for all . Then there exist a linear subspace and a norm such that is a Banach space and the equation admits resolvent family of contractions on , where is defined in Lemma 3.4.
Taking or , we obtain from the preceding corollary the following.
Corollary 4.2 (see ). Let be a linear operator on such that . Then there exist a linear subspace and a norm such that is a Banach space and the restriction of to is the infinitesimal generator of a -semigroup of contractions on .
Corollary 4.3 (see ). Let be a linear operator on such that . Let be the operator in defined as above. Then is the infinitesimal generator of a strongly continuous cosine family of contractions on .
Remark 4.4. Applying Theorem 3.5 with we obtain corresponding results for -times integrated semigroups and -times integrated cosine families taking and , respectively.
Suppose that and . In the following examples we search conditions under which the function is bounded. This ensure the non-triviality of the subspace .
Example 4.5. Let us consider and . Let be a closed linear and densely defined operator on a Banach space such that (a), (b),
(c) and .
Let . Then applying Laplace transform, we have that
It follows from (2.4) that
Take with , then . By Remark 2.4, for each , there is a constant such that Therefore From here and part , it follows that is a bounded function for . In particular, by Proposition 3.6. Since , we have that for . It follows from Theorem 3.5 that is the infinitesimal generator of a strongly continuous Laplace transformable -resolvent family of contractions on . Note in particular that if and , then is the generator of -times integrated semigroup on each subspace for . It means that the initial value problem is well posed in the sense that there exists a strongly continuous family of linear operators on a subspace of such that for all initial values there exists a unique classical solution of (4.6).
Example 4.6. Let us consider and when , that is, . Let be a closed linear and densely defined operator on a Banach space such that (a),(b),
Let . Then
Let . First, we consider . In this case, if we take inverse Laplace transform, we obtain
Obviously, this function is bounded for and .
Now, we consider . In this case
Remember that the incomplete Gamma function is defined by
About the asymptotic behavior, we know that (see [23, formula (8.357)]) as and .
It can be verified that the following formulas holds (see [23, formula (2.632)]):
Note that and , so we can apply (4.11). Then, for sufficiently large, we obtain that
The boundedness follows from the fact that . Therefore, for sufficiently large where depends on and . Since , we have that for . It follows from Theorem 3.5 that is the infinitesimal generator of -times integrated cosine function of contractions on for . As in the above example, it now means that the initial value problem is well posed in the sense that there exists a strongly continuous family of linear operators on a nontrivial subspace of such that for all initial values there exists a unique classical solution of (4.15).
The first author is partially supported by a research grant of Banco Santander and Universidad Tecnológica de Bolívar. The second author is partially supported by Proyecto FONDECYT 1100485.
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