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Journal of Function Spaces and Applications
Volume 2013 (2013), Article ID 579326, 8 pages
http://dx.doi.org/10.1155/2013/579326
Research Article

On Perturbation of Convoluted -Regularized Operator Families

1School of Mathematics, Yunnan Normal University, Kunming 650092, China
2Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China
3Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical Sciences, Fudan University, Shanghai 200433, China
4Department of Mathematics, Central China Normal University, Wuhan 430079, China

Received 3 February 2013; Accepted 28 March 2013

Academic Editor: James H. Liu

Copyright © 2013 Fang Li 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

Of concern are two classes of convoluted -regularized operator families: convoluted -cosine operator families and convoluted -semigroups. We obtain new and general multiplicative and additive perturbation theorems for these convoluted -regularized operator families. Two examples are given to illustrate our abstract results.

1. Introduction

It is well known that the cosine operator families (resp., the semigroups) and the fractionally integrated -cosine operator families (resp., integrated -semigroups) are important tools in studying incomplete second-order (resp., first-order) abstract Cauchy problems (cf., e.g., [117]). As an extension of the cosine operator families (resp., the semigroups) as well as the fractionally integrated -cosine operator families (resp., integrated -semigroups), the convoluted -cosine operator families (resp., convoluted -semigroups) (cf., e.g., [15, 18, 19]) are also good operator families in dealing with ill-posed incomplete second order (resp. first order) abstract Cauchy problems.

In last two decades, there are many works on the perturbations on the -regularized operator families (cf., e.g., [16, 2024]). In the present paper, we will study the multiplicative and additive perturbation for two classes of convoluted -regularized operator families: convoluted -cosine operator families and convoluted -semigroups, and our purpose is to obtain some new and general perturbation theorems for these convoluted -regularized operator families and to make the results new even for convoluted -times integrated -cosine operator families (resp., convoluted -times integrated -semigroups) (, where denotes the nonnegative integers).

Throughout this paper, , , denote the set of positive integers, the real numbers, and the complex plane, respectively. denotes a nontrivial complex Banach space, and denotes the space of bounded linear operators from into . In the sequel, we assume that is an injective operator. denotes the space of all continuous functions from to . For a closed linear operator on , its domain, range, resolvent set, and the -resolvent set are denoted by , , , and , respectively, where is defined by is an exponentially bounded function, and for , where is the Laplace transform of as in the monograph [15]. We define

Next, we recall some notations and basic results from [15, 19] about the convoluted -cosine operator families and convoluted -semigroups.

The following definition is the convoluted version of [15, Chapter 1, Definition 4.1].

Definition 1. Let and . Let be a strongly continuous operator family such that for some , and Then, is called a subgenerator of the exponentially bounded -convoluted -cosine operator family . Moreover, the operator is called the generator of the .

Proposition 2. Let be a closed operator and a strongly continuous, exponentially bounded operator family. Then is the subgenerator of a -convoluted -cosine operator family if and only if(1), ;(2), , and

Remark 3. If is the subgenerator of a -convoluted -cosine operator family, then .

Definition 4. Let and . Let be a strongly continuous operator family such that for some , and Then, is called a subgenerator of an exponentially bounded -convoluted -semigroup . Moreover, the operator is called the generator of the .

Proposition 5. Let be a closed operator, and a strongly continuous, exponentially bounded operator family. Then, is the subgenerator of a -convoluted -semigroup if and only if(1), ;(2), , and

Remark 6. From [15], we know that the -cosine operator families (resp., -semigroups) are exactly the -times integrated -cosine operator families (resp., the -times integrated -semigroups). Let be the well-known Gamma function, and Then, by Propositions 2 and 5, we get results for the -times integrated -cosine operator families (resp., -times integrated -semigroups) as well as -cosine operator families (resp., -semigroups). For more information on various operator families, we refer the reader to, for example, [3, 68, 14, 15, 17, 22] and references therein.

2. Multiplicative Perturbation Theorems

Lemma 7. Suppose that is a subgenerator of an exponentially bounded -convoluted -cosine operator family on . If , then .

Proof. For any and , let Then, Therefore, This means that . Thus, by Remark 3, we see that .

Theorem 8. Let be a closed linear operator on and . Assume that there exists an injective operator on satisfying , . Then, the following statements hold.(1)If subgenerates an exponentially bounded -convoluted -cosine operator family on , then subgenerates an exponentially bounded -convoluted -cosine operator family on .(2)If subgenerates an exponentially bounded -convoluted -cosine operator family on and , then generates an exponentially bounded -convoluted -cosine operator family on .

Proof. Assume that subgenerates an exponentially bounded -convoluted -cosine operator family on .
In this case, it is easy to see that for any , the operator is bounded, since where . Now, for each , we define a bounded linear operator as follows: Clearly, the graph norms of and are equivalent. Therefore, noting that subgenerates an exponentially bounded -convoluted -cosine operator family on , we obtain, for every , and , that there exists a constant such that Hence, is strongly continuous.
Similarly, we can prove that is exponentially bounded; that is, there exists a constant such that
As in the monograph [15], we write Then, by (16), we have Hence, Furthermore, On the other hand, for each , , we obtain Therefore, It follows from (20) that Thus, by Definition 1, we know that subgenerates an exponentially bounded -convoluted -cosine operator family on .
(2) Assume that subgenerates an exponentially bounded -convoluted -cosine operator family on and , and let It is not hard to see that is closed operator on and Since subgenerates an exponentially bounded -convoluted -cosine operator family on , we know from that the operator subgenerates an exponentially bounded -convoluted -cosine operator family on .
Noting that and in view of Lemma 7, we see that generates an exponentially bounded -convoluted -cosine operator family on .

Theorem 9. Let be a subgenerator of an exponentially bounded -convoluted -cosine operator family on , , and . Suppose that(H1) there exists an operator such that is Laplace transformable, and (H2) for any , , and where is a constant;(H3) there exists an injective operator such that and . Then,(1) subgenerates an exponentially bounded -convoluted -cosine operator family,(2)if , then generates an exponentially bounded -convoluted -cosine operator family;(3)if and , , then generates an exponentially bounded -convoluted -cosine operator family on .

Proof. For each , , define Then, the operator family has the following properties:(i)for any , ; (ii). Therefore, the following series is uniformly convergent on every compact interval in , and we set Clearly, where , and Moreover, As in the monograph [15], we write, for sufficiently large , Thus, by (5), we have This implies that Let Then, for large , we have So, for sufficiently large , This means that the operator is invertible.
On the other hand, since and are injective, and we infer that is injective. This together with (40) implies that By Definition 1, we know that subgenerates an exponentially bounded -convoluted -cosine operator family on .
By the proof of , we see that the operator is invertible, and implies that In view of Lemma 7, we get
By virtue of Theorem 8 (2), we have the conclusion.

Remark 10. It is easy to see that if we take then (H1) is satisfied.
(2) In Theorem 9, if we take then we obtain the perturbations for -times integrated -cosine operator families.
(3) In Theorem 9, if we take and , then we have the multiplicative perturbations on the exponentially bounded -cosine operator families.
By Theorem 9, we can immediately deduce the following theorem on -convoluted -semigroups.

Theorem 11. Let be a subgenerator of an exponentially bounded -convoluted -semigroup on , and . Suppose that(H1) there exists an operator such that is Laplace transformable, and (H2) for any , , and where is a constant;(H3) there exists an injective operator such that and .Then,(1) subgenerates an exponentially bounded -convoluted -semigroup on ;(2)if , then generates an exponentially bounded -convoluted -semigroup on .(3)if , then generates an exponentially bounded -convoluted -semigroup on .

Remark 12. (1) In Theorem 11, if we take then we obtain the perturbations for -times integrated -semigroups.
(2) In Theorem 11, if we take and , then we have the multiplicative perturbations on the exponentially bounded -semigroups.

3. Additive Perturbation Theorem

Theorem 13. Let , , and there exists an injective operator such that and .(i) Suppose that is a subgenerator of an exponentially bounded -convoluted -cosine operator family on . If there exists an operator such that is Laplace transformable, and then subgenerates an exponentially bounded -convoluted -cosine operator family on , where (ii) Suppose that is a subgenerator of an exponentially bounded -convoluted -semigroup on . If there exists an operator such that is Laplace transformable, and then subgenerates an exponentially bounded -convoluted -semigroup on , where

Proof. Replacing (37) with the following equality: and by the arguments similar to those in the proof of Theorem 9, we can prove (i).
Point (ii) can also be deduced by a similar way.

Remark 14. In Theorem 13, if we take then we obtain an additive perturbation theorem for the exponentially bounded -times integrated -cosine operator families (resp., -times integrated -semigroups) as well as -cosine operator families (resp., -times integrated -semigroup).

4. Examples

Example 1. Let where As in [22, Examples 8.1 and 8.2], we can prove that is the generator of an exponentially bounded once integrated -semigroup ([15]).
Define for every , and . Then, we can simply verify , , and Therefore, taking and using Remark 12 (1), we know that subgenerates an exponentially bounded once integrated -semigroup on .

Example 2. Let , , It follows from [15] that generates an exponentially bounded -cosine operator family on , where . Moreover, it is well known that generates a strongly continuous cosine operator family on .
Let and define , as follows: Set , Clearly, and . Take Then, generates an exponentially bounded -cosine operator family on , where Hence, where Therefore, we have, for each , Since we see that there exist , such that Consequently, if there exists an injective operator such that and , then taking and using Remark 14, we know that subgenerates a -cosine operator family on .
Moreover, it is not hard to see that there exist , such that Hence, if there exists an injective operator such that and , then by Remark 10 (3) (), we know that generates a -cosine operator family on .

Acknowledgments

The authors would like to thank the referees very much for helpful suggestions. The work was supported partly by the NSF of China (11201413, 11071042, and 11171210), the Educational Commission of Yunnan Province (2012Z010), and the Research Fund for Shanghai Key Laboratory for Contemporary Applied Mathematics (08DZ2271900).

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