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

Fang Li,¹ Jin Liang,² Ti-Jun Xiao,³ and Jun Zhang⁴

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., [1–17]). 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, 20–24]). 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, 6–8, 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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