#### Abstract

We obtain the multiplicative perturbation theorems for convoluted -cosine functions (resp., convoluted -semigroups) and -times integrated -cosine functions (resp., -times integrated -semigroups) for . Moreover, we obtain some new results for perturbations on -cosine functions (resp., -semigroups). Some examples are presented.

#### 1. Introduction and Preliminaries

The -times integrated -semigroups, -times integrated -cosine functions () [1–6], -times integrated semigroups (i.e., -semigroups), and -times integrated -cosine functions (i.e., -cosine functions) [5, 7–11] are powerful tools in studying ill-posed abstract Cauchy problems. The convoluted -cosine functions (resp., convoluted -semigroups) are the extension of -times integrated -cosine functions (resp., -times integrated -semigroups), they can be used to deal with more complicated ill-posed abstract Cauchy problems of evolution equations [5, 12–16].

Many researchers studied the perturbations on -cosine functions and -semigroups [17–22]. In [16], Kostić studied the additive perturbations of convoluted -cosine functions and convoluted -semigroups. However, to the authors’ knowledge, few papers can be found in the literature for the multiplicative perturbations on the convoluted -cosine functions (resp., convoluted -semigroups).

In this paper, based on the previously mentioned works we study the multiplicative perturbations on the convoluted -cosine functions and convoluted -semigroups. Moreover, we obtain the corresponding new results for -times integrated -semigroups (resp., -times integrated -cosine functions) (, denotes the nonnegative integers).

Throughout this paper, , , , and denote the positive integers, the nonnegative integers, the real numbers, the complex plane, respectively. denotes a nontrivial complex Banach space, 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 , 0 , where is the Laplace transform of . We define .

The next definition is the convoluted version of Definition 4.1 in Chapter 1 of [5].

*Definition 1 (see [5, 13, 15]). *Let . If and there exists a strongly continuous operator family such that for some , and
then it is said that is a subgenerator of an exponentially bounded -convoluted -cosine function . The operator is called the generator of .

Theorem 2 (see [13–15]). *Let be a strongly continuous, exponentially bounded operator family, and let be a closed operator. Then the statements (i) and (ii) are equivalent, where*(i)*is the subgenerator of a -convoluted -cosine function ,*(ii)* , ,* * , and
*

*Definition 3. *Let . If and there exists a strongly continuous operator family such that for some , and
then it is said that is a subgenerator of an exponentially bounded -convoluted -semigroup . The operator is called the generator of .

Theorem 4. *Let be a strongly continuous, exponentially bounded operator family, and let be a closed operator. Then the assertions (i) and (ii) are equivalent, where*(i)* is the subgenerator of a -convoluted -semigroup ,*(ii)*(1) , ,(2), and
*

*Remark 5 (see [16]). *In Theorems 2 and 4, putting , where denotes the Gamma function, one obtains the classes of -times integrated -cosine functions and -times integrated -semigroups; a -times integrated -cosine function (resp., -times integrated -semigroup) is defined to be a -cosine function (resp., -semigroup). More knowledge for them, we refer the reader to, for example, [1–3, 5, 7–11, 18] and references there in.

Next, we recall the definitions of -times integrated -semigroup and -times integrated -cosine functions ().

*Definition 6 (see [5]). *Let and let . If (resp., ) and there exists a strongly continuous operator family (resp., ) such that (resp., ) for some , and
then it is said that is a subgenerator of an exponentially bounded -times integrated -cosine function (resp., -times integrated -semigroup ) on . If , then (resp., ) is called an exponentially bounded -times integrated -cosine function (resp., -times integrated -semigroup).

We present the definition of -cosine functions which will be used in the proof of Theorem 12.

*Definition 7 (see [1, 5]). *A strongly continuous family of bounded linear operators on is called a -cosine function on , if , and , for all .

#### 2. Main Results

Suppose that is a subgenerator of an exponentially bounded -convoluted -cosine function on , , for any with , we set for some and , where is some function and , with .

We have the following multiplicative perturbation theorem.

Theorem 8. *Suppose that is a subgenerator of an exponentially bounded -convoluted -cosine function on . Let , and is dense in ,
**
If for all , then subgenerates an exponentially bounded -convoluted -cosine function on . *

*Proof. * For all , , is large enough and is small enough, we have

Let be any strongly continuous function with ; we define
Obviously, is continuous on , from (9) and the denseness of , maps into .

It follows from (9) that is bounded. For each , set
Then, , and there exists a constant such that ,
For sufficiently large , we set
Taking Laplace transform of (12), we have
Therefore for ,
Noting (8), for , we have
that is
Then from Definition 1, subgenerates an exponentially bounded -convoluted -cosine function .

Theorem 9. *Suppose is a subgenerator of an exponentially bounded -convoluted -cosine function on , . Let with and let , and is dense in . If for any ,
**
where is a constant, then for some (and all) , , subgenerates an exponentially bounded -convoluted -cosine function on . *

* Proof. * Define the operator functions as follows:

By induction, we obtain (i)for any , , (ii), , , for all .

Define the operator function

Noting that the series is uniformly converge on every compact interval in , we can see that the series (20) is uniformly converge on every compact interval in , so does the operator . It is obvious that and is continuous on for any . Moreover,
For sufficiently large, we set
Next, we show that the following equalities hold:
By induction, it is not difficult to see that
Let
By hypothesis, can be extended to and satisfies
Set
Then from (25) and (27), for sufficiently large. Therefore, the series
converges.

For and , from (25), we have
Similarly, we can prove (24). Now, from Definition 1, we conclude that subgenerates an exponentially bounded -convoluted -cosine function on .

By the proof of Theorems 8 and 9, we immediately obtain the following results for -convoluted -semigroups.

Theorem 10. *Suppose that is a subgenerator of an exponentially bounded -convoluted -semigroup on . is dense in . Let with and let . *(i)*One sets
for some and , where is a function and . If , then subgenerates an exponentially bounded -convoluted -semigroup on provided that for all . *(ii)*If for any ,
where is a constant, then for some (and all) , , subgenerates an exponentially bounded -convoluted -semigroup on . *

*Proof. * (i) For any with , sufficiently large and sufficiently small , we have
where is a constant. The rest part of the proof is exactly the same as the corresponding part of the proof of Theorem 8.

The proof of (ii) is similar to the one of Theorem 9.

In Theorems 8–10, take , we have the following result for -times integrated -cosine function (resp., -times integrated -semigroup).

Theorem 11. *Suppose is a subgenerator of an exponentially bounded -times integrated -cosine function (resp., -times integrated -semigroup ) on . Let with and let , and is dense in . *(i)*One sets
for any with ,
for some and , where is a function. If (resp., ), then subgenerates an exponentially bounded -times integrated -cosine function (resp., -times integrated -semigroup) on provided that for all .*(ii)*If for any ,
where and is a constant,
then for some (and all) , , subgenerates an exponentially bounded -times integrated -cosine function (resp., -times integrated -semigroup) on . *

When , from Theorem 11(ii), we immediately obtain the result of -times integrated -cosine function (resp., -times integrated -semigroup).

Theorem 12. *Let with and let , and is dense in . Suppose that is an exponentially bounded generator of a -cosine function (resp., -semigroup ) on . If for any ,
**
where .
**
for some and , then subgenerates an exponentially bounded -cosine function (resp., -semigroup) on . *

Noting the Definition 7 and the special properties of -cosine functions (resp., -semigroups), we obtain a different result from Theorem 11(i) (when ).

Theorem 13. *Let with and let , and is dense in , (resp., ). Suppose that is an exponentially bounded generator of a -cosine function (resp., -semigroup ) on . If
**
for some and , letting , then for any , subgenerates an exponentially bounded -cosine function (resp., -semigroup) on . *

*Proof. *We prove only for -cosine functions. Choose such that . For any , pick a large enough such that , and then pick a small enough such that , then for all , , we have
where .

Let be any strongly continuous function; we define
Obviously, is continuous on , from (42) and the denseness of , maps into .

It follows from (42) that is bounded. For each , set
Then, , and there exists a constant such that :
For , , we define inductively
Next, we will prove by induction that for any , , for , and that for every , is strongly continuous in and
Indeed for , this is true. Assume that (47) holds for . Then for , we get
Then for , ,
Hence, , , and is continuous in for each . From (48), we have
Therefore, is strongly continuous in and (47) holds for all . Taking Laplace transform of (47), then the conclusion can be proved in a similar way in the proof of Theorem 8.

We can prove the case of -semigroups in a similar way.

#### 3. Examples

*Example 14. *Let

It is well known that there exist positive real numbers and such that

Moreover, generates an exponentially bounded -convoluted semigroup for some , where , then [14, 23]. We set
Obviously, and . Then from Theorem 10(ii) (), subgenerates an exponentially bounded -convoluted semigroup on .

*Example 15. *Let ,
and . Arguing as in [3, Examples 8.1 and 8.2], one gets that is a generator of an exponentially bounded once integrated -semigroup [16].

For and , we set
Then one can simply verify that , and , . Then from Theorem 11(i), subgenerates an exponentially bounded once integrated -semigroup on .

*Example 16. *Let , (,
Then generates a strongly continuous cosine function on . It follows from [5] that generates an exponentially bounded -cosine function on , where .

Set , , , . Define bounded linear operators , as follows:
Let ,
Taking and putting , then generates an exponentially bounded -cosine function on , where
We denote , , , then
and for any , ,
It follows from and that there exist , such that
then
and then (40) is satisfied.

#### Acknowledgments

The authors are grateful to the referee for his/her valuable suggestions. This work was partly supported by the NSF of China (11201413), the NSF of Yunnan Province (2009ZC054M), the Educational Commission of Yunnan Province (2012Z010), and the Foundation of Key Program of Yunnan Normal University.