Open Access

Some Results on -Times Integrated -Regularized Semigroups

Advances in Difference Equations20102011:394584

https://doi.org/10.1155/2011/394584

Received: 21 October 2010

Accepted: 13 December 2010

Published: 20 December 2010

Abstract

We present a generation theorem of -times integrated -regularized semigroups and clarify the relation between differentiable -times integrated -regularized semigroups and singular -times integrated -regularized semigroups.

1. Introduction and Preliminaries

In 1987, Arendt [1] studied the -times integrated semigroups, which are more general than semigroups (there exist many operators that generate -times integrated semigroups but not semigroups).

In recent years, the -times integrated -regularized semigroups have received much attention because they can be used to deal with ill-posed abstract Cauchy problems and characterize the "weak" well-posedness of many important differential equations (cf., e.g., [218]).

Stimulated by the works in [2, 57, 9, 1218], in this paper, we present a generation theorem of the -times integrated -regularized semigroups for the case that the domain of generator and the range of regularizing operator are not necessarily dense, and prove that the subgenerator of an exponentially bounded, differentiable -times integrated -regularized semigroup is also a subgenerator of a singular -times integrated -regularized semigroup.

Throughout this paper, is a Banach space; denotes the dual space of ; denotes the space of all linear and bounded operators from to , it will be abbreviated to ; denotes the dual space of . By we denote the space of all continuously differentiable -valued functions on . is the space of all continuous -valued functions on .

All operators are linear. For a closed linear operator , we write , , for the domain, the range, the resolvent set of in a Banach space , respectively.

We denote by the part of in , that is,
(1.1)
The -resolvent set of is defined as:
(1.2)

We abbreviate -times integrated -regularized semigroup to -times integrated -semigroup.

Definition 1.1.

Let be a nonnegative integer. Then is the subgenerator of an exponentially bounded -times integrated -semigroup if for some and there exists a strongly continuous family with for some such that
(1.3)

In this case, is called the exponentially bounded -times integrated -semigroup generated by .

If (resp., ), then is called a generator of an exponentially bounded -times integrated semigroup (resp., -semigroup).

We recall some properties of -times integrated -semigroup.

Lemma 1.2 (see [10, Lemma 3.2]).

Assume that is a subgenerator of an -times integrated -semigroup . Then

  1. (i)

    ,

     
  2. (ii)

    , and ,

     
  3. (iii)

    .

     

In particular, .

Definition 1.3.

Let . If and there exists such that

  1. (i)

    and is strongly continuous,

     
  2. (ii)

    for , ,

     
  3. (iii)

    , , ,

     

then we say that is a singular -times integrated -semigroup with subgenerator .

Remark 1.4.

Clearly, an exponentially bounded -times integrated -semigroup is a singular -times integrated -semigroup. But the converse is not true.

2. The Main Results

Theorem 2.1.

Let , be constants, and let be a closed operator satisfying . Assume that is the nonnegative measurable function on . A necessary and sufficient condition for is the subgenerator of an -times integrated -semigroup satisfying

  1. (A1)

    ,

     
  2. (A2)
    , , is that for ,
    1. (i)

      ,

       
    2. (ii)

      , .

       
     

Proof.

Sufficiency. Let . Set
(2.1)
For , we have
(2.2)
 Using this fact together with Widder's classical theorem, it is not difficult to see that the existence of a measurable function with , a.e., ( ) such that
(2.3)
Let , , . In view of the convolution theorem for Laplace transforms and from (2.3), we have
(2.4)
Using the uniqueness of Laplace transforms and the linearity of for each , , we can see that for each , is linear and
(2.5)
Hence for all , there exists such that
(2.6)
(2.7)
(2.8)
Denote by the quotient mapping. Since , we deduce
(2.9)

It follows from the uniqueness theorem for Laplace transforms that , that is, .

Combining (2.7) and (2.8) yields that is strongly continuous and
(2.10)

Now, we conclude that is an -times integrated -semigroup satisfying (A2). Assertion (A1) is immediate, by (2.8) and (i).

Necessity.

Let . Since is an -times integrated -semigroup on , we have
(2.11)
for . Noting that and , we find
(2.12)
Then for any and , we obtain
(2.13)
Therefore, there exists a measurable function on with (a.e.) such that
(2.14)
Furthermore, by calculation, we have
(2.15)

Assertion (i) is an immediate consequence of (2.11) and (A1).

Remark 2.2.

If and , then is an integrated semigroup in the sense of Bobrowski [2].

Theorem 2.3.

Let , be constants, and let be a closed operator satisfying . Assume that is a subgenerator of an -times integrated -semigroup and satisfies (ii) of Theorem 2.1 and . If is a subgenerator of an n-times integrated -semigroup on , then for , ,
(2.16)
(2.17)

Proof.

For , , set as follows:
(2.18)

Since is strongly continuous on , is strongly continuous on .

Fixing , we have
(2.19)
It follows from the uniqueness of Laplace transforms that , . So we get (2.16). By the hypothesis , we see
(2.20)

and the proof is completed.

Now, we study the relation between differentiable -times integrated -semigroups and singular -times integrated -semigroups.

Theorem 2.4.

Let , and let be a closed operator satisfying . Assume that is the nonnegative measurable function on . The following two assertions are equivalent:

  1. (1)

    is the subgenerator of a singular -times integrated -semigroup satisfying .

     
  2. (2)
    is the subgenerator of an exponentially bounded -times integrated -semigroup satisfying
    (2.21)

    Proof.

     
(1)(2): we set
(2.22)

Since is locally integrable on is well-defined for any . It is easy to check that belongs to .

For every , since
(2.23)

we deduce that is exponentially bounded.

Moreover, for , we have
(2.24)

Thus is the desired semigroup in (2).

(2)(1): for any , we set
(2.25)

Then and .

Noting that
(2.26)
we find
(2.27)
Since is continuously differentiable for , we get
(2.28)
Moreover, for , we have
(2.29)

Thus, is a singular -times integrated -semigroup with subgenerator .

Theorem 2.5.

Let , be constants, and let be a closed operator satisfying . Let be the function in Theorem 2.4. If is the subgenerator of a singular -times integrated -semigroup , satisfying , and satisfies
(2.30)

then

  1. (1)

    for , , ,

     
  2. (2)

    for , ,

     
  3. (3)

    for , , ,

     
  4. (4)

    for ,   if and only if   ,

     

where and are the symbols mentioned in Theorem 2.3.

Proof.

It follows from Theorems 2.3 and 2.4 that subgenerates an -times integrated -semigroup , which is continuously differentiable for and satisfies (2.16) and (2.17).

Differentiating (2.16) with respect to , we obtain
(2.31)

This completes the proof of (1).

To show (2), for , we have
(2.32)
Letting , we get
(2.33)
To show (3), for , since , it follows from (2.17) that is continuous for , thus, we have
(2.34)

Obviously, the equality above is true for .

Noting that
(2.35)
we can deduce that implies , and from
(2.36)

assertion (4) is immediate if we note that implies .

Declarations

Acknowledgments

The authors are grateful to the referees for their valuable suggestions. This work is supported by the NSF of Yunnan Province (2009ZC054M).

Authors’ Affiliations

(1)
School of Mathematics, Yunnan Normal University

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© Fang Li et al. 2011

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