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Some Results on -Times Integrated -Regularized Semigroups

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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,


The -resolvent set of is defined as:


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


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)

      , .


Sufficiency. Let . Set


For , we have


 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


Let , , . In view of the convolution theorem for Laplace transforms and from (2.3), we have


Using the uniqueness of Laplace transforms and the linearity of for each , , we can see that for each , is linear and


Hence for all , there exists such that


Denote by the quotient mapping. Since , we deduce


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

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


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


Let . Since is an -times integrated -semigroup on , we have


for . Noting that and , we find


Then for any and , we obtain


Therefore, there exists a measurable function on with (a.e.) such that


Furthermore, by calculation, we have


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 , ,



For , , set as follows:


Since is strongly continuous on , is strongly continuous on .

Fixing , we have


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


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



(1)(2): we set


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

For every , since


we deduce that is exponentially bounded.

Moreover, for , we have


Thus is the desired semigroup in (2).

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


Then and .

Noting that


we find


Since is continuously differentiable for , we get


Moreover, for , we have


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



  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.


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


This completes the proof of (1).

To show (2), for , we have


Letting , we get


To show (3), for , since , it follows from (2.17) that is continuous for , thus, we have


Obviously, the equality above is true for .

Noting that


we can deduce that implies , and from


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


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

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Correspondence to Huiwen Wang.

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Li, F., Wang, H. & Qu, Z. Some Results on -Times Integrated -Regularized Semigroups. Adv Differ Equ 2011, 394584 (2011) doi:10.1155/2011/394584

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  • Differential Equation
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