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Some Results on -Times Integrated -Regularized Semigroups

Advances in Difference Equations20102011:394584

  • Received: 21 October 2010
  • Accepted: 13 December 2010
  • Published:


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.


  • Differential Equation
  • Partial Differential Equation
  • Ordinary Differential Equation
  • Functional Analysis
  • Cauchy Problem

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 .



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

Authors’ Affiliations

School of Mathematics, Yunnan Normal University, Kunming, 650092, China


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© Fang Li et al. 2011

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