- Research Article
- Open Access

# Some Results on -Times Integrated -Regularized Semigroups

- Fang Li
^{1}, - Huiwen Wang
^{1}Email author and - Zihai Qu
^{1}

**2011**:394584

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

© Fang Li et al. 2011

**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.

## Keywords

- 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., [2–18]).

Stimulated by the works in [2, 5–7, 9, 12–18], 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 abbreviate -times integrated -regularized semigroup to -times integrated -semigroup.

Definition 1.1.

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

- (i)
,

- (ii)
, and ,

- (iii)
.

In particular, .

Definition 1.3.

Let . If and there exists such that

- (i)
and is strongly continuous,

- (ii)
for , ,

- (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

- (A1)
,

- (A2), , is that for ,
- (i)
,

- (ii)
, .

- (i)

Proof.

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

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

Necessity.

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.

Proof.

Since is strongly continuous on , is strongly continuous on .

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

- (2)
Proof.

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

we deduce that is exponentially bounded.

Thus is the desired semigroup in (2).

Then and .

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

Theorem 2.5.

then

- (1)
for , , ,

- (2)
for , ,

- (3)
for , , ,

- (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).

This completes the proof of (1).

Obviously, the equality above is true for .

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

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