- 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

Â© 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., [2â€“18]).

Stimulated by the works in [2, 5â€“7, 9, 12â€“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 , â€‰â€‰if and only ifâ€‰â€‰ ,

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).

## Authorsâ€™ Affiliations

## References

- Arendt W:
**Vector-valued Laplace transforms and Cauchy problems.***Israel Journal of Mathematics*1987,**59**(3):327-352. 10.1007/BF02774144MathSciNetView ArticleMATHGoogle Scholar - Bobrowski A:
**On the generation of non-continuous semigroups.***Semigroup Forum*1997,**54**(2):237-252.MathSciNetView ArticleMATHGoogle Scholar - Li Y-C, Shaw S-Y:
**On local**-times integrated**-semigroups.***Abstract and Applied Analysis*2007,**2007:**-18.Google Scholar - Li Y-C, Shaw S-Y:
**On characterization and perturbation of local****-semigroups.***Proceedings of the American Mathematical Society*2007,**135**(4):1097-1106. 10.1090/S0002-9939-06-08549-2MathSciNetView ArticleMATHGoogle Scholar - Liang J, Xiao T-J:
**Integrated semigroups and higher order abstract equations.***Journal of Mathematical Analysis and Applications*1998,**222**(1):110-125. 10.1006/jmaa.1997.5909MathSciNetView ArticleMATHGoogle Scholar - Liang J, Xiao T-J:
**Wellposedness results for certain classes of higher order abstract Cauchy problems connected with integrated semigroups.***Semigroup Forum*1998,**56**(1):84-103. 10.1007/s00233-002-7007-1MathSciNetView ArticleMATHGoogle Scholar - Liang J, Xiao T-J:
**Norm continuity for****of linear operator families.***Chinese Science Bulletin*1998,**43**(9):719-723. 10.1007/BF02898945MathSciNetView ArticleMATHGoogle Scholar - Nagaoka K:
**Generation of the integrated semigroups by superelliptic differential operators.***Journal of Mathematical Analysis and Applications*2008,**341**(2):1143-1154. 10.1016/j.jmaa.2007.10.045MathSciNetView ArticleMATHGoogle Scholar - Tanaka N:
**Locally Lipschitz continuous integrated semigroups.***Studia Mathematica*2005,**167**(1):1-16. 10.4064/sm167-1-1MathSciNetView ArticleMATHGoogle Scholar - Thieme HR:
**"Integrated semigroups" and integrated solutions to abstract Cauchy problems.***Journal of Mathematical Analysis and Applications*1990,**152**(2):416-447. 10.1016/0022-247X(90)90074-PMathSciNetView ArticleMATHGoogle Scholar - Thieme HR:
**Differentiability of convolutions, integrated semigroups of bounded semi-variation, and the inhomogeneous Cauchy problem.***Journal of Evolution Equations*2008,**8**(2):283-305. 10.1007/s00028-007-0355-2MathSciNetView ArticleMATHGoogle Scholar - Xiao T-J, Liang J:
**Integrated semigroups, cosine families and higher order abstract Cauchy problems.**In*Functional Analysis in China, Mathematics and Its Applications*.*Volume 356*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1996:351-365.View ArticleGoogle Scholar - Xiao T-J, Liang J:
**Widder-Arendt theorem and integrated semigroups in locally convex space.***Science in China. Series A*1996,**39**(11):1121-1130.MathSciNetMATHGoogle Scholar - Xiao T-J, Liang J:
**Laplace transforms and integrated, regularized semigroups in locally convex spaces.***Journal of Functional Analysis*1997,**148**(2):448-479. 10.1006/jfan.1996.3096MathSciNetView ArticleMATHGoogle Scholar - Xiao T-J, Liang J:
*The Cauchy Problem for Higher-Order Abstract Differential Equations, Lecture Notes in Mathematics*.*Volume 1701*. Springer, Berlin, Germany; 1998:xii+301.View ArticleGoogle Scholar - Xiao T-J, Liang J:
**Approximations of Laplace transforms and integrated semigroups.***Journal of Functional Analysis*2000,**172**(1):202-220. 10.1006/jfan.1999.3545MathSciNetView ArticleMATHGoogle Scholar - Xiao T-J, Liang J:
**Higher order abstract Cauchy problems: their existence and uniqueness families.***Journal of the London Mathematical Society*2003,**67**(1):149-164. 10.1112/S0024610702003794MathSciNetView ArticleMATHGoogle Scholar - Xiao T-J, Liang J:
**Second order differential operators with Feller-Wentzell type boundary conditions.***Journal of Functional Analysis*2008,**254**(6):1467-1486. 10.1016/j.jfa.2007.12.012MathSciNetView ArticleMATHGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.