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Some Results on Times Integrated Regularized Semigroups
Advances in Difference Equations volume 2011, Article number: 394584 (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 illposed abstract Cauchy problems and characterize the "weak" wellposedness 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 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

(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.
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).
Necessity.
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 ntimes integrated semigroup on , then for , ,
Proof.
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)
is the subgenerator of a singular times integrated semigroup satisfying .

(2)
is the subgenerator of an exponentially bounded times integrated semigroup satisfying
(2.21)Proof.
(1)⇒(2): we set
Since is locally integrable on is welldefined 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
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).
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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Acknowledgments
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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Keywords
 Differential Equation
 Partial Differential Equation
 Ordinary Differential Equation
 Functional Analysis
 Cauchy Problem