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Some Results on -Times Integrated -Regularized Semigroups
Advances in Difference Equations volume 2011, Article number: 394584 (2010)
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  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 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.
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
, and ,
In particular, .
Let . If and there exists such that
and is strongly continuous,
for , ,
, , ,
then we say that is a singular -times integrated -semigroup with subgenerator .
Clearly, an exponentially bounded -times integrated -semigroup is a singular -times integrated -semigroup. But the converse is not true.
2. The Main Results
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
, , is that for ,
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).
If and , then is an integrated semigroup in the sense of Bobrowski .
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.
Let , and let be a closed operator satisfying . Assume that is the nonnegative measurable function on . The following two assertions are equivalent:
is the subgenerator of a singular -times integrated -semigroup satisfying .
is the subgenerator of an exponentially bounded -times integrated -semigroup satisfying(2.21)
(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 .
Since is continuously differentiable for , we get
Moreover, for , we have
Thus, is a singular -times integrated -semigroup with subgenerator .
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
for , , ,
for , ,
for , , ,
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 .
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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Li, F., Wang, H. & Qu, Z. Some Results on -Times Integrated -Regularized Semigroups. Adv Differ Equ 2011, 394584 (2011) doi:10.1155/2011/394584
- Differential Equation
- Partial Differential Equation
- Ordinary Differential Equation
- Functional Analysis
- Cauchy Problem