- Research Article
- Open Access
Some Results on -Times Integrated -Regularized Semigroups
© Fang Li et al. 2011
- Received: 21 October 2010
- Accepted: 13 December 2010
- Published: 20 December 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.
- Differential Equation
- Partial Differential Equation
- Ordinary Differential Equation
- Functional Analysis
- Cauchy Problem
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 abbreviate -times integrated -regularized semigroup to -times integrated -semigroup.
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.
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
- (A2), , is that for ,
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).
Assertion (i) is an immediate consequence of (2.11) and (A1).
If and , then is an integrated semigroup in the sense of Bobrowski .
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.
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 .
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 .
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).
This completes the proof of (1).
Obviously, the equality above is true for .
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).
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