- Research Article
- Open Access
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 .
Arendt W: Vector-valued Laplace transforms and Cauchy problems. Israel Journal of Mathematics 1987,59(3):327-352. 10.1007/BF02774144
Bobrowski A: On the generation of non-continuous semigroups. Semigroup Forum 1997,54(2):237-252.
Li Y-C, Shaw S-Y:On local -times integrated -semigroups. Abstract and Applied Analysis 2007, 2007:-18.
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/S0002-9939-06-08549-2
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.1997.5909
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/s00233-002-7007-1
Liang J, Xiao T-J:Norm continuity for of linear operator families. Chinese Science Bulletin 1998,43(9):719-723. 10.1007/BF02898945
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.2007.10.045
Tanaka N: Locally Lipschitz continuous integrated semigroups. Studia Mathematica 2005,167(1):1-16. 10.4064/sm167-1-1
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-P
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/s00028-007-0355-2
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.
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.
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.1996.3096
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.
Xiao T-J, Liang J: Approximations of Laplace transforms and integrated semigroups. Journal of Functional Analysis 2000,172(1):202-220. 10.1006/jfan.1999.3545
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/S0024610702003794
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.2007.12.012
The authors are grateful to the referees for their valuable suggestions. This work is supported by the NSF of Yunnan Province (2009ZC054M).