Some Results on -Times Integrated -Regularized Semigroups

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 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,

(1.1)

The -resolvent set of is defined as:

(1.2)

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

(1.3)

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) , .

Proof.

Sufficiency. Let . Set

(2.1)

For , we have

(2.2)

 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

(2.3)

Let , , . In view of the convolution theorem for Laplace transforms and from (2.3), we have

(2.4)

Using the uniqueness of Laplace transforms and the linearity of for each , , we can see that for each , is linear and

(2.5)

Hence for all , there exists such that

(2.6)

(2.7)

(2.8)

Denote by the quotient mapping. Since , we deduce

(2.9)

It follows from the uniqueness theorem for Laplace transforms that , that is, .

Combining (2.7) and (2.8) yields that is strongly continuous and

(2.10)

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

(2.11)

for . Noting that and , we find

(2.12)

Then for any and , we obtain

(2.13)

Therefore, there exists a measurable function on with (a.e.) such that

(2.14)

Furthermore, by calculation, we have

(2.15)

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 n-times integrated -semigroup on , then for , ,

(2.16)

(2.17)

Proof.

For , , set as follows:

(2.18)

Since is strongly continuous on , is strongly continuous on .

Fixing , we have

(2.19)

It follows from the uniqueness of Laplace transforms that , . So we get (2.16). By the hypothesis , we see

(2.20)

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

(2.22)

Since is locally integrable on is well-defined for any . It is easy to check that belongs to .

For every , since

(2.23)

we deduce that is exponentially bounded.

Moreover, for , we have

(2.24)

Thus is the desired semigroup in (2).

(2)⇒(1): for any , we set

(2.25)

Then and .

Noting that

(2.26)

we find

(2.27)

Since is continuously differentiable for , we get

(2.28)

Moreover, for , we have

(2.29)

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

(2.30)

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

(2.31)

This completes the proof of (1).

To show (2), for , we have

(2.32)

Letting , we get

(2.33)

To show (3), for , since , it follows from (2.17) that is continuous for , thus, we have

(2.34)

Obviously, the equality above is true for .

Noting that

(2.35)

we can deduce that implies , and from

(2.36)

assertion (4) is immediate if we note that implies .