#### Abstract

Suppose that is a Banach space and is an injective operator in , the space of all bounded linear operators on . In this note, a two-parameter -semigroup (regularized semigroup) of operators is introduced, and some of its properties are discussed. As an application we show that the existence and uniqueness of solution of the 2-abstract Cauchy problem , , , is closely related to the two-parameter -semigroups of operators.

#### 1. Introduction and Preliminaries

Suppose that is a Banach space and is a linear operator in with domain and range . For a given , the abstract Cauchy problem for with the initial value consists of finding a solution to the initial value problem where by a solution we mean a function , which is continuous for , continuously differentiable for , for , and is satisfied.

If , the space of all bounded linear operators on X, is injective, then a one-parameter -semigroup (regularized semigroup) of operators is a family for which , , and for each , the mapping is continuous. An operator with and where, for , is called the infinitesimal generator of .

Regularized semigroups and their connection with the have been studied in [1β6] and some other papers. Also the concept of local -semigroups and their relation with the have been considered in [7β10].

In Section 2, we introduce the concept of two-parameter regularized semigroups of operators and their generator. Some basic properties of two-parameter regularized semigroups and their relation with the generators are studied in this section.

In Section 3, two-parameter abstract Cauchy problems are considered. It is proved that the existence and uniqueness of its solutions is closely related with two-parameter regularized semigroups of operators.

#### 2. Two-Parameter Regularized Semigroups

In this section we introduce two-parameter regularized semigroup and its generator on Banach spaces. Then some properties of two-parameter regularized semigroups are studied.

*Definition 2.1. *Suppose that is a Banach space and is an injective operator. A family is called a two-parameter regularized semigroup (or two parameter -semigroup) if(i),(ii), for all ,(iii), for all .It is called exponentially bounded if , for some .

Suppose that is a two-parameter -semigroup. Put and , then it is easy to see that these families are two commuting one-parameter -semigroups such that . Also and commute with . If and are their generators, respectively, then we will think of as the generator of .

From the one-parameter case (see [8]), one can prove that , and , .

Also if and are two commuting one-parameter -semigroups, then one can see that is a two-parameter -semigroup of operators.

The following is an example of a two-parameter -semigroup which is not exponentially bounded.

*Example 2.2. *Let , and , , then is a two-parameter -semigroup which is not exponentially bounded.

In the following theorem we can see some elementary properties of a two-parameter -semigroup.

Theorem 2.3. *Suppose that is a two-parameter -semigroup with the infinitesimal generator . Then, one has the following.*(i)*For each and for every , , is in . Also
*(ii)*For each , and for every , and ; furthermore
*(iii)* and and are closed.*(iv)*For any , and each , and are in . Also for this , and ,
*(v)*For any , is a one-parameter -semigroup whose generator is an extension of .*

*Proof. *To prove (i), suppose . First we note that for any ,
Thus
which tends to as . This implies that is in and
A similar argument implies that it is in and
For the second part, from the continuity of we have
Now the fact that is injective completes the proof of this part.

The proof of (ii) has a process similar to the first part of (i).

To prove (iii), we first note that and are closed as a trivial consequence of the one-parameter case (see [2]). For any we saw that
which tends to , as . This implies that .

To prove (iv), we let . If and , there is such that
Hence
which is in the , and this implies that is in , similarly it is in .

Now from [2, Theorem β2.4(b)], for , from the fact that is in ,
On the other hand from the part (ii) and closedness of ,
which implies that exists. Hence from the continuity of
But is injective so
The second one is similar.

To prove (v), first we note that is a one-parameter -semigroup. Now if ,
Now the fact that is injective implies that

For an exponentially bounded one-parameter -semigroup with the generator , from [1] the existence of is guaranteed for sufficiently large . Now we have the following lemma for one-parameter -semigroups of operators which is similar to the Yosida-approximation theorem for strongly continuous semigroups. This will be applied in our study of two-parameter regularized semigroups.

Lemma 2.4. *Let be a one-parameter -semigroup satisfying the condition , for some and , with the generator . If for , , then one has the following.*(i)*For any , , , and so is bounded. Also is a one-parameter -semigroup which is exponentially bounded.*(ii)*For any , and for all , . Also if is dense in , then the first equality holds on .*(iii)*For any , .*

*Proof. *The first inequality of (i) is trivial. From [2, Lemma β2.8], we know that for any , ; thus,
This implies our desired equality.

For the second part, first we show that . For this we note that
This and the first part imply that . Now we prove the -semigroup properties of . Trivially . Also from the last equality,
The fact that , , is a bounded operator trivially implies that is exponentially bounded. Now the continuity of the mapping at zero implies the strongly continuity of .

To prove (ii), for , from (i) and the fact that is closed, we have
The continuity of and implies that for any , .

Now for ,
For the last part of (ii), if has a dense range, then by [8, Lemma β1.1.3], , and so , which means that .

To prove (iii), for any , we have
This and the previous part prove the existence of

Using this theorem we may find the following approximation theorem for two-parameter regularized semigroups.

Corollary 2.5. *Suppose that is the infinitesimal generator of an exponentially bounded two-parameter -semigroup , then for each ,
*

For exponentially bounded -semigroup satisfying , with the infinitesimal generator , define and , where . From the previous Lemma and are bounded operators.

Theorem 2.6. *(i) Let be the generator of an exponentially bounded two-parameter -semigroup, then for large enough , *(ii)*Let be the generator of an exponentially bounded two-parameter -semigroup, then , and for ,
*(iii)*Suppose that and are the generators of two exponentially bounded one-parameter -semigroups and , respectively. If their resolvents commute and is dense in , then is a two-parameter -semigroup.*

*Proof. *The proof of (i) follows trivially from the properties of two-parameter -semigroups.

To prove (ii), we let ; from the strongly continuity of and the fact that is closed, we have
However, is injective, and this completes the proof of (i).

To prove (iii), from our hypothesis, for sufficiently large , , we know that
By Lemma 2.4, and , thus . From (iii) of Lemma 2.4, for each ,
So
Now the continuity of and and the fact that imply that for each , . Thus
On the other hand , which completes the proof.

If and are two closed operators on , then with , , is a Banach space.

Proposition 2.7. *Suppose that is injective and is a two-parameter -semigroup with the generator . Then defines a two-parameter -semigroup, with the generator , where , and , are the part of and on , respectively.*

*Proof. *The -semigroup properties of are obvious. Let be the generator of ; we show that and . First we note that
Let . So we have
These show that in , that is, and . Hence . Conversely, if , then
so . Hence and .

A similar argument shows that , which completes the proof.

#### 3. Two-Parameter Abstract Cauchy Problems

Suppose that , , is linear operator. Consider the following two-parameter Cauchy problem: We mean by a solution a continuous Banach-valued function which has continuous partial derivative and satisfies 2-.

In this section first we prove that if is the infinitesimal generator of a two-parameter -semigroup of operators, then 2- has a unique solution for any . Next it is proved that under some condition on , existence and uniqueness of solutions of 2-, for every , imply that this unique solution is induced by a two-parameter regularized semigroup.

Theorem 3.1. *Suppose that an extension of is the generator of a two-parameter -semigroup , then 2- has the unique solution , for all .*

*Proof. *The fact that is a solution of 2- is obvious from Theorem 2.3. It is enough to show that 2- has the unique solution , for the initial value . From one-parameter case (see [2]), we know that the systems
have the unique solution zero. Now if is a solution of 2-, then
are two solutions of (3.2), for the initial value , since
The uniqueness of solution in one-parameter case implies that . So
Also and are two solutions of (3.3) for the initial value . From the uniqueness of solution in (3.3), , for all . Thus

The uniqueness of solution 2-, for all , also leads us to a two-parameter -semigroup. This will be shown in the following theorem.

In this theorem and have their meaning in Proposition 2.7.

Theorem 3.2. *Suppose that is injective and are two closed operators satisfying
**
If, for each , the Cauchy problem 2- has a unique solution , then there exists a two-parameter -semigroup on such that . Moreover, the infinitesimal generator of is a restriction of , where and are the part of and on , respectively.*

*Proof. *Suppose that, for any , 2- has a unique solution . For and , define .

From the uniqueness of solution is a well-defined and linear operator on and
By uniqueness of solutions one can see that

We are going to show that is a bounded operator on . Let . Define the mapping by . Obviously is linear. We claim that this mapping is closed. Suppose that , and in with its usual supremum norm. From the Cauchy problem we know that
Letting , we obtain
for any . Now define on by
One can see that is a solution of 2-. Indeed from (3.12)
Also (3.12) and the fact that commutes with and imply that
Similarly
Uniqueness of the solution implies that
In particular for and ,
The fact that is injective implies that , which shows that is closed operator.

By the Closed Graph Theorem is a continuous operator from Banach space into the Banach space . So if in , then in ; thus for each ,
But and were arbitrary; hence is continuous for any . Also for every , is continuous on ; that is, is strongly continuous family of operators.

Now let be its infinitesimal generator and , then
which implies that , but
Hence . The injectivity of implies that . Thus and . This shows that is a restriction of . Similarly one can see that is a restriction of , which completes the proof.

We conclude this section with a simple example as an application of our discussion. Consider the following sequence of initial value problems:

Suppose that , the space of all complex sequences in which vanish at infinity. Now define linear operators and in and operator on as follows: Using these operators the initial value problem (3.22) can be rewrite as follows: where and . One can easily see that is the generator of the following two-parameter -semigroup: on . Hence for every , by Theorem 3.1, the abstract Cauchy problem (3.24) has the unique solution

This implies that for each , is a solution of (3.22).

#### Acknowledgment

The author is grateful to the referees for their very useful suggestions which helped him to improve the presentation considerably.