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 [1–6] and some other papers. Also the concept of local -semigroups and their relation with the have been considered in [7–10].

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  2.4(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  2.8], 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  1.1.3], , 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