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Abstract and Applied Analysis
Volume 2010, Article ID 785428, 13 pages
http://dx.doi.org/10.1155/2010/785428
Research Article

On Regularized Quasi-Semigroups and Evolution Equations

Department of Mathematics, Sabzevar Tarbiat Moallem University, P.O. Box 397, Sabzevar, Iran

Received 26 November 2009; Accepted 16 April 2010

Academic Editor: Wolfgang Ruess

Copyright © 2010 M. Janfada. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We introduce the notion of regularized quasi-semigroup of bounded linear operators on Banach spaces and its infinitesimal generator, as a generalization of regularized semigroups of operators. After some examples of such quasi-semigroups, the properties of this family of operators will be studied. Also some applications of regularized quasi-semigroups in the abstract evolution equations will be considered. Next some elementary perturbation results on regularized quasi-semigroups will be discussed.

1. Introduction and Preliminaries

The theory of quasi-semigroups of bounded linear operators, as a generalization of strongly continuous semigroups of operators, was introduced in 1991 [1], in a preprint of Barcenas and Leiva. This notion, its elementary properties, exponentially stability, and some of its applications in abstract evolution equations are studied in [25]. The dual quasi-semigroups and the controllability of evolution equations are also discussed in [6].

Given a Banach space , we denote by the space of all bounded linear operators on . A biparametric commutative family is called a quasi-semigroup of operators if for every and , it satisfies(1), the identity operator on , (2), (3)(4), for some continuous increasing mapping .

Also regularized semigroups and their connection with abstract Cauchy problems are introduced in [7] and have been studied in [812] and many other papers.

We mention that if is an injective operator, then a one-parameter family is called a -semigroup if for any it satisfies and .

In this paper we are going to introduce regularized quasi-semigroups of operators.

In Section 2, some useful examples are discussed and elementary properties of regularized quasi-semigroups are studied.

In Section 3 regularized quasi-semigroups are applied to find solutions of the abstract evolution equations. Also perturbations of the generator of regularized quasi-semigroups are also considered in this section. Our results are mainly based on the work of Barcenas and Leiva [1].

2. Regularized Quasi-Semigroups

Suppose is a Banach space and is a two-parameter family of operators in . This family is called commutative if for any ,

Definition 2.1. Suppose is an injective bounded linear operator on Banach space . A commutative two-parameter family in is called a regularized quasi-semigroups (or -quasi-semigroups) if (), for any ; (), ; () is strongly continuous, that is, ()there exists a continuous and increasing function , such that for any , . For a -quasi-semigroups on Banach space , let be the set of all for which the following limits exist in the range of : Now for and , define is called the infinitesimal generator of the regularized quasi-semigroup . Somewhere we briefly apply generator instead of infinitesimal generator.

Here are some useful examples of regularized quasi-semigroups.

Example 2.2. Let be an exponentially bounded strongly continuous -semigroup on Banach space , with the generator . Then defines a -quasi-semigroup with the generator , , and so .

Example 2.3. Let , the space of all bounded uniformly continuous functions on with the supremum-norm. Define , by One can see that is a regularized -quasi-semigroup of operators on , with the infinitesimal generator on , where .

Example 2.4. Let be a strongly continuous semigroup of operators on Banach space , with the generator . If is injective and commutes with , , then is a -quasi-semigroup with the generator . Thus . In fact, for , boundedness of implies that Now injectivity of implies that , and so .

Example 2.5. Let be a strongly continuous exponentially bounded -semigroup of operators on Banach space , with the generator . For , define where , and , with . We have and the -semigroup properties of imply that So is a -quasi-semigroup (the other properties can be also verified easily). Also and for ,

Some elementary properties of regularized quasi-semigroups can be seen in the following theorem.

Theorem 2.6. Suppose is a -quasi-semigroup with the generator on Banach space . Then(i)for any and , and (ii)for each , (iii)if is locally integrable, then for each and , (iv)let be a continuous function; then for every , (v)Let be injective and for any , . Then is a -quasi-semigroup with the generator ,(vi)Suppose is a quasi-semigroup of operators on Banach space with the generator , and commutes with every , . Then is a -quasi-semigroup of operators on with the generator .

Proof. First we note that from the commutativity of ; Also implies that Thus from continuity of , we have Thus and .
To prove (ii), consider the quotient which tends to as .
Also for , Now the strongly continuity of implies that Thus Hence by the strongly continuity of , Thus . The second equality holds by (i).
Now integrating of this equation, we have Hence injectivity of implies (iii).()is trivial from continuity of and strongly continuity of . In (v), obviously is a -quasi-semigroup. For , we have which tends to , as . This proves .()can be seen easily.

3. Evolution Equations and Regularized Quasi-Semigroups

Suppose is an injective bounded linear operator on Banach space and . In this section, we study the solutions of the following abstract evolution equation using the regularized quasi-semigroups: One can see [13, 14] for a comprehensive studying of abstract evolution equations.

Theorem 3.1. Let be the infinitesimal generator of a -quasi-semigroups on Banach space , with domain . Then for each and , the initial value problem (3.1) admits a unique solution.

Proof. Let . By Theorem 2.6(ii), is a solution of (3.1).
Now we show that this solution is unique. Suppose is another solution of (3.1). Trivially . Let . For and , define From -quasi-semigroup properties, for small enough , we have So This means that Therefore, from this, the fact that satisfies (3.1), and , we obtain that Hence for every , . Consequently, is a constant function on . In particular, . So from , we have Hence . Now injectivity of implies that , which proves the uniqueness of the solution.

Now with the above notation, we consider the inhomogeneous evolution equation The following theorem guarantees the existence and uniqueness of solutions of (3.8) with some sufficient conditions on .

Theorem 3.2. Let be a -quasi-semigroup on Banach space , with the generator whose domain is . If is a continuous function, each operator is closed, and then the initial value equation (3.8) admits a unique solution

Proof. For the existence of the solution, it is enough to show that in (3.10) is continuously differentiable and satisfies (3.8).
Trivially . We know that is a solution of (3.1) by Theorem 3.1. Define which is in by our hypothesis. We have On the other hand, the -quasi-semigroup properties imply that So Since the range of is in , passing to the limit when , and using Theorem 2.6(v), we have Therefore, exists. Also by our hypothesis is closed, and , thus Consequently, Hence This completes the proof.

We conclude this section with two simple perturbation theorems and some examples, as applications of our discussion.

Theorem 3.3. (a) Suppose is the infinitesimal generator of a strongly continuous semigroup and with domain is the generator of a regularized -quasi-semigroup , which commutes with . Then with domain is the infinitesimal generator of a regularized -quasi-semigroup.
(b) Suppose is the infinitesimal generator of an exponentially bounded -semigroup and with domain is the generator of a quasi-semigroup (resp., regularized -quasi-semigroup) , which commutes with . Then with domain is the infinitesimal generator of a -regularized quasi-semigroup (resp., regularized -quasi-semigroup).

Proof. In (a) and (b), define One can see that is a -regularized quasi-semigroup (in (b), resp., regularized -quasi-semigroup). We just prove that is its generator. In (a), let be the infinitesimal generator of and . Hence exist in and the range of , respectively. Now the fact that commutes with and strongly continuity of implies that exists in the range of . So exists in the range of and By injectivity of , .
The proof the other parts is similar.

Theorem 3.4. Let be a -quasi-semigroup of operator on Banach space with the generator on domain . If commutes with , , and , then is the infinitesimal generator of -regularized quasi-semigroup

Proof. The -quasi-semigroup properties of can be easily verified. We just prove that its generator is . Let ; we have which tends to , as . This completes the proof.

Example 3.5. Let . Consider the following initial value problem: Let , with the supremum-norm. Define by , . Also define by , where . It is well known that is the infinitesimal generator of -regularized semigroup , defined by . Now with , if is defined by , then by Example 2.3, is the infinitesimal generator of the regularized -quasi-semigroup . Using Theorem 3.3 and the fact that ,  , we obtain that is the infinitesimal generator of regularized -quasi-semigroup . Also using these operators, (3.26) can be written as Thus by Theorem 3.1 for any , (3.26) has the unique solution

Example 3.6. For a given sequence of complex numbers with nonzero elements and , consider the following equation: Let be the space , the set of all complex sequence with zero limit at infinity. For a bounded sequence , define and on by One can easily see that and is a bounded linear operator which is injective. It is well known that is the infinitesimal generator of strongly continuous semigroup Thus by Example 2.4, , defined by is the infinitesimal generator of the -quasi-semigroup Using these operators, one can rewrite (3.29) as where . Trivially commutes with , for any . Now using Theorem 3.3 we obtain that is the infinitesimal generator of of -quasi-semigroup Also from Theorem 3.1, with , for any , (3.34) has a unique solution But from definition of , for a given , So the solution of (3.34) is or equivalently the solution of (3.29) is

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