On Regularized Quasi-Semigroups and Evolution Equations

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 [2–5]. 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 [8–12] 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