On Two-Parameter Regularized Semigroups and the Cauchy Problem

Mohammad Janfada

doi:10.1155/2009/415847

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.

Highlights

Introduction and PreliminariesSuppose that X is a Banach space and A is a linear operator in X with domain D A and range R A
It is proved that the existence and uniqueness of its solutions is closely related with two-parameter regularized semigroups of operators
We have the following lemma for one-parameter C-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

Summary

Introduction and Preliminaries

Suppose that X is a Banach space and A is a linear operator in X with domain D A and range R A. 1.2 and where, for x ∈ D A , Ax : C−1limt → 0 T t x − Cx /t is called the infinitesimal generator of T t Regularized semigroups and their connection with the ACP A; x have been studied in 1–6 and some other papers. The concept of local C-semigroups and their relation with the ACP A; x have been considered in 7–10. Some basic properties of two-parameter regularized semigroups and their relation with the generators are studied . It is proved that the existence and uniqueness of its solutions is closely related with two-parameter regularized semigroups of operators

Two-Parameter Regularized Semigroups

Two-Parameter Abstract Cauchy Problems