C-regular Semigroups On Complete Random Norms

In this paper,we generalize the related theorems and abstract Cauchy problems of C-regularized semigroups to complete random normed modules,then prove the related conclusions of kernel spaceN(f)of L~0-linear functional f in the framework of complete random normed modules,and finally generalize the abstract Cauchy initial value problem to complete random normed modules under the assumption of L~0-Lipschitzcondition.This article is divided into five chapters:In Chapter 1 we briefly introduces the development process of stochastic metric theory and C-regularized semigroups,its historical background,and the main research content of this article.Then it gives the terms and their meanings that we need to prove later;In Chapter 2,as a preliminary knowledge,we briefly introduce the basic concepts such as random normed modules,random linear operators and some basic applications under random normed modules to be used in our later proofs;In Chapter 3,we show that on the basis of the random normed module hierarchy,a kernel spaceN(f)of an L~0-linear functional f from a randomly normed moduleX to an algebra L~0(F,K)is either a closed submodule orN(f)in some specific layers are dense submodules of X;In Chapter 4,we introduce the definition of C-regularized semigroups and their infinitesimal generators,as well as other properties and results,and generalize them to a complete random normed module,and further for each almost surely bounded C-regularized semigroups have established differential and integral formulas and also give the main results of some almost surely bounded C-regularized semigroups and their infinitesimal generators;In Chapter 5,we prove that C-regularized semigroups and their associated abstract Cauchy problems can also be extended to the framework of complete random normed modules.