Generating Theore, Yosida Approximation And The Laplace Inverse Transformation Of Two Parameter C Semi-groups

This article, based on the definition of one parameter C semi-group and its generatingtheorem, Yosida approximation and the Laplace inverse transformation, making use ofclassical theory researched by the predecessors, combining with the researching methodof two parameter C semi-group, discusses basic properties of two parameters C semi-group and its infinitesimal generator, studies the re-solvent and re solvent equation of tw-o parameters C semi-group, as well as the Hille-Yosida theorem and approximation,and obtains inverter Laplace transformation of two parameters C semi-group, and itscore proposition-generation theorem. This paper consists of the following parts:(1) C semi-group is the meaningful extention of strongly continuous semi-group ofbounded linear operators. Some properties of two parameters C semi-group by Xu Qi-ang, and the convergence of one parameter C Semi-group is extended to two paramet-ers C Semi-group. And based on this, the related properties of two parameters C semi-group and infinitesimal generator are given in this part.(2) The re-solvent of two parameters C semi-group is given by Rong Rong et al. Onthis basis, the re-solvent and re-solvent equation of two parameters C semi-group are st-udied in this part.(3) Lang Kailu gave the Hille-Yosida theorem to compress C Semi-group and disc- ussed properties of Hille-Yosida, and discussed properties of Hille-Yosida C spaces ofall arbitrary operators in a Banach spaces by using or Hille-Yosida theorem which is tocompress Semi-group. On this basis, this paper further explores the Hille-yosida theoremof two parameters C semi-group and researches the Yosida approximation of two para-meters C Semi-group.(4)In this part, some properties of one parameter C semi-group is extended to twoparameters C semi-group, getting two parameters C semi-group, generating elementsand some of its basic properties, researching generation theorem on the two parametersC semi-group.(5)Based on the above discussion of infinitesimal generators of two parameters Csemi-group, that is, preliminary solution type of infinitesimal generator is Laplace tran-sformation of two parameters C semi-group. At last, the article focuses on thestudy of Laplace inverter transformation of infinitesimal generator preliminary solutionto obtain semi-group in this part.