Study On The Perturbation And Probabilistic Approximation Theory Of M-times Integrated Semigroups

In this article,based on the perturbation and approximation theory of integratedC-semigroups and m integrated C-semigroups. Firstly,we give the basic concept ofbi-continuous m times integrated C-semigroups and cosine operator functions andproperties,and discusses its perturbation and approximation theorems.The first chapter, given the research background, current research scholars athome and abroad and the major research content.The second chapter, mainly introduced the basic concepts and nature of m timesintegrated C-semigroup, exponentially bounded C-cosine operator functions andm times integratedC-cosine operator functions, and given three types of semigroupscorresponding perturbation results.The third chapter, studied if the linear operator A is generator of the bi-continuous m integrated C-semigroups, and if operator A in the perturbation ofbounded operator B, then A Bcan still generate a new semigroup, and we given thespecific basic form; proved that if the operator A generated semigroup is exponentialor local Lipschitz continuous, then also the operator A Bgenerated semigroup isexponential or local Lipschitz continuous.The fourth chapter, the probabilistic representations for bi-continuous mintegrated C-semigroups on Banach space were discussed in the light of operatorvalued mathematical expectation and Riemann-Stieltjes integral. Then,with Riemann-Stieltjes integral, Taylor expansion, H lder inequality and estimations of momentgenerating functions of some suitable random variables were used, some probabilisticapproximations and estimations of convergent rates were presented for bi-continuousm integratedC-semigroups and the general conclusion of probabilistic approximationwas drawn.The fifth chapter, discussed if the operator A is generator of the bi-continuous mintegrated C-cosine operator function, and for each one1, boundedoperator B satisfy two different perturbation conditions, then A Bcan still thegenerator of bi-continuous m integrated C-cosine operator function respectively andnew operator still generate a new semigroup, and we given the specific basic form.