The Study Of The Related Problems To Two-parameter N-th Order α-Times Integral C-Group And Bi-Continuous N-th Order α-Times Integral C-Group

In this thesis,the definition of exponentially bounded two-parameter n-th order α-times integral C-group and the definition of exponentially bounded bi-continuous n-th order α-times integral C-group are introduced by using the research methods in the classical operator semigroups theory and combining the characteristics of exponentially bounded two-parameter n-th order α-times integral C-semigroup and exponentially bounded bi-continuous n-th order α-times integral C-semigroup.Their sub-generators and properties are studied according to these two definitions.The theorems related to their point spectrum,residual spectrum and continuous spectrum are also explored.The relationship between semigroups and groups are discussed,the theory of semigroups and groups is enriched.This thesis consists of the following four parts:In the first part,exponentially bounded two-parameter n-th order α-times integral C-group are presented.Exponentially bounded two-parameter n-th order α-times integral C-group and its sub-generator are defined.The properties of exponentially bounded two-parameter n-th order α-times integral C-group are studied.In the second part,the spectral mapping theorem of exponentially bounded two-parameter n-th order α-times integral C-group is presented.The definition of the resolvent set and the spectrum of exponentially bounded two-parameter n-th order α-times integral C-group are given.The point spectrum,the continuous spectrum,and the residual spectrum of exponentially bounded two-parameter n-th order α-times integral C-group are studied.In the third part,exponentially bounded bi-continuous n-th order α-times integral C-group are presented.Exponentially bounded bi-continuous n-th order α-times integral C-group and its sub-generator are defined.Sub-generator and properties of exponentially bounded bi-continuous n-th order α-times integral C-group are studied.In the fourth part,the spectral mapping theorem of exponentially bounded bi-continuous n-th order α-times integral C-group is presented.The definition of the resolvent set and the spectrum of exponentially bounded bi-continuous n-th order α-times integral C-group are given.The point spectrum,the continuous spectrum,and the residual spectrum of exponentially bounded bi-continuous n-th order α-times integral C-group are studied.