Qualitative Research And Application Of N-th Order α-times Integrated C-semigroups

The theory of linear operator semigroup in Banach space is an important tool of solvingthe abstract Cauchy problem, and it plays an important role in researching the theory offunctional analysis. Many scholars have made further research onn α-times integratedC-semigroups since deLaubenfels, Shengwang Wang and other people introducted thedefinition of it.This thesis makes use of the related theory of functional analysis, operator semigroups tomake a promotion to α-times integrated C-semigroups, and introduce the concept of n-thorder α-times integrated C-semigroups, then discuss the basic properties and equivalencerelation of the solution to high-order abstract Cauchy problem in Banach space. Then itmakes a definition of exponential bounded bicontinuousn-th order α-times integratedC-semigroups and obtains their basic properties and inverse transform of Laplace. Thisthesis is divided into the following chapters:The first chapter of this thesis introduces the research background, significance and thepreliminary knowledge.The second chapter gives the concept ofn-th order α-times integrated C-semigroups,and discusses the basic properties ofn-th order α-times integrated C-semigroups,including the relations with existence and uniqueness of solutions to equation and theproperties of its subgenerator at first. Then it gives the definitions of resolvent set andresolvent of the subgenerator ofn-th order α-times integrated C-semigroups, andresearches the equivalence relation betweenn-th order α-times integrated C-semigroupsand the integral representation of resolvent of its subgenerator. It gets the resolvent identity ofsubgenerator ofn-thorder α-timesintegrated C-semigroups at last.The third chapter researches the equivalence relation betweenn-th order α-timesintegrated C-semigroups and high order abstract Cauchy problem existing a unique solution,which suggests that the stability of solution can be released by the existence and uniquenessof solutions of equation. So we can get the equivalence relation betweenn-th order α-times integrated C-semigroups and the C well-posedness of abstract Cauchy problem.The fourth chapter gives the definition of bicontinuous, equicontinuity and exponentialbounded bicontinuousn-th order α-times integrated C-semigroups at first. Then itdiscusses the properties of exponential bounded bicontinuousn-th order α-times integratedC-semigroups, including the boundedness of its resolvent. It researches expression of Laplace inverse transformation of exponential bounded bicontinuousn-th order α-timesintegrated C-semigroups at last.The fifth chapter summarizes the main conclusions and gives some considerableproblems for further study.