The Properties And Applications Of Bi-continuous N-times Integrated C-semigroups

This article is the further study on basis of basic theory of bi-continuous n-times integrated C- semigroups, which receive the spectrum theorem, perturbation theorem and Abstract Cauchy problem of bi-continuous n- times integrated C-semigroups.Firstly,in the Spectral theory we mainly discuss that how obtain the spectrum of semigroupsσ(T(t)) by the spectrum of generatorσ(A) and relationship between them,and by Laplace inverse transformation, we get a expressed of the semigroup T(t) by used which have great significance for researchiing Abstract Cauchy problem;Secondly,in the Perturbation theory we chiefly discusse that when the gener-ators A and A+B are very close in some sense, the relationship between semi-group T(t) generated by A and semigroup S(t) generated by B, as well as the necessary and sufficient conditions of bi-continuous n- times integrated C- semi-group{Tk(t):t≥0} converges to bi-continuous n- times integrated C- semigroup C-semigroups{T(t):t≥0}, we get a genaral approximation theorem;Thirdly,the main issue in the Abstract Cauchy problem discussion is bi-continuous n- times integrated C- semigroup's relations with solutions of two points Abstract Cauchy problem,thus enriching the theory of semigroup.