Completely Continuous Generalized C₀-semigroup

Based on the definitions of the completely continuouity and completely continuousgeneralized C_osemigroup, this paper gives the concept of completely continuousgeneralized C_osemigroup in Banach space, defines its generating element, and gets toknow some important properties of the completely continuous generalized semigroup.Through the study of the nature of generalized completely continuous C_osemigroups,and its generating element, exponential formula, Laplace inversion and theapproximation theorem, this paper gets some important conclusions about completelycontinuous generalized C_osemigroup, which enriches the content of the semigroup ofoperators.First of all, according to the definitions of the completely continuity and thecontinuous generalized C_osemigroup, this paper gives the definition of completelycontinuous generalized C_osemigroup, defines its generator, and studies the nature thesemigroup itself.Secondly, this paper gives the concept of completely continuous generalized C_osemigroup and its nature, obtains the connections between the generalized C_osemigroupand the completely continuous generalized C_osemigroup.Thirdly, this paper discusses the conditions in which operator A would generatecompletely continuous generalized C_osemigroup, that is also called the generationtheorem. Also, this paper discusses the perturbation problem of the generator and itsfeasibility of the implementation of disturbance conditions under the assumption that C isgenerally available operator.Finally, this paper discusses exponential formula, Laplace inversion and theapproximation theorem of completely continuous generalized C_osemigroup.