Regularized functional calculi for pseudo-differential operators, integro-differential equations, and integrated semigroups

This dissertation involves functional calculus properties of pseudo-differential operators, well-posedness of integro-differential equations, functional differential equations with Hille-Yosida operators and perturbations of integrated semi-groups.; We start by studying the functional calculus property of the generators of commuting bounded {dollar}Csb0{dollar} groups, and then apply the abstract results to differential operators in {dollar}Lsp{lcub}p{rcub}(Rsp{lcub}n{rcub}){dollar} spaces. The regularized semigroups generated by the differential operators are also constructed by using these functional calculi.; For the integro-differential equations and functional differential equations, we use a reduction method and reduce these equations to abstract Cauchy problems of differential equations on product Banach spaces. The well-posedness results we obtain here improve and generalize some known results. Applications to partial differential equations are considered.; For the perturbation problem of integrated semigroups, we prove a new multiplicative perturbation theorem, and use it to obtain some additive perturbation results. Applications to cosine functions are also investigated.