Bi-Continuous N-times Integrated C-Semigroups

In the last years,the study of semigroups on spaces of bounded continuous (or uniformly continuous) functions led to consider semigroups for which the usual strong continuity fails to hold on Banach spaces.Kuhnemund considered bi-continuous semigroups,i.e., semigroups are strongly continuous with respect to an additional locally convex topology on a Banach space which is coarser than the norm topology. In this paper,we introduce bi-continuous n-times integrated C-semigroups by combining bi-continuous semigroups and n-times integrated C-semigroups.And we define their generators and C-resolvents. On the basis of their properties,the generation theorem of bi-continuous n-times integrated C-semigroups is obtained.we introduce the concepts of uniformly bi-continuous n-times integrated C-semigroups and combine the relations of generators and C-resolvents,so we gain the approximation theorem of bi-continuous n-times integrated C-semigroups.We debate the representation theorem of bi-continuous n-times integrated C-semigroups enlightened by A.Pazy's C₀ semigroups exponential formulas and other literatures.