Bi-continuous α-times Integrated C-semigroups

In this paper, an additional locally convex topology is endowed which is coarser than the norm topology on Banach space. And the bi-continuousα-times inte-grated C-semigroups is introduced by combining bi-continuous semigroups andα-times integrated C-semigroups. And their generators and subgenerators are de-fined. By studing the properties and Laplace transforms of the bi-continuousα-times integrated C-semigroups, the relations among generators, subgenerators and C-pseudoresolvents, the generation theorem of bi-continuousα-times integrated C-semigroups is concluded. The concepts of uniformly bi-continuousα-times integrated C-semigroups is introduced with the relations of generators and C-resolvents.So the approximaion theorem of bi-continuousα-times integrated C-semigroups is gained. The representation theorem of bi-continuousα-times integrated C-semigroups is de-bated by A.Pazy's Co-emigroups exponential formulas and other literatures. In the first part, the bi-continuousα-times integrated C-semigroups is introduced. In the second part, the C-pseudoresolvents and Laplace transforms of bi-continuousα-times integrated C-semigroups are debated. In the third part, the generators and subgenerators of bi-continuousα-times integrated C-semigroups are discussed. Fi-nally,three theorem which are important theorem of bi-continuousα-times integrated C-semigroups are studied.