Bi-continuous N-times Integrated C-semigroups

In this paper,we endow with an additional locally convex topology which is coarser than the norm topology on Banach space.And introduce bi-continuous n-times integrated C-semigroups by combining bi- continuous semigroups and n-times integrated C-semigroups,And then define their generators and subgenerators. By studing the properties and Laplace transforms of bi-continuous n-times integrated C-semigroups,and the relation among generators,subgenerators and C-pseudoresolvents, we conclude some important properties of bi-continuous n-times integrated C-semigroups,So the theory of operator semigroup are enriched.Firstly,we endow with an additional locally convex topologyΥ,which is coarser than the norm topology in Banach space,and bi-continuous n-times integrated C-semigroups are defined,the properties of semigroup itself are also studied.Secondly,C-pseudoresolvents of bi-continuous n-times integrated C-semigroups are provided.Based on the concept,Laplace transforms are provided and its properties are discussed.Thirdly,the generators of bi-continuous n-times integrated C-semigroups are defined on the conditions of the exponentially bounded and the locally bounded,and their properties are discussed.Moreover,the definition of subgenerators of bi-continuous n-times integrated C-semigroups is provided and the relationships between generators and subgenerators are discussed.Lastly,the representation theorem which is an important theorem of bi-continuous n-times integrated C-semigroups is provided.