Abstract:

This paper is concered with the local C-semigroups and local C-coscine functions without the assumption that the image of  C is dense in a Banach space X. First, the sufficient and necessary conditions for  a local C-semiproup and a C-cosine function to be the restriction of a global C-semigroup and a  global C-cosine function to an interval are given,respectively.Secondaly, it is characterized for a closed operator to be the generator of a global  C-semigroup and a local C-cosine function,respectively.

Key words: Local C-semigroups, local C-cosine functions, asymptotic C-resolvent