On α-times integrated C-semigroups and the abstract Cauchy problem

This paper is concerned with or-times integrated C-semigroups for alpha > 0 and the associated abstract Cauchy problem: u'(t) = Au(t)+ t(alpha -1)/Gamma(alpha )x, t > 0; u(0) = 0. We first investigate basic properties of an alpha -times integrated C-semigroup which may not be exponentially bounded. We then characterize the generator A of an exponentially bounded alpha -times integrated C-semigroup, either in terms of its Laplace transforms or in terms of existence of a unique solution of the above abstract Cauchy problem for every alpha in (lambda - A)C-1(X).