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

This paper is concerned with α-times integrated C-semigroups for α > 0 and the associated abstract Cauchy problem: ${\displaystyle u\prime \left(t\right)=Au\left(t\right)+\frac{t^{\alpha -1}}{\Gamma (\alpha )}x}$, t >0; u(0) = 0. We first investigate basic properties of an α-times integrated C-semigroup which may not be exponentially bounded. We then characterize the generator A of an exponentially bounded α-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 x in ${\displaystyle {(\lambda -A)}^{-1}C\left(X\right)}$.