A novel characteristic of solution operator for the fractional abstract Cauchy problem

Motivated by an equality of the Mittag-Leffler function proved recently by the authors, this paper develops an operator theory for the fractional abstract Cauchy problem (FACP) with order ∈(0,1). The notion of fractional semigroup is introduced. It is proved that a family of bounded linear operator is a solution operator for (FACP) if and only if it is a fractional semigroup. Moreover, the well-posedness of the problem (FACP) is also discussed. It is shown that the problem (FACP) is well-posed if and only if its coefficient operator generates a fractional semigroup.