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 α∈(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.