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A novel characteristic of solution operator for the fractional abstract Cauchy problem. (English) Zbl 1231.47039
This paper introduces the notion of fractional semigroups and develops an operator theory for the fractional abstract Cauchy problem (FACP). It is proved that a family of bounded linear operators is a solution operator for the (FACP) if and only if it is a fractional semigroup. Several important properties of solution operators are established by use of an equality for the Mittag-Leffler function. Based on these properties, the authors discuss uniqueness of solutions to the (FACP) and show that the (FACP) is well-posed if and only if its coefficient operator generates a fractional semigroup.

##### MSC:
 47D06 One-parameter semigroups and linear evolution equations 34A08 Fractional ordinary differential equations and fractional differential inclusions
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##### References:
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