Summary: Here, if , the author studies the Cauchy problem in a Banach space for fractional evolution equations of the form
where is a closed linear operator defined on a dense set in into , which generates a semigroup and is a family of closed linear operators defined on a dense set in into . The existence and uniqueness of a solution to the considered Cauchy problem is studied for a wide class of the family of operators . The solution is given in terms of some probability densities. An application is given for the theory of integro-partial differential equations of fractional orders.