Some probability densities and fundamental solutions of fractional evolution equations

In the present paper, if 0<α 1, we shall study the Cauchy problem in a Banach space E for fractional evolution equations of the form where A is a closed linear operator defined on a dense set in E into E, which generates a semigroup and {B(t):t 0} is a family of a closed linear operators defined on a dense set in E into E. The existence and uniqueness of the solution of the considered Cauchy problem is studied for a wide class of the family of operators {B(t):t 0}. 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.