Some probability densities and fundamental solutions of fractional evolution equations

In the present paper, if 0 < alpha less than or equal to 1, we shall study the Cauchy problem in a Banach space E for fractional evolution equations of the form d(alpha)u/dt(alpha) = Au(t) + B(t)u(t), where A is a closed linear operator defined on a dense set in E into E, which generates a semigroup and {B(t) : t greater than or equal to 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 greater than or equal to 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. (C) 2002 Elsevier Science Ltd. All rights reserved.