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Some probability densities and fundamental solutions of fractional evolution equations. (English) Zbl 1005.34051

Summary: Here, if 0<α1, the author studies the Cauchy problem in a Banach space E for fractional evolution equations of the form

d α u dt α =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):t0} is a family of closed linear operators defined on a dense set in E into E. The existence and uniqueness of a solution to the considered Cauchy problem is studied for a wide class of the family of operators {B(t):t0}. 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.

MSC:
34G20Nonlinear ODE in abstract spaces
35K90Abstract parabolic equations
45K05Integro-partial differential equations