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

Summary: Here, if \(0<\alpha\leq 1\), the author studies the Cauchy problem in a Banach space \(E\) for fractional evolution equations of the form \[ {d^\alpha u\over dt^\alpha} =Au(t)+B(t)u(t), \] where \(A\) is a closed linear operator defined on a dense set in \(E\) into \(E\), which generates a semigroup and \(\{B(t):t\geq 0\}\) 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 \(\{B(t):t\geq 0\}\). 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.

MSC:

34G20 Nonlinear differential equations in abstract spaces
35K90 Abstract parabolic equations
45K05 Integro-partial differential equations
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