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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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##### References:
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