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

Summary: Here, if $0<\alpha \le 1$, the author studies the Cauchy problem in a Banach space $E$ for fractional evolution equations of the form

$\frac{{d}^{\alpha }u}{d{t}^{\alpha }}=Au\left(t\right)+B\left(t\right)u\left(t\right),$

where $A$ is a closed linear operator defined on a dense set in $E$ into $E$, which generates a semigroup and $\left\{B\left(t\right):t\ge 0\right\}$ 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 $\left\{B\left(t\right):t\ge 0\right\}$. 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 ODE in abstract spaces 35K90 Abstract parabolic equations 45K05 Integro-partial differential equations