Some probability densities and fundamental solutions of fractional evolution equations.

*(English)*Zbl 1005.34051Summary: 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 $\left\{B\right(t):t\ge 0\}$ 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 $\left\{B\right(t):t\ge 0\}$. 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 ODE in abstract spaces |

35K90 | Abstract parabolic equations |

45K05 | Integro-partial differential equations |