Inverse problems for a generalized fractional diffusion equation with unknown history

Inverse problems for a diffusion equation containing a generalized fractional derivative are studied. The equation holds in a time interval $(0,T)$ and it is assumed that a state $u$ (solution of diffusion equation) and a source $f$ are known for $t\in (t_0,T)$ where $t_0$ is some number in $(0,T)$. Provided that $f$ satisfies certain restrictions, it is proved that product of a kernel of the derivative with an elliptic operator as well as the history of $f$ for $t\in (0,t_0)$ are uniquely recovered. In case of less restrictions on $f$ the uniqueness of the kernel and the history of $f$ is shown. Moreover, in a case when a functional of $u$ for $t\in (t_0,T)$ is given the uniqueness of the kernel is proved under unknown history of $f$.