On the uniqueness of solutions of two inverse problems for the subdiffusion equation

Let $A$ be an arbitrary positive selfadjoint operator, defined in a separable Hilbert space $H$. The inverse problems of determining the right-hand side of the equation and the function $\phi$ in the non-local boundary value problem $D_t^{\rho} u(t) + Au(t) = f(t)$ ($0 < \rho < 1, 0 < t \leq T$), $u(\xi) = \alpha u(0) + \phi$, ($\alpha$ is a constant and $0 < \xi \leq T)$, is considered. Operator $D_t$ on the left-hand side of the equation expresses the Caputo derivative. For both inverse problems $u(\xi_1) = V$ is taken as the over-determination condition. Existence and uniqueness theorems for solutions of the problems under consideration are proved. The influence of the constant $\alpha$ on the existence and uniqueness of a solution to problems is investigated. An interesting effect was discovered: when solving the forward problem, the uniqueness of the solution $u(t)$ was violated, while when solving the inverse problem for the same values of $\alpha$, the solution $u(t)$ became unique.