  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.
