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

Abstract

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 $ϕ$ in the non-local boundary value problem $D_t^ρ u(t) + Au(t) = f(t)$ ($0 < ρ< 1, 0 < t \leq T$), $u(ξ) = αu(0) + ϕ$, ($α$ is a constant and $0 < ξ\leq T)$, is considered. Operator $D_t$ on the left-hand side of the equation expresses the Caputo derivative. For both inverse problems $u(ξ_1) = V$ 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 $α$ 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 $α$, the solution $u(t)$ became unique.