Inverse problems for the subdiffusion equation

In the Hilbert space H consider the subdiffusion equation Dtρu(t)+Au(t) = fg(t), ρ 2 (0; 1), t 2 (0; T]. In this equation A is an unbounded positive self-adjoint operator, Dtρ stands for the Caputo fractional derivative, g(t) 2 C[0; T] is a given function and f 2 H is an unknown element. It is assumed that A has a compact inverse A−1. The subdiffusion equation is studied with initial u(0) = ’ and non-local u(0) = u(T) conditions. To solve the inverse problem, an additional condition in the form u(t0) = , t0 2 (0; T) is taken. Here is a given element of H. The existence of a solution to these inverse problems is proved for an arbitrary choice of t0 2 (0; T).