Inverse source problem for the abstract fractional differential equation

In a Banach space, the inverse source problem for a fractional differential equation with Caputo-Dzhrbashyan derivative is considered. The initial and observation conditions are given by elements from D(A), and the operator function on the right side is sufficiently smooth. Two types of the observation operator are considered: integral and at the final point. Under the assumptions that operator A is a generator of positive and compact semigroup the uniqueness, existence and stability of the solution are proved.