Identification Problem for Strongly Degenerate Evolution Equations with the Gerasimov–Caputo Derivative

For a linear differential equation in a Banach space with a degenerate operator multiplying the fractional Gerasimov–Caputo derivative, we prove a theorem on the existence of a unique solution of the inverse problem with an unknown time-dependent coefficient. It is assumed that the relative boundedness condition is satisfied for the pair of operators occurring in the equation and an overdetermation condition with an operator whose kernel contains the degeneracy subspace of the equation under study is specified. The result obtained is illustrated with an example of a model system of partial differential equations unsolved for the fractional time derivative, containing an unknown coefficient, and equipped with initial and boundary conditions and an overdetermination condition.