Inverse problem for evolutionary equation with the Gerasimov–Caputo fractional derivative in the sectorial case

We investigates the unique solvability of a class of linear inverse problems with a time-independent unknown coefficient in an evolution equation in Banach space, which is resolved with respect to the fractional Gerasimov–Caputo derivative. We assume that the operator in the right-hand side of the equation generates a family of resolving operators of the corresponding homogeneous equation, which is exponentially bounded and analytic in a sector containing the positive semiaxis. It is shown that for the well-posedness of the inverse problem it is necessary to choose as the space of initial data the definition domain of the generating operator endowed with its graph norm. Sufficient conditions for the unique solvability of the inverse problem are found. The obtained abstract results are applied to the unique solvability study of an inverse problem for a class of time-fractional order partial differential equations. The considered example, in particular, shows that when choosing as the source data space not the domain of definition of the generating operator, but the whole space, the inverse problem is ill-posed.