Direct and inverse problems for linear equations with Caputo — Fabrizio derivative and a bounded operator

The unique solvability of linear inverse coefficient problems for the evolutionary equation in a Banach space with the Caputo — Fabrizio derivative is studied. An operator at the unknown function in the equation is assumed to be bounded, the equation is endowed with the Cauchy condition. For the inverse problem with a constant unknown coefficient and with an integral overdefinition condition in the sense of Riemann — Stieltjes, which includes the condition of final overdefinition as a special case, a well-posedness criterion is obtained. Sufficient conditions for unique solvability and an estimate of the well-posedness for the solution are obtained for a linear inverse problem with a time-dependent unknown coefficient. The abstract results obtained are used in the study of inverse problems with an unknown coefficient depending only on spatial variables or only on time, for equations with polynomials of a self-adjoint elliptic differential operator with respect to spatial variables.