Reconstruction of the nonhomogeneous term in an abstract evolution equation

A nonhomogeneous evolution equation the right-hand side of which is unknown is considered in a Banach space. For the determination of the unknown right-hand side, in addition to the Cauchy condition, a special overdetermination is given. In the paper a systematic theory for the investigation of the problem indicated is constructed, and sufficient conditions for the existence and uniqueness of solution are given. Special attention is paid to certain problems of perturbation theory.