An identification problem for a linear evolution equation in a Banach space and applications

In this paper we prove both the existence and uniqueness of a solution to an identification problem for a first order linear differential equation in a general Banach space. Namely, we extend the explicit representation for the solution of this problem previously obtained by Anikonov and Lorenzi to the case of an infinitesimal generator of an analytic C_0-semigroup of contractions to the general nonanalytic case and also to the case of a restriction expressed in terms of an operator-valued measure. So, our abstract result handles both parabolic and hyperbolic equations and systems.