Identification for a semilinear evolution equation in a Banach space

In this paper we prove an existence result for strict solutions to an identification problem for a first order semilinear differential equation, in a general Banach space, subjected to an overdetermination expressed by means of a Lebesgue integral. Under the additional assumption that the $C_0$-semigroup generated by the linear part has a sufficiently fast exponential decay, we prove the uniqueness and the continuous dependence of the solution on the data.