Existence and uniqueness properties for solutions of a class of Banach space valued evolution equations

In this note we provide a self-contained proof of an existence and uniqueness result for a class of Banach space valued evolution equations with an additive forcing term. The framework of our abstract result includes, for example, finite dimensional ordinary differential equations (ODEs), semilinear deterministic partial differential equations (PDEs), as well as certain additive noise driven stochastic partial differential equations (SPDEs) as special cases. The framework of our general result assumes somehow mild regularity conditions on the involved semigroup and also allows the involved semigroup operators to be nonlinear. The techniques used in the proofs of our results are essentially well-known in the relevant literature. The contribution of this note is to provide a rather general existence and uniqueness result which covers several situations as special cases and also to provide a self-contained proof for this existence and uniqueness result.