  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.
