“…[35,48,47,70]), is devoted to the problem of existence and uniqueness of a bounded (µ-pseudo) almost periodic mild solution to (52) in a Banach space X. The adopted approach is based on superposition theorems in the Banach space S p AP(R, X) (or S p PAP(R, X, µ)) combined with the Banach's fixed-point principle, applied to the nonlinear operator (Γu)(t) = To our knowledge, all existing results use the fact that Γ maps AP(R, X) into itself, but Γ does not map S p AP(R, X) into AP(R, X) nor into S p AP(R, X) (see in particular [35,48,47,70]). The proposed proofs may be summarized as follows: if u ∈ AP(R, X), then u satisfies the compactness condition (Com) of Subsection 2.4, and u ∈ S p AP(R, X).…”