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Comments on almost automorphic and almost periodic functions in Banach spaces. (English) Zbl 1096.34036
The paper provides an introductory discussion of almost-periodic and (uniformly continuous) Bochner almost automorphic functions, a formulation (without proofs) of two theorems about the existence of solutions of \(x'=Ax+f\) on \(\mathbb{R}\) which are almost automorphic resp. almost-periodic, with \(A\) an infinitesimal generator of an exponentially stable \(C_0\)-semigroup and \(f:\mathbb{R}\to X\) almost automorphic resp. Stepanoff \(S^2\)-almost-periodic, \(X\) reflexive resp. Hilbert space. (For theorem 2.6 the proof of (2.9) is not correct.)

34G10 Linear differential equations in abstract spaces
43A60 Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions
47D06 One-parameter semigroups and linear evolution equations
35B15 Almost and pseudo-almost periodic solutions to PDEs