Generalized almost periodic and ergodic solutions of linear differential equations on the half-line in Banach spaces

The Bohl–Bohr–Amerio–Kadets theorem states that the indefinite integral y = Pφ of an almost periodic (ap) φ :R→X is again ap if y is bounded and the Banach space X does not contain a subspace isomorphic to c0. This is here generalized in several directions: Instead of R it holds also for φ defined only on a half-line J, instead of ap functions abstract classesAwith suitable properties are admissible, φ ∈ A can be weakened to φ in some “mean” classMq+1A, then Pφ ∈MqA; hereMA contains all f ∈ L1 loc with (1/h) h 0 f (· + s)ds in A for all h>0 (usually A ⊂MA⊂M2A⊂··· strictly); furthermore, instead of boundedness of y mean boundedness, y in some MkL∞, or in MkE, E = ergodic functions, suffices. The Loomis–Doss result on the almost periodicity of a bounded Ψ for which all differences Ψ(t +h)−Ψ(t) are ap for h>0 is extended analogously, also to higher order differences. Studying “difference spaces” ΔA in this connection, we obtain decompositions of the form: Any bounded measurable function is the sum of a bounded ergodic function and the indefinite integral of a bounded ergodic function. The Bohr–Neugebauer result on the almost periodicity of bounded solutions y of linear differential equations P(D)y = φ of degree m with ap φ is extended similarly for φ ∈Mq+mA; then y ∈MqA provided, for example, y is in someMkU with U = L∞ or is totally ergodic and, for the half-line, Re λ 0 for all eigenvalues P(λ) = 0. Analogous results hold for systems of linear differential equations. Special case: φ bounded and Pφ ergodic implies Pφ bounded. If all Reλ >0, there exists a unique solution y growing not too fast; this y is inMqA if φ ∈Mq+mA, for quite general A.  2003 Elsevier Inc. All rights reserved.