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 = Pphi of an almost periodic (ap) phi: R --> X is again ap if y is bounded and the Banach space X does not contain a subspace isomorphic to c(0). This is here generalized in several directions: Instead of R it holds also for 0 defined only on a half-line J, instead of ap, functions abstract classes A with suitable properties are admissible, phi is an element of A can be weakened to phi in some "mean" class Mq+1 A, then Pphi is an element of M(q)A; here MA contains all f is an element of L-loc(1) with (1/ h) integral(0)(h) f ((.) + s) ds in A for all h > 0 (usually A subset of MA subset of M-2 A subset of(...) strictly); furthermore, instead of boundedness of y mean boundedness, y in some (MLinfinity)-L-k, or in (ME)-E-k, E = ergodic functions, suffices. The Loomis-Doss result on the almost periodicity of a bounded P for which all differences Psi (t + h) - Psi (t) are ap for h > 0 is extended analogously, also to higher order differences. Studying "difference spaces" AA 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 = phi of degree m with ap phi is extended similarly for phi is an element of Mq+m A; then y is an element of M-q A provided, for example, y is in some (MU)-U-k with U = Linfinity or is totally ergodic and, for the half-line, Relambda greater than or equal to 0 for all eigenvalues P(lambda) = 0. Analogous results hold for systems of linear differential equations. Special case: phi bounded and Pphi ergodic implies P0 bounded. If all Re lambda > 0, there exists a unique solution y growing not too fast; this y is in M-q A if phi is an element of Mq+m, for quite general A. (C) 2003 Elsevier Inc. All rights reserved.