Almost Periodic Solutions of First- and Second-Order Cauchy Problems

Almost periodicity of solutions of first- and second-order Cauchy problems on the real line is proved under the assumption that the imaginary (resp. real) spectrum of the underlying operator is countable. Related results have been obtained by Ruess–V and Basit. Our proof uses a new idea. It is based on a factorisation method which also gives a short proof (of the vector-valued version) of Loomisʹ classical theorem, saying that a bounded uniformly continuous function from into a Banach spaceXwith countable spectrum is almost periodic ifc0 X. Our method can also be used for solutions on the half-line. This is done in a separate paper.