New Spectral Criteria for Almost Periodic Solutions of Evolution Equations

We present a general spectral decomposition technique for bounded solutions to inhomogeneous linear periodic evolution equations of the form _ x = A(t)x+f(t) (), with f having precompact range, which will be then applied to nd new spectral criteria for the existence of almost periodic solutions with speciic spectral properties in the resonnant case where e isp(f) may intersect the spectrum of the monodromy operator P of () (here sp(f) denotes the Carleman spectrum of f). We show that if () has a bounded uniformly continuous mild solution u and ? (P)ne isp(f) is closed, where ? (P) denotes the part of (P) on the unit circle, then () has a bounded uniformly continuous mild solution w such that e isp(w) = e isp(f). Moreover, w is a "spectral component" of u. This allows to solve the general Massera-typed problem for almost periodic solutions. Various spectral criteria for the existence of almost periodic, quasi-periodic mild solutions to () are given.