We present a general spectral decomposition technique for bounded solutions to inhomogeneous linear periodic evolution equations of the form cursive Greek chi̇ = A(t)cursive Greek chi+f(t) (*), with f having precompact range, which is then applied to find new spectral criteria for the existence of almost periodic solutions with specific spectral properties in the resonant 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 σΓ(P)\e^isp(f) is closed, where σΓ(P) denotes the part of σ(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 us to solve the general Massera-type problem for almost periodic solutions. Various spectral criteria for the existence of almost periodic and quasi-periodic mild solutions to (*) are given.

Naito T., Van Minh N., Shin J.S.

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