Evolution semigroups and spectral criteria for almost periodic solutions of periodic evolution equations

We investigate spectral criteria for the existence of(almost) periodic solu tions to linear I-periodic evolution equations of the Form dx/dt = A (t) x + f(t) with (in general, unbounded) A(t) and (almost) periodic f. Using the evolution semigroup associated with the evolutionary process generated by the equation under consideration we show that if the spectrum of the monodr omy operator does not intersect the set <(e(isp(f)))over bar>, then the abo ve equation has an almost periodic (mild) solution x(f) which is unique if one requires sp(x(f)) subset of <(lambda+2 pi k, k is an element of Z, lamb da is an element of sp(f)})over bar>. We emphasize that our method allows u s to treat the equations without assumption on the existence of Floquet rep resentation. This improves recent results on the subject. In addition we di scuss some particular cases, in which the spectrum of monodromy operator do es not intersect the unit circle, and apply the obtained results to study t he asymptotic behavior of solutions. Finally, an application to parabolic e quations is considered. (C) 1999 Academic Press.