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Evolution semigroups and spectral criteria for almost periodic solutions of periodic evolution equations. (English) Zbl 0924.34038
The authors consider mild solutions of nonhomogeneous 1-periodic evolution equations on the line. They show that for each almost-periodic inhomogeneity $$f$$ there exists a unique almost-periodic mild solution if and only if the spectrum of the monodromy operator $$P$$ does not intersect the unit circle. Under an additional technical condition (which holds e.g. in the parabolic case), they obtain an almost-periodic solution if $$\sigma(P)$$ does not intersect the closure of $$\exp(i sp(f))$$, where $$sp(f)$$ is the spectrum of $$f$$. The approach is based on spectral properties of the associated evolution semigroup. Applications to semilinear equations and abstract parabolic problems are given.

##### MSC:
 34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations 47D06 One-parameter semigroups and linear evolution equations 34G10 Linear differential equations in abstract spaces
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