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2001 | 145 | 2 | 97-111
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New spectral criteria for almost periodic solutions of evolution equations

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We present a general spectral decomposition technique for bounded solutions to inhomogeneous linear periodic evolution equations of the form ẋ = A(t)x + 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 $\overline{e^{i sp(f)}}$ 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 $σ_{Γ}(P) ∖ \overline{e^{i sp(f)}}$ is closed, where $σ_{Γ}(P)$ denotes the part of σ(P) on the unit circle, then (*) has a bounded uniformly continuous mild solution w such that $\overline{e^{i sp(w)}} = \overline{e^{i sp(f)}}$. 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.
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  • Department of Mathematics, University of Electro-Communications, Chofu, Tokyo 182-8585, Japan
  • Department of Mathematics, Hanoi University of Science, 334 Nguyen Trai, Hanoi, Vietnam
  • Department of Mathematics, University of Electro-Communications, Chofu, Tokyo 182-8585, Japan
  • Department of Mathematics, Korea University, Kodaira, Tokyo 187-8560, Japan
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bwmeta1.element.bwnjournal-article-doi-10_4064-sm145-2-1
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