New spectral criteria for almost periodic solutions of evolution equations
Volume 145 / 2001
Abstract
We present a general spectral decomposition technique for bounded solutions to inhomogeneous linear periodic evolution equations of the form $\dot {x}=A(t)x+f(t) \ (*)$, 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 $\overline {e^{i\hskip 1pt{\rm sp}(f)}}$ may intersect the spectrum of the monodromy operator $P$ of $(*)$ (here ${\rm sp}(f)$ denotes the Carleman spectrum of $f$). We show that if $(*)$ has a bounded uniformly continuous mild solution $u$ and $\sigma _{\mit \Gamma } (P) {\setminus} \overline {e^{i\hskip 1pt{\rm sp}(f)}}$ is closed, where $\sigma _{\mit \Gamma } (P)$ denotes the part of $\sigma (P)$ on the unit circle, then $(*)$ has a bounded uniformly continuous mild solution $w$ such that $\overline {e^{i\hskip 1pt{\rm sp}(w)}} =\overline {e^{i\hskip 1pt{\rm sp}(f)}}$. 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.
