New spectral criteria for almost periodic solutions of evolution equations
Tom 145 / 2001
Streszczenie
We present a general spectral decomposition technique for bounded solutions to inhomogeneous linear periodic evolution equations of the form , 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.
