New spectral criteria for almost periodic solutions of evolution equations

Toshiki Naito¹ ; Nguyen Van Minh² ; Jong Son Shin³

¹ Department of Mathematics University of Electro-Communications Chofu, Tokyo 182-8585, Japan
² Department of Mathematics Hanoi University of Science 334 Nguyen Trai, Hanoi, Vietnam and Department of Mathematics University of Electro-Communications Chofu, Tokyo 182-8585, Japan
³ Department of Mathematics Korea University Kodaira, Tokyo 187-8560, Japan

Studia Mathematica, Tome 145 (2001) no. 2, pp. 97-111

Résumé

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.

Détail

DOI : 10.4064/sm145-2-1

Keywords: present general spectral decomposition technique bounded solutions inhomogeneous linear periodic evolution equations form dot f * having precompact range which applied spectral criteria existence almost periodic solutions specific spectral properties resonant where overline hskip may intersect spectrum monodromy operator * here denotes carleman spectrum * has bounded uniformly continuous mild solution sigma mit gamma setminus overline hskip closed where sigma mit gamma denotes part sigma unit circle * has bounded uniformly continuous mild solution overline hskip overline hskip moreover spectral component allows solve general massera type problem almost periodic solutions various spectral criteria existence almost periodic quasi periodic mild solutions * given