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New spectral criteria for almost periodic solutions of evolution equations

Tom 145 / 2001

Toshiki Naito, Nguyen Van Minh, Jong Son Shin Studia Mathematica 145 (2001), 97-111 MSC: Primary 34G10, 47D06; Secondary 34C27, 35B15, 45M15. DOI: 10.4064/sm145-2-1

Streszczenie

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.

Autorzy

  • Toshiki NaitoDepartment of Mathematics
    University of Electro-Communications
    Chofu, Tokyo 182-8585, Japan
    e-mail
  • Nguyen Van MinhDepartment of Mathematics
    Hanoi University of Science
    334 Nguyen Trai, Hanoi, Vietnam
    and
    Department of Mathematics
    University of Electro-Communications
    Chofu, Tokyo 182-8585, Japan
    e-mail
  • Jong Son ShinDepartment of Mathematics
    Korea University
    Kodaira, Tokyo 187-8560, Japan
    e-mail

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