New spectral criteria for almost periodic solutions of evolution equations

Toshiki Naito ; Nguyen Van Minh ; Jong Son Shin

Studia Mathematica, Tome 147 (2001), p. 97-111 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

Détail

Publié le : 2001-01-01

Zbl 0982.34074

EUDML-ID : urn:eudml:doc:284429