New spectral criteria for almost periodic solutions of evolution equations

• Autorzy: Toshiki Naito , Nguyen Van Minh , Jong Son Shin
• Języki publikacji: EN

Abstrakty

EN

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.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2001
• Tom: 145
• Numer: 2
• Strony: 97-111

Daty

wydano

2001

Twórcy

autor

Toshiki Naito

• Department of Mathematics, University of Electro-Communications, Chofu, Tokyo 182-8585, Japan

autor

Nguyen Van Minh

• Department of Mathematics, Hanoi University of Science, 334 Nguyen Trai, Hanoi, Vietnam
• Department of Mathematics, University of Electro-Communications, Chofu, Tokyo 182-8585, Japan

autor

Jong Son Shin

• Department of Mathematics, Korea University, Kodaira, Tokyo 187-8560, Japan

Identyfikatory

DOI

10.4064/sm145-2-1