New spectral criteria for almost periodic solutions of evolution equations

@article{ToshikiNaito2001,
abstract = {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.},
author = {Toshiki Naito, Nguyen Van Minh, Jong Son Shin},
journal = {Studia Mathematica},
keywords = {periodic evolution equation; spectrum of functions; spectral decomposition; almost-periodic mild solution; quasi-periodic mild solution; Fourier coefficient; Carleman spectrum},
language = {eng},
number = {2},
pages = {97-111},
title = {New spectral criteria for almost periodic solutions of evolution equations},
url = {http://eudml.org/doc/284429},
volume = {145},
year = {2001},
}

TY - JOUR
AU - Toshiki Naito
AU - Nguyen Van Minh
AU - Jong Son Shin
TI - New spectral criteria for almost periodic solutions of evolution equations
JO - Studia Mathematica
PY - 2001
VL - 145
IS - 2
SP - 97
EP - 111
AB - 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.
LA - eng
KW - periodic evolution equation; spectrum of functions; spectral decomposition; almost-periodic mild solution; quasi-periodic mild solution; Fourier coefficient; Carleman spectrum
UR - http://eudml.org/doc/284429
ER -