New spectral criteria for almost periodic solutions of evolution equations

Toshiki Naito, Nguyen Van Minh, and Jong Son Shin. "New spectral criteria for almost periodic solutions of evolution equations." Studia Mathematica 145.2 (2001): 97-111. <http://eudml.org/doc/284429>.

@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 -