Partially irregular almost periodic solutions of ordinary differential systems

Demenchuk, Alexandr. "Partially irregular almost periodic solutions of ordinary differential systems." Mathematica Bohemica 126.1 (2001): 221-228. <http://eudml.org/doc/248844>.

@article{Demenchuk2001,
abstract = {Let $f(t,x)$ be a vector valued function almost periodic in $t$ uniformly for $x$, and let $\{\mathrm \{m\}od\}(f)=L_1\oplus L_2$ be its frequency module. We say that an almost periodic solution $x(t)$ of the system \[ \dot\{x\} = f (t, x), \quad t\in \mathbb \{R\}, \ \ x\in D \subset \mathbb \{R\}^n \] is irregular with respect to $L_2$ (or partially irregular) if $(\{\mathrm \{m\}od\}(x)+L_1) \cap L_2 = \lbrace 0\rbrace $. Suppose that $ f(t,x) = A(t)x + X(t, x), $ where $A(t)$ is an almost periodic $(n\times n)$-matrix and $\{\mathrm \{m\}od\} (A)\cap \{\mathrm \{m\}od\}(X)= \lbrace 0\rbrace .$ We consider the existence problem for almost periodic irregular with respect to $\{\mathrm \{m\}od\} (A)$ solutions of such system. This problem is reduced to a similar problem for a system of smaller dimension, and sufficient conditions for existence of such solutions are obtained.},
author = {Demenchuk, Alexandr},
journal = {Mathematica Bohemica},
keywords = {almost periodic differential systems; almost periodic solutions; almost-periodic differential systems; almost-periodic solutions},
language = {eng},
number = {1},
pages = {221-228},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Partially irregular almost periodic solutions of ordinary differential systems},
url = {http://eudml.org/doc/248844},
volume = {126},
year = {2001},
}

TY - JOUR
AU - Demenchuk, Alexandr
TI - Partially irregular almost periodic solutions of ordinary differential systems
JO - Mathematica Bohemica
PY - 2001
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 126
IS - 1
SP - 221
EP - 228
AB - Let $f(t,x)$ be a vector valued function almost periodic in $t$ uniformly for $x$, and let ${\mathrm {m}od}(f)=L_1\oplus L_2$ be its frequency module. We say that an almost periodic solution $x(t)$ of the system \[ \dot{x} = f (t, x), \quad t\in \mathbb {R}, \ \ x\in D \subset \mathbb {R}^n \] is irregular with respect to $L_2$ (or partially irregular) if $({\mathrm {m}od}(x)+L_1) \cap L_2 = \lbrace 0\rbrace $. Suppose that $ f(t,x) = A(t)x + X(t, x), $ where $A(t)$ is an almost periodic $(n\times n)$-matrix and ${\mathrm {m}od} (A)\cap {\mathrm {m}od}(X)= \lbrace 0\rbrace .$ We consider the existence problem for almost periodic irregular with respect to ${\mathrm {m}od} (A)$ solutions of such system. This problem is reduced to a similar problem for a system of smaller dimension, and sufficient conditions for existence of such solutions are obtained.
LA - eng
KW - almost periodic differential systems; almost periodic solutions; almost-periodic differential systems; almost-periodic solutions
UR - http://eudml.org/doc/248844
ER -