Partially irregular almost periodic solutions of ordinary differential systems

Demenchuk, Alexandr

Mathematica Bohemica, Tome 126 (2001) no. 1, pp. 221-228.

Résumé

Let $f(t,x)$ be a vector valued function almost periodic in $t$ uniformly for $x$, and let ${\mathrm mod}(f)=L_1\oplus L_2$ 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 mod}(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 ${\mathrm mod} (A)\cap {\mathrm mod}(X)= \lbrace 0\rbrace .$ We consider the existence problem for almost periodic irregular with respect to ${\mathrm mod} (A)$ 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.

Détail

DOI : 10.21136/MB.2001.133916

Classification : 34C27
Keywords: almost periodic differential systems; almost periodic solutions