MATHEMATICA BOHEMICA, Vol. 126, No. 1, pp. 221-228 (2001)

Partially irregular almost periodic solutions of ordinary differential systems

Alexandr Demenchuk

Alexandr Demenchuk, Department of Differential Equations, Institute of Mathematics, National Academy of Sciences of Belarus, Surganova 11, 220072 Minsk, Belarus, e-mail:

Abstract: Let $f(t,x)$ be a vector valued function almost periodic in $t$ uniformly for $x$, and let $ mod(f)=L_1\oplus L_2$ be its frequency module. We say that an almost periodic solution $x(t)$ of the system $$ \dot x = f (t, x), \quad t\in \R , x\in D \subset \R ^n $$ is irregular with respect to $L_2$ (or partially irregular) if $( mod(x)+L_1) \cap L_2 = \{0\}$. \endgraf Suppose that $ f(t,x) = A(t)x + X(t, x), $ where $A(t)$ is an almost periodic $(n\times n)$-matrix and $ mod (A)\cap mod(X)= \{0\}.$ We consider the existence problem for almost periodic irregular with respect to $ 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.

Keywords: almost periodic differential systems, almost periodic solutions

Classification (MSC2000): 34C27

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