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Partially irregular almost-periodic solutions to ordinary differential systems. (English) Zbl 0987.34039
Summary: 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. The author says that an almost-periodic solution $$x(t)$$ to 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 $$(\text{mod}(x)+L_1) \cap L_2 = \{0\}$$.
Suppose that $$f(t,x) = A(t)x + X(t, x),$$ where $$A(t)$$ is an almost-periodic $$n\times n$$-matrix and $$\mod(A)\cap \text{mod}(X)= \{0\}.$$ The author considers the existence of almost-periodic irregular solutions with respect to $$\text{mod}(A)$$ to such a system. This problem is reduced to a similar problem for a system of smaller dimension, and sufficient conditions for the existence of such solutions are obtained.

##### MSC:
 34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations
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