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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.

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