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Di erential Equations, Vol. 36, No. 12, 2000, pp. 1770{1776. Translated from Di erentsial'nye Uravneniya, Vol. 36, No. 12, 2000, pp. 1615{1620. Original Russian Text Copyright c 2000 by Lipnitskii. ORDINARY DIFFERENTIAL EQUATIONS A. V. Lipnitskii Institute of Mathematics, National Academy of Sciences, Belarus Received November 10, 1999 Erugin [1, p. 137] posed the following problem: are all systems x _ = A(t)x; x 2 R;t0; (1) with almost periodic [2, p. 20] coecient matrices Lyapunov proper? It was solved by Million- shchikov in his classical papers [3{5]. It was shown in [3] that almost all (in the sense of an arbi- trary measure translaion-invariant with respect to t) systems with bounded uniformly continuous coecient matrices are proper. The existence of improper systems of the form (1) with a limit- periodic [2, p. 114] coecient matrix of an arbitrary smoothness class up to C was provedin[4]. As was shown in [5], A(t) can be chosen to be quasiperiodic with arbitrary smoothness. A non- almost reducible di erential system in an appropriate class was explicitly constructed in [4, 5], and then, on the basis of Theorem 1 and Lemma 1 in [6], it was shown that the closure of the
Differential Equations – Springer Journals
Published: Oct 3, 2004
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