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On Sturm type theorems for linear systems of differential equations. (Russian) Zbl 0594.34010

The author deals with two hypotheses of Mingarelli for the equations (1): \(d^ 2x/dt^ 2+P(t)x=0\) and (2): \(d^ 2x/dt^ 2+Q(t)x=0\) with continuous \(n\times n\) matrices P(t), Q(t) for \(t\in [a,b]\) provided all proper values are real (the greatest of them being denoted \(\lambda_{P(t)}\), \(\lambda_{Q(t)}):\) \(1^ o\) if \(\lambda_{P(t)}\leq 0\) for all \(t\in [a,b]\) then (1) is disconjugate; \(2^ o\) if \(\lambda_{P(t)}\leq \lambda_{Q(t)}\) for all \(t\in [a,b]\), \(\lambda_ P\neq \lambda_ Q\) and a,b are conjugate points for (1), then there exists \(c\in]a,b[\) such that a,c are conjugate points for (2). By means of three geometric lemmas concerning \({\mathbb{R}}^ 3\) the author proves that the hypotheses \(1^ o\) and \(2^ o\) are false for \(n=3\).
Reviewer: E.Barvinek

MSC:

34A30 Linear ordinary differential equations and systems
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