Oscillation and nonoscillation criteria for two-dimensional systems of first-order linear ordinary differential equations. (English) Zbl 0930.34025

This article contains oscillation and nonoscillation criteria for the linear system \[ u'= p(x)v,\quad v'=- q(x)u,\quad x\in\mathbb{R}_+,\tag{1} \] with \(p,q\in L^1_{\text{loc}}(\mathbb{R}_+, \mathbb{R}_+)\), \(\int^\infty p=+\infty\), and \(\int^\infty q<\infty\). The two main theorems establish sufficient conditions of Hille’s type for all nontrivial solutions to (1) to be oscillatory at \(\infty\) (componentwise). Additional results are nonoscillation criteria for (1), in which \(q(x)\) is permitted to change sign. The theorems generalize classic ones of E. Hille [Trans. Am. Math. Soc. 64, 234-252 (1948; Zbl 0031.35402)] and Z. Nehari [Trans. Am. Math. Soc. 85, 428-445 (1957; Zbl 0078.07602)]. Analogues for linear and semilinear scalar differential equations were obtained by A. Lomtatidze [Arch. Math., Brno, 32, No. 3, 181-193 (1996; Zbl 0908.34023); Georgian Math. J. 4, No. 2, 129-138 (1997; Zbl 0877.34029)].


34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34A30 Linear ordinary differential equations and systems
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