## 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)].

### MSC:

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

### Keywords:

oscillation; nonoscillation; linear system

### Citations:

Zbl 0031.35402; Zbl 0078.07602; Zbl 0908.34023; Zbl 0877.34029
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