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Oscillation and nonoscillation for two-dimensional nonlinear systems of ordinary differential equations. (English) Zbl 1521.34032

The author considers two-dimensional non-linear systems of ordinary differential equations in the form \begin{align*} u^\prime &= a(t) \left|v\right|^{\frac{1}{\alpha}} \text{sgn}\,v, \\ v^\prime &= -b(t) \left|u\right|^{{\alpha}} \text{sgn}\,u, \end{align*} where \(\alpha \in (0, \infty)\) is a constant, \(a, b\) are real-valued continuous functions, and \(a\) is non-negative and not identically zero in a neighbourhood of infinity. The paper contains three main results – two oscillation criteria (in fact, one of them consists of two results) and one non-oscillation criterion. In these criteria, the both cases \[ \int^\infty a(\tau) \, \mathrm{d}\tau = \infty, \qquad \int^\infty a(\tau) \, \mathrm{d}\tau < \infty \] are included. Since the treated systems are generalizations of half-linear equations, there exists a clear motivation for the study of these systems. In addition, for half-linear equations, many useful consequences of the main results are collected in Section 3. Then, interesting illustrative examples are mentioned in Section 4. Note that results from [H. J. Li and C. C. Yeh, Hiroshima Math. J. 25, No. 3, 585–594 (1995; Zbl 0872.34019)] are improved in the paper under review.

MSC:

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

Citations:

Zbl 0872.34019
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References:

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