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Oscillatory properties of solutions to certain two-dimensional systems of non-linear ordinary differential equations. (English) Zbl 1336.34053
The authors derive new oscillation criteria for the nonlinear system
\[ u'=g(t)\,|v|^{1/\alpha}{\text{sgn}}v, \quad v'=-p(t)\,|u|^\alpha{\text{sgn}}u, \tag{*} \] where \(\alpha>0\), the coefficients \(g\) and \(p\) are locally integrable on \([0,\infty)\), and \(g(t)\geq0\) with \(\int_0^\infty g(t)\,dt=\infty\) or \(\int_0^\infty g(t)\,dt<\infty\). No sign condition on \(p(t)\) is imposed. The authors establish new Kamenev and Hartman-Wintner type criteria for system (\(*\)), which extend known results in the literature. The proofs are based on a sophisticated analysis of inequalities involving the (assumed) nonoscillatory solutions of (\(*\)).

34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
Full Text: DOI
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