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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 (\(*\)).

MSC:
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
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[1] Došlý, O.; Řehák, P., (Half-Linear Differential Equations, North-Holland Mathematics Studies, vol. 202, (2005), Elsevier Amsterdam)
[2] Hardy, G. H.; Littlewood, J. E.; Pólya, G., Inequalities, (1951), Cambridge University Press London · Zbl 0634.26008
[3] Hoshino, H.; Imabayashi, R.; Kusano, T.; Tanigawa, T., On second-order half-linear oscillations, Adv. Math. Sci. Appl., 8, 1, 199-216, (1998) · Zbl 0898.34036
[4] Kamenev, I. V., An integral criterion for oscillation of linear differential equations of second order, Math. Notes, 23, 2, 136-138, (1978) · Zbl 0408.34031
[5] Kandelaki, N.; Lomtatidze, A.; Ugulava, D., On oscillation and nonoscillation of a second order half-linear equation, Georgian Math. J., 7, 2, 329-346, (2000) · Zbl 0957.34032
[6] Kusano, T.; Naito, Y., Oscillation and nonoscillation criteria for second order quasilinear differential equations, Acta Math. Hungar., 76, 1-2, 81-99, (1997) · Zbl 0906.34024
[7] Kusano, T.; Wang, J., Oscillation properties of half-linear functional-differential equations of the second order, Hiroshima Math. J., 25, 2, 371-385, (1995) · Zbl 0840.34079
[8] Li, H. J., Oscillation criteria for half-linear second order differential equations, Hiroshima Math. J., 25, 3, 571-583, (1995) · Zbl 0872.34018
[9] Li, H. J.; Yeh, C. C., An integral criterion for oscillation of nonlinear differential equations, Math. Japon., 41, 1, 185-188, (1995) · Zbl 0816.34024
[10] Li, H. J.; Yeh, C. C., Oscillations of half-linear second order differential equations, Hiroshima Math. J., 25, 3, 585-594, (1995) · Zbl 0872.34019
[11] Lomtatidze, A., Oscillation and nonoscillation of Emden-Fowler type equation of second-order, Arch. Math. (Brno), 32, 3, 181-193, (1996) · Zbl 0908.34023
[12] Lomtatidze, A.; Šremr, J., On oscillation and nonoscillation of two-dimensional linear differential system, Georgian Math. J., 20, 3, 573-600, (2013) · Zbl 1296.34082
[13] Manojlović, J. V., Oscillation criteria for second-order half-linear differential equations, Math. Comput. Modelling, 30, 5-6, 109-119, (1999) · Zbl 1042.34532
[14] Mirzov, J. D., On some analogs of sturm’s and kneser’s theorems for nonlinear systems, J. Math. Anal. Appl., 53, 2, 418-425, (1976) · Zbl 0327.34027
[15] Mirzov, J. D., (Asymptotic properties of solutions of systems of nonlinear nonautonomous ordinary differential equations, Folia Facul. Sci. Natur. Univ. Masar. Brun., Mathematica, vol. 14, (2004), Masaryk University Brno) · Zbl 1154.34300
[16] Řehák, P., A Riccati technique for proving oscillation of a half-linear equation, Electron. J. Differential Equations, 2008, 105, 1-8, (2008) · Zbl 1170.34317
[17] Zhou, Y.; Chen, X. W., Oscillation and nonoscillation of second order half-linear differential equations, J. Math. Sci. (NY), 191, 3, 344-353, (2013) · Zbl 1310.34044
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