##
**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.

Reviewer: Michal Veselý (Brno)

### MSC:

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

### Citations:

Zbl 0872.34019
Full Text:
DOI

### References:

[1] | O. Došlý and P. Řehák, Half-linear Differential Equations, North-Holland Mathematics Studies 202, Elsevier Science B.V., Amsterdam, 2005. · Zbl 1090.34001 |

[2] | M. Dosoudilová, A. Lomtatidze and J. Šremr, Oscillatory properties of solutions to certain two-dimensional systems of non-linear ordinary differential equations, Nonlinear Anal. 120 (2015), 57-75. · Zbl 1336.34053 |

[3] | P. Hartman, Ordinary Differential Equations, John Wiley & Sons, New York, 1964; Classics in Applied Mathematics 38, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. |

[4] | H. Hoshino, R. Imabayashi, T. Kusano and T. Tanigawa, On second-order half-linear oscillations, Adv. Math. Sci. Appl. 8 (1998), no. 1, 199-216. · Zbl 0898.34036 |

[5] | J. Jaroš, K. Takaŝi and T. Tanigawa, Nonoscillatory half-linear differential equations and generalized Karamata functions, Nonlinear Anal. 64 (2006), no. 4, 762-787. · Zbl 1103.34017 |

[6] | ____, Nonoscillatory solutions of planar half-linear differential systems: a Riccati equation approach, Electron. J. Qual. Theory Differ. Equ. 2018, Paper No. 92, 28 pp. · Zbl 1413.34139 |

[7] | T. Kusano and J. V. Manojlović, Precise asymptotic behavior of regularly varying solutions of second order half-linear differential equations, Electron. J. Qual. Theory Differ. Equ. 2016, Paper No. 62, 24 pp. · Zbl 1389.34164 |

[8] | T. Kusano, A. Ogata and H. Usami, Oscillation theory for a class of second order quasilinear ordinary differential equations with application to partial differential equations, Japan. J. Math. (N.S.) 19 (1993), no. 1, 131-147. · Zbl 0792.34033 |

[9] | H. J. Li and C. C. Yeh, Oscillations of half-linear second order differential equations, Hiroshima Math. J. 25 (1995), no. 3, 585-594. · Zbl 0872.34019 |

[10] | ____, Sturmian comparison theorem for half-linear second-order differential equations, Proc. Roy. Soc. Edinburgh Sect. A 125 (1995), no. 6, 1193-1204. · Zbl 0873.34020 |

[11] | A. Lomtatidze and N. Partsvania, Oscillation and nonoscillation criteria for two-dimensional systems of first order linear ordinary differential equations, Georgian Math. J. 6 (1999), no. 3, 285-298. · Zbl 0930.34025 |

[12] | A. Lomtatidze and J. Šremr, On oscillation and nonoscillation of two-dimensional linear differential systems, Georgian Math. J. 20 (2013), no. 3, 573-600. · Zbl 1296.34082 |

[13] | J. D. Mirzov, On some analogs of Sturm’s and Kneser’s theorems for nonlinear systems, J. Math. Anal. Appl. 53 (1976), no. 2, 418-425. · Zbl 0327.34027 |

[14] | D. D. Mirzov, The oscillation of the solutions of a certain system of differential equations, Math. Zametki 23 (1978), no. 3, 401-404. · Zbl 0407.34034 |

[15] | M. Naito, Asymptotic behavior of nonoscillatory solutions of half-linear ordinary differential equations, Arch. Math. (Basel) 116 (2021), no. 5, 559-570. · Zbl 1468.34076 |

[16] | ____, Remarks on the existence of nonoscillatory solutions of half-linear ordinary differential equations: I, Opuscula Math. 41 (2021), no. 1, 71-94. · Zbl 1478.34064 |

[17] | ____, Remarks on the existence of nonoscillatory solutions of half-linear ordinary differential equations: II, Arch. Math. (Brno) 57 (2021), no. 1, 41-60. · Zbl 07332703 |

[18] | M. Naito and H. Usami, On the existence and asymptotic behavior of solutions of half-linear ordinary differential equations, J. Differential Equations 318 (2022), 359-383. · Zbl 1497.34075 |

[19] | Z. Opluštil, Oscillation criteria for two dimensional system of non-linear ordinary differential equations, Electron. J. Qual. Theory Differ. Equ. 2016, Paper No. 52, 17 pp. · Zbl 1363.34098 |

[20] | ____, On non-oscillation for certain system of non-linear ordinary differential equations, Georgian Math. J. 24 (2017), no. 2, 277-285. · Zbl 1367.34035 |

[21] | ____, Oscillatory properties of certain system of non-linear ordinary differential equations, Miskolc Math. Notes 19 (2018), no. 1, 439-459. · Zbl 1463.34149 |

[22] | P. Řehák, Asymptotic formulae for solutions of half-linear differential equations, Appl. Math. Comput. 292 (2017), 165-177. · Zbl 1410.34104 |

[23] | P. Řehák and V. Taddei, Solutions of half-linear differential equations in the classes gamma and pi, Differential Integral Equations 29 (2016), no. 7-8, 683-714. · Zbl 1374.34206 |

[24] | K. Takaŝi and J. V. Manojlović, Asymptotic behavior of solutions of half-linear differential equations and generalized Karamata functions, Georgian Math. J. 28 (2021), no. 4, 611-636. · Zbl 1476.34115 |

[25] | D. Willett, On the oscillatory behavior of the solutions of second order linear differential equations, Ann. Polon. Math. 21 (1969), 175-194. · Zbl 0174.13701 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.