##
**Asymptotic representations of solutions of essentially nonlinear cyclic systems of ordinary differential equations.**
*(English.
Russian original)*
Zbl 1271.34055

Differ. Equ. 48, No. 5, 630-646 (2012); translation from Differ. Uravn. 48, No. 5, 622-639 (2012).

The authors consider the system
\[
y^\prime_{i} = \alpha _{i}p_{i}(t)\varphi _{i+1}(y_{i+1}) , \qquad i=1,2,\dotsc, n,
\]
and all functions with subscript \(n+1\) are assumed to coincide with those with subscript 1.

Here, \(\alpha _{i}\in \{ -1,1\} \) \(p_{i}: [a,\omega )\rightarrow (0,\infty )\) are continuous functions. \(-\infty <a<\omega \leq +\infty \), \(Y^{o}_{i}\) is equal to either zero or \(\pm \infty \), and \(\varphi _{i}(y)\) are normalized regularly varying functions of order \(\sigma _{i}\) at \(Y^{o}_{i}\) [E. Seneta, Lect. Notes Math. 508 (1976; Zbl 0324.26002)].

By applying a more general result of V. M. Evtukhov and A. M. Samoilenko [Ukr. Mat. Zh. 62, No. 1, 52–80 (2010); translation in Ukr. Math. J. 62, No. 1, 56–86 (2010; Zbl 1224.35033)], the existence of solutions \(y_{i}(t)\) belonging to a special class of functions is proved.

Also, the asymptotic representation as \(t\rightarrow \omega \) is obtained first for \(\frac{y_{i}(t)}{\varphi _{i+1}(y_{i+1}(t))}\) and under some additional condition on \(\varphi _{i}\), also for \(y_{i}(t)\).

Reviewer’s remark: The title of the above mentioned book by E. Seneta is in the References erroneously cited as “Properly varying function” and this false terminology is then propagated throughout the paper.

Here, \(\alpha _{i}\in \{ -1,1\} \) \(p_{i}: [a,\omega )\rightarrow (0,\infty )\) are continuous functions. \(-\infty <a<\omega \leq +\infty \), \(Y^{o}_{i}\) is equal to either zero or \(\pm \infty \), and \(\varphi _{i}(y)\) are normalized regularly varying functions of order \(\sigma _{i}\) at \(Y^{o}_{i}\) [E. Seneta, Lect. Notes Math. 508 (1976; Zbl 0324.26002)].

By applying a more general result of V. M. Evtukhov and A. M. Samoilenko [Ukr. Mat. Zh. 62, No. 1, 52–80 (2010); translation in Ukr. Math. J. 62, No. 1, 56–86 (2010; Zbl 1224.35033)], the existence of solutions \(y_{i}(t)\) belonging to a special class of functions is proved.

Also, the asymptotic representation as \(t\rightarrow \omega \) is obtained first for \(\frac{y_{i}(t)}{\varphi _{i+1}(y_{i+1}(t))}\) and under some additional condition on \(\varphi _{i}\), also for \(y_{i}(t)\).

Reviewer’s remark: The title of the above mentioned book by E. Seneta is in the References erroneously cited as “Properly varying function” and this false terminology is then propagated throughout the paper.

Reviewer: Vojislav Marić (Novi Sad)

### MSC:

34D05 | Asymptotic properties of solutions to ordinary differential equations |

26A12 | Rate of growth of functions, orders of infinity, slowly varying functions |

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\textit{V. M. Evtukhov} and \textit{E. S. Vladova}, Differ. Equ. 48, No. 5, 630--646 (2012; Zbl 1271.34055); translation from Differ. Uravn. 48, No. 5, 622--639 (2012)

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### References:

[1] | Mirzov, D.D., Asymptotic Properties of Solutions of a System of Emden-Fowler Type, Differ. Uravn., 1985, vol. 21, no. 9, pp. 1498–1504. · Zbl 0591.34047 |

[2] | Mirzov, D.D., Some Asymptotic Properties of the Solutions of a System of Emden-Fowler Type, Differ. Uravn., 1987, vol. 23, no. 9, pp. 1519–1532. |

[3] | Mirzov, D.D., Asimptoticheskie svoistva reshenii sistem nelineinykh neavtonomnykh obyknovennykh differentsial’nykh uravnenii (Asymptotic Properties of Solutions of Systems of Nonlinear Nonautonomous Ordinary Differential Equations), Maikop, 1993. |

[4] | Evtukhov, V.M., Asymptotic Representations of Regular Solutions of a Two-Dimensional System of Differential Equations, Dopov. Nats. Akad. Nauk Ukr., 2002, no. 4, pp. 11–17. · Zbl 1015.34024 |

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[8] | Belozerova, M.A., Asymptotic Properties of a Class of Solutions of Essentially Nonlinear Second-Order Differential Equations, Mat. Stud., 2008, vol. 29, no. 1, pp. 52–62. · Zbl 1164.34425 |

[9] | Bilozerova, M.O., Nauk. Visn. Cherniv. Univ., Chernivtsi, 2008, vol. 374, pp. 34–43. |

[10] | Belozerova, M.A., Asymptotic Representations of Solutions of Second-Order Nonautonomous Differential Equations with Nonlinearities Close to Power Type, Nelin. Koleb., 2009, vol. 12, no. 1, pp. 3–15. · Zbl 1277.34061 |

[11] | Evtukhov, V.M. and Belozerova, M.A., Asymptotic Representations of Solutions of Essentially Nonlinear Nonautonomous Differential Equations of the Second Order, Ukrain. Mat. Zh., 2008, vol. 60, no. 3, pp. 310–331. · Zbl 1164.34427 · doi:10.1007/s11253-008-0063-2 |

[12] | Evtukhov, V.M., Asymptotic Representations of Solutions of Nonautonomous Ordinary Differential Equations, Doctoral (Phys.-Math.) Dissertation, Kiev, 1998. |

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