## 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.

### MSC:

 34D05 Asymptotic properties of solutions to ordinary differential equations 26A12 Rate of growth of functions, orders of infinity, slowly varying functions

### Citations:

Zbl 0324.26002; Zbl 1224.35033
Full Text:

### 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 [5] Evtukhov, V.M., Asymptotic Representations of Regular Solutions of a Semilinear Two-Dimensional System of Differential Equations, Dopov. Nats. Akad. Nauk Ukr., 2002, no. 5, pp. 11–17. · Zbl 1015.34024 [6] Seneta, E., Pravil’no menyayushchiesya funktsii (Properly Varying Functions), Moscow, 1985. · Zbl 0563.26002 [7] Evtukhov, V.M. and Samoilenko, A.M., Existence Conditions for Solutions Vanishing at Singular Point for Real Nonautonomous Systems of Quasilinear Differential Equations, Ukrain. Mat. Zh., 2010, vol. 62, no. 1, pp. 52–80. · Zbl 1224.35033 · doi:10.1007/s11253-010-0333-7 [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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