×

Asymptotic representations for the solutions of second-order differential equations with rapidly and regularly varying nonlinearities. (English. Ukrainian original) Zbl 1428.34071

J. Math. Sci., New York 238, No. 3, 316-332 (2019); translation from Neliniĭni Kolyvannya 20, No. 4, 549-563 (2017).
The author considers the equation \[ y''=\alpha_0p(t)\varphi_0(y)\varphi_1(y'), \] where \(\alpha_0=\pm 1\), \(p: [a,\omega)\to(0,\infty)\) with \(-\infty< a <\omega\le+\infty\), \(\varphi_1\) is a regularly varaing function, and the following limit holds \[ \lim_{y\to Y_0}\frac{\varphi_0(y)\varphi''_0(y)}{(\varphi'_0(y))^2}=1, \] in which \(Y_0\in\{0,\pm\infty\}\). Asymptotic representations for solutions are obtained. An illustrative example is given as well.

MSC:

34D05 Asymptotic properties of solutions to ordinary differential equations
26A12 Rate of growth of functions, orders of infinity, slowly varying functions
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Evtukhov, VM; Drik, NG, Asymptotic behavior of solutions of a second-order nonlinear differential equation, Georg. Math. J., 3, 123-151, (1996) · Zbl 0853.34050
[2] Evtukhov, VM; Kharkov, VM, Asymptotic representation of solutions of essentially nonlinear differential equations of second order, Differents. Uravn., 43, 1311-1323, (2007) · Zbl 1161.34024
[3] E. Seneta, Regularly Varying Functions [Russian translation], Nauka, Moscow (1985). · Zbl 0563.26002
[4] V. Maric, Regular Variation and Differential Equations, Springer Science & Business Media, Berlin (2000). · Zbl 0946.34001
[5] N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular Variation, Cambridge Univ. Press, Cambridge (1987). · Zbl 0617.26001
[6] Evtukhov, VM; Klopot, AM, Asymptotic behavior of solutions of ordinary differential equations of order \(n\) with regularly varying nonlinearities, Differents. Uravn., 50, 584-600, (2014) · Zbl 1323.34071
[7] V. M. Evtukhov and A. M. Samoilenko, “Conditions for the existence of solutions of real nonautonomous systems of quasilinear differential equations vanishing at a singular point,” Ukr. Mat. Zh., 62, No. 1, 52-80 (2010); English translation: Ukr. Math. J., 62, No. 1, 56-86 (2010). · Zbl 1224.35033
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.