We consider a differential equation of the n th order containing a sum of terms with regularly and rapidly varying nonlinearities on the right-hand side and determine the asymptotic behavior of some types of solutions to this equation.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
V. M. Evtukhov, Asymptotic Representations of Solutions of Nonautonomous Ordinary Differential Equations [in Russian], Doctoral-Degree Thesis (Physics and Mathematics), Kyiv (1998).
V. M. Evtukhov and L. A. Kirillova, “On the asymptotics of solutions of nonlinear second-order differential equations,” Differents. Uravn., 41, No. 8, 1053–1061 (2005); English translation: Different. Equat., 41, No. 8, 1105–1114 (2005); https://doi.org/10.1007/s10625-005-0256-5.
V. M. Evtukhov and A. M. Klopot, “Asymptotic representations for some classes of solutions of ordinary differential equations of order n with regularly varying nonlinearities,” Ukr. Mat. Zh., 65, No. 3, 354–380 (2013); English translation: Ukr. Math. J., 65, No. 3, 393–422 (2013); https://doi.org/10.1007/s11253-013-0785-7.
V. M. Evtukhov and A. M. Klopot, “Asymptotic behavior of solutions of n th-order ordinary differential equations with regularly varying nonlinearities,” Differents. Uravn., 50, No. 5, 584–600 (2014); English translation: Different. Equat., 50, No. 5, 581–597 (2014); https://doi.org/10.1134/S0012266114050024.
V. M. Evtukhov and N. P. Kolun, “Asymptotics of the solutions of second-order differential equations with regularly and rapidly varying nonlinearities,” Nelin. Kolyv., 21, No. 3, 323–346 (2018); English translation: J. Math. Sci., 243, No. 3, 381–408 (2019); https://doi.org/10.1007/s10958-019-04546-w.
V. M. Evtukhov and N. P. Kolun, “Asymptotic representations of the solutions of differential equations with regularly and rapidly varying nonlinearities,” Mat. Met. Fiz.-Mekh. Polya, 60, No. 1, 32–42 (2017); English translation: J. Math. Sci., 240, No. 1, 34–47 (2019); https://doi.org/10.1007/s10958-019-04334-6.
V. M. Evtukhov and N. P. Kolun, “Rapidly varying solutions of a second-order differential equation with regularly and rapidly varying nonlinearities,” Ukr. Mat. Visn., 15, No. 1, 18–42 (2018); English translation: J. Math. Sci., 235, No. 1, 15–34 (2018); https://doi.org/10.1007/s10958-018-4055-y.
V. M. Evtukhov and A. M. Samoilenko, “Asymptotic representations of solutions of nonautonomous ordinary differential equations with regularly varying nonlinearities,” Differ. Uravn., 47, No. 5, 628–650 (2011); English translation: Differ. Equat., 47, No. 5, 627–649 (2011); https://doi.org/10.1134/S001226611105003X.
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); https://doi.org/10.1007/s11253-010-0333-7.
A. M. Klopot, “Asymptotic behavior of solutions of nonautonomous ordinary differential equations of the n th order with regularly varying nonlinearities,” Visn. Odes’k. Nats. Univ. Mat. Mekh., 18, Issue 3 (19), 16–34 (2013).
A. M. Klopot, “On the asymptotics of solutions of nonautonomous differential equations of order n ,” Nelin. Kolyv., 15, No. 4, 447–465 (2012); English translation: J. Math. Sci., 194, No. 4, 354–373 (2013); https://doi.org/10.1007/s10958-013-1534-z.
N. P. Kolun, “Asymptotics of slowly varying solutions of the second-order differential equations with regularly and rapidly varying nonlinearities,” Doslidzh. Mat. Mekh., 23, Issue 2 (32), 54–67 (2018).
N. P. Kolun, “Asymptotic behavior of the solutions of differential equations of the second order with nonlinearities of different kinds,” Nauk. Visn. Uzhgorod. Nats. Univ., Ser. Mat. Inform., Issue 1 (34), 26–41 (2019).
N. P. Kolun, “Asymptotic representations of slowly varying solutions of second-order differential equations with nonlinearities of different types on the right-hand side,” Bukov. Mat. Zh., 6, Nos. 3-4, 89–102 (2018).
E. Seneta, Regularly Varying Functions, Ser.: Lecture Notes in Mathematics, Vol. 508, Springer (1976).
S. Cano-Casanova, “Decay rate at infinity of the positive solutions of a generalized class of Thomas–Fermi equations,” in: W. Feng, Z. Feng, M. Grasselli, A. Ibragimov, X. Lu, S. Siegmund, and J. Voigt (editors), Proc. of the 8th AIMS Conf. on Discrete and Continuous Dynamical Systems (Dresden, Germany), Supplement 2011, Vol. 1, American Institute of Mathematical Sciences, pp. 240–249.
V. M. Evtukhov and A. M. Klopot, “Asymptotic behavior of solutions of ordinary differential equations of n th order with regularly varying nonlinearities,” Mem. Differ. Equat. Math. Phys., 61, 37–61 (2014).
V. M. Evtukhov and N. P. Kolun, “Asymptotic behaviour of solutions of second-order nonlinear differential equations,” Mem. Differ. Equat. Math. Phys., 75, 105–114 (2018).
T. Kusano, J. V. Manojlović, and V. Marić, “Increasing solutions of Thomas–Fermi type differential equations—The sublinear case,” Bull. Acad. Serbe Sci. Arts, Cl. Sci. Math. Natur., Sci. Math., CXLIII, No. 36, 21–36 (2011).
J. V. Manojlović and V. Marić, “An asymptotic analysis of positive solutions of Thomas–Fermi type sublinear differential equations,” Mem. Differ. Equat. Math. Phys., 57, 75–94 (2012).
V. Marić, Regular Variation and Differential Equations, Series: Lecture Notes in Mathematics, Vol. 1726, Springer, Berlin–Heidelberg (2000).
V. Marić and Z. Radašin, “Asymptotic behavior of solutions of the equation y′′ = f (t)φ(ψ(y)) ,” Glas. Mat., 23 (43), No. 1, 27–34 (1988).
V. Marić and M. Tomić, “Asymptotics of solutions of a generalized Thomas–Fermi equation,” J. Differ. Equat., 35, No. 1, 36–44 (1980); https://doi.org/10.1016/0022-0396(80)90047-9.
S. D. Taliaferro, “Asymptotic behavior of positive decreasing solutions of y′′ = F(t, y, y′),” in: I. J. Bakelman (editor), Geometric Analysis and Nonlinear PDEs, Series: Lecture Notes in Pure and Applied Mathematics, Vol. 144, Marcel Dekker, New York (1993), pp. 105–127.
S. D. Taliaferro, “Asymptotic behavior of solutions of y′′ = φ(t) f (y),” SIAM J. Math. Anal., 12, No. 6, 853–865 (1981); https://doi.org/10.1137/0512071.
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 63, No. 4, pp. 34–45, October–December, 2020.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Kolun, N.P. Asymptotic Behavior of Some Types of Solutions of Differential Equations with Different Types of Nonlinearities. J Math Sci 273, 924–938 (2023). https://doi.org/10.1007/s10958-023-06554-3
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-023-06554-3