×

Asymptotic representations of solutions of nonautonomous ordinary differential equations with regularly varying nonlinearities. (English. Russian original) Zbl 1242.34092

Differ. Equ. 47, No. 5, 627-649 (2011); translation from Differ. Uravn. 47, No. 5, 627-649 (2011).
The authors obtain asymptotic representations of solutions of a class of nonautonomous ordinary differential equations with regularly varying nonlinearities.

MSC:

34D05 Asymptotic properties of solutions to ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Seneta, E., Regularly Varying Functions, Berlin: Springer-Verlag, 1976. Translated under the title Pravil’no menyayushchiesya funktsii, Moscow: Nauka, 1985.
[2] Adamovic, D.D., Sur quelques propertietes des fonctions a croissance lente de Karamata. I, II, Mat. Vesnik (Beograd), 1966, vol. 3, pp. 123–136, 161–172.
[3] Kiguradze, I.T. and Chanturiya, T.A., Asimptoticheskie svoistva reshenii neavtonomnykh obyknovennykh differentsial’nykh uravnenii (Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations), Moscow, 1990.
[4] Kostin, A.V., The Asymptotic Behavior of the Extendable Solutions of Equations of Emden-Fowler Type, Dokl. Akad. Nauk SSSR, 1971, vol. 200, no. 1, pp. 28–31.
[5] Kostin, A.V. and Evtukhov, V.M., Asymptotic Behavior of the Solutions of a Certain Nonlinear Differential Equation, Dokl. Akad. Nauk SSSR, 1976, vol. 231, no. 5, pp. 1059–1062.
[6] Evtukhov, V.M., Asymptotic Representations of Solutions of a Class of Second-Order Nonlinear Differential Equations, Soobshch. Akad. Nauk GSSR, 1982, vol. 106, no. 3, pp. 473–476. · Zbl 0502.34042
[7] Evtukhov, V.M., Asymptotic Properties of the Solutions of a Certain Class of Second-Order Differential Equations, Math. Nachr., 1984, vol. 15, pp. 215–236.
[8] Kostin, A.V., Asymptotics of the Regular Solutions of Nonlinear Ordinary Differential Equations, Differ. Uravn., 1987, vol. 23, no. 3, pp. 524–526. · Zbl 0654.34048
[9] Evtukhov, V.M., Asymptotic Properties of Monotone Solutions of a Class of Nonlinear Differential Equations of the nth-Order, Dokl. Rasshir. Zased. Sem. Inst. Prikl. Mat., 1988, vol. 3, no. 3, pp. 62–65.
[10] Evtukhov, V.M., Asymptotic Representations of Monotone Solutions of an nth-Order Nonlinear Differential Equation of Emden-Fowler Type, Dokl. Akad. Nauk, 1992, vol. 234, no. 2, pp. 258–260.
[11] Evtukhov, V.M., A Class of Monotone Solutions of an nth-Order Nonlinear Differential Equation of Emden-Fowler Type, Soobshch. Akad. Nauk Gruzii, 1992, vol. 145, no. 2, pp. 269–273. · Zbl 0771.34009
[12] Wong, P.K., Existence and Asymptotic Behavior of Proper Solutions of a Class of Second-Order Nonlinear Differential Equations, Pacific J. Math., 1963, vol. 13, pp. 737–760. · Zbl 0115.07203 · doi:10.2140/pjm.1963.13.737
[13] Marić, V. and Tomić, M., Asymptotic Properties of Solutions of the Equation y” = f(x){\(\Phi\)}(y), Math. Z., 1976, vol. 149, pp. 261–266. · Zbl 0316.34054 · doi:10.1007/BF01175588
[14] Talliaferro, S.D., Asymptotic Behavior of the Solutions of the Equation y” = {\(\Phi\)}(t)f(y), SIAM J. Math. Anal., 1981, vol. 12, no. 6, pp. 47–59. · Zbl 0474.34048 · doi:10.1137/0512071
[15] Evtukhov, V.M. and Kirillova, L.A., Asymptotic Representations for Unbounded Solutions of Second Order Nonlinear Differential Equations Close to Equations of Emden-Fowler Type, Mem. Differential Equations Math. Phys., 2003, vol. 30, pp. 153–158. · Zbl 1062.34051
[16] Evtukhov, V.M. and Kirillova, L.A., On the Asymptotic Behavior of Solutions of Nonlinear Differential Equations, Differ. Uravn., 2005, vol. 41, no. 8, pp. 1105–1114. · Zbl 1139.34039 · doi:10.1007/s10625-005-0256-5
[17] Kirillova, L.A., On the Asymptotic Behavior of Solutions of Second-Order Nonlinear Differential Equations, Nelin. Koliv., 2005, vol. 8, no. 1, pp. 18–28. · Zbl 1115.34307
[18] Evtukhov, V.M. and Khar’kov, V.M., Asymptotic Representations of Solutions of Second-Order Essentially Nonlinear Differential Equations, Differ. Uravn., 2007, vol. 43, no. 10, pp. 1311–1323.
[19] Evtukhov, V.M. and Stekhun, A.A., Asymptotic Representations of Unbounded Solutions of Third-Order Nonlinear Differential Equations, Mat. Metodi Fiz.-Mekh. Polya, 2004, vol. 47, no. 4, pp. 82–87. · Zbl 1089.34526
[20] Evtukhov, V.M. and Stekhun, A.A., Asymptotic Representations of Solutions of a Class of Third-Order Nonlinear Nonautonomous Differential Equations, Ukrain. Mat. Zh., 2007, vol. 59, no. 10, pp. 1363–1375. · Zbl 1164.34428
[21] Evtukhov, V.M., Asymptotic Representations of Solutions of Nonautonomous Ordinary Differential Equations, Doctoral (Phys.-Math.) Dissertation, Kiev, 1998.
[22] Evtukhov, V.M. and Samoilenko, A.M., Conditions for the Existence of Solutions Vanishing at Infinity for Real Autonomous 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
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.