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Title: On asymptotic behavior of solutions to Emden-Fowler type higher-order differential equations (English)
Author: Astashova, Irina
Language: English
Journal: Mathematica Bohemica
ISSN: 0862-7959
Volume: 140
Issue: 4
Year: 2015
Pages: 479-488
Summary lang: English
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Category: math
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Summary: For the equation $$ y^{(n)}+|y|^{k}\mathop {\rm sgn} y=0,\quad k>1,\ n=3,4, $$ existence of oscillatory solutions $$ y=(x^*-x)^{-\alpha } h(\log (x^*-x)),\quad \alpha =\frac {n}{k-1},\ x<x^*, $$ is proved, where $x^*$ is an arbitrary point and $h$ is a periodic non-constant function on $\mathbb {R}$. The result on existence of such solutions with a positive periodic non-constant function $h$ on $\mathbb {R}$ is formulated for the equation $$ y^{(n)}=|y|^{k}\mathop {\rm sgn} y, \quad k>1,\ n=12,13,14. $$ (English)
Keyword: nonlinear ordinary differential equation of higher order
Keyword: asymptotic behavior of solutions
Keyword: oscillatory solution
MSC: 34C10
MSC: 34C15
idZBL: Zbl 06537678
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Date available: 2015-11-17T20:54:17Z
Last updated: 2017-01-02
Stable URL: http://hdl.handle.net/10338.dmlcz/144464
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Reference: [1] Astashova, I. V.: On power and non-power asymptotic behavior of positive solutions to Emden-{F}owler type higher-order equations.Adv. Difference Equ. 2013 (2013), Article No. 2013:220, 15 pages. MR 3092838
Reference: [2] Astashova, I. V.: Qualitative properties of solutions to quasilinear ordinary differential equations.Qualitative Properties of Solutions to Differential Equations and Related Topics of Spectral Analysis: scientific edition UNITY-DANA (2012), Russian 22-290 I. V. Astashova.
Reference: [3] Astashova, I. V.: Application of dynamical systems to the study of asymptotic properties of solutions to nonlinear higher-order differential equations.J. Math. Sci., New York 126 (2005), 1361-1391 translated from \kern 3sp Sovrem. Mat. Prilozh. 8 (2003), 3-33 Russian. MR 2157611, 10.1007/s10958-005-0066-6
Reference: [4] Astashova, I. V.: On asymptotic behavior of oscillatory solutions of some nonlinear differential equations of the third and forth order.Reports of extended session of a seminar of the I. N. Vekua Institute of Applied Mathematics 3 Tbilisi 9-12 (1988), Russian.
Reference: [5] Astashova, I. V.: Asymptotic behavior of solutions of certain nonlinear differential equations.Reports of the extended sessions of a seminar of the I. N. Vekua Institute of Applied Mathematics I Tbilis. Gos. Univ. Tbilisi (1985), 9-11 Russian I. T. Kiguradze. MR 0861629
Reference: [6] Astashova, I. V., Vyun, S. A.: On positive solutions with non-power asymptotic behavior to Emden-Fowler type twelfth order differential equation.Differ. Equ. 48 (2012), 1568-1569 Russian.
Reference: [7] Kiguradze, I. T., Chanturia, T. A.: Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations.translated from the Russian original, Nauka, Moskva, 1985 Mathematics and Its Applications (Soviet Series) 89 Kluwer Academic Publishers, Dordrecht (1993). Zbl 0782.34002, MR 1220223
Reference: [8] Kozlov, V. A.: On Kneser solutions of higher order nonlinear ordinary differential equations.Ark. Mat. 37 (1999), 305-322. Zbl 1118.34317, MR 1714766, 10.1007/BF02412217
Reference: [9] Marsden, J. E., McCracken, M.: The Hopf Bifurcation and Its Applications. With contributions by P. Chernoff et al.Applied Mathematical Sciences 19 Springer, New York (1976). MR 0494309
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