Journal of Mathematical Sciences

, Volume 230, Issue 5, pp 651–655 | Cite as

Asymptotics of Oscillating Solutions to Equations with Power Nonlinearities

  • I. V. Astashova


We present results on the existence of oscillating solutions of specific form (“quasiperiodic solutions) for a nonlinear differential equation with power nonlinearity. For oscillating solutions to third-order equations of this type, we obtain an asymptotics of extremums, which is expressed through the asymptotics of extremums of a “quasiperiodic” solution. These results clarify the asymptotic formulas for the modules of extremums of solutions obtained by the author earlier.

Keywords and phrases

oscillating solution asymptotic behavior nonlinear equation 

AMS Subject Classification

34C10 34E10 


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  1. 1.
    I. V. Astashova, “On the asymptotic behavior of sign-alternating solutions to some nonlinear third- and fourth-order differential equations,” in: Proc. I. N. Vekua Seminar [in Russian], 3, No. 3, Tbilisi (1988), pp. 9–12.Google Scholar
  2. 2.
    I. V. Astashova, “Qualitative properties of solutions to quasilinear ordinary differential equations,” in: Qualitative Properties of Solutions to Differential Equations and Related Topics of Spectral Analysis (I. V. Astashova, ed.) [in Russian], Moscow (2012), pp. 22–288.Google Scholar
  3. 3.
    I. V. Astashova, “On positive solutions with nonpower asymptotics and quasiperiodic solutions of higher-order Emden–Fowler-type equations,” Probl. Mat. Anal., 79, 17–31 (2015).zbMATHGoogle Scholar
  4. 4.
    I. V. Astashova, “On the existence of positive solutions with a nonpower asymptotics of an equation of the Emden–Fowler-type of 13th and 14th order,” Differ. Equ., 49, No. 6, 775–777 (2013).Google Scholar
  5. 5.
    I. V. Astashova, “On power and nonpower asymptotic behavior of positive solutions to Emden–Fowler-type higher-order equations,” Adv. Differ. Equ., 220 (2013), DOI:
  6. 6.
    I. V. Astashova, “On quasi-periodic solutions to a higher-order Emden–Fowler-type differential equation,” Boundary-Value Probl., 174, 1–8 (2014).MathSciNetzbMATHGoogle Scholar
  7. 7.
    I. V. Astashova, “On asymptotic classification of solutions to nonlinear third- and fourth-order differential equations with power nonlinearity,” Vestn. Bauman Mosk. Tekh. Univ. Ser. Estestv. Nauk, 2, 3–25 (2015).Google Scholar
  8. 8.
    I. V. Astashova, “On asymptotic classification of solutions to fourth-order differential equations with singular power nonlinearity,” Math. Model. Anal., 21, No. 4, 502–521 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    I. V. Astashova, “On asymptotic behavior of solutions to Emden–Fowler-type higher-order differential equations,” Math. Bohem., 140, No. 4, 479–488 (2015).MathSciNetzbMATHGoogle Scholar
  10. 10.
    I. T. Kiguradze and T. A. Chanturia, Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations, Kluwer Academic, Dordrecht (1993).CrossRefGoogle Scholar
  11. 11.
    V. A. Kozlov, “On Kneser solutions of higher-order nonlinear ordinary differential equations,” Ark. Mat., 37, No. 2, 305–322 (1999).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.M. V. Lomonosov Moscow State UniversityMoscowRussia
  2. 2.G. V. Plekhanov Russian University of EconomicsMoscowRussia

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