Hiroshima Mathematical Journal

Asymptotic analysis of positive solutions of third order nonlinear differential equations

Jaroslav Jaroš, Takaŝi Kusano, and Tomoyuki Tanigawa

Full-text: Open access

Abstract

It is shown that an application of the theory of regular variation (in the sense of Karamata) gives the possibility of determining the existence and precise asymptotic behavior of positive solutions of the third-order nonlinear differential equation $(|x''|^{\alpha-1}x'')' + q(t)|x|^\beta x = 0$, where $\alpha > \beta > 0$ are constants and $q:[a,\infty)\to(0,\infty)$ is a continuous regularly varying function.

Article information

Source
Hiroshima Math. J., Volume 44, Number 1 (2014), 1-34.

Dates
First available in Project Euclid: 17 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.hmj/1395061555

Digital Object Identifier
doi:10.32917/hmj/1395061555

Mathematical Reviews number (MathSciNet)
MR3178434

Zentralblatt MATH identifier
1300.34124

Subjects
Primary: 34C11: Growth, boundedness 26A12: Rate of growth of functions, orders of infinity, slowly varying functions [See also 26A48]

Keywords
Third order nonlinear differential equation positive solutions asymptotic behavior regularly varying functions

Citation

Jaroš, Jaroslav; Kusano, Takaŝi; Tanigawa, Tomoyuki. Asymptotic analysis of positive solutions of third order nonlinear differential equations. Hiroshima Math. J. 44 (2014), no. 1, 1--34. doi:10.32917/hmj/1395061555. https://projecteuclid.org/euclid.hmj/1395061555


Export citation

References

  • N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, Encyclopedia of Mathematics and its Applications, 27, Cambridge University Press, Cambridge, 1987. \smallskip
  • O. Haupt and G. Aumann, Differential- und Integralrechnung, 2, W. de Gruyter, Berlin, 1938. \smallskip
  • J. Jaroš, T. Kusano and T. Tanigawa, Asymptotic analysis of positive solutions of a class of third-order nonlinear differential equations in the framework of regular variation, Math. Nachr. 286 (2013), 205-223. \smallskip
  • T. Kusano and J. Manojlović, Precise asymptotic behavior of solutions of the sublinear Emden-Fowler differential equation, Appl. Math. Comput., 217 (2011), 4382-4396. \smallskip
  • T. Kusano and J. Manojlović, Asymptotic behavior of positive solutions of sublinear differential equations of Emden-Fowler type, Comput. Math. Appl., 62 (2011), 551-565. \smallskip
  • V. Marić, Regular Variation and Differential Equations, Lecture Notes in Mathematics 1726, Springer-Verlag, Berlin, 2000. \smallskip
  • M. Naito and F. Wu, On the existence of eventually positive solutions of fourth-order quasilinear differential equations, Nonlinear Analysis, 57 (2004), 253-263.