Taiwanese Journal of Mathematics

ASYMPTOTIC ANALYSIS OF FOURTH ORDER QUASILINEAR DIFFERENTIAL EQUATIONS IN THE FRAMEWORK OF REGULAR VARIATION

Jelena Milošević and Jelena V. Manojlović

Full-text: Open access

Abstract

Under the assumptions that $p(t),\;q(t)$ are regularly varying functions satisfying condition $$\int_a^\infty\frac{dt}{p(t)^{\frac{1}{\alpha}}}=\infty,$$ existence and asymptotic form of regularly varying intermediate solutions  are studied for a fourth-order quasilinear differential equation $$\left(p(t)|x''(t)|^{\alpha-1}\,x''(t)\right)^{\prime\prime}+q(t)|x(t)|^{\beta-1}\,x(t)=0,\quad \alpha \gt \beta\gt 0.$$ It is shown that the asymptotic behavior of all such solutions is governed by a unique explicit law.

Article information

Source
Taiwanese J. Math., Volume 19, Number 5 (2015), 1415-1456.

Dates
First available in Project Euclid: 4 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1499133717

Digital Object Identifier
doi:10.11650/tjm.19.2015.5048

Mathematical Reviews number (MathSciNet)
MR3412014

Zentralblatt MATH identifier
1357.34070

Subjects
Primary: 34A34: Nonlinear equations and systems, general
Secondary: 26A12: Rate of growth of functions, orders of infinity, slowly varying functions [See also 26A48]

Keywords
fourth order differential equation regularly varying function slowely varying function asymptotic behavior of solutions positive solutions

Citation

Milošević, Jelena; Manojlović, Jelena V. ASYMPTOTIC ANALYSIS OF FOURTH ORDER QUASILINEAR DIFFERENTIAL EQUATIONS IN THE FRAMEWORK OF REGULAR VARIATION. Taiwanese J. Math. 19 (2015), no. 5, 1415--1456. doi:10.11650/tjm.19.2015.5048. https://projecteuclid.org/euclid.twjm/1499133717


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, 1987.
  • O. Haupt and G. Aumann, Differential- und Integralrechnung, Walter de Gruyter, Berlin, 1938.
  • J. Jaro\ptmrsš and T. Kusano, Self-adjoint differential equations and generalized Karamata functions, Bull. Classe Sci. Mat. Nat., Sci. Math., Acad. Serbe Sci. Arts, CXXIX, No. 29 (2004), 25-60.
  • J. Jaroš, T. Kusano and J. Manojlovi\' c, Asymptotic analysis of positive solutions of generalized Emden-Fowler differential equations in the framework of regular variation, Cent. Eur. J. Math., 11(12) (2013), 2215-2233.
  • M. Naito and F. Wu, A note on the existance and asymptotic behavior of nonoscillatory solutions of fourth order quasilinear differential equations, Acta Math. Hungar, 102(3) (2004), 177-202.
  • V. Marić, Regular Variation and Differential Equations, Lecture Notes in Mathematics 1726, Springer-Verlar, Berlin-Heidelberg, 2000.
  • 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.
  • T. Kusano, J. Manojlovi\' c and V. Mari\' c, Increasing solutions of Thomas-Fermi type differential equations - the sublinear case, Bull. T. de Acad. Serbe Sci. Arts, Classe Sci. Mat. Nat., Sci. Math., CXLIII, No. 36 (2011), 21-36.
  • T. Kusano and J. Manojlović, Positive solutions of fourth order Emden-Fowler type differential equations in the framework of regular variation, Applied Mathematics and Computation, 218 (2012), 6684-6701.
  • T. Kusano and J. Manojlović, Positive solutions of fourth order Thomas-Fermi type differential equations in the framework of regular variation, Acta Applicandae Mathematicae, 121 (2012), 81-103.
  • T. Kusano, J. Manojlovi\' c and J. Milo\ptmrsševi\' c, Intermediate solutions of second order quasilinear differential equation in the framework of regular variation, Applied Mathematics and Computation, 219 (2013), 8178-8191.
  • T. Kusano, J. Manojlović and T. Tanigawa, Existance and asymptotic behavior of positive solutions of fourth order quasilinear differential equations, Taiwanese Journal of Mathematics, 17(3) (2013), 999-1030.
  • T. Kusano, V. Marić and T. Tanigawa, An asymptotic analysis of positive solutions of generalized Thomas-Fermi differential equations - the sub-half-linear case, Nonlinear Anal., 75 (2012), 2474-2485.
  • J. V. Manojlovi\' c and V. Mari\' c, An asymptotic analysis of positive solutions of Thomas-Fermi type differential equations - sublinear case, Memoirs on Differential Equations and Mathematical Physics, 57 (2012), 75-94.
  • S. Matucci and P. \ptmrs\v Rehák, Asymptotics of decreasing solutions of coupled $p-$Laplacian systems in the framework of regular variation, Annali di Mat. Pura ed Appl., DOI: 10.1007/s10231-012-0303-9.
  • P. \ptmrs\v Rehák, Asymptotic behavior of increasing solutions to a system of n nonlinear differential equations, Nonlinear Analysis, 77 (2013), 45-58.
  • E. Seneta, Regularly Varying Functions, Lecture Notes in Mathematics, 508, Springer Verlag, Berlin-Heidelberg-New York, 1976
  • F. Wu, Nonoscillatory solutions of fourth order quasilinear differential equations, Funkcialaj Ekvacioj, 45 (2002), 71-88.