Moderately growing solutions of third-order differential equations with a singular nonlinearity and regularly varying coefficients

Kučerová, Ivana

doi:10.1515/jamsi-2015-0003

Abstract

This paper is concerned with asymptotic analysis of moderately growing solutions of the third-order differential equation with singular nonlinerity

((((x′)^α₁*)′)^α₂*)′ +q(t)x^-β= 0;

(A) where α₁, α₂ and β are positive constants and σ : [α;∞)→(0;∞) is a continuous regularly varying function of index σ, α > 0 and u^γ* = |u|^γ sgnu. An application of the theory of regular variation allows us to establish necessary and sufficient conditions for the existence of regularly varying solutions of (A) which are moderately growing and to acquire precise information about the asymptotic behavior at infinity of these solutions. The Schauder-Tychonoff fixed point technique is used.

References

[1] Bingham, N. H., Goldie, C. M., Teugels, J. L.: Regular variation. Encyklopedia of Mathematics and its Applications 27, Cambridge University Press, Cambridge, 1987.
Search in Google Scholar
[2] Jaroš, J., Kusano, T., Tanigawa, T.: 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.
Search in Google Scholar
[3] Jaroš, J., Kusano, T., Tanigawa, T.: Existence and Precise Asymptotics of Positive Solutions for a Class of Nonlinear Differenctial Equations of the Third Order. Georgian Math. J. 20 (2013), 493-531.
Search in Google Scholar
[4] Kamo, K., Usami, H.: Asymptotic forms of positive solutions of quasilinear ordinary differential equations with singular nonlinearities. Nonlinear Anal. 68 (2008), 1627-1639.
Search in Google Scholar
[5] Kučerová, I.: Decaying regularly varying solutions of third-order differential equations with a singular nonlinearity, Acta Univ. Palack. Olomuc., Fac. Rerum. Natur. Math. (in print).
Search in Google Scholar
[6] Kusano, T., Manojlović, J.: Asymptotic behavior of positive solutions of odd order Emden-Fowler type differential equations in the framework of regular variation. Electron. J. Qual. Theory Differ. Equ. 45 (2012), 1-23.
Search in Google Scholar
[7] Kusano, T., Manojlović, J.: Asymptotic behavior of positive solutions of sublinear differential equations of Emden-Fowler type. Comput. Math. Appl. 62 (2011), 551-565.
Search in Google Scholar
[8] Kusano, T., Tanigawa, T.: Positive Solutions to a Class of Second Order Differential Equations with Singular Nonlinearities. Appl. Anal. Vol 69(1998), 315-331.
Search in Google Scholar
[9] Marić, V.: Regular variation and differential equations. Lecture notes in Mathematics 1726, Springer-Verlag, Berlin-Heidelberg, 2000. 60 I. Kuce rová ˇ10.1007/BFb0103952
Search in Google Scholar
[10] Tanigawa, T.: Positive solutions to second order singular differential equations involving the onedimensional M-Laplace operator. Georgian Math. J. 6 (1999) 347-362.
Search in Google Scholar
[11] Taylor, A. E.: L’Hospital’s rule. Amer. Math. Monthly 59 (1952), 20-24.
Search in Google Scholar
[12] Bingham, N. H., Goldie, C.M., Teugels, J.L.: Regular variation. Encyklopedia of Mathematics and its Applications 27, Cambridge University Press, Cambridge, 1987.
Search in Google Scholar
[13] Jaroš, J., Kusano, L., Tanigawa, T.: Asymptotic analysis of positive solutions of a class of third order nonlinear differential equations in the framework of regular variation. Math. Nachr., Vol. 286, Issue 2-3(2013), 205-223.10.1002/mana.201100296
Search in Google Scholar
[14] Jaroš, J., Kusano, L., Tanigawa, T.: Existence and Precise Asymptotics of Positive Solutions for a Class of Nonlinear Differenctial Equations of the Third Order. Georgian Math. J., Vol. 20, Issue 3 (2013), 493-531.
Search in Google Scholar
[15] Kamo, K., Usami, H.: Asymptotic forms of positive solutions of quasilinear ordinary differential equations with singular nonlinearities. Nonlinear Anal. 01/2008; 68(6), 1627-1639.10.1016/j.na.2006.12.045
Search in Google Scholar
[16] Kusano, T., Manojlović, J.: Asymptotic behavior of positive solutions of odd order Emden-Fowler type differential equations in the framework of regular variation. Electron. J. Qual. Theory Differ. Equ. 2012, No. 45 (2012), 1-23.
Search in Google Scholar
[17] Kusano, T., Manojlović, J.: Asymptotic behavior of positive solutions of sublinear differential equations of Emden-Fowler type. Comput. Math. Appl., Vol. 62, Issue 2 (2011), 551-565.
Search in Google Scholar
[18] Kusano, T., Tanigawa, T.: Positive Solutions to a Class of Second Order Differential Equations with Singular Nonlinearities. Appl. Anal. Vol 69(3-4) (1998), 315-331.10.1080/00036819808840665
Search in Google Scholar
[19] Marić, V.: Regular variation and differential equations. Lecture notes in Mathematics Vol 1726, Springer-Verlag, Berlin-Heidelberg, 2000.10.1007/BFb0103952
Search in Google Scholar
[20] Tanigawa, T.: Positive solutions to second order singular differential equations involving the onedimensional M-Laplace operator. Georgian Math. J., Vol. 6, No. 4 (1999) 347-362.
Search in Google Scholar
[21] Taylor, A. E. L’Hospital’s rule. The american Mathematical Monthly 59, No. 1 (1952), 20-24.
Search in Google Scholar

Moderately growing solutions of third-order differential equations with a singular nonlinearity and regularly varying coefficients

Abstract

Paradigm

My account