# Article

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Keywords:
third order nonlinear differential equation; singular nonlinearity; positive solution; decaying solution; asymptotic behavior; regularly varying functions
Summary:
This paper is concerned with asymptotic analysis of strongly decaying solutions of the third-order singular differential equation $x^{\prime \prime \prime }+q(t)x^{-\gamma }=0$, by means of regularly varying functions, where $\gamma$ is a positive constant and $q$ is a positive continuous function on $[a,\infty )$. It is shown that if $q$ is a regularly varying function, then it is possible to establish necessary and sufficient conditions for the existence of slowly varying solutions and regularly varying solutions of (A) which decrease to $0$ as $t\rightarrow \infty$ and to acquire precise information about the asymptotic behavior at infinity of these solutions. The main tool is the Schauder–Tychonoff fixed point theorem combined with the basic theory of regular variation.
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. MR 0898871 | Zbl 0617.26001
[2] Evtukhov, V. M., Samoilenko, A. M.: Asymptotic representations of solutions of nonautonomous ordinary differential equations with regularly varying nonlinearities. Differ. Uravn. 47 (2011), 628–650, (in Russian); translation in Differ. Equ. 47 (2011), 627–649. DOI 10.1134/S001226611105003X | MR 2918280 | Zbl 1242.34092
[3] 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. DOI 10.1002/mana.201100296 | MR 3021476 | Zbl 1269.34054
[4] 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. DOI 10.1515/gmj-2013-0027 | MR 3100967
[5] Kamo, K., Usami, H.: Asymptotic forms of positive solutions of quasilinear ordinary differential equations with singular nonlinearities. Nonlinear Anal. 68 (2008), 1627–1639. DOI 10.1016/j.na.2006.12.045 | MR 2388837 | Zbl 1140.34023
[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. DOI 10.14232/ejqtde.2012.1.45 | MR 2943099
[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. DOI 10.1016/j.camwa.2011.05.019 | MR 2817892 | Zbl 1228.34072
[8] Kusano, T., Tanigawa, T.: Positive solutions to a class of second order differential equations with singular nonlinearities. Appl. Anal. 69 (1998), 315–331. MR 1706534 | Zbl 0923.34032
[9] Marić, V.: Regular Variation and Differential Equations. Lecture notes in Mathematics 1726, Springer-Verlag, Berlin–Heidelberg, 2000. MR 1753584
[10] Tanigawa, T.: Positive solutions to second order singular differential equations involving the one-dimensional M-Laplace operator. Georgian Math. J. 6 (1999), 347–362. DOI 10.1023/A:1022965616534 | MR 1693224 | Zbl 0933.34028
[11] Taylor, A. E.: L’Hospital’s rule. Amer. Math. Monthly 59 (1952), 20–24. MR 0044602 | Zbl 0046.06202

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