Asymptotic analysis of positive solutions of generalized Emden-Fowler differential equations in the framework of regular variation

Jaroslav Jaroš ; Kusano Takaŝi ; Jelena Manojlović

Open Mathematics, Tome 11 (2013), p. 2215-2233 / Harvested from The Polish Digital Mathematics Library

Résumé

Positive solutions of the nonlinear second-order differential equation $(p(t)|x^{\prime }|^{\alpha - 1} x^{\prime })^{\prime } + q(t)|x|^{\beta - 1} x = 0,\alpha > \beta > 0,$ are studied under the assumption that p, q are generalized regularly varying functions. An application of the theory of regular variation gives the possibility of obtaining necessary and sufficient conditions for existence of three possible types of intermediate solutions, together with the precise information about asymptotic behavior at infinity of all solutions belonging to each type of solution classes.

Détail

Publié le : 2013-01-01

Zbl 1329.34097

EUDML-ID : urn:eudml:doc:269520

Bibliographie

[1] Bingham N.H., Goldie C.M., Teugels J.L., Regular Variation, Encyclopedia Math. Appl., 27, Cambridge University Press, Cambridge, 1987 http://dx.doi.org/10.1017/CBO9780511721434 | Zbl 0617.26001

[2] Elbert Á., Kusano T., Oscillation and nonoscillation theorems for a class of second order quasilinear differential equations, Acta Math. Hungar., 1990, 56(3–4), 325–336 http://dx.doi.org/10.1007/BF01903849

[3] Haupt O., Differential- und Integralrechnung, Walter de Gruyter, Berlin, 1938

[4] Howard H.C., Maric V., Regularity and nonoscillation of solutions of second order linear differential equations, Bull. Cl. Sci. Math. Nat. Sci. Math., 1997, 22, 85–98 | Zbl 0947.34015

[5] Jaroš J., Kusano T., Self-adjoint differential equations and generalized Karamata functions, Bull. Cl. Sci. Math. Nat. Sci. Math., 2004, 29, 25–60 | Zbl 1091.34521

[6] Jaroš J., Kusano T., Tanigawa T., Nonoscillatory half-linear differential equations and generalized Karamata functions, Nonlinear Anal., 2006, 64(4), 762–787 http://dx.doi.org/10.1016/j.na.2005.05.045 | Zbl 1103.34017

[7] Kamo K., Usami H., Asymptotic forms of weakly increasing positive solutions for quasilinear ordinary differential equations, Electron. J. Differential Equations, 2007, #126 | Zbl 1153.34023

[8] Kusano T., Manojlovic J., Asymptotic behavior of positive solutions of sublinear differential equations of Emden-Fowler type, Comput. Math. Appl., 2011, 62(2), 551–565 http://dx.doi.org/10.1016/j.camwa.2011.05.019 | Zbl 1228.34072

[9] Kusano T., Manojlovic J.V., Maric V., Increasing solutions of Thomas-Fermi type differential equations - the sublinear case, Bull. Cl. Sci. Math. Nat. Sci. Math., 2011, 36, 21–36 | Zbl 1299.34044

[10] Kusano T., Maric V., Tanigawa T., An asymptotic analysis of positive solutions of generalized Thomas-Fermi differential equations - the sub-half-linear case, Nonlinear Anal., 2012, 75(4), 2474–2485 http://dx.doi.org/10.1016/j.na.2011.10.039 | Zbl 1248.34075

[11] Kusano T., Ogata A., Usami H., Oscillation theory for a class of second order quasilinear ordinary differential equations with application to partial differential equations, Japan. J. Math., 1993, 19(1), 131–147 | Zbl 0792.34033

[12] Manojlovic J., Maric V., An asymptotic analysis of positive solutions of Thomas-Fermi type sublinear differential equations, Mem. Differential Equations Math. Phys., 2012, 57, 75–94 | Zbl 1298.34090

[13] Maric V., Regular Variation and Differential Equations, Lecture Notes in Math., 1726, Springer, Berlin, 2000 http://dx.doi.org/10.1007/BFb0103952 | Zbl 0946.34001

[14] Maric V., Tomic M., A classification of solutions of second order linear differential equations by means of regularly varying functions, Publ. Inst. Math. (Beograd), 1990, 48(62), 199–207 | Zbl 0722.34007

[15] Naito M., On the asymptotic behavior of nonoscillatory solutions of second order quasilinear ordinary differential equations, J. Math. Anal. Appl., 2011, 381(1), 315–327 http://dx.doi.org/10.1016/j.jmaa.2011.04.006 | Zbl 1223.34045

[16] Seneta E., Regularly Varying Functions, Lecture Notes in Math., 508, Springer, Berlin-New York, 1976 http://dx.doi.org/10.1007/BFb0079658 | Zbl 0324.26002