Previous |  Up |  Next


Title: On Existence and Asymptotic Properties of Kneser Solutions to Singular Second Order ODE. (English)
Author: Vampolová, Jana
Language: English
Journal: Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica
ISSN: 0231-9721
Volume: 52
Issue: 1
Year: 2013
Pages: 135-152
Summary lang: English
Category: math
Summary: We investigate an asymptotic behaviour of damped non-oscillatory solutions of the initial value problem with a time singularity $\left( p(t)u^{\prime }(t) \right)^{\prime } + p(t)f ( u(t) )=0$, $u(0)=u_0$, $u^{\prime }(0)=0$ on the unbounded domain $[0,\infty )$. Function $f$ is locally Lipschitz continuous on $\mathbb {R}$ and has at least three zeros $L_0 <0$, $0$ and $L>0$. The initial value $u_0\in (L_0, L)\setminus \lbrace 0\rbrace $. Function $p$ is continuous on $[0,\infty ),$ has a positive continuous derivative on $(0,\infty )$ and $p(0)=0$. Asymptotic formulas for damped non-oscillatory solutions and their first derivatives are derived under some additional assumptions. Further, we provide conditions for functions $p$ and $f$, which guarantee the existence of Kneser solutions. (English)
Keyword: singular ordinary differential equation of the second order
Keyword: time singularities
Keyword: unbounded domain
Keyword: asymptotic properties
Keyword: Kneser solutions
Keyword: damped solutions
Keyword: non-oscillatory solutions
MSC: 34A12
MSC: 34D05
idZBL: Zbl 06285760
idMR: MR3202755
Date available: 2013-08-02T08:05:21Z
Last updated: 2014-07-30
Stable URL:
Reference: [1] Abraham, F. F.: Homogeneous Nucleation Theory. Acad. Press, New York, 1974.
Reference: [2] Bartušek, M., Cecchi, M., Došlá, Z., Marini, M.: On oscillatory solutions of quasilinear differential equations. J. Math. Anal. Appl. 320 (2006), 108–120. Zbl 1103.34016, MR 2230460, 10.1016/j.jmaa.2005.06.057
Reference: [3] Bongiorno, V., Scriven, L. E., Davis, H. T.: Molecular theory of fluid interfaces. J. Colloid and Interface Science 57 (1967), 462–475. 10.1016/0021-9797(76)90225-3
Reference: [4] Cecchi, M., Marini, M., Villari, G.: On some classes of continuable solutions of a nonlinear differential equation. J. Differential Equations 118 (1995), 403–419. Zbl 0827.34020, MR 1330834, 10.1006/jdeq.1995.1079
Reference: [5] Cecchi, M., Marini, M., Villari, G.: Comparison results for oscillation of nonlinear differential equations. NoDea 6 (1999), 173–190. Zbl 0927.34023, MR 1694795, 10.1007/s000300050071
Reference: [6] Derrick, G. H.: Comments on nonlinear wave equations as models for elementary particles. J. Math. Physics 5 (1965), 1252–1254. MR 0174304, 10.1063/1.1704233
Reference: [7] Fife, P. C.: Mathematical Aspects of Reacting and Diffusing Systems. Lecture notes in Biomathematics Springer 28 (1979), 223–224. Zbl 0403.92004, MR 0527914
Reference: [8] Fischer, R. A.: The wave of advance of advantageous genes. Journ. of Eugenics 7 (1937), 355–369. 10.1111/j.1469-1809.1937.tb02153.x
Reference: [9] Gouin, H., Rotoli, G.: An analytical approximation of density profile and surface tension of microscopic bubbles for Van der Waals fluids. Mech. Research Communic. 24 (1997), 255–260. Zbl 0899.76064, 10.1016/S0093-6413(97)00022-0
Reference: [10] Ho, L. F.: Asymptotic behavior of radial oscillatory solutions of a quasilinear elliptic equation. Nonlinear Analysis 41 (2000), 573–589. Zbl 0962.34019, MR 1780633, 10.1016/S0362-546X(98)00298-3
Reference: [11] Jaroš, J., Kusano, T., Tanigawa, T.: Nonoscillatory half-linear differential equations and generalized Karamata functions. Nonlinear Analysis 64 (2006), 762–787. Zbl 1103.34017, MR 2197094, 10.1016/
Reference: [12] Kiguradze, I., Chanturia, T.: Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations. Kluwer Acad. Publ., Dordrecht, 1993. Zbl 0782.34002, MR 1220223
Reference: [13] Kulenović, M. R. S., Ljubović, Ć.: All solutions of the equilibrium capillary surface equation are oscillatory. Applied Mathematics Letters 13 (2000), 107–110. MR 1760271, 10.1016/S0893-9659(00)00041-0
Reference: [14] Kusano, T., Manojlović, J. V.: Asymptotic analysis of Emden-Fowler differential equations in the framework of regular variation. Annali di Matematica Pura ed Applicata 190 (2011), 619–644. Zbl 1245.34039, MR 2861062, 10.1007/s10231-010-0166-x
Reference: [15] Kwong, M. K., Wong, J. S. W.: A nonoscillation theorem for sublinear Emden-Fowler equations. Nonlinear Analysis 64 (2006), 1641–1646. Zbl 1099.34033, MR 2200164, 10.1016/
Reference: [16] Kwong, M. K., Wong, J. S. W.: A nonoscillation theorem for superlinear Emden–Fowler equations with near-critical coefficients. J. Differential Equations 238 (2007), 18–42. Zbl 1125.34023, MR 2334590, 10.1016/j.jde.2007.03.021
Reference: [17] Li, W. T.: Oscillation of certain second-order nonlinear differential equations. J. Math. Anal. Appl. 217 (1998), 1–14. Zbl 0893.34023, 10.1006/jmaa.1997.5680
Reference: [18] Lima, P. M., Chemetov, N. V., Konyukhova, N. B., Sukov, A. I.: Analytical–numerical investigation of bubble-type solutions of nonlinear singular problems. J. Comp. Appl. Math. 189 (2006), 260–273. Zbl 1100.65066, MR 2202978, 10.1016/
Reference: [19] Linde, A. P.: Particle Physics and Inflationary Cosmology. Harwood Academic, Chur, Switzerland, 1990.
Reference: [20] Ou, C. H., Wong, J. S. W.: On existence of oscillatory solutions of second order Emden–Fowler equations. J. Math. Anal. Appl. 277 (2003), 670–680. Zbl 1027.34039, MR 1961253, 10.1016/S0022-247X(02)00617-0
Reference: [21] O’Regan, D.: Existence theory for nonlinear ordinary differential equations. Kluwer, Dordrecht, 1997. Zbl 1077.34505, MR 1449397
Reference: [22] Rachůnková, I., Rachůnek, L.: Asymptotic formula for oscillatory solutions of some singular nonlinear differential equation. Abstract and Applied Analysis 2011 (2011), 1–9. Zbl 1222.34034
Reference: [23] Rachůnková, I., Tomeček, J.: Bubble-type solutions of nonlinear singular problem. Mathematical and Computer Modelling 51 (2010), 658–669. 10.1016/j.mcm.2009.10.042
Reference: [24] Rachůnková, I., Rachůnek, L., Tomeček, J.: Existence of oscillatory solutions of singular nonlinear differential equations. Abstract and Applied Analysis 2011 (2011), 20 pages. Zbl 1222.34035, MR 2795071
Reference: [25] Rachůnková, I., Tomeček, J.: Strictly increasing solutions of a nonlinear singular differential equation arising in hydrodynamics. Nonlinear Analysis 72 (2010), 2114–2118. Zbl 1186.34014, MR 2577608, 10.1016/
Reference: [26] Rachůnková, I., Tomeček, J.: Homoclinic solutions of singular nonautonomous second order differential equations. Boundary Value Problems 2009 (2009), 1–21. Zbl 1190.34028
Reference: [27] Rohleder, M.: On the existence of oscillatory solutions of the second order nonlinear ODE. Acta Univ. Palacki. Olomuc., Fac. rer. nat., Math. 51, 2 (2012), 107–127. Zbl 1279.34050, MR 3058877
Reference: [28] van der Waals, J. D., Kohnstamm, R.: Lehrbuch der Thermodynamik. 1, Leipzig, 1908.
Reference: [29] Wong, J. S. W.: Second–order nonlinear oscillations: A case history. In: Proceedings of the Conference on Differential & Difference Equations and Applications Hindawi (2006), 1131–1138. Zbl 1147.34024, MR 2309447
Reference: [30] Wong, P. J. Y., Agarwal, R. P.: Oscillatory behavior of solutions of certain second order nonlinear differential equations. J. Math. Anal. Appl. 198 (1996), 337–354. Zbl 0855.34039, MR 1376268, 10.1006/jmaa.1996.0086
Reference: [31] Wong, P. J. Y., Agarwal, R. P.: The oscillation and asymptotically monotone solutions of second order quasilinear differential equations. Appl. Math. Comput. 79 (1996),207–237. MR 1407599


Files Size Format View
ActaOlom_52-2013-1_11.pdf 318.5Kb application/pdf View/Open
Back to standard record
Partner of
EuDML logo