zbMATH — the first resource for mathematics

Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Asymptotic properties of third-order delay trinomial differential equations. (English) Zbl 1210.34107
Summary: The aim of this paper is to study properties of the third-order delay trinomial differential equation $$((1/r(t))y''(t))'+p(t)y'(t)+q(t)y(\sigma(t))=0$$ by transforming this equation into a second-/third-order binomial differential equation. Using suitable comparison theorems, we establish new results on the asymptotic behavior of solutions. The obtained criteria improve and generalize earlier ones.

MSC:
 34K25 Asymptotic theory of functional-differential equations
Full Text:
References:
 [1] B. Baculíková, E. M. Elabbasy, S. H. Saker, and J. D\vzurina, “Oscillation criteria for third-order nonlinear differential equations,” Mathematica Slovaca, vol. 58, no. 2, pp. 201-220, 2008. · Zbl 1174.34052 · doi:10.2478/s12175-008-0068-1 [2] B. Baculíková, “Oscillation criteria for second order nonlinear differential equations,” Archivum Mathematicum, vol. 42, no. 2, pp. 141-149, 2006. · Zbl 1164.34499 · emis:journals/AM/06-2/index.html · eudml:130146 [3] B. Baculíková, “Oscillation of certain class of the third order neutral differential equations,” Abstract and Applied Analysis. In press. [4] B. Baculíková and D. Lacková, “Oscillation criteria for second order retarded differential equations,” Studies of the University of \vZilina. Mathematical Series, vol. 20, no. 1, pp. 11-18, 2006. · Zbl 05375291 [5] M. Bartu\vsek, M. Cecchi, Z. Do\vslá, and M. Marini, “Oscillation for third-order nonlinear differential equations with deviating argument,” Abstract and Applied Analysis, vol. 2010, Article ID 278962, 19 pages, 2010. · Zbl 1192.34073 · doi:10.1155/2010/278962 [6] R. Bellman, Stability Theory of Differential Equations, McGraw-Hill, New York, NY, USA, 1953. · Zbl 0053.24705 [7] T. A. Chanturija and I. T. Kiguradze, Asymptotic Properties of Nonautonomous Ordinary Differential Equations, Nauka, Moscow, Russia, 1990. · Zbl 0719.34003 [8] J. D\vzurina, “Asymptotic properties of the third order delay differential equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 26, no. 1, pp. 33-39, 1996. · Zbl 0840.34076 · doi:10.1016/0362-546X(94)00239-E [9] J. D\vzurina, “Comparison theorems for nonlinear ODEs,” Mathematica Slovaca, vol. 42, no. 3, pp. 299-315, 1992. [10] J. D\vzurina and R. Kotorová, “Properties of the third order trinomial differential equations with delay argument,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 5-6, pp. 1995-2002, 2009. · Zbl 1173.34348 · doi:10.1016/j.na.2009.01.070 [11] L. Erbe, “Existence of oscillatory solutions and asymptotic behavior for a class of third order linear differential equations,” Pacific Journal of Mathematics, vol. 64, no. 2, pp. 369-385, 1976. · Zbl 0339.34030 · doi:10.2140/pjm.1976.64.369 [12] P. Hartman, Ordinary Differential Equations, John Wiley & Sons, New York, NY, USA, 1964. · Zbl 0125.32102 [13] G. D. Jones, “An asymptotic property of solutions of y$^{\prime}$$^{\prime}$$^{\prime}$+p(x)y$^{\prime}$+q(x)y=0,” Pacific Journal of Mathematics, vol. 47, pp. 135-138, 1973. · Zbl 0264.34040 · doi:10.2140/pjm.1973.47.135 [14] T. Kusano and M. Naito, “Comparison theorems for functional-differential equations with deviating arguments,” Journal of the Mathematical Society of Japan, vol. 33, no. 3, pp. 509-532, 1981. · Zbl 0494.34049 · doi:10.2969/jmsj/03330509 [15] T. Kusano, M. Naito, and K. Tanaka, “Oscillatory and asymptotic behaviour of solutions of a class of linear ordinary differential equations,” Proceedings of the Royal Society of Edinburgh A, vol. 90, no. 1-2, pp. 25-40, 1981. · Zbl 0486.34021 · doi:10.1017/S0308210500015328 [16] A. C. Lazer, “The behavior of solutions of the differential equation y$^{\prime}$$^{\prime}$$^{\prime}$+p(x)y$^{\prime}$+q(x)y=0,” Pacific Journal of Mathematics, vol. 17, pp. 435-466, 1966. · Zbl 0143.31501 · doi:10.2140/pjm.1966.17.435 [17] W. E. Mahfoud, “Comparison theorems for delay differential equations,” Pacific Journal of Mathematics, vol. 83, no. 1, pp. 187-197, 1979. · Zbl 0441.34053 · doi:10.2140/pjm.1979.83.187 [18] N. Parhi and S. Padhi, “On asymptotic behavior of delay-differential equations of third order,” Nonlinear Analysis: Theory, Methods & Applications, vol. 34, no. 3, pp. 391-403, 1998. · Zbl 0935.34063 · doi:10.1016/S0362-546X(97)00600-7 [19] N. Parhi and S. Padhi, “Asymptotic behaviour of solutions of third order delay-differential equations,” Indian Journal of Pure and Applied Mathematics, vol. 33, no. 10, pp. 1609-1620, 2002. · Zbl 1025.34068 [20] A. \vSkerlík, “Integral criteria of oscillation for a third order linear differential equation,” Mathematica Slovaca, vol. 45, no. 4, pp. 403-412, 1995. · Zbl 0855.34038 · eudml:31668 [21] K. Tanaka, “Asymptotic analysis of odd order ordinary differential equations,” Hiroshima Mathematical Journal, vol. 10, no. 2, pp. 391-408, 1980. · Zbl 0453.34033 [22] W. F. Trench, “Canonical forms and principal systems for general disconjugate equations,” Transactions of the American Mathematical Society, vol. 189, pp. 319-327, 1973. · Zbl 0289.34051 · doi:10.2307/1996862