×

Oscillation and asymptotic behavior of higher-order nonlinear differential equations. (English) Zbl 1254.34090

Summary: The aim of this paper is to offer a generalization of the Philos and Staikos lemma. As a possible application of the lemma in the oscillation theory, we study the asymptotic properties and the oscillation of the \(n\)th-order delay differential equation \[ (r(t)[x^{(n-1)}(t)]^\gamma)' + q(t)x^\gamma(\tau(t)) = 0 \] . The results obtained utilize also comparison theorems.

MSC:

34K11 Oscillation theory of functional-differential equations
34K25 Asymptotic theory of functional-differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] B. Baculíková and J. D\vzurina, “Oscillation of third-order neutral differential equations,” Mathematical and Computer Modelling, vol. 52, no. 1-2, pp. 215-226, 2010. · Zbl 1201.34097 · doi:10.1016/j.mcm.2010.02.011
[2] B. Baculíková, J. Graef, and J. Dzurina, “On the oscillation of higher order delay differential equations,” Nonlinear Oscillations, vol. 15, no. 1, pp. 13-24, 2012.
[3] B. Baculíková and J. D\vzurina, “Oscillation of third-order nonlinear differential equations,” Applied Mathematics Letters, vol. 24, no. 4, pp. 466-470, 2011. · Zbl 1209.34042 · doi:10.1016/j.aml.2010.10.043
[4] B. Baculíková, “Properties of third-order nonlinear functional differential equations with mixed arguments,” Abstract and Applied Analysis, vol. 2011, Article ID 857860, 15 pages, 2011. · Zbl 1217.34109 · doi:10.1155/2011/857860
[5] B. Baculíková and J. D\vzurina, “Oscillation of third-order functional differential equations,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 43, pp. 1-10, 2010. · Zbl 1211.34077
[6] J. D\vzurina, “Comparison theorems for nonlinear ODEs,” Mathematica Slovaca, vol. 42, no. 3, pp. 299-315, 1992.
[7] L. H. Erbe, Q. Kong, and B. G. Zhang, Oscillation Theory for Functional-Differential Equations, vol. 190, Marcel Dekker, New York, NY, USA, 1994. · Zbl 0821.34067
[8] S. R. Grace, R. P. Agarwal, R. Pavani, and E. Thandapani, “On the oscillation of certain third order nonlinear functional differential equations,” Applied Mathematics and Computation, vol. 202, no. 1, pp. 102-112, 2008. · Zbl 1154.34368 · doi:10.1016/j.amc.2008.01.025
[9] S. R. Grace and B. S. Lalli, “Oscillation of even order differential equations with deviating arguments,” Journal of Mathematical Analysis and Applications, vol. 147, no. 2, pp. 569-579, 1990. · Zbl 0711.34085 · doi:10.1016/0022-247X(90)90371-L
[10] T. Li, Z. Han, P. Zhao, and S. Sun, “Oscillation of even-order neutral delay differential equations,” Advances in Difference Equations, vol. 2010, Article ID 184180, 9 pages, 2010. · Zbl 1209.34082 · doi:10.1155/2010/184180
[11] I. T. Kiguradze and T. A. Chaturia, Asymptotic Properties of Solutions of Nonatunomous Ordinary Dierential Equations, Kluwer Academic, Dordrecht, The Netherlands, 1993. · Zbl 0782.34002
[12] 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
[13] G. S. Ladde, V. Lakshmikantham, and B. G. Zhang, Oscillation Theory of Differential Equations with Deviating Arguments, vol. 110, Marcel Dekker, New York, NY, USA, 1987. · Zbl 0622.34071
[14] W. E. Mahfoud, “Oscillation and asymptotic behavior of solutions of Nth order nonlinear delay differential equations,” Journal of Differential Equations, vol. 24, no. 1, pp. 75-98, 1977. · Zbl 0341.34065 · doi:10.1016/0022-0396(77)90171-1
[15] C. G. Philos, “On the existence of nonoscillatory solutions tending to zero at \infty for differential equations with positive delays,” Archiv der Mathematik, vol. 36, no. 2, pp. 168-178, 1981. · Zbl 0463.34050 · doi:10.1007/BF01223686
[16] C. G. Philos, “Oscillation and asymptotic behavior of linear retarded dierential equations of arbitrary order,” Tech. Rep. 57, University of Ioannina, 1981.
[17] C. G. Philos, “On the existence of nonoscillatory solutions tending to zero at 1 for differential equations with positive delay,” Journal of the Australian Mathematical Society, vol. 36, pp. 176-186, 1984. · doi:10.1017/S1446788700024630
[18] C. Zhang, T. Li, B. Sun, and E. Thandapani, “On the oscillation of higher-order half-linear delay differential equations,” Applied Mathematics Letters, vol. 24, no. 9, pp. 1618-1621, 2011. · Zbl 1223.34095 · doi:10.1016/j.aml.2011.04.015
[19] Q. Zhang, J. Yan, and L. Gao, “Oscillation behavior of even-order nonlinear neutral differential equations with variable coefficients,” Computers & Mathematics with Applications, vol. 59, no. 1, pp. 426-430, 2010. · Zbl 1189.34135 · doi:10.1016/j.camwa.2009.06.027
[20] G. Xing, T. Li, and C. Zhang, “Oscillation of higher-order quasi-linear neutral differential equations,” Advances in Difference Equations, vol. 2011, pp. 1-10, 2011. · Zbl 1272.34095
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.