×

zbMATH — the first resource for mathematics

Oscillation for third-order nonlinear differential equations with deviating argument. (English) Zbl 1192.34073
Summary: We study necessary and sufficient conditions for the oscillation of the third-order nonlinear ordinary differential equation with damping term and deviating argument
\[ x'''(t)+q(t)x'(t)+r(t)f(x(\varphi(t)))=0. \] Motivated by the work of I. T. Kiguradze [Differ. Uravn. 28, No. 2, 207–219 (1992; Zbl 0768.34018)], the existence and asymptotic properties of nonoscillatory solutions are investigated in case when the differential operator \({\mathcal L}x=x'''+q(t)x'\) is oscillatory.

MSC:
34K11 Oscillation theory of functional-differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] M. Bartu\vsek, M. Cecchi, Z. Do, and M. Marini, “On nonoscillatory solutions of third order nonlinear differential equations,” Dynamic Systems and Applications, vol. 9, no. 4, pp. 483-499, 2000. · Zbl 1016.34036
[2] 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
[3] M. Cecchi, Z. Do, and M. Marini, “On third order differential equations with property A and B,” Journal of Mathematical Analysis and Applications, vol. 231, no. 2, pp. 509-525, 1999. · Zbl 0926.34025 · doi:10.1006/jmaa.1998.6247
[4] M. Cecchi, Z. Do, and M. Marini, “Asymptotic behavior of solutions of third order delay differential equations,” Archivum Mathematicum, vol. 33, no. 1-2, pp. 99-108, 1997. · Zbl 0916.34059 · eudml:226088
[5] I. Mojsej, “Asymptotic properties of solutions of third-order nonlinear differential equations with deviating argument,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 11, pp. 3581-3591, 2008. · Zbl 1151.34053 · doi:10.1016/j.na.2007.04.001
[6] S. H. Saker, “Oscillation criteria of third-order nonlinear delay differential equations,” Mathematica Slovaca, vol. 56, no. 4, pp. 433-450, 2006. · Zbl 1141.34040 · eudml:32119
[7] S. H. Saker, “Oscillation criteria of Hille and Nehari types for third-order delay differential equations,” Communications in Applied Analysis, vol. 11, no. 3-4, pp. 451-468, 2007. · Zbl 1139.34049
[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] A. Tiryaki and M. F. Akta\cs, “Oscillation criteria of a certain class of third order nonlinear delay differential equations with damping,” Journal of Mathematical Analysis and Applications, vol. 325, no. 1, pp. 54-68, 2007. · Zbl 1110.34048 · doi:10.1016/j.jmaa.2006.01.001
[10] L. Erbe, “Oscillation, nonoscillation, and asymptotic behavior for third order nonlinear differential equations,” Annali di Matematica Pura ed Applicata, Series 4, vol. 110, pp. 373-391, 1976. · Zbl 0345.34023 · doi:10.1007/BF02418014
[11] J. W. Heidel, “Qualitative behavior of solutions of a third order nonlinear differential equation,” Pacific Journal of Mathematics, vol. 27, pp. 507-526, 1968. · Zbl 0172.11703 · doi:10.2140/pjm.1968.27.507
[12] Y. Feng, “Solution and positive solution of a semilinear third-order equation,” Journal of Applied Mathematics and Computing, vol. 29, no. 1-2, pp. 153-161, 2009. · Zbl 1179.34021 · doi:10.1007/s12190-008-0121-9
[13] F. M. Minhós, “On some third order nonlinear boundary value problems: existence, location and multiplicity results,” Journal of Mathematical Analysis and Applications, vol. 339, no. 2, pp. 1342-1353, 2008. · Zbl 1144.34009 · doi:10.1016/j.jmaa.2007.08.005
[14] M. Bartu\vsek, M. Cecchi, and M. Marini, “On Kneser solutions of nonlinear third order differential equations,” Journal of Mathematical Analysis and Applications, vol. 261, no. 1, pp. 72-84, 2001. · Zbl 0995.34025 · doi:10.1006/jmaa.2000.7473
[15] P. K. Palamides and R. P. Agarwal, “An existence theorem for a singular third-order boundary value problem on [0,+\infty ),” Applied Mathematics Letters, vol. 21, no. 12, pp. 1254-1259, 2008. · Zbl 1206.34042 · doi:10.1016/j.aml.2007.11.001
[16] I. T. Kiguradze, “An oscillation criterion for a class of ordinary differential equations,” Differentsial’nye Uravneniya, vol. 28, no. 2, pp. 207-219, 1992. · Zbl 0768.34018
[17] O. Boruvka, Linear Differential Transformationen 2. Ordung, VEB, Berlin, Germany, 1967. · Zbl 0153.11201
[18] M. Marini, “Criteri di limitatezza per le soluzioni dell’equazione lineare del secondo ordine,” Bollettino della Unione Matematica Italiana, vol. 4, pp. 225-231, 1971.
[19] P. Hartman, Ordinary Differential Equations, John Wiley & Sons, New York, NY, USA, 1964. · Zbl 0125.32102
[20] E. F. Beckenbach and R. Bellman, Inequalities, vol. 30 of Ergebnisse der Mathematik und ihrer Grenzgebiete, N. F., Springer, Berlin, Germany, 1961. · Zbl 0186.09606
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.