Properties of third-order nonlinear functional differential equations with mixed arguments. (English) Zbl 1217.34109
Summary: The aim of this paper is to offer sufficient conditions for property (B) and/or the oscillation of the third-order nonlinear functional differential equation with mixed arguments
\[ [a(t)[x''(t)]^\gamma]'=q(t)f(x[\tau(t)])+p(t)h(x[\sigma(t)]). \]
Both cases \(\int^\infty a^{-1/\gamma}(s)\, ds=\infty\) and \(\int^\infty a^{-1/\gamma}(s)\, ds<\infty\) are considered. We deduce properties of the studied equations via new comparison theorems. The results obtained essentially improve and complement earlier ones.
\[ [a(t)[x''(t)]^\gamma]'=q(t)f(x[\tau(t)])+p(t)h(x[\sigma(t)]). \]
Both cases \(\int^\infty a^{-1/\gamma}(s)\, ds=\infty\) and \(\int^\infty a^{-1/\gamma}(s)\, ds<\infty\) are considered. We deduce properties of the studied equations via new comparison theorems. The results obtained essentially improve and complement earlier ones.
MSC:
| 34K11 | Oscillation theory of functional-differential equations |
References:
| [1] | R. P. Agarwal, S.-L. Shieh, and C.-C. Yeh, “Oscillation criteria for second-order retarded differential equations,” Mathematical and Computer Modelling, vol. 26, no. 4, pp. 1-11, 1997. · Zbl 0902.34061 · doi:10.1016/S0895-7177(97)00141-6 |
| [2] | R. P. Agarwal, S. R. Grace, and D. O’Regan, “On the oscillation of certain functional differential equations via comparison methods,” Journal of Mathematical Analysis and Applications, vol. 286, no. 2, pp. 577-600, 2003. · Zbl 1057.34072 · doi:10.1016/S0022-247X(03)00494-3 |
| [3] | R. P. Agarwal, S. R. Grace, and T. Smith, “Oscillation of certain third order functional differential equations,” Advances in Mathematical Sciences and Applications, vol. 16, no. 1, pp. 69-94, 2006. · Zbl 1116.34050 |
| [4] | 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 |
| [5] | B. Baculíková, R. P. Agarwal, T. Li, and J. D\vzurina, “Oscillation of third-order nonlinear functional differential equations with mixed arguments,” to appear in Acta Mathematica Hungarica. · Zbl 1265.34231 |
| [6] | 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 |
| [7] | J. D\vzurina, “Asymptotic properties of third order delay differential equations,” Czechoslovak Mathematical Journal, vol. 45(120), no. 3, pp. 443-448, 1995. · Zbl 0842.34073 |
| [8] | J. D\vzurina, “Comparison theorems for functional-differential equations with advanced argument,” Unione Matematica Italiana. Bollettino, vol. 7, no. 3, pp. 461-470, 1993. · Zbl 0803.34066 |
| [9] | L. H. Erbe, Q. Kong, and B. G. Zhang, Oscillation Theory for Functional-Differential Equations, vol. 190 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1994. · Zbl 0821.34067 |
| [10] | 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 |
| [11] | I. Györi and G. Ladas, Oscillation Theory of Delay Differential Equations, Oxford Mathematical Monographs, The Clarendon Press, New York, NY, USA, 1991. · Zbl 0780.34048 |
| [12] | T. S. Hassan, “Oscillation of third order nonlinear delay dynamic equations on time scales,” Mathematical and Computer Modelling, vol. 49, no. 7-8, pp. 1573-1586, 2009. · Zbl 1175.34086 · doi:10.1016/j.mcm.2008.12.011 |
| [13] | 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 |
| [14] | G. S. Ladde, V. Lakshmikantham, and B. G. Zhang, Oscillation Theory of Differential Equations with Deviating Arguments, vol. 110 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1987. · Zbl 0622.34071 |
| [15] | N. Parhi and S. Pardi, “On oscillation and asymptotic property of a class of third order differential equations,” Czechoslovak Mathematical Journal, vol. 49(124), no. 1, pp. 21-33, 1999. · Zbl 0955.34023 · doi:10.1023/A:1022495705823 |
| [16] | Ch. 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 |
| [17] | 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 |
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.
