Oscillation results for third order nonlinear neutral differential equations of mixed type

Ethiraju Thandapani; Renu Rama

doi:10.26637/mjm101/005

Abstract

Some oscillation results are obtained for the third order nonlinear mixed type neutral differential equations of the form
\begin{align*}
&\left(\left(x(t)+b(t) x\left(t-\tau_1\right)+c(t) x\left(t+\tau_2\right)\right)^\alpha\right)^{\prime \prime \prime}\\&=q(t) x^\beta\left(t-\sigma_1\right)+p(t) x^\gamma\left(t+\sigma_2\right), t \geq t_0
\end{align*}
where \(\alpha, \beta\) and \(\gamma\) are ratios of odd positive integers \(\tau_1, \tau_2, \sigma_1\) and \(\sigma_2\) are positive constants.

Keywords:

Oscillation, neutral differential equations

Mathematics Subject Classification:

34C15

Authors Imprint References Funding Related Articles Downloads

Ethiraju Thandapani Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai -600 005, India.
Renu Rama Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai -600 005, India.

Pages: 38-49
Date Published: 01-01-2013
Vol. 1 No. 01 (2013): Malaya Journal of Matematik (MJM)

R. P. Agarwal, M. F. Aktas, and A. Tiryaki, On oscillation criteria for third order nonlinear delay differential equations, Arch. Math., 45(2009), 1-18.

R. P. Agarwal, B. Baculáková, J. Dˇ zurina and T.Li, Oscillation of third-order nonlinear functional differential equations with mixed arguments, Acta Math. Hungar., 134(2011), 54-67. DOI: https://doi.org/10.1007/s10474-011-0120-4

R. P. Agarwal, S. R. Grace and D. O’Regan, Oscillation Theory for Difference and Functional Differential Equation, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000. DOI: https://doi.org/10.1007/978-94-015-9401-1_2

B. Bacul´ıková and J. Dˇ zurina, Oscillation of third-order functional differential equations, Elec. J.Qual. Theo. Diff. Equ., 43(2010), 1-10. DOI: https://doi.org/10.14232/ejqtde.2010.1.43

B. Bacul´ıková and J. Dˇ zurina, Oscillation of third-order neutral differential equations, Math. Comput.Modelling, 52(2010), 215-226. DOI: https://doi.org/10.1016/j.mcm.2010.02.011

B. Bacul´ıková and J. Dˇ zurina, On the aymptotic behavior of a class of third order nonlinear neutral differential equations, Cent. European J. Math., 8(2010), 1091-1103. DOI: https://doi.org/10.2478/s11533-010-0072-x

B. Bacul´ıková and J. Dˇ zurina, Oscillation of third-order nonlinear differential equations, Appl. Math.Lett., (2011), 466-470. DOI: https://doi.org/10.1016/j.aml.2010.10.043

B. Bacul´ıková, E. M. Elabbasy, S. H. Saker, and J. Dˇ zurina, Oscillation criteria for third-order non-linear differential euqations, Math. Slovaca, 58(2008), 201-220. DOI: https://doi.org/10.2478/s12175-008-0068-1

T. Candan and R. S. Dahiya, Functional differential equations of third order, Elec. J. Diff. Equ.,12(2005), 47-56.

P. Das, Oscillation criteria for odd order neutral equations, J. Math. Anal. Appl., 188(1994), 245-257. DOI: https://doi.org/10.1006/jmaa.1994.1425

K. Gopalsamy, B. S. Lalli and B. G. Zhang, Oscillation of odd order neutral differential equations, Czech. Math. J., 42(1992), 313-323. DOI: https://doi.org/10.21136/CMJ.1992.128330

S.R.Grace, Oscillation criteria for nth order neutral functional differential equations, J. Math. Anal.Appl., 184(1994), 44-55. DOI: https://doi.org/10.1006/jmaa.1994.1182

S. R. Grace, On oscillation of mixed neutral equations, J. Math. Anal. Appl., 194(1995), 377-388. DOI: https://doi.org/10.1006/jmaa.1995.1306

S. R. Grace, Oscillation of mixed neutral fractional differential equations, Appl. Math. Comp.,68(1995), 1-13. DOI: https://doi.org/10.1016/0096-3003(94)00075-F

S. R. Grace, R. P. Agarwal, R. Pavani, and E. Thandapani, On the oscillation of certain third order nonlinear functional differential equations, Appl. Math. Comput., 202(2008), 102-112. DOI: https://doi.org/10.1016/j.amc.2008.01.025

S.R.Grace and B.S.Lalli, Oscillation theorems for certain neutral functional differential equations with periodic cofficients, Dynam. Systems Appl., 3(1994), 85-93.

J. R. Graef, R. Savithri, and E. Thandapani, Oscillatory properties of third order neutral delay differential equations, Disc. Cont. Dyn. Sys. A, 2003, 342-350.

T. Li and E. Thandapani, Oscillation of solutions to odd-order nonlinear neutral functional differential equations, Elec. J. Diff. Equ., 23(2011), 1-12. DOI: https://doi.org/10.14232/ejqtde.2011.1.76

T.Li, C.Zhang and G.Xing, Oscillation of third order neutral delay differential equations, Abstract and Applied Analysis, 2012(2012), 1-11. DOI: https://doi.org/10.1155/2012/569201

S. H. Saker and J. Dˇ zurina, On the oscillation of certain class of third-order nonlinear delay differential equations, Math. Bohemica, 135(2010), 225-237. DOI: https://doi.org/10.21136/MB.2010.140700

E. Thandapani and T. Li, On the oscillation of third-order quasi-linear neutral functional differential equations, Arch. Math., 47(2011), 181-199. DOI: https://doi.org/10.1186/1687-1847-2011-45

J. R. Yau, Oscillation of higher order neutral differential equations of mixed type, Israel J. Math.,115(2000), 125-136. DOI: https://doi.org/10.1007/BF02810583

C. Zhang, T. Li, B. Sun and E. Thandapani, On the oscillation of higher-order half-linear delay differential equations, Appl. Math. Lett., 24(2011), 1618-1621. DOI: https://doi.org/10.1016/j.aml.2011.04.015