Oscillatory behavior of solutions of certain third order mixed neutral differential equations

Main Article Content

Ethiraj Thandapani
Renu Rama

Abstract

The objective of this paper is to study the oscillatory and asymptotic properties of third order mixed neutral differential equation of the form

 $$ (a(t) [x(t) + b(t) x(t - \tau_{1}) + c(t) x(t + \tau_{2})]'')' + q(t) x^{\alpha}(t - \sigma_{1}) + p(t) x^{\beta}(t + \sigma_{2}) = 0 $$

where $a(t), b(t), c(t), q(t)$ and $p(t)$ are positive continuous functions, $\alpha$ and $\beta$ are ratios of odd positive integers, $\tau_{1}, \tau_{2}, \sigma_{1}$ and $\sigma_{2}$ are positive constants. We establish some sufficient conditions which ensure that all solutions are either oscillatory or converge to zero. Some examples are provided to illustrate the main results.

Article Details

How to Cite
Thandapani, E., & Rama, R. (2013). Oscillatory behavior of solutions of certain third order mixed neutral differential equations. Tamkang Journal of Mathematics, 44(1), 99–112. https://doi.org/10.5556/j.tkjm.44.2013.1150
Section
Papers
Author Biographies

Ethiraj Thandapani

Ramanujan Institute for Advanced Study inMathematics, University ofMadras, Chennai -600 005, India.

Renu Rama

Ramanujan Institute for Advanced, Study inMathematics, University ofMadras, Chennai - 600 005, India.

References

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. Baculakova, J. Dvzurina and T.Li, Oscillation of third-order nonlinear functional differential equations with mixed arguments, Acta Math. Hungar., 134(2011), 54--67.

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.

R. P. Agarwal, S. R. Grace and D. O'Regan, Oscillation criteria for certain nth order differential equations with devating arguments, J. Math. Anal. Appl., 262(2001), 601--622.

R. P. Agarwal, S. R. Grace and D. O'Regan, The oscillation of certain higher-order functional differential equations, Math. Comput. Modelling, 37(2003), 705--728.

B. Baculikova and J. Dvzurina, Oscillation of third-order functional differential equations, Elec. J. Qual. Theo. Diff. Equ., 43(2010), 1--10.

B. Baculikova and J. Dvzurina, Oscillation of third-order neutral differential equations, Math. Comput. Modelling, 52 (2010), 215--226.

B. Baculikova and J. Dvzurina, On the aymptotic behavior of a class of third order nonlinear neutral differential equations, Cent. European J. Math., 8(2010), 1091-1103.

B. Baculikova and J. Dvzurina, Oscillation of third-order nonlinear differential equations, Appl. Math. Lett. 2011, 466--470.

B. Baculikova, E. M. Elabbasy, S. H. Saker, and J. Dvzurina, Oscillation criteria for third-order non-linear differential euqations, Math. Slovaca, 58(2008), 201--220.

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.

K. Gopalsamy, B. S. Lalli and B. G. Zhang, Oscillation of odd order neutral differential equations, Czech. Math. J., 42(1992), 313--323.

S. R. Grace, On oscillation of mixed neutral equations, J. Math. Anal. Appl., 194(1995), 377--388.

S. R. Grace, Oscillation of mixed neutral fractional differential equations, Appl. Math. Comp., 68(1995), 1--13.

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.

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.

Z.Har, T.Li, S.Sun and W.Chen, On the oscillation of second order neutral delay differential equations, Adv. Diff. Eqns. Vol(2010) Article ID 289340, 8pages, 2010.

T. Li and E. Thandapani, Oscillation of solutions to odd-order nonlinear neutral functional differential equations,Elec. J. Diff. Equ., 23(2011), 1--12.

T.Li, C.Zhang and G.Xing, Oscillation of third order neutral delay differential equations, Abstract and Applied Analysis, 2012(2012), 1--11.

S. H. Saker and J. Dvzurina, On the oscillation of certain class of third-order nonlinear delay differential equations, Math. Bohemica, 135(2010), 225--237.

E. Thandapani and T. Li, On the oscillation of third-order quasi-linear neutral functional differentiall equations, Arch. Math., 47(2011), 181--199.

J. R. Yau, Oscillation of higher order neutral differential equations of mixed type, Israel J. Math., 115(2000), 125--136.

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.