On the oscillation of third order quasilinear delay differential equations with Maxima

R. Arul; M. Mani

doi:10.26637/mjm204/016

Abstract

In this paper, we study the oscillation and asymptotic properties of third order quasilinear neutral delay differential equation
$\left(a(t)\left((x(t)+p(t) x(\tau(t)))^{\prime \prime}\right)^\alpha\right)^{\prime}+q(t) \max _{[\sigma(t), t]} x^\alpha(s)=0, t \geq t_0 \geq 0$
where $\alpha$ is a ratio of odd positive integers and $\int_{t_0}^{\infty} \frac{1}{a^{1 / \alpha}(t)} d t=\infty$ . We establish a new condition which guarantees that every solution is either oscillatory or converges to zero. There results extend some known results in the literature without "maxima". Examples are given to illustrate the main results.

Keywords:

Oscillation, quasilinear, neutral, delay, third order, differential equations with maxima

Mathematics Subject Classification:

34K15

Authors Imprint References Funding Related Articles Downloads

R. Arul Department of Mathematics, Kandaswami Kandar’s College, Velur–638 182, Tamil Nadu, India.
M. Mani Department of Mathematics, Kandaswami Kandar’s College, Velur–638 182, Tamil Nadu, India.

Pages: 489-496
Date Published: 01-10-2014
Vol. 2 No. 04 (2014): Malaya Journal of Matematik (MJM)

B.Baculikova and J.Dzurina, 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

D.D.Bainov and S.G.Hristova, Differential Equations with Maxima, CRC Press, Taylor and Francis Group, New York. 2011. DOI: https://doi.org/10.1201/b10877

D.Bainov, V.Petrov and V.Proytcheva, Oscillatory and asymptotic behaviour of second order neutral differential equations with 'Maxima', Dyn. Sys. Appl., 4 (1993), 135-146.

D.Bainov, V.Petrov and V.Proytcheva, Oscillation and nonoscillation of first order neutral differential equations with 'Maxima', SUTJ. Math., 31(1995), 17-28. DOI: https://doi.org/10.55937/sut/1262208353

D.Bainov, V.Petrov and V.Proytcheva, Existence and asymptotic behaviour of nonoscillatory solutions of second order neutral differential equations with 'Maxima', J. Comput. Appl. Math., 83 (1997), 237-249. DOI: https://doi.org/10.1016/S0377-0427(97)00105-2

D.D.Bainov and A.I.Zahariev, Oscillatory and asymptotic properties of a class of functional differential equations with 'Maxima', Czech. Math. J., 34(1984), 247-251. DOI: https://doi.org/10.21136/CMJ.1984.101947

D.Bainov, V.Petrov and V.Proicheva, Oscillation of neutral differential equations with 'Maxima', Rev. Math., 8(1995), 171-180. DOI: https://doi.org/10.5209/rev_REMA.1995.v8.n1.17715

Z.Han, T.Li, S.Sun and W.Chen, Oscillation of second order quasilinear neutral delay differential equations, J. Appl. Math. Comput., 40(2012), 143-152. DOI: https://doi.org/10.1007/s12190-012-0562-z

T.Li, Z.Han, C.Zhang and S.Sun, On the oscillation of second order Emden-Fowler neutral differential equations, J. Appl. Math. Comput., 42(2)(2013), 131-138.

T.Li, S.Sun, Z.Han, B.Han and Y.Sun, Oscillation results for second order quasilinear neutral delay differential equations, Hacettepe J. of Math. and Stat., 37 (2011), 601-610. DOI: https://doi.org/10.1007/s12190-010-0453-0

A.R.Magomedev, On some problems of differential equations with 'Maxima', Izv. Acad. Sci. Azerb. SSr, Ser. Phys-Techn. and Math. Sci., 108(1977), 104-108.

V.A.Petrov, Nonoscillatory solutions of neutral differential equations with 'Maxima', Commun. Appl. Anal., 2(1998), 129-142.

E.P.Popov, Automic Regulation and Control, Nauka, Moscow., 1996.

E.Thandapani and V.Ganesan, Oscillatory and asymptotic behavior of solution of second order neutral delay differential equations with "maxiam", Inter. J. of Pure and Appl. Math., 78(7)(2012), 1029-1039.

B.G.Zhang and G.Zhang, Qualitative properties of functional differential equations with 'Maxima', Rocky Mountain J. of Math., 29(1999), 357-367. DOI: https://doi.org/10.1216/rmjm/1181071696