Oscillation theorems for higher order neutral nonlinear dynamic equations on time scales

A. Benaissa Cherif; F. Z. Ladrani; A. Hammoudi

doi:10.26637/mjm404/007

Abstract

In this paper, we will establish some oscillation criteria for the even-order nonlinear dynamic equation
$$
\left(a\left(x^{\Delta^{n-2}}\right)^\gamma\right)^{\Delta^2}(t)+f\left(t, x^\alpha(t)\right)=0, \quad t \in\left[t_0, \infty\right)_{\mathbb{T}}
$$
on a time scales $\mathbb{T}$ with $n$ is an even integer $\geq 3$, where $\gamma$ and $\alpha$ are the ratios of positive odd integer and $a$ is a real valued rd-continuous function defined on $\mathbb{T}$.

Keywords:

Time scale, Oscillation, Neutral delay differential equation

Mathematics Subject Classification:

34K11, 39A10, 39A99

Authors Imprint References Funding Related Articles Downloads

A. Benaissa Cherif Department of Mathematics, University of Ain Temouchent, BP 284, 46000 Ain Temouchent, Algeria.
F. Z. Ladrani Department of Mathematics, Higher normal school of Oran, BP 1523, 31000 Oran, Algeria. https://orcid.org/0000-0001-7205-5682
A. Hammoudi Department of Mathematics, University of Ain Temouchent, BP 284, 46000 Ain Temouchent, Algeria.

Pages: 599-605
Date Published: 01-10-2016
Vol. 4 No. 04 (2016): Malaya Journal of Matematik (MJM)

S. Hilger, Analysis on measure chains-a unified approach to continuous and discrete calculus, Results Math, $8(1990), 18-56$ DOI: https://doi.org/10.1007/BF03323153

M. Bohner, and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston, (2001).

M. Bohne, and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, (2001). DOI: https://doi.org/10.1007/978-1-4612-0201-1

B. Karpuz, Sufficient conditions for the oscillation and asymptotic beaviour of higher-order dynamic equations of neutral type, Applied Mathematics and Computation, 221(2013), $453-462$. DOI: https://doi.org/10.1016/j.amc.2013.06.090

B. Baculíkovà, and J. Dzǔurina, Oscillation theorems for higher order neutral differential equations, Applied Mathematics and Computation, 219(2012), $3769-3778$. DOI: https://doi.org/10.1016/j.amc.2012.10.006

R. P. Agarwal, M. Bohner, T. Li, and C. Zhang, A new approach in the study of oscillatory behavior of even-order neutral delay differential equations, Applied Mathematics and Computation, 225(2013), 787 - 794. DOI: https://doi.org/10.1016/j.amc.2013.09.037

S.R. Grace, On the Oscillation of nth Order Dynamic Equations on Time-Scales, Mediterr. J. Math, 10(2013), $147-156$ DOI: https://doi.org/10.1007/s00009-012-0201-9

Y. Shi, Oscillation criteria for nth order nonlinear neutral differential equations, Applied Mathematics and Computation, 235(2014), $423-429$. DOI: https://doi.org/10.1016/j.amc.2014.03.018

C. Zhanga, R. P. Agarwal , M. Bohner, and T. Li, New results for oscillatory behavior of even-order half-linear delay differential equations, Applied Mathematics Letters, 26(2013), $179-183$. DOI: https://doi.org/10.1016/j.aml.2012.08.004

S. H. Saker, Oscillation of second-order nonlinear neutral delay dynamic equations on time scales, Journal of Computational and Applied Mathematics, 187(2006), 123 - 141.

S. H. Saker, Oscillation of second-order nonlinear neutral delay dynamic equations on time scales, Journal of Computational and Applied Mathematics, 187(2006), 123 - 141. DOI: https://doi.org/10.1016/j.cam.2005.03.039

L. Erbe, A. Peterson, and S. H. Saker, Hille and Nehari type criteria for third dorder dynamic equations, J. Math. Anal. Appl, 329(2007), $112-131$ DOI: https://doi.org/10.1016/j.jmaa.2006.06.033

L. Erbe, T.S. Hassan, and A. Peterson, Oscillation criteria for nonlinear damped dynamic equations on time scales, Appl. Math. Comput, 203(2008), $343-357$. DOI: https://doi.org/10.1016/j.amc.2008.04.038

S. R. Grace, M. Bohner, and S. Sun, Oscillation of fourth-order dynamic equations. Hacettepe Journal of Mathematics and Statistics, 39(2010), 545 - 553.

F. Z. Ladrani, A. Hammoudi, and A. Benaissa Cherif, Oscillation theorems for fourth-order nonlinear dynamic equations on time scales, Electronic Journal of Mathematical Analysis and Applications, 3(2015), 46 - 58.

T. Li, E. Thandapani, and S. Tang, Oscillation theorems for fourth-order delay dynamic equations on time scales, Bulletin of Mathematical Analysis and Applications, 3(2011), 190 - 199.

Y. Qi, and J. Yu, Oscillation criteria for fourth-order nonlinear delay dynamic equations, Electronic Journal of Differential Equations, 79(2013), 1 - 17.