Oscillation criteria for third order nonlinear delay dynamic equations on time scales

Zhenlai Han¹ ; Tongxing Li² ; Shurong Sun³ ; Fengjuan Cao²

¹ School of Science University of Jinan Jinan, Shandong 250022, P.R. China and School of Control Science and Engineering Shandong University Jinan, Shandong 250061, P.R. China
² School of Science University of Jinan Jinan, Shandong 250022, P.R. China
³ School of Science University of Jinan Jinan, Shandong 250022, P.R. China and Department of Mathematics and Statistics Missouri University of Science and Technology Rolla, MO 65409-0020, U.S.A.

Annales Polonici Mathematici, Tome 99 (2010) no. 2, pp. 143-156

Résumé

By means of Riccati transformation technique, we establish some new oscillation criteria for third-order nonlinear delay dynamic equations $$ ((x^{\Delta\Delta}(t))^\gamma)^\Delta+p(t)x^\gamma(\tau(t))=0 $$ on a time scale $\mathbb{T};$ here $\gamma>0$ is a quotient of odd positive integers and $p$ a real-valued positive rd-continuous function defined on $\mathbb{T}.$ Our results not only extend and improve the results of T. S. Hassan [Math. Comput. Modelling 49 (2009)] but also unify the results on oscillation of third-order delay differential equations and third-order delay difference equations.

Détail

DOI : 10.4064/ap99-2-3

Keywords: means riccati transformation technique establish oscillation criteria third order nonlinear delay dynamic equations delta delta gamma delta gamma tau time scale mathbb here gamma quotient odd positive integers real valued positive rd continuous function defined mathbb results only extend improve results hassan math comput modelling unify results oscillation third order delay differential equations third order delay difference equations