Advances in Difference Equations
Volume 2010 (2010), Article ID 450264, 24 pages
doi:10.1155/2010/450264
Research Article

Oscillatory Behavior of Quasilinear Neutral Delay Dynamic Equations on Time Scales

Zhenlai Han,1,2 Shurong Sun,1,3 Tongxing Li,1 and Chenghui Zhang2

1School of Science, University of Jinan, Jinan, Shandong 250022, China
2School of Control Science and Engineering, Shandong University, Jinan, Shandong 250061, China
3Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO 65409-0020, USA

Received 6 December 2009; Accepted 4 March 2010

Academic Editor: Leonid Berezansky

Copyright © 2010 Zhenlai Han et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

By means of the averaging technique and the generalized Riccati transformation technique, we establish some oscillation criteria for the second-order quasilinear neutral delay dynamic equations [r(t)|xΔ(t)|γ-1xΔ(t)]Δ+q1(t)|y(δ1(t))|α-1y(δ1(t))+q2(t)|y(δ2(t))|β-1y(δ2(t))=0, t∈[t0,∞)𝕋, where x(t)=y(t)+p(t)y(τ(t)), and the time scale interval is [t0,∞)𝕋:=[t0,∞)∩𝕋. Our results in this paper not only extend the results given by Agarwal et al. (2005) but also unify the oscillation of the second-order neutral delay differential equations and the second-order neutral delay difference equations.