Oscillatory Behavior of Quasilinear Neutral Delay Dynamic Equations on Time Scales
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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 1
School of Science, University of Jinan, Jinan, Shandong 250022, China School of Control Science and Engineering, Shandong University, Jinan, Shandong 250061, China 3 Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO 65409-0020, USA 2
Correspondence should be addressed to Zhenlai Han, [email protected] Received 6 December 2009; Accepted 4 March 2010 Academic Editor: Leonid Berezansky Copyright q 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. 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 rt|xΔ t|γ−1 xΔ tΔ q1 t|yδ1 t|α−1 yδ1 t q2 t|yδ2 t|β−1 yδ2 t 0, t ∈ t0 , ∞T , where xt yt ptyτt, 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.
1. Introduction The theory of time scales, which has recently received a lot of attention, was introduced by Hilger in his Ph.D. Thesis in 1988 in order to unify continuous and discrete analysis see Hilger 1. Several authors have expounded on various aspects of this new theory and references cited therein. A book on the subject of time scale, by Bohner and Peterson 2, summarizes and organizes much of the time scale calculus; we refer also the last book by Bohner and Peterson 3 for advances in dynamic equations on time scales. A time scale T is an arbitrary closed subset of the reals, and the cases when this time scale is equal to the reals or to the integers represent the classical theories of differential and of difference equations. Many other interesting time scales exist see Bohner and Peterson 2.
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Advances in Difference Equations
In the last few years, there has been much research activity concerning the oscillation and nonoscillation of solutions of various dynamic equations on time scales which attempts to harmonize the oscillation theory for the continuous and the discrete to include them in one comprehensive theory and to eliminate obscurity form both, for instance, the papers 4–20 and the reference cited therein. For oscillation of delay dynamic equations on time scales, see recently papers 21–32. However, there are very few results dealing with the oscillation of the solutions of neutral delay dynamic equations on time scales; we refer the reader to 33–44. Agarwal et al. 33 and Saker 37 consider the second-order nonlinear neutral
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