International Journal of Differential Equations
Volume 2011 (2011), Article ID 840569, 16 pages
http://dx.doi.org/10.1155/2011/840569
Research Article

Oscillation Theorems for Second-Order Half-Linear Advanced Dynamic Equations on Time Scales

Shuhong Tang,1 Tongxing Li,2,3 and Ethiraju Thandapani4

1School of Information and Control Engineering, Weifang University, Shandong, Weifang 261061, China
2School of Control Science and Engineering, Shandong University, Shandong, Jinan 250061, China
3School of Mathematical Science, University of Jinan, Shandong, Jinan 250022, China
4Ramanujan Institute for Advanced Study in Mathematics, University of Madras, 600 005 Chennai, India

Received 7 May 2011; Revised 7 July 2011; Accepted 26 July 2011

Academic Editor: Dumitru Baleanu

Copyright © 2011 Shuhong Tang 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

This paper is concerned with the oscillatory behavior of the second-order half-linear advanced dynamic equation ( 𝑟 ( 𝑡 ) ( 𝑥 Δ ( 𝑡 ) ) 𝛾 ) Δ + 𝑝 ( 𝑡 ) 𝑥 𝛾 ( 𝑔 ( 𝑡 ) ) = 0 on an arbitrary time scale 𝕋 with sup 𝕋 = ∞ , where 𝑔 ( 𝑡 ) ≥ 𝑡 and ∫ ∞ 𝑡 𝑜 ( Δ 𝑠 / ( 𝑟 1 / 𝛾 ( 𝑠 ) ) ) < ∞ . Some sufficient conditions for oscillation of the studied equation are established. Our results not only improve and complement those results in the literature but also unify the oscillation of the second-order half-linear advanced differential equation and the second-order half-linear advanced difference equation. Three examples are included to illustrate the main results.