International Journal of Differential Equations
Volume 2011 (2011), Article ID 863801, 15 pages
http://dx.doi.org/10.1155/2011/863801
Research Article

Oscillation of Second-Order Nonlinear Delay Dynamic Equations on Time Scales

H. A. Agwa, A. M. M. Khodier, and Heba A. Hassan

Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo, Egypt

Received 3 May 2011; Accepted 6 June 2011

Academic Editor: Elena Braverman

Copyright © 2011 H. A. Agwa 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

In this work, we use the generalized Riccati transformation and the inequality technique to establish some new oscillation criteria for the second-order nonlinear delay dynamic equation ( 𝑝 ( 𝑡 ) ( 𝑥 Δ ( 𝑡 ) ) 𝛾 ) Δ + 𝑞 ( 𝑡 ) 𝑓 ( 𝑥 ( 𝜏 ( 𝑡 ) ) ) = 0 , on a time scale 𝕋 , where 𝛾 is the quotient of odd positive integers and p(t) and q(t) are positive right-dense continuous (rd-continuous) functions on 𝕋 . Our results improve and extend some results established by Sun et al. 2009. Also our results unify the oscillation of the second-order nonlinear delay differential equation and the second-order nonlinear delay difference equation. Finally, we give some examples to illustrate our main results.