Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 137471, 14 pages
http://dx.doi.org/10.1155/2012/137471
Research Article

Oscillation and Nonoscillation Criteria for Nonlinear Dynamic Systems on Time Scales

Shanliang Zhu and Chunyun Sheng

College of Mathematics and Physics, Qingdao University of Science and Technology, Qingdao 266061, China

Received 10 April 2012; Accepted 23 June 2012

Academic Editor: Jinde Cao

Copyright © 2012 Shanliang Zhu and Chunyun Sheng. 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

We consider the nonlinear dynamic system 𝑥 Δ ( 𝑡 ) = 𝑎 ( 𝑡 ) 𝑔 ( 𝑦 ( 𝑡 ) ) , 𝑦 Δ ( 𝑡 ) = − 𝑓 ( 𝑡 , 𝑥 𝜎 ( 𝑡 ) ) . We establish some necessary and sufficient conditions for the existence of oscillatory and nonoscillatory solutions with special asymptotic properties for the system. We generalize the known results in the literature. Some examples are included to illustrate the results.