Advances in Difference Equations
Volume 2010 (2010), Article ID 103065, 11 pages
doi:10.1155/2010/103065
Research Article

Oscillation of Second-Order Sublinear Dynamic Equations with Damping on Isolated Time Scales

Quanwen Lin1,2 and Baoguo Jia1,2

1Department of Mathematics, Maoming University, Maoming 525000, China
2School of Mathematics and Computer Science, Zhongshan University, Guangzhou 510275, China

Received 8 October 2010; Accepted 27 December 2010

Academic Editor: M. Cecchi

Copyright © 2010 Quanwen Lin and Baoguo Jia. 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 concerns the oscillation of solutions to the second sublinear dynamic equation with damping 𝑥 Δ Δ ( 𝑡 ) + 𝑞 ( 𝑡 ) 𝑥 Δ 𝜎 ( 𝑡 ) + 𝑝 ( 𝑡 ) 𝑥 𝛼 ( 𝜎 ( 𝑡 ) ) = 0 , on an isolated time scale 𝕋 which is unbounded above. In 0 < 𝛼 < 1 , α is the quotient of odd positive integers. As an application, we get the difference equation Δ 2 𝑥 ( 𝑛 ) + 𝑛 − 𝛾 Δ 𝑥 ( 𝑛 + 1 ) + [ ( 1 / 𝑛 ( l n 𝑛 ) 𝛽 ) + 𝑏 ( ( − 1 ) 𝑛 / ( l n 𝑛 ) 𝛽 ) ] 𝑥 𝛼 ( 𝑛 + 1 ) = 0 , where 𝛾 > 0 , 𝛽 > 0 , and 𝑏 is any real number, is oscillatory.