International Journal of Mathematics and Mathematical Sciences
Volume 2011 (2011), Article ID 769893, 11 pages
http://dx.doi.org/10.1155/2011/769893
Research Article

Necessary and Sufficient Conditions for Positive Solutions of Second-Order Nonlinear Dynamic Equations on Time Scales

Zhi-Qiang Zhu

Department of Computer Science, Guangdong Polytechnic Normal University, Guangzhou 510665, China

Received 28 December 2010; Revised 22 March 2011; Accepted 13 April 2011

Academic Editor: Mihai Putinar

Copyright © 2011 Zhi-Qiang Zhu. 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 existence of nonoscillatory solutions for the nonlinear dynamic equation ( 𝑝 ( 𝑡 ) ( 𝜓 ∘ 𝑥 ) ( 𝑥 𝜎 ) Δ ) Δ + 𝑞 ( 𝑡 ) 𝑓 ∘ 𝑥 𝜎 = 0 on time scales. By making use of the generalized Riccati transformation technique, we establish some necessary and sufficient criteria to guarantee the existence. The last examples show that our results can be applied on the differential equations, the difference equations, and the 𝑞 -difference equations.