International Journal of Differential Equations
Volume 2009 (2009), Article ID 714357, 15 pages
doi:10.1155/2009/714357
Research Article

Oscillation Theorems for Second-Order Damped Nonlinear Differential Equations

Hui-Zeng Qin1 and Yongsheng Ren2

1Institute of Applied Mathematics, Shandong University of Technology, Zibo, Shandong 255049, China
2Collage of Mechanical and Electronic Engineering, Shandong University of Science and Technology, Qindao 266510, China

Received 26 September 2008; Revised 28 January 2009; Accepted 23 March 2009

Academic Editor: Qingkai Kong

Copyright © 2009 Hui-Zeng Qin and Yongsheng Ren. 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 present new oscillation criteria for the differential equation of the form [r(t)U(t)]′+p(t)k2(x(t), x′(t))|x(t)|νU(t)+q(t)ϕ(x(g1(t)),x′(g2(t)))f(x(t))=0, where U(t)=k1(x(t),x′(t))|x′(t)|α−1x′(t), α≤β,  ν=(β−α)/(α+1). Our research is different from most known ones in the sense that H function is not employed in our results, though Riccati's substitution and its generalized forms are used. Our criteria which are established under quite general assumptions are an extension for previous results. In particular, by taking β=α, the above-mentioned equation can be reduced into the various types of equations concerned by people currently.