Abstract and Applied Analysis
Volume 2010 (2010), Article ID 564068, 11 pages
doi:10.1155/2010/564068
Research Article

New Oscillation Criteria for Second-Order Neutral Delay Differential Equations with Positive and Negative Coefficients

Yuzhen Bai and Lihua Liu

School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China

Received 23 November 2009; Revised 8 May 2010; Accepted 11 June 2010

Academic Editor: Ağacık Zafer

Copyright © 2010 Yuzhen Bai and Lihua Liu. 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 establish the oscillation and asymptotic criteria for the second-order neutral delay differential equations with positive and negative coefficients having the forms [x(t)+∑i∈Rci(t)x(αi(t))]′′+r(t)[x(t)+∑i∈Rci(t)x(αi(t))]′+∑i∈Ppi(t)x(βi(t))-∑i∈Qqi(t)x(γi(t))=0 and [x(t)+∑i∈Rci(t)x(αi(t))]′′+r(t)[x(t)+∑i∈Rci(t)x(αi(t))]′+∑i∈Ppi(t)x(βi(t))-∑i∈Qqi(t)x(γi(t))=f(t). The obtained new oscillation criteria extend and improve the recent results given in the paperof B. Karpuz et al. (2009).