New Oscillation Criteria for Second-Order Delay Differential Equations with Mixed Nonlinearities

Yuzhen Bai and Lihua Liu School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China Correspondence should be addressed to Yuzhen Bai, [email protected] Received 10 January 2010; Accepted 17 June 2010 Academic Editor: Binggen Zhang Copyright q 2010 Y. Bai and L. 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. We establish new oscillation criteria for second-order delay differential equations with mixed nonlinearities of the form p t x′ t ′ ∑n i 1 pi t x t−τi ∑n i 1 qi t |x t−τi |αisgnx t−τi e t , t ≥ 0, where p t , pi t , qi t , and e t are continuous functions defined on 0,∞ , and p t > 0, p′ t ≥ 0, and α1 > · · · > αm > 1 > αm 1 > · · · > αn > 0. No restriction is imposed on the potentials pi t , qi t , and e t to be nonnegative. These oscillation criteria extend and improve the results given in the recent papers. An interesting example illustrating the sharpness of our results is also provided.