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Discrete Dynamics in Nature and Society
Volume 2010, Article ID 796256, 9 pages
Research Article

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

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

Received 10 January 2010; Accepted 17 June 2010

Academic Editor: Binggen Zhang

Copyright Β© 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.


We establish new oscillation criteria for second-order delay differential equations with mixed nonlinearities of the form βˆ‘(𝑝(𝑑)π‘₯β€²(𝑑))β€²+𝑛𝑖=1𝑝𝑖(𝑑)π‘₯(π‘‘βˆ’πœπ‘–βˆ‘)+𝑛𝑖=1π‘žπ‘–(𝑑)|π‘₯(π‘‘βˆ’πœπ‘–)|𝛼𝑖sgnπ‘₯(π‘‘βˆ’πœπ‘–)=𝑒(𝑑),𝑑β‰₯0, where 𝑝(𝑑), 𝑝𝑖(𝑑), π‘žπ‘–(𝑑), and 𝑒(𝑑) are continuous functions defined on [0,∞), and 𝑝(𝑑)>0, π‘ξ…ž(𝑑)β‰₯0, and 𝛼1>β‹―>π›Όπ‘š>1>π›Όπ‘š+1>β‹―>𝛼𝑛>0. No restriction is imposed on the potentials 𝑝𝑖(𝑑), π‘žπ‘–(𝑑), and 𝑒(𝑑) 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.