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

Yuzhen Bai and Lihua Liu

Discrete Dynamics in Nature and Society, 2010, vol. 2010, 1-9

Abstract:

We establish new oscillation criteria for second-order delay differential equations with mixed nonlinearities of the form ∑ ( ð ‘ ( ð ‘¡ ) ð ‘¥ ′ ( ð ‘¡ ) ) ′ + ð ‘› ð ‘– = 1 ð ‘ ð ‘– ( ð ‘¡ ) ð ‘¥ ( ð ‘¡ − ð œ ð ‘– ∑ ) + ð ‘› ð ‘– = 1 ð ‘ž ð ‘– ( ð ‘¡ ) | ð ‘¥ ( ð ‘¡ − ð œ ð ‘– ) | ð ›¼ ð ‘– s g n ð ‘¥ ( ð ‘¡ − ð œ ð ‘– ) = ð ‘’ ( ð ‘¡ ) , ð ‘¡ ≥ 0 , where ð ‘ ( ð ‘¡ ) , ð ‘ ð ‘– ( ð ‘¡ ) , ð ‘ž ð ‘– ( ð ‘¡ ) , and ð ‘’ ( ð ‘¡ ) are continuous functions defined on [ 0 , ∞ ) , and ð ‘ ( ð ‘¡ ) > 0 , ð ‘ î…ž ( ð ‘¡ ) ≥ 0 , and ð ›¼ 1 > ⋯ > ð ›¼ ð ‘š > 1 > ð ›¼ ð ‘š + 1 > ⋯ > ð ›¼ ð ‘› > 0 . No restriction is imposed on the potentials ð ‘ ð ‘– ( ð ‘¡ ) , ð ‘ž ð ‘– ( ð ‘¡ ) , and ð ‘’ ( ð ‘¡ ) 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.

Date: 2010
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnddns:796256

DOI: 10.1155/2010/796256

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