Advances in Difference Equations
Volume 2011 (2011), Article ID 513757, 14 pages
doi:10.1155/2011/513757
Research Article

Oscillation Criteria for Second-Order Neutral Delay Dynamic Equations with Mixed Nonlinearities

Ethiraju Thandapani,1 Veeraraghavan Piramanantham,2 and Sandra Pinelas3

1Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai 600 005, India
2Department of Mathematics, Bharathidasan University, Tiruchirappalli 620 024, India
3Departamento de Matemática, Universidade dos Açores, 9501-801 Ponta Delgada, Azores, Portugal

Received 20 September 2010; Revised 30 November 2010; Accepted 23 January 2011

Academic Editor: Istvan Gyori

Copyright © 2011 Ethiraju Thandapani et al. 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

This paper is concerned with some oscillation criteria for the second order neutral delay dynamic equations with mixed nonlinearities of the form ( 𝑟 ( 𝑡 ) 𝑢 ( 𝑡 ) ) Δ + 𝑞 ( 𝑡 ) | 𝑥 ( 𝜏 ( 𝑡 ) ) | 𝛼 − 1 ∑ 𝑥 ( 𝜏 ( 𝑡 ) ) + 𝑛 𝑖 = 1 𝑞 𝑖 ( 𝑡 ) | 𝑥 ( 𝜏 𝑖 ( 𝑡 ) ) | 𝛼 𝑖 − 1 𝑥 ( 𝜏 𝑖 ( 𝑡 ) ) = 0 , where 𝑡 ∈ 𝕋 and 𝑢 ( 𝑡 ) = | ( 𝑥 ( 𝑡 ) + 𝑝 ( 𝑡 ) 𝑥 ( 𝛿 ( 𝑡 ) ) ) Δ | 𝛼 − 1 ( 𝑥 ( 𝑡 ) + 𝑝 ( 𝑡 ) 𝑥 ( 𝛿 ( 𝑡 ) ) ) Δ with 𝛼 1 > 𝛼 2 > ⋯ > 𝛼 𝑚 > 𝛼 > 𝛼 𝑚 + 1 > ⋯ > 𝛼 𝑛 > 0 . Further the results obtained here generalize and complement to the results obtained by Han et al. (2010). Examples are provided to illustrate the results.