Báo cáo hóa học: Research Article Oscillation Criteria for Second-Order Neutral Delay Dynamic Equations with Mixed

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Báo cáo hóa học: " Research Article Oscillation Criteria for Second-Order Neutral Delay Dynamic Equations with Mixed "Hindawi Publishing CorporationAdvances in Difference EquationsVolume 2011, Article ID 513757, 14 pagesdoi:10.1155/2011/513757Research ArticleOscillation Criteria forSecond-Order Neutral Delay Dynamic Equationswith Mixed Nonlinearities Ethiraju Thandapani,1 Veeraraghavan Piramanantham,2 and Sandra Pinelas3 1 Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai 600 005, India 2 Department of Mathematics, Bharathidasan University, Tiruchirappalli 620 024, India 3 Departamento de Matem´ tica, Universidade dos Acores, 9501-801 Ponta Delgada, Azores, Portugal a ¸ Correspondence should be addressed to Sandra Pinelas, sandra.pinelas@clix.pt Received 20 September 2010; Revised 30 November 2010; Accepted 23 January 2011 Academic Editor: Istvan Gyori Copyright q 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. 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 p t x δ t Δ |α−1 x t αi − 1 0, where t ∈ T and u t n i 1 qi t |x τi t | |xt x τi t Δ with α1 > α2 > · · · > αm > α > αm 1 > · · · > αn > 0. Further the results obtained here ptxδt generalize and complement to the results obtained by Han et al. 2010 . Examples are provided to illustrate the results.1. IntroductionSince the introduction of time scale calculus by Stefan Hilger in 1988, there has been greatinterest in studying the qualitative behavior of dynamic equations on time scales, see, forexample, 1–3 and the references cited therein. In the last few years, the research activityconcerning the oscillation and nonoscillation of solutions of ordinary and neutral dynamicequations on time scales has been received considerable attention, see, for example, 4–8and the references cited therein. Moreover the oscillatory behavior of solutions of secondorder differential and dynamic equations with mixed nonlinearities is discussed in 9–16 . In 2004, Agarwal et al. 5 have obtained some sufficient conditions for the oscillationof all solutions of the second order nonlinear neutral delay dynamic equation Δ Δγ p t y t−τ f t, y t − δ rt yt 0 1.1 Advances in Difference Equations2on time scale T, where t ∈ T, γ is a quotient of odd positive integers such that γ ≥ 1, r t ,p t are real valued rd-continuous functions defined on T such that r t > 0, 0 ≤ p t < 1, andf t, u ≥ q t |u|γ . In 2009, Tripathy 17 has considered the nonlinear neutral dynamic equation of theform Δ Δγ γ p t y t−τ q t y t−δ sgn y t − δ t ∈ T, rt yt 0, 1.2where γ > 0 is a quotient of odd positive integers, r t , q t are positive real valued rd-continuous functions on T, p t is a nonnegative real valued rd-continuous function on Tand established sufficient conditions for the oscillation of all solutions of 1.2 using Ricattitransformation. ¸ı Saker et al. 18 , Sah´ner 19 , and Wu et al. 20 established various oscillation resultsfor the second order neutral delay dynamic equations of the form Δ Δγ t ∈ T, rt yt ptyτt f t, y δ t 0, ...