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New oscillation results for second-order neutral delay dynamic equations

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This paper is concerned with oscillatory behavior of a certain class of second-order neutral delay dynamic equations ( r ( t ) [ x ( t ) + p ( t ) x ( τ ( t ) ) ] Δ ) Δ + q ( t ) x ( δ ( t ) ) = 0 , on a time scale T with sup T = ∞ , where 0 ≤ p ( t ) ≤ p 0 < ∞ . Some new results are presented that not only complement and improve those related results in the literature, but also improve some known results for a second-order delay dynamic equation without a neutral term. Further, the main results improve some related results for second-order neutral differential equations. MSC: 34K11, 34N05, 39A10.

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Publié par
Publié le 01 janvier 2012
Nombre de lectures 5
Langue English

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Zhang et al. Advances in Difference Equations 2012, 2012 :227 http://www.advancesindifferenceequations.com/content/2012/1/227
Open Access
R E S E A R C H New oscillation results for second-order neutral delay dynamic equations Chenghui Zhang 1 , Ravi P Agarwal 2 , Martin Bohner 3* and Tongxing Li 1 * Correspondence: bohner@mst.edu 3 Department of Mathematics and Abstract Statistics, Missouri S&T, Rolla, MO This paper is concern d with oscillatory behavior of a certain class of second-order 65409-0020, USA e Full list of author information is neutral delay dynamic equations available at the end of the article ( r ( t ) x ( t ) + p ( t ) x ( t ) ) ) + q ( t ) x ( t ) ) = 0, on a time scale T with sup T = , where 0 p ( t ) p 0 < . Some new results are presented that not only complement and improve those related results in the literature, but also improve some known results for a second-order delay dynamic equation without a neutral term. Further, the main results improve some related results for second-order neutral differential equations. MSC: 34K11; 34N05; 39A10 Keywords: oscillation; neutral delay dynamic equation; second-order equation; time scale
1 Introduction In this paper, we are concerned with oscillation of a class of second-order neutral delay dynamic equations, r ( t ) x ( t ) + p ( t ) x τ ( t )  + q ( t ) x δ ( t ) = , (.)
where t [ t , ) T := [ t , ) T , and ( H ) r , p , q C rd ( [ t , ) T , R ) , r ( t ) >  , p ( t ) p < , q ( t ) >  ; ( H ) δ C rd ( [ t , ) T , T ) , δ ( t ) t , lim t →∞ δ ( t ) = , τ δ = δ τ ; ( H ) τ C rd ( [ t , ) T , T ) , τ ( t ) t , τ ( t ) τ >  , τ ( [ t , ) T ) = [ τ ( t ), ) T , where τ is a constant. Throughout this paper, we assume that solutions of ( .) exist for any t [ t , ) T . A solu-tion x of (.) is called oscillatory if it is neither eventually positive nor eventually negative; otherwise, we call it nonoscillatory. Equation ( .) is said to be oscillatory if all its solutions oscillate. A time scale T is an arbitrary nonempty closed subset of the real numbers R . Since we are interested in oscillatory behavior, we suppose that the time scale under consideration is not bounded above and is a time scale interval of the form [ t , ) T . For some concepts related to the notion of time scales, see [ , ]. © 2012 Zhang et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribu-tion License (http://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.