New oscillation results for second-order neutral delay dynamic equations

Zhang Chenghui, Agarwal, Bohner Martin, Li, Li Tongxing

Description

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.

Sujets

• Oscillation
• Time scale

Informations

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

Extrait

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: [e-mail] 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 (τ ( t ) )   )  + 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 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 r  d ( [ 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.