18 pages
English
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Oscillation of higher order nonlinear dynamic equations on time scales

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18 pages
English

Description

Some new criteria for the oscillation of n th order nonlinear dynamic equations of the form x Δ n t + q t x σ ξ t λ = 0 are established in delay ξ ( t ) ≤ t and non-delay ξ ( t ) = t cases, where n ≥ 2 is a positive integer, λ is the ratio of positive odd integers. Many of the results are new for the corresponding higher order difference equations and differential equations are as special cases. Mathematics Subject Classification (2011): 34C10; 34C15.

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Publié le 01 janvier 2012
Nombre de lectures 13
Langue English

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Graceet al.Advances in Difference Equations2012,2012:67 http://www.advancesindifferenceequations.com/content/2012/1/67
R E S E A R C HOpen Access Oscillation of higher order nonlinear dynamic equations on time scales 1* 23 Said R Grace, Ravi P Agarwaland Ağacık Zafer
* Correspondence: srgrace@eng.cu. edu.eg 1 Department of Engineering Mathematics, Faculty of Engineering, Cairo University, Oman, Giza 12221, Egypt Full list of author information is available at the end of the article
Abstract Some new criteria for the oscillation ofnth order nonlinear dynamic equations of the form n  λ  σ x(t)+q(t)x(ξ (t))= 0
are established in delayξ(t)tand nondelayξ(t) =tcases, wheren2 is a positive integer,lis the ratio of positive odd integers. Many of the results are new for the corresponding higher order difference equations and differential equations are as special cases. Mathematics Subject Classification (2011):34C10; 34C15. Keywords:oscillation, neutral, time scale, higher order
1. Introduction Consider thenth order nonlinear delay dynamic equation n  λ  σ x(t)+q(t)x(ξ (t))= 0(1:1) on an arbitrary timescaleTRwith supT=and0T, wheren2 is a posi + tive integer,lis the ratio of positive odd integers,q:TR=(0,)and Δ ξ:TTare realvalued rdcontinuous functions,ξ(t)t,ξ(t)0, and limt®ξ(t) = [ . Throughout the article bytsfort,sl meaTwe shalnt[s,)T:=s,)T. s For the forward jump operators, we use the usual notationx=xs. We recall that a solutionxof Equation (1.1) is said to be nonoscillatory if there exists at0Tsuch thatx(t)x(s(t)) > 0 for alltt0; otherwise, it is said to be oscilla tory. Equation (1.1) is said to be oscillatory if all its solutions are oscillatory. Recently, there has been an increasing interest in studying the oscillatory behavior of firstand secondorder dynamic equations on timescales, see [17]. However, there are very few results regarding the oscillation of higher order equations. Therefore, the pur pose of this article is to obtain new criteria for the oscillation of Equation (1.1). This topic is fairly new for dynamic equations on time scales. For a general background on time scale calculus, we may refer to [8,9]. The article is organized as follows: In Section 2, some preliminary lemmas and nota tions are given, while Section 3 is devoted to the study of Equation (1.1) via compari son with a set of secondorder dynamic equations whose oscillatory character is
© 2012 Grace et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution 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.