Oscillation of higher order nonlinear dynamic equations on time scales

Grace, Agarwal, Zafer, Zafer Aäÿacä±K

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.

Sujets

• Oscillation
• Neutral
• Time scale

Informations

• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 13
• Langue: English

Extrait

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: [e-mail]. 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, wheren≥2 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scale T ⊆ R with sup T =∞ and 0∈ T , wheren≥2 is a posi + tive integer,lis the ratio of positive odd integers, q: T → R = ( 0,∞ ) and Δ ξ: T → T are realvalued rdcontinuous functions,ξ(t)≤t,ξ(t)≥0, and lim t®∞ ξ(t) = [ ∞. Throughout the article byt≥sfort, s l mea ∈ ∈ T we shaln t[s,∞ ) ∩ T :=s,∞ ) T . s For the forward jump operators, we use the usual notationx=x○s. We recall that a solutionxof Equation (1.1) is said to be nonoscillatory if there exists a t 0 ∈ T such thatx(t)x(s(t)) > 0 for allt≥t 0 ; 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

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