11 pages
English
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Oscillation of higher-order neutral dynamic equations on time scales

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11 pages
English

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In this article, using comparison with second-order dynamic equations, we establish sufficient conditions for oscillatory solutions of an n th-order neutral dynamic equation with distributed deviating arguments. The arguments are based on Taylor monomials on time scales. 2000 Mathematics Subject Classification: 34K11; 39A10; 39A99. In this article, using comparison with second-order dynamic equations, we establish sufficient conditions for oscillatory solutions of an n th-order neutral dynamic equation with distributed deviating arguments. The arguments are based on Taylor monomials on time scales. 2000 Mathematics Subject Classification: 34K11; 39A10; 39A99.

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Publié le 01 janvier 2012
Nombre de lectures 8
Langue English

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MertAdvances in Difference Equations2012,2012:68 http://www.advancesindifferenceequations.com/content/2012/1/68
R E S E A R C HOpen Access Oscillation of higherorder neutral dynamic equations on time scales Raziye Mert
Correspondence: raziyemert@cankaya.edu.tr Department of Mathematics and Computer Science, Çankaya University, 06810 Ankara, Turkey
Abstract In this article, using comparison with secondorder dynamic equations, we establish sufficient conditions for oscillatory solutions of annthorder neutral dynamic equation with distributed deviating arguments. The arguments are based on Taylor monomials on time scales. 2000 Mathematics Subject Classification:34K11; 39A10; 39A99. Keywords:time scale, higher order, oscillation, Taylors formula
1. Introduction In this article, we investigate the oscillatory and asymptotic behavior of solutions of higherorder neutral dynamic equations with forcing term of the form d 2 n α  [x(t) +p(t)x(τ(t))] +λiqi(t,ξ)fi(x(ηi(t,ξ)))ξ=g(t),t[t0,)T,(1:1) i=1 c wherea1 is the quotient of two odd positive integers,l1,l2Î{1, 0, 1},c, d, t0T, and[t0,)T:= [t0,)Tdenotes a time scale interval with supT=. In recent years, there has been much research activity concerning the oscillation and nonoscillation of solutions of dynamic equations on time scales. We refer the reader to the monographs [13], the articles [411], and the references cited therein. However, most of the obtained results are concerned with secondorder dynamic equations whereas for higher order equations results are very seldom. Motivated by Candan and Dahiya [12], the main purpose of this article is to derive some oscillation and asymptotic criteria for Equation (1.1) via comparison with sec ondorder dynamic equations whose oscillatory character are known. Recall that atime scaleTis an arbitrary nonempty closed subset of the real numbers. The most wellknown examples areT=R,T=Z, and Zn , whereq> 1. The forward and backward jump operators T=q:={q:nZ} ∪ {0} are defined by σ(t) := inf{sT:s>t}andρ(t) := sup{sT:s<t}, respectively, whereinf:= supTandsup:= infT. A pointtTis said to be leftdense ift>infTandr(t) =t, rightdense ift<supTands(t) =t, leftscattered ifr(t)< t, and rightscattered ifs(t)> t. A functionfthat is defined on a time scale is © 2012 Mert; 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.