Oscillation of higher-order neutral dynamic equations on time scales

Description

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.

Sujets

• Documents
• Savoirs
• Time scale
• Oscillation
• Taylor series

Informations

• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 8
• Langue: English

Exrait

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: [e-mail] 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, Taylor’s 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))] +λ i q i (t,ξ)f i (x(η i (t,ξ)))ξ=g(t),t∈[t 0 ,∞) T , (1:1) i=1 c wherea≥1 is the quotient of two odd positive integers,l 1 ,l 2 Î{1, 0, 1},c, d, t 0 ∈ T , and [t 0 ,∞) T := [t 0 ,∞)∩ T denotes a time scale interval with sup T =∞ . 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 scale T is an arbitrary nonempty closed subset of the real numbersℝ. The most wellknown examples are T = R , T = Z , and Z n , whereq> 1. The forward and backward jump operators T =q:={q:n∈ Z } ∪ {0} are defined by σ(t) := inf{s∈ T :s>t}andρ(t) := sup{s∈ T :s<t}, respectively, where inf∅:= sup T and sup∅:= inf T . A point t∈ T is said to be leftdense if t>inf T andr(t) =t, rightdense if t<sup T ands(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.