Oscillation of higher-order quasi-linear neutral differential equations

Xing Guojing, Li Tongxing, Zhang, Zhang Chenghui

Description

In this note, we establish some oscillation criteria for certain higher-order quasi-linear neutral differential equation. These criteria improve those results in the literature. Some examples are given to illustrate the importance of our results. 2010 Mathematics Subject Classification 34C10; 34K11. In this note, we establish some oscillation criteria for certain higher-order quasi-linear neutral differential equation. These criteria improve those results in the literature. Some examples are given to illustrate the importance of our results. 2010 Mathematics Subject Classification 34C10; 34K11.

Sujets

• Oscillation

Informations

• Publié par: biomed
• Publié le: 01 janvier 2011
• Nombre de lectures: 34
• Langue: English

Extrait

Xinget al.Advances in Difference Equations2011,2011:45 http://www.advancesindifferenceequations.com/content/2011/1/45

R E S E A R C H

Oscillation of higherorder differential equations 1 1,2 1* Guojing Xing , Tongxing Li and Chenghui Zhang

* Correspondence: [e-mail]. cn 1 Shandong University, School of Control Science and Engineering, Jinan, Shandong 250061, People’s Republic of China Full list of author information is available at the end of the article

quasilinear

Open Access

neutral

Abstract In this note, we establish some oscillation criteria for certain higherorder quasilinear neutral differential equation. These criteria improve those results in the literature. Some examples are given to illustrate the importance of our results. 2010 Mathematics Subject Classification34C10; 34K11. Keywords:Oscillation, neutral differential equation, higherorder, quasilinear

1. Introduction The neutral differential equations find numerous applications in natural science and technology. For example, they are frequently used for the study of distributed networks containing lossless transmission lines, see Hale [1]. In the past few years, many studies have been carried out on the oscillation and nonoscillation of solutions of various types of neutral functional differential equations. We refer the reader to the papers [222] and the references cited therein. In this work, we restrict our attention to the oscillation of higherorder quasilinear neutral differential equation of the form γ  (n−1) γ r(t) (x(t) +p(t)x(τ(t))) +q(t)x(σ(t)) = 0,n≥2 (1:1)

Throughout this paper, we assume that:

(C 1 )g≤1 is the quotient of odd positive integers; (C 2 )pÎC ([t 0 ,∞), [0,∞)); (C 3 )qÎC ([t 0 ,∞), [0,∞)), andqis not eventually zero on any half line [t * ,∞) for t * ≥t 0 ; 1 1 (C 4 )r,τ,sÎC ([t 0 ,∞),ℝ),r(t) > 0,r’(t)≥0, lim t®∞ τ(t) = lim t®∞ s(t) =∞,s 1 1 exists andsis continuously differentiable, wheresdenotes the inverse function ofs.

We consider only those solutionsxof equation (1.1) which satisfy sup {|x(t)| :t≥T} > 0 for allT≥t 0 . We assume that equation (1.1) possesses such a solution. As usual, a solution of equation (1.1) is called oscillatory if it has arbitrarily large zeros on [t 0 ,∞); otherwise, it is called nonoscillatory. Equation (1.1) is said to be oscillatory if all its solutions are oscillatory.

© 2011 Xing 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.