Journal of Applied Mathematics
Volume 2011 (2011), Article ID 158060, 8 pages
http://dx.doi.org/10.1155/2011/158060
Research Article

Oscillation and Asymptotic Behaviour of a Higher-Order Nonlinear Neutral-Type Functional Differential Equation with Oscillating Coefficients

Mustafa Kemal Yildiz,1 Emrah Karaman,2 and Hülya Durur3

1Department of Mathematics, Faculty of Science and Arts, Afyon Kocatepe University, ANS Campus, 03200 Afyon, Turkey
2Department of Mathematics, Faculty of Science and Arts, Karabük University, 78050 Karabük, Turkey
3Department of Technical Programs, Vocational High School of Ardahan, Ardahan University, 75000 Ardahan, Turkey

Received 6 August 2010; Accepted 16 April 2011

Academic Editor: J. C. Butcher

Copyright © 2011 Mustafa Kemal Yildiz et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We will study oscillation of bounded solutions of higher-order nonlinear neutral delay differential equations of the following type: [ 𝑦 ( 𝑡 ) + 𝑝 ( 𝑡 ) 𝑓 ( 𝑦 ( 𝜏 ( 𝑡 ) ) ) ] ( 𝑛 ) + 𝑞 ( 𝑡 ) ℎ ( 𝑦 ( 𝜎 ( 𝑡 ) ) ) = 0 , 𝑡 ≥ 𝑡 0 , 𝑡 ∈ ℝ , where 𝑝 ∈ 𝐶 ( [ 𝑡 0 , ∞ ) , ℝ ) , l i m 𝑡 → ∞ 𝑝 ( 𝑡 ) = 0 , 𝑞 ∈ 𝐶 ( [ 𝑡 0 , ∞ ) , ℝ + ) , 𝜏 ( 𝑡 ) , 𝜎 ( 𝑡 ) ∈ 𝐶 ( [ 𝑡 0 , ∞ ) , ℝ ) , 𝜏 ( 𝑡 ) , 𝜎 ( 𝑡 ) < 𝑡 , l i m 𝑡 → ∞ 𝜏 ( 𝑡 ) , 𝜎 ( 𝑡 ) = ∞ , and 𝑓 , ℎ ∈ 𝐶 ( ℝ , ℝ ) . We obtain sufficient conditions for the oscillation of all solutions of this equation.