International Journal of Mathematics and Mathematical Sciences
Volume 2012 (2012), Article ID 951898, 9 pages
http://dx.doi.org/10.1155/2012/951898
Research Article

Oscillation and Asymptotic Behavior of Higher-Order Nonlinear Differential Equations

J. Džurina and B. Baculíková

Department of Mathematics, Faculty of Electrical Engineering and Informatics, Technical University of Košice, Letná 9, 042 00 Košice, Slovakia

Received 20 March 2012; Accepted 11 June 2012

Academic Editor: Feng Qi

Copyright © 2012 J. Džurina and B. Baculíková. 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

The aim of this paper is to offer a generalization of the Philos and Staikos lemma. As a possible application of the lemma in the oscillation theory, we study the asymptotic properties and oscillation of the 𝑛 th order delay differential equations ( 𝐸 ) ( 𝑟 ( 𝑡 ) [ 𝑥 ( 𝑛 − 1 ) ( 𝑡 ) ] 𝛾 ) + 𝑞 ( 𝑡 ) 𝑥 𝛾 ( 𝜏 ( 𝑡 ) ) = 0 . The results obtained utilize also the comparison theorems.