Abstract and Applied Analysis
Volume 2012 (2012), Article ID 185242, 17 pages
http://dx.doi.org/10.1155/2012/185242
Research Article

Fourth-Order Differential Equation with Deviating Argument

M. Bartušek,1 M. Cecchi,2 Z. Došlá,1 and M. Marini2

1Department of Mathematics and Statistics, Masaryk University, 61137 Brno, Czech Republic
2Department of Electronics and Telecommunications, University of Florence, 50139 Florence, Italy

Received 6 December 2011; Accepted 13 January 2012

Academic Editor: Paul Eloe

Copyright © 2012 M. Bartušek 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 consider the fourth-order differential equation with middle-term and deviating argument 𝑥 ( 4 ) ( 𝑡 ) + 𝑞 ( 𝑡 ) 𝑥 ( 2 ) ( 𝑡 ) + 𝑟 ( 𝑡 ) 𝑓 ( 𝑥 ( 𝜑 ( 𝑡 ) ) ) = 0 , in case when the corresponding second-order equation ℎ + 𝑞 ( 𝑡 ) ℎ = 0 is oscillatory. Necessary and sufficient conditions for the existence of bounded and unbounded asymptotically linear solutions are given. The roles of the deviating argument and the nonlinearity are explained, too.