International Journal of Mathematics and Mathematical Sciences
Volume 2012 (2012), Article ID 975298, 10 pages
http://dx.doi.org/10.1155/2012/975298
Research Article

On Properties of Third-Order Differential Equations via Comparison Principles

B. Baculíková and J. Džurina

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

Received 21 March 2012; Revised 18 June 2012; Accepted 19 June 2012

Academic Editor: Christian Corda

Copyright © 2012 B. Baculíková and J. Džurina. 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 objective of this paper is to offer sufficient conditions for certain asymptotic properties of the third-order functional differential equation [ 𝑟 ( 𝑡 ) [ 𝑥 ′ ( 𝑡 ) ] 𝛾 ] + 𝑝 ( 𝑡 ) 𝑥 ( 𝜏 ( 𝑡 ) ) = 0 , where studied equation is in a canonical form, that is, ∫ ∞ 𝑟 − 1 / 𝛾 ( 𝑠 ) d 𝑠 = ∞ . Employing Trench theory of canonical operators, we deduce properties of the studied equations via new comparison theorems. The results obtained essentially improve and complement earlier ones.