Abstract and Applied Analysis
Volume 2011 (2011), Article ID 857860, 15 pages
http://dx.doi.org/10.1155/2011/857860
Research Article

Properties of Third-Order Nonlinear Functional Differential Equations with Mixed Arguments

B. Baculíková

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

Received 14 December 2010; Accepted 20 January 2011

Academic Editor: Josef Diblík

Copyright © 2011 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 sufficient conditions for property (B) and/or the oscillation of the third-order nonlinear functional differential equation with mixed arguments [ 𝑎 ( 𝑡 ) [ 𝑥 ″ ( 𝑡 ) ] 𝛾 ] ′ = 𝑞 ( 𝑡 ) 𝑓 ( 𝑥 [ 𝜏 ( 𝑡 ) ] ) + 𝑝 ( 𝑡 ) ℎ ( 𝑥 [ 𝜎 ( 𝑡 ) ] ) . Both cases ∫ ∞ 𝑎 − 1 / 𝛾 ( 𝑠 ) d 𝑠 = ∞ and ∫ ∞ 𝑎 − 1 / 𝛾 ( 𝑠 ) d 𝑠 < ∞ are considered. We deduce properties of the studied equations via new comparison theorems. The results obtained essentially improve and complement earlier ones.