Abstract and Applied Analysis
Volume 2010 (2010), Article ID 417869, 20 pages
doi:10.1155/2010/417869
Research Article

On the Critical Case in Oscillation for Differential Equations with a Single Delay and with Several Delays

Jaromír Baštinec,1 Leonid Berezansky,2 Josef Diblík,1,3 and Zdeněk Šmarda1

1Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, 61600 Brno, Czech Republic
2Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel
3Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Brno University of Technology, 66237 Brno, Czech Republic

Received 1 July 2010; Accepted 26 August 2010

Academic Editor: Allan C. Peterson

Copyright © 2010 Jaromír Baštinec 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

New nonoscillation and oscillation criteria are derived for scalar delay differential equations x˙(t)+a(t)x(h(t))=0,a(t)≥0,h(t)≤t,t≥t0, and x˙(t)+∑k=1mak(t)x(hk(t))=0,ak(t)≥0,hk(t)≤t, and t≥t0, in the critical case including equations with several unbounded delays, without the usual assumption that the parameters a,h,ak, and hk of the equations are continuous functions. These conditions improve and extend some known oscillation results in the critical case for delay differential equations.