Advances in Difference Equations
Volume 2010 (2010), Article ID 749852, 12 pages
doi:10.1155/2010/749852
Research Article

Asymptotical Convergence of the Solutions of a Linear Differential Equation with Delays

Josef Diblík, Miroslava Růžičková, and Zuzana Šutá

Department of Mathematics, University of Žilina, Faculty of Science, Univerzitná 8215/1, 010 26 Žilina, Slovakia

Received 1 January 2010; Accepted 23 April 2010

Academic Editor: Ağacik Zafer

Copyright © 2010 Josef Diblík 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

The asymptotic behavior of the solutions of the first-order differential equation ẏ(t)=∑i=1nβi(t)[y(t-δi)-y(t-τi)] containing delays is studied with βi:[t0-τ,∞)→[0,∞), τ=max⁡{τ1,…,τn}, ∑i=1nβi(t)>0, τi>δi>0. The attention is focused on an analysis of the asymptotical convergence of solutions. A criterion for the asymptotical convergence of all solutions, characterized by the existence of a strictly increasing bounded solution, is proved. Relationships with the previous results are discussed, too.