Abstract and Applied Analysis
Volume 2011 (2011), Article ID 754701, 16 pages
http://dx.doi.org/10.1155/2011/754701
Research Article

Existence and Asymptotic Behavior of Positive Solutions of Functional Differential Equations of Delayed Type

J. Diblík and M. Kúdelčíková

Department of Mathematics, Faculty of Science, University of Žilina, 010 26 Žilina, Slovakia

Received 30 September 2010; Accepted 14 October 2010

Academic Editor: Elena Braverman

Copyright © 2011 J. Diblík and M. Kúdelčíková. 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

Solutions of the equation ̇ 𝑦 ( 𝑡 ) =   − 𝑓 ( 𝑡 , 𝑦 𝑡 ) are considered for 𝑡 → ∞ . The existence of two classes of positive solutions which are asymptotically different is proved using the retract method combined with Razumikhin's technique. With the aid of two auxiliary linear equations, which are constructed using upper and lower linear functional estimates of the right-hand side of the equation considered, inequalities for both types of positive solutions are given as well.