Abstract and Applied Analysis
Volume 2011 (2011), Article ID 671967, 24 pages
http://dx.doi.org/10.1155/2011/671967
Research Article

Asymptotic Behavior of Solutions of Delayed Difference Equations

J. Diblík1,2 and I. Hlavičková2

1Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, University of Technology, 602 00 Brno, Czech Republic
2Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, 616 00 Brno, Czech Republic

Received 24 January 2011; Accepted 9 May 2011

Academic Editor: Miroslava Růžičková

Copyright © 2011 J. Diblík and I. Hlavičková. 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

This contribution is devoted to the investigation of the asymptotic behavior of delayed difference equations with an integer delay. We prove that under appropriate conditions there exists at least one solution with its graph staying in a prescribed domain. This is achieved by the application of a more general theorem which deals with systems of first-order difference equations. In the proof of this theorem we show that a good way is to connect two techniques—the so-called retract-type technique and Liapunov-type approach. In the end, we study a special class of delayed discrete equations and we show that there exists a positive and vanishing solution of such equations.