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Abstract and Applied Analysis
Volume 2004 (2004), Issue 6, Pages 461-470
http://dx.doi.org/10.1155/S1085337504306056

Subdominant positive solutions of the discrete equation Δu(k+n)=p(k)u(k)

1Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, Technick 8, Brno 616 00, Czech Republic
2Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Brno University of Technology, Žižkova 17, Brno 662 37, Czech Republic

Received 8 October 2002

Copyright © 2004 Hindawi Publishing Corporation. 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

A delayed discrete equation Δu(k+n)=p(k)u(k) with positive coefficient p is considered. Sufficient conditions with respect to p are formulated in order to guarantee the existence of positive solutions if k. As a tool of the proof of corresponding result, the method described in the author's previous papers is used. Except for the fact of the existence of positive solutions, their upper estimation is given. The analysis shows that every positive solution of the indicated family of positive solutions tends to zero (if k) with the speednot smaller than the speed characterized by the function k·(n/(n+1))k. A comparison with the known results is given and some open questions are discussed.