Abstract and Applied Analysis
Volume 2004 (2004), Issue 6, Pages 461-470
doi:10.1155/S1085337504306056
Subdominant positive solutions of the discrete equation Δu(k+n)=−p(k)u(k)
Jaromír Baštinec1
and Josef Diblík2
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, {Z}i {z}kova 17, Brno 662 37, Czech Republic
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.