Discrete Dynamics in Nature and Society
Volume 3 (1999), Issue 1, Pages 57-63
doi:10.1155/S1026022699000084

Stability of a time discrete perturbed dynamical system with delay

Michael I. Gil'1 and Sui Sun Cheng2

1Department of Mathematics, Ben Gurion University, P.O. Box 653, Beer Sheva 84105, Israel
2Department of Mathematics, Tsing Hua University, ROC, Hsinchu 30043, Taiwan

Received 11 February 1999

Copyright © 1999 Michael I. Gil' and Sui Sun Cheng. 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

Let Cn be the set of n complex vectors endowed with a norm ‖⋅‖Cn. Let A,B be two complex n×n matrices and τ a positive integer. In the present paper we consider the nonlinear difference equation with delay of the type uk+1=Auk+Buk−τ+Fk(uk,uk−τ),      k=0,1,2,…, where Fk:Cn×Cn→Cn satisfies the condition ‖Fk(x,y)‖Cn≤p‖x‖Cn+q‖y‖Cn,       k=0,1,2,…, where p and q are positive constants. In this paper, absolute stability conditions for this equation are established.