Discrete Dynamics in Nature and Society
Volume 2007 (2007), Article ID 89413, 9 pages
doi:10.1155/2007/89413
Research Article
Permanence for a Generalized Discrete Neural Network System
Mathematical Institute of the Serbian Academy of Science, Knez Mihailova 35/I, Beograd 11000, Serbia
Received 7 November 2006; Revised 31 January 2007; Accepted 5 February 2007
Copyright © 2007 Stevo Stević. 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
We prove that the system of difference equations xn+1(i)=λixn(i)+fi(αixn(i+1)−βixn−1(i+1)), i∈{1,2,…,k}, n∈ℕ, (we regard that xn(k+1)=xn(1)) is permanent, provided that αi≥βi, λi+1∈[0,βi/αi), i∈{1,2,…,k}, fi:ℝ→ℝ, i∈{1,2,…,k}, are nondecreasing functions bounded from below and such that there are
δi∈(0,1) and M>0 such that fi(αix)≤δix, i∈{1,2,…,k}, for all x≥M. This result considerably extends the results existing in the literature. The above system is an extension of a two-dimensional discrete neural network system.