Discrete Dynamics in Nature and Society
Volume 2007 (2007), Article ID 89413, 9 pages
Research Article

Permanence for a Generalized Discrete Neural Network System

Stevo Stević

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.


We prove that the system of difference equations xn+1(i)=λixn(i)+fi(αixn(i+1)βixn1(i+1)), i{1,2,,k}, n, (we regard that xn(k+1)=xn(1)) is permanent, provided that αiβi, λi+1[0,βi/αi), i{1,2,,k}, fi:, 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 xM. This result considerably extends the results existing in the literature. The above system is an extension of a two-dimensional discrete neural network system.