Discrete Dynamics in Nature and Society
Volume 2007 (2007), Article ID 89413, 9 pages
doi:10.1155/2007/89413

Permanence for a generalized discrete neural network system

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.