Permanence for a Generalized Discrete Neural Network System

Stevic, Stevo

doi:https://doi.org/10.1155/2007/89413

Discrete Dynamics in Nature and Society

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Research Article | Open Access

Volume 2007 | Article ID 089413 | https://doi.org/10.1155/2007/89413

Permanence for a Generalized Discrete Neural Network System

Stevo Stevic¹

Received07 Nov 2006

Revised31 Jan 2007

Accepted05 Feb 2007

Published14 Mar 2007

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.

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Copyright

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.

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