Table of Contents
Advances in Artificial Neural Systems
Volume 2014, Article ID 252674, 7 pages
http://dx.doi.org/10.1155/2014/252674
Research Article

Long Time Behavior for a System of Differential Equations with Non-Lipschitzian Nonlinearities

Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia

Received 10 May 2014; Revised 7 September 2014; Accepted 8 September 2014; Published 14 September 2014

Academic Editor: Ozgur Kisi

Copyright © 2014 Nasser-Eddine Tatar. 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 consider a general system of nonlinear ordinary differential equations of first order. The nonlinearities involve distributed delays in addition to the states. In turn, the distributed delays involve nonlinear functions of the different variables and states. An explicit bound for solutions is obtained under some rather reasonable conditions. Several special cases of this system may be found in neural network theory. As a direct application of our result it is shown how to obtain global existence and, more importantly, convergence to zero at an exponential rate in a certain norm. All these nonlinearities (including the activation functions) may be non-Lipschitz and unbounded.