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

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.