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. 1. Introduction Of concern is the following system: with continuous data , , coefficients , and inputs , . The functions and are nonlinear continuous functions. This is a general nonlinear version of several systems that arise in many applications (see [1–9] and Section 4 below). The literature is very rich of works on the asymptotic behavior of solutions for special cases of system (1) (see for instance [10–19]). Here the integral terms represent some kind of distributed delays but discrete delays may be recovered as well by considering delta Dirac distributions. Different sufficient conditions on the coefficients, the functions, and the kernels have been established ensuring convergence to equilibrium or (uniform, global, and asymptotic) stability. In applications it is important to have global asymptotic stability at a very rapid rate like the exponential rate. Roughly speaking, it has been assumed that the coefficients must dominate the coefficients of some “bad” similar terms that appear in the estimations. For the nonlinearities (activation functions), the first assumptions of boundedness, monotonicity, and differentiability have been all weakened to a Lipschitz condition. According to [8, 20] and other references, even this condition needs to be weakened further. Unfortunately, we can find only few papers on continuous but not Lipschitz continuous activation functions. Assumptions like partially Lipschitz and linear growth, -inverse H？lder continuous or inverse Lipschitz, non-Lipschitz but bounded were used (see [16, 21, 22]). For H？lder continuous activation functions we refer the reader to [23], where exponential stability was proved under some boundedness and monotonicity conditions on the activation functions and the coefficients form a Lyapunov diagonally stable matrix (see also [24, 25] for other results without these conditions). There are, however, a good number of