Global convergence of delayed dynamical systems

In this paper, we discuss some delayed dynamical systems, investigating their stability and convergence in critical case. To ensure the stability, the coefficients of the dynamical system must satisfy some inequalities. In most existing literatures, the restrictions on the coefficients are strict inequalities. The tough question is what will happen in the case (critical case) the strict inequalities are replaced by nonstrict inequalities (i.e., "<" is replaced by "less than or equal to"). The purpose of this paper is to discuss this critical case and give an affirmative answer in the case that the activation functions are hyperbolic tangent.