Existence of stable equilibrium point for dynamical systems of Volterra type

This paper presents sufficient conditions for the existence of a nonnegative and stable equilibrium point of a dynamical system of Volterra type, (1) $\text{(}\text{d}\text{dt}\text{) x}{}_{\mathrm{i}}\text{(t) = ?x}{}_{\mathrm{i}}\text{(t)f}{}_{\mathrm{i}}\text{(x}{}_1\text{(t),…, x}{}_{\mathrm{n}}\text{(t)) ? q}{}_{\mathrm{i}}\text{], i = 1,…, n}$, for every q = (q₁,…, q_n)^T?Rⁿ. Results of a nonlinear complementarity problem are applied to obtain the conditions. System (1) has a nonnegative and stable equilibrium point if (i) f(x) = (f₁(x),…,f_n(x))^T is a continuous and differentiable M-function and it satisfies a certain surjectivity property, or (ii), f(x) is continuous and strongly monotone on R₊₀ⁿ.