Gap functions and Lyapunov functions

Equilibrium problems play a central role in the study of complex and competitive systems. Many variational formulations of these problems have been presented in these years. So, variational inequalities are very useful tools for the study of equilibrium solutions and their stability. More recently a dynamical model of equilibrium problems based on projection operators was proposed. It is designated as globally projected dynamical system (GPDS). The equilibrium points of this system are the solutions to the associated variational inequality (VI) problem. A very popular approach for finding solution of these VI and for studying its stability consists in introducing the so-called "gap-functions", while stability analysis of an equilibrium point of dynamical systems can be made by means of Lyapunov functions. In this paper we show strict relationships between gap functions and Lyapunov functions.