Variational Inequalities , Complementarity Problems and Pseudo - Monotonicity

It is well known that variational inequalities are systematically used in the theory of many practical problems related to ”equilibrium”. The equilibrium is an important state considered in Physics, Engineering, Sciences and Economics and even in Biology, [5], [37], [38], [24], [56]. When a variational inequality is defined on a closed convex set, which is in particular a closed convex cone, in this case our variational inequality is a complementarity problem. We have this situation in many problems considered in Economics. Because of this, the Complementarity Theory is strongly related to the study of equilibrium [16], [24], [36], [37], [38], [49]. The problem to find the equilibrium of an economical system is exactly the problem to find the solutions of a variational inequality or of a complementarity problem. It is known that the solvability of a variational inequality or of a complementarity problem is not an evident problem. Because of this fact, there exist many existence theorems [5], [11], [16], [36]-[49], [56], [73]-[77]. The solutions of a variational inequality or a complementarity problem give us a static information about an equilibrium state. In many problems in Economics, in Mechanics, or in Engineering we are interested to know the evolution of equilibrium with respect to the parameter ”time”. Certainly, we must study the evolution of equilibrium under the constraints used in the definition of the set with respect to which the variational inequality or the complementarity problem is considered. This fact is now possible by using the notion of ”local projected dynamical system”, defined in [22] and studied in [23], [63], [57]-[63], [72] and [21]. When we associate to a variational inequality, or to a complementarity problem, a local projected dynamical system, the critical points (i.e. the equilibrium points) of this dynamical system are exactly the solutions of the variational inequality (respectively of the complementarity problem) and therefore are the equilibrium points of considered practical problems. Obviously, by this method, we can apply the theory of dynamical systems to the study of variational inequalities or to the study of complementarity problems from the dynamical point of view. In this paper we will present several existence theorems and we will study the stability of equilibrium given by a variational inequality or by a complementarity problem in a general Hilbert space, using the notion of local projected dynamical system. We will show that the pseudo-monotonicity plays an important role. The paper will be finished with some comments and open problems.