Some developments in Pre-equilibrium and Equilibrium analysis of Variational Inequalities

Abstract

In the present work, the notion of equilibrium and pre-equilibrium of variational inequalities (but also some for some quasi-variational inequalities)is developed in Weighted Hilbert spaces, in strictly convex and smooth Banach spaces and in reflexive Banach spaces. The concept of Weighted variational inequality is introduced, some associated questions as regularity,delayed equilibrium and Lagrangian duality are developed and applied to the traffic equilibrium problem. The more recent notion of pre-equilibrium very important in time dependent equilibrium must be understood as the optimal path from an arbitrarily point to reach the equilibrium (critical point of the system). The notion of Non pivot and Implicit Dynamical system is introduced, an existence result is given (in Hilbert spaces with linear duality mapping) as application an existence result is given also for a specific quasi-variational inequality (translated set) without using the classical assumption for the projection (Lipschitz) [This assumption is wrong a very simple case and a counter example is provided]. The notion of projected dynamical systems is extended to strictly convex and smooth Banach spaces and reflexive Banach spaces and the equivalence between critical points of such PDS and equilibrium of Variational inequalities is proved. Some applications will also be given to the traffic equilibrium problem, an elementary design of an industrial application will be also illustrated.