Uma teoria de dualidade para o problema de equilíbrio

Abstract

In this paper, we present a theory of duality for the equilibrium problem (PE) based on the concept of conjugated functions. Thus, some elements of convex analysis theory were addressed, mainly the concept of conjugation that underlies the results presented. In addition, we show that the formulation (PE) includes, for example, as a particular case the problem of classical equilibrium studied by Blum and Oettli and the equilibrium problem studied by Flores-Baz´an. Also, we show some problems that can be written under the formulation (PE), such as the convex optimization problem, the problem of fixed point, the variational inequality studied by Mosco and the generalized quasi-variational inequality studied by Morgan and Romaniello. We apply the duality theory of (PE) to these problems and in the case of applying the dual scheme of (PE) associated with the convex optimization problem we obtain the classical Lagrangian dual.