A Note on Fenchel Duality for Equilibrium Problems

By means of the Fenchel duality, we introduce a dual problem associated to an equilibrium problem that turns out to be an equilibrium problem itself in the dual space. We present conditions which entail the solvability of the primal and dual problem. Moreover, we introduce the notion of robust and optimistic solution for parametric equilibrium problems, and we show that the solutions of the dual of the robust problem coincide with the optimistic solutions of the dual parametric equilibrium problem.