Ekeland’s principle for cyclically antimonotone equilibrium problems

In this paper, we deal with equilibrium problems without convexity assumptions either for the domain or for the function involved. First we give an Ekeland’s variational principle for equilibrium problems on complete metric spaces which generalizes some recent results. The main improvement consists in widening the class of bifunctions for which the variational principle holds: instead of a triangle inequality property a suitable approximation from below of the bifunction ff is required and we show that this condition is equivalent to the cyclic antimonotonicity of ff. We furnish a simple and direct proof of the existence of equilibria for such equilibrium problems in presence of a compact feasible set without passing through the existence of approximate solutions. This fact allows us to remove the metric structure on the topological space and additional technical conditions. Subsequently, a weak coercivity condition is introduced to deal with the non weakly compact case in reflexive Banach space. Finally, applications both to uncountable systems of equilibrium problems and to the quasiequilibrium problems are also discussed.