A note on equilibrium problems with properly quasimonotone bifunctions

In this paper, we consider some well-known equilibrium problems and their d uals in a topological Hausdorff vector space X for a bifunction F defined o n K x K,where K is a convex subset of X. Some necessary conditions are inve stigated, proving different results depending on the behaviour of F on the diagonal set. The concept of proper quasimonotonicity for bifunctions is de fined, and the relationship with generalized monotonicity is investigated. The main result proves that the condition of proper quasimonotonicity is sh arp in order to solve the dual equilibrium problem on every convex set.