Equilibrium existence under generalized convexity

We introduce, in the first part, the notion of weakly convex pair of correspondences, we give its economic interpretation, we state a fixed point and a selection theorem. Then, by using a tehnique based on a continuous selection, we prove existence theorems of quilibrium for an abstract economy. In the second part, we define the weakly biconvex correspondences, we prove a selection theorem and we also demonstrate the existence of equilibrium for a generalized quasi-game (2003 Kim's model). In the last part of the paper, we give other applications in the game theory, finding equilibrium for abstract economies having correspondences with weakly convex graph. We show that the equilibrium exists without continuity assumptions.