  In this paper, we introduce several types of correspondences: weakly
naturally quasiconvex, *-weakly naturally quasiconvex, weakly biconvex and
correspondences with *--weakly convex graph and we prove some fixed point
theorems for these kinds of correspondences. As a consequence, using a version
of W. K. Kim's quasi-point theorem, we obtain the existence of equilibria for a
quasi-game.
