  We introduce the notions of weakly *-concave and weakly naturally
quasi-concave correspondence and prove fixed point theorems and continuous
selection theorems for these kind of correspondences. As applications in the
game theory, by using a tehnique based on a continuous selection, we establish
new existence results for the equilibrium of the abstract economies. The
constraint correspondences are weakly naturally quasi-concave. We show that the
equilibrium exists without continuity assumptions.
