Some results on approximate continuous selections, fixed points and minimax inequalities

Abstract

We introduce the class of multi-valued D-mappings from a non-empty compact subset K of a locally convex Hausdorff topological vector space into a vector space (which, for K convex, includes the class of convex mappings) and we prove that they have a special approximate continuous selection property, from which an approximate fixed point theorem is derived. These results are applied, in particular, to Darboux continuous functions defined on a closed real interval. Furthermore, we extend sorne game theoretic statements concerning continuous decision rules to the case when the strategies of one player are constrained by those of the other and we obtain sorne results related to Ky Fan's inequality involving severa! functions.