A common fixed point theorem with applications (Q481770)

Let \(X\) be a nonempty compact convex subset of a locally convex Hausdorff topological vector space. The main result is that a family \(\{T_i: X\rightrightarrows X\}_{i\in I}\) of closed set-valued mappings with nonempty convex values has a common fixed point if, for each nonempty finite subset \(J\) of the index set \(I\), the mapping \(S_J: X\rightrightarrows X\) defined by \[ S_J(x)=\text{conv}\, \Big(\bigcup_{j\in J}T_j(x)\Big)\setminus \bigcup_{j\in J}T_j(x) \] has no fixed point, where the proof is based on the Kakutani-Fan-Glicksberg fixed point theorem. As consequences, a variational inequality of Stampacchia type and some Ky Fan-type minimax inequalities are investigated.