Extensions of two fixed point theorems of Ky Fan (Q801282)

Let E, F be Hausdorff topological vector spaces, where F has sufficiently many continuous linear functionals, \(X\subset E\) be a nonempty compact convex subset. The following two results, each extends a well-known fixed point theorem of Ky Fan, are proved. Let F be a upper semi-continuous set-valued map on X whose values are nonempty closed convex subsets of F, g:X\(\to F\) be a continuous map satisfying \((a)\quad f(x)\cap g(X)\neq \emptyset\) for all \(x\in X\); (b) \(g^{-1}(C)\) is convex (or empty) for any closed convex set C in F. Then there exists a point \(x_ 0\in X\) such that \(g(x_ 0)\in f(x_ 0).\) Theorem 3. Let f,g:X\(\to F\) be two continuous maps, where g satisfies (c) g(X) is convex and \(g^{-1}(y)\) is convex for any \(y\in g(X)\). Then either there exists a point \(x_ 0\in X\) such that \(g(x_ 0)=f(x_ 0),\) or there exist a point \(x_ 0\in X\) and a continuous seminorm p on F such that for all \(y\in g(X)\), \(0<p(g(x_ 0)-f(x_ 0))\leq p(y-f(x_ 0)).\)