Existence of maximal elements and applications (Q2746991)

If \(C\) is a nonempty subset of a vector space \(X\) and \(F:C \to 2^X\) is a set-valued map from \(C\) to \(X\), then \(F\) is said to be a KKM-map if for each finite subset \(S \subset C\), \(conv(S) \subset \bigcup_{x \in S} Fx\). Ky Fan's Lemma says that if \(C\) is a nonempty convex compact subset of \(\mathbb{R}^n\) and \(F:C \to 2^C\) is a set-valued function with convex values and an open graph such that \(x \notin Fx\) for each \(x \in C\), then there exists an \(x \in C\) such that \(Fx= \emptyset\). The author proves a number of results related to Ky Fan's KKM-principle, such as the following one. NEWLINENEWLINENEWLINETheorem. \(C\) is a nonempty closed convex subset of a topological Hausdorff vector space \(X\) and \(F:C \to 2^C\) be such that NEWLINENEWLINENEWLINE(i) \(x \notin Fx\) for each \(x \in C\); NEWLINENEWLINENEWLINE(ii) \(Fx\) is closed for each \(x \in C\); NEWLINENEWLINENEWLINE(iii) \(F^{-1}(y)=\{x \in C \mid y \in Fx\}\) is convex for each \(y \in C\); NEWLINENEWLINENEWLINE(iv) \(C\) can be covered by a finite number of values of \(F\). NEWLINENEWLINENEWLINEThen there exists an \(x \in C\) such that \(Fx= \emptyset\). NEWLINENEWLINENEWLINEThe author uses his results to prove some known theorems on the existence of fixed points and of zeroes for single valued maps.