An extension of Ky Fan's best approximation theorem (Q2747002)

An extension of Ky Fan's best approximation theorem is proved. The most general result is as follows: Theorem. Let \(C\) be a nonempty closed convex subset of a normed linear space \(X\) and let \(F:C \times C \to R\) be a function such that NEWLINENEWLINENEWLINE(i) \(F\) is continuous; NEWLINENEWLINENEWLINE(ii) for every \(y \in C\), the set \(\{x \in C |F(x,y)=0\}\) is nonempty and convex; NEWLINENEWLINENEWLINE(iii) \(\bigcup_{x \in C}\{x \in C |F(x,y)=0\}\) is contained in a compact subset of \(C\). NEWLINENEWLINENEWLINEThen there exists an \(x_0 \in C\) such that \(F(x_0,x_0)=0\). NEWLINENEWLINENEWLINEThis result is applied to prove some fixed point theorems.