Essentiality for Mönch type maps (Q2701576)

Let \(E\) be a separated topological vector space and \(U\subset E\) an open neighbourhood of zero. A continuous map \(F:\overline{U}\to E\) is said to be Mönch map if for a countable set \(C\subset\overline{U}\) we can have \(C\subset\overline{\text{conv}}(\{0\}\cup F(C))\) only if \(\overline{C}\) is compact. A Mönch map \(F:\overline{U}\to E\) with \(x\not=F(x)\) for \(x\in\partial U\) is said to be essential if every Mönch map \(G:\overline{U}\to E\) with \(F|\partial U=G|\partial U\) has a fixed point. NEWLINENEWLINENEWLINEA typical result of the article reads as follows: Let \(E\) be a quasicomplete metrizable locally convex vector space, \(U\subset E\) an open neighbourhood of zero and \(F:\overline{U}\to E\) a Mönch map. If \(x\not=\lambda F(x)\) whenever \(0<\lambda\leq 1\) and \(x\in\partial U\) then \(F\) is essential.