A note on fixed point properties in abstract convex spaces (Q2753313)

A pair \((X,F)\) of a topological space \(X\) and a mapping \(F: \langle X\rangle\to X\), where \(\langle X\rangle\) denotes the family of all finite subsets of \(X\), is called \(c\)-space if:NEWLINENEWLINENEWLINE1. \(F(A)\) is not empty and contractible for all \(A\in\langle X\rangle\);NEWLINENEWLINENEWLINE2. \(A\subset B\) implies \(F(A) \subset F(B)\) for all \(A,B\in\langle X\rangle\),NEWLINENEWLINENEWLINEA subset \(C\) of \(X\) is called an \(F\)-set if \(F(A)\subset C\) for all \(A\in\langle C\rangle\).NEWLINENEWLINENEWLINEA \(c\)-space \((X,F)\) is called an l.c. metric space if \(X\) is a metric space such that the set \(\{x\in X:d(x,E) <\varepsilon\}\) is an \(F\)-set and \(\varepsilon>0\). A \(c\)-space \((X,F)\) is said to have the fixed point property if any continuous function on \(X\) into itself has a fixed point.NEWLINENEWLINENEWLINEA multivalued mapping \(\varphi\) is said to be a Fan-Browder map if \(\varphi(x)\) is a nonempty \(F\)-set for each \(x\in X\) and \(\varphi^{-1}(y)\) is open for each \(y\in X\). A multivalued mapping \(\gamma\) on \(c\)-space \((X,F)\) into itself is a Kakutani map if \(\gamma(x)\) is a nonempty closed \(F\)-set for each \(x\in X\) and \(\gamma\) is upper semi-continuous. A \(c\)-space \((X,F)\) is said to have the \(K\)-fixed point property if any Kakutani map on \((X,F)\) into itself has a fixed point, and to have the \(FB\)-fixed point property if any Fan-Browder map on \((X,F)\) into itself has a fixed point.NEWLINENEWLINENEWLINEIn this paper the author proved the following theorem: Let \((X,F)\) be an l.c. metric space all of whose one-point sets are \(F\)-sets. Then the following are equivalent:NEWLINENEWLINENEWLINE1. \((X,F)\) has the fixed point property;NEWLINENEWLINENEWLINE2. \((X,F)\) has the \(FB\)-fixed point property;NEWLINENEWLINENEWLINE3. \((X,F)\) has the \(K\)-fixed point property.NEWLINENEWLINEFor the entire collection see [Zbl 0969.00060].