Generic existence of fixed points for set-valued mappings
The authors consider set-valued mappings acting on a closed convex subset of a Banach space which are nonexpansive with respect to the Hausdorff metric \(H\), and they study situations where, generically, such maps will have fixed points. More specifically, let \(K\) be a closed convex set with nonempty interior in a Banach space \(X\) and call \(S(K)\) the set of nonempty closed convex subsets of \(K\). Choose a point \(\theta\in K\) and endow the set \(M\) of nonexpansive mappings \(K\to S(K)\) with the uniformity \[ M(n)=\bigcap_{x\in K\cap B(\theta;n)}\{(T_1,T_2)\in M\times M\mid H(T_1x,T_2x)\leq 1/n\} \] for \(n\geq 1\). The authors show that there is an open dense set \(F\subset M\) such that for each \(T\in F\) there is an \(x\in K\) and a neighbourhood \(U\) of \(T\) in \(F\) such that \(x\) is a fixed point for each \(S\in U\). If \(K\) is bounded there is a similar result for mappings with convex graph.
scientific article; zbMATH DE number 1880351