Lambda-hyperconvexity in metric spaces
It is shown that every non-expansive map \(f:M\to M\) on a bounded \(\lambda\)-hyperconvex metric space \(M\), \(\lambda<2\), has a fixed point. A metric space \(M\) is defined to be \(\lambda\)-hyperconvex if for any non-empty set \(A\subset M\) which is an intersection of closed balls and any collection \(\{B(a_i,r_i)\}_{i\in\mathcal I}\) of pairwise intersecting closed balls of radii \(r_i\) centered at points \(a_i\in A\) the intersection \(A\cap\bigcap_{i\in\mathcal I}B(a_i,\lambda r_i)\) is not empty. If this condition holds for any finite (resp. countable) index set \(\mathcal I\), then the space \(M\) is called finitely (resp. sequentially) \(\lambda\)-hyperconvex. It is shown that each complete metric space is 2-hyperconvex and each uniformly convex Banach space is \(\lambda\)-hyperconvex for some \(\lambda<2\). The Banach space \(c_0\) is finitely 1-hyperconvex but not 1-hyperconvex while the Banach space \(\ell_\infty^c([0,1])\) (of all bounded real-valued functions on [0,1] with countable support) is sequentially 1-hyperconvex but not 1-hyperconvex.
scientific article; zbMATH DE number 1539288
scientific article