ON THE FIXED-POINTS OF COMMUTING NONEXPANSIVE MAPS IN HYPERCONVEX SPACES

Let {T1, ..., TN} be a finite commuting family of nonexpansive maps of a hyperconvex space such that each Ti has bounded orbits. We show: (i) Each point has a bounded orbit under the semigroup generated by {Ti}; (ii) There is a common fixed point for the family if (and only if) T = T1T2··· TN has a fixed point; (iii) For each ε > 0, there is a nonempty set of common ε-approximate fixed points for the family. Some additional related results are also given. © 1992.