Common fixed points for commuting mappings in hyperconvex spaces

In a bounded hyperconvex metric space M we prove common fixed point results for nonexpansive mappings f:M→M and F:M→2M such that the mappings either commute or commute weakly. Our results provide hyperconvex space analogues of similar common fixed point theorems in Banach and CAT(0) spaces. Our method for weakly commuting mappings uses the hyperconvexity of N(M) the space of nonexpansive mappings of M with the sup metric. We show that if F:M→2M has externally hyperconvex values then the set of all nonexpansive selections of F is an externally hyperconvex subset of N(M).