Let M be a bounded metric space and supposesatisfies for some fixedfor all x,y€M. It is shown that under this assumption, for certain spaces M there will always exist z€M such that. If M is a convex subset of a Banach space, then weak compactness and normal structure suffice. If M is a hyperconvex metric space (in particular, if M is an intersection of closed balls in l∞) then there always exists z €M such that. Mappings of the type considered arise naturally. For example if K is a convex subset of a Banach space and if h,p€(0,1], then mappingswhich are Hölder continuous in the senseobviously satisfy the weaker condition.