An abstract common fixed point principle and its applications

Abstract

We establish a common fixed point principle for a commutative family of self-maps on an abstract set. This principle easily yields the Markoff-Kakutani theorem for affine maps, Kirk's theorem for nonexpansive maps and Cano's theorem for maps on the unit interval. As another application we obtain a new common fixed point theorem for a commutative family of maps on an arbitrary interval, which generalises an earlier result of Mitchell.