Iterations of the mean-type mappings and uniqueness of invariant means

Abstract. It is known, that under some natural conditions, the iterates of any continuous mean-type mapping converge to a continuous invariant mean-type mapping, and, moreover, the invariant continuous mean-type mapping is unique. In this paper we show that the assumption of the continuity in the "moreover" part of this result is superfluous. We also show that every increasing and homogeneous mean is continuous, and we give some new conditions on convergence of the sequence of iterates of, not necessarily continuous, mean-type mapping to a unique invariant mean-type mapping. Some examples and applications are presented.