  In the finite dimensional case, mean-type mappings, their invariant means,
relations between the uniqueness of invariant means and convergence of orbits
of the mapping, are considered. In particular it is shown, that the uniqueness
of an invariance mean implies the convergence of all orbits. A strongly
irregular mean-type mapping is constructed and its unique invariant mean is
determined. An application in solving a functional equation is presented.

Classical Analysis and ODEs