There is at most one continuous invariant mean

Abstract

AbstractWe show that, for a (not necessarily continuous) weakly contractive mean-type mapping $$\mathbf {M} :I^p\rightarrow I^p$$ M : I p → I p (where I is an interval and $$p \in \mathbb {N}$$ p ∈ N ), the functional equation $$K \circ \mathbf {M}=K$$ K ∘ M = K has at most one solution in the family of continuous means $$K :I^p \rightarrow I$$ K : I p → I . Some general approach to this equation is also given.