Invariant graphs of functions for the mean-type mappings

Abstract

Let I be a real interval, J a subinterval of I , p ≥ 2 an integer number, and M 1 , ... , M p : I p → I the continuous means. We consider the problem of invariance of the graphs of functions ϕ : J p −1 → I with respect to the mean-type mapping M = (M 1 , ... , M p ).Applying a result on the existence and uniqueness of an M -invariant mean [7], we prove that if the graph of a continuous function ϕ : J p −1 → I is M -invariant, then ϕ satisfies a simple functional equation. As a conclusion we obtain a theorem which, in particular, allows to determine all the continuous and decreasing in each variable functions ϕ of the M -invariant graphs. This improves some recent results on invariant curves [8] where the case p = 2 is considered.