Invariant Graphs of Functions for the Mean-type Mappings

Let I be a real interval, J a subinterval of I, p ≥ 2 an integer number, and M1, . . . ,Mp : I → I the continuous means. We consider the problem of invariance of the graphs of functions φ : Jp−1 → I with respect to the mean-type mapping M = (M1, . . . ,Mp) . 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 : Jp−1 → I is M-invariant, then φ satis es 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. AMS 2010 Mathematics Subject Classi cation: Primary: 26A18, 26E60, 39B22, Secondary: 37D10.