Geodesic
Invariant graphs of functions for the mean-type mappings
Janusz Matkowski1
1 Faculty of Mathematics, Computer Science and Econometric, University of Zielona Góra, Podgórna 50, PL-65246 Zielona Góra, Poland ; Institute of Mathematics, Silesian University, Bankowa 14, PL-40007 Katowice, Poland
ESAIM. Proceedings, Tome 36 (2012), pp. 209-216.

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Let I be a real interval, J a subinterval of I, p ≥ 2 an integer number, and M1, ... , Mp : Ip → 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 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.
DOI : 10.1051/proc/201236017

Janusz Matkowski 1

1 Faculty of Mathematics, Computer Science and Econometric, University of Zielona Góra, Podgórna 50, PL-65246 Zielona Góra, Poland ; Institute of Mathematics, Silesian University, Bankowa 14, PL-40007 Katowice, Poland 
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     title = {Invariant graphs of functions for the mean-type mappings},
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     doi = {10.1051/proc/201236017},
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Janusz Matkowski. Invariant graphs of functions for the mean-type mappings. ESAIM. Proceedings, Tome 36 (2012), pp. 209-216. doi : 10.1051/proc/201236017. http://geodesic.mathdoc.fr/articles/10.1051/proc/201236017/

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