# Invariant graphs of functions for the mean-type mappings

ESAIM: Proceedings (2012)

• Volume: 36, page 209-216
• ISSN: 1270-900X

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## Abstract

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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.

## How to cite

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Matkowski, Janusz. Fournier-Prunaret, D., Gardini, L., and Reich, L., eds. " Invariant graphs of functions for the mean-type mappings ." ESAIM: Proceedings 36 (2012): 209-216. <http://eudml.org/doc/251195>.

@article{Matkowski2012,
abstract = {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.},
author = {Matkowski, Janusz},
editor = {Fournier-Prunaret, D., Gardini, L., Reich, L.},
journal = {ESAIM: Proceedings},
keywords = {mean; mean-type mapping; invariant mean; function of invariant graph; iteration; functional equation},
language = {eng},
month = {8},
pages = {209-216},
publisher = {EDP Sciences},
title = { Invariant graphs of functions for the mean-type mappings },
url = {http://eudml.org/doc/251195},
volume = {36},
year = {2012},
}

TY - JOUR
AU - Matkowski, Janusz
AU - Fournier-Prunaret, D.
AU - Gardini, L.
AU - Reich, L.
TI - Invariant graphs of functions for the mean-type mappings
JO - ESAIM: Proceedings
DA - 2012/8//
PB - EDP Sciences
VL - 36
SP - 209
EP - 216
AB - 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.
LA - eng
KW - mean; mean-type mapping; invariant mean; function of invariant graph; iteration; functional equation
UR - http://eudml.org/doc/251195
ER -

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