Fixed points and iterations of mean-type mappings
Janusz Matkowski
Open Mathematics, Tome 10 (2012), p. 2215-2228 / Harvested from The Polish Digital Mathematics Library
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Let (X, d) be a metric space and T: X → X a continuous map. If the sequence (T n)n∈ℕ of iterates of T is pointwise convergent in X, then for any x ∈ X, the limit μT(x)=limn→∞Tn(x) is a fixed point of T. The problem of determining the form of µT leads to the invariance equation µT ○ T = µT, which is difficult to solve in general if the set of fixed points of T is not a singleton. We consider this problem assuming that X = I p, where I is a real interval, p ≥ 2 a fixed positive integer and T is the mean-type mapping M =(M 1,...,M p) of I p. In this paper we give the explicit forms of µM for some classes of mean-type mappings. In particular, the classical Pythagorean harmony proportion can be interpreted as an important invariance equality. Some applications are presented. We show that, in general, the mean-type mappings are not non-expansive.

Publié le : 2012-01-01
EUDML-ID : urn:eudml:doc:269119 
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     author = {Janusz Matkowski},
     title = {Fixed points and iterations of mean-type mappings},
     journal = {Open Mathematics},
     volume = {10},
     year = {2012},
     pages = {2215-2228},
     zbl = {1260.26036},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0106-7}
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Janusz Matkowski. Fixed points and iterations of mean-type mappings. Open Mathematics, Tome 10 (2012) pp. 2215-2228. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0106-7/

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