Fixed points and iterations of mean-type mappings

• Autorzy: Janusz Matkowski
• Języki publikacji: EN

Abstrakty

EN

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 $$\mu _T (x) = \mathop {\lim }\limits_{n \to \infty } T^n (x)$$ 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.

Słowa kluczowe

EN

Mean; Homogeneous mean; Mean-type mapping; Invariant mean; Iteration; Fixed point

Kategorie tematyczne

• 26A18: Iteration
• 26E60: Means

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2012
• Tom: 10
• Numer: 6
• Strony: 2215-2228

Daty

wydano

2012-12-01

online

2012-10-12

Twórcy

autor

Janusz Matkowski

• Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, Szafrana 4a, 65-516, Zielona Góra, Poland

Bibliografia

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• [8] Matkowski J., Iterations of the mean-type mappings, In: Iteration theory, Yalta, September 7–13, 2008, Grazer Math. Ber., 354, Karl-Franzens-Universität Graz, Graz, 2009, 158–179
• [9] Matkowski J., Invariance of a quasi-arithmetic mean with respect to a system of generalized Bajraktarević means, Appl. Math. Lett., 2012, 25(11), 1651–1655 http://dx.doi.org/10.1016/j.aml.2012.01.030[WoS][Crossref]
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Identyfikatory

DOI

10.2478/s11533-012-0106-7