Fixed points and iterations of mean-type mappings

Janusz Matkowski

Open Mathematics (2012)

  • Volume: 10, Issue: 6, page 2215-2228
  • ISSN: 2391-5455

Abstract

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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 μ T ( x ) = lim n → ∞ 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.

How to cite

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Janusz Matkowski. "Fixed points and iterations of mean-type mappings." Open Mathematics 10.6 (2012): 2215-2228. <http://eudml.org/doc/269119>.

@article{JanuszMatkowski2012,
abstract = {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 \[\mu \_T (x) = \mathop \{\lim \}\limits \_\{n \rightarrow \infty \} T^n (x)\] 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.},
author = {Janusz Matkowski},
journal = {Open Mathematics},
keywords = {Mean; Homogeneous mean; Mean-type mapping; Invariant mean; Iteration; Fixed point; mean; homogeneous mean; mean-type mapping; invariant mean; iteration; fixed point},
language = {eng},
number = {6},
pages = {2215-2228},
title = {Fixed points and iterations of mean-type mappings},
url = {http://eudml.org/doc/269119},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Janusz Matkowski
TI - Fixed points and iterations of mean-type mappings
JO - Open Mathematics
PY - 2012
VL - 10
IS - 6
SP - 2215
EP - 2228
AB - 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 \[\mu _T (x) = \mathop {\lim }\limits _{n \rightarrow \infty } T^n (x)\] 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.
LA - eng
KW - Mean; Homogeneous mean; Mean-type mapping; Invariant mean; Iteration; Fixed point; mean; homogeneous mean; mean-type mapping; invariant mean; iteration; fixed point
UR - http://eudml.org/doc/269119
ER -

References

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  2. [2] Borwein J.M., Borwein P.B., Pi and the AGM, Canad. Math. Soc. Ser. Monogr. Adv. Texts, 4, John Wiley & Sons, New York, 1998 
  3. [3] Bullen P.S., Handbook of Means and their Inequalities, Math. Appl., 560, Kluwer, Dordrecht-Boston-London, 2003 Zbl1035.26024
  4. [4] Gauss C.F., Bestimmung der Anziehung eines Elliptischen Ringen, Ostwalds Klassiker Exakt. Wiss., 225, Akademische Verlagsgesellschaft, Leipzig, 1927 
  5. [5] Matkowski J., Iterations of mean-type mappings and invariant means, Ann. Math. Sil., 1999, 13, 211–226 Zbl0954.26015
  6. [6] Matkowski J., Invariant and complementary quasi-arithmetic means, Aequationes Math., 1999, 57(1), 87–107 http://dx.doi.org/10.1007/s000100050072[Crossref] Zbl0930.26014
  7. [7] Matkowski J., Lagrangian mean-type mappings for which the arithmetic mean is invariant, J. Math. Anal. Appl., 2005, 309(1), 15–24 http://dx.doi.org/10.1016/j.jmaa.2004.10.033[Crossref] 
  8. [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 Zbl1220.26003
  9. [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] Zbl1268.26035
  10. [10] Ng C.T., Functions generating Schur-convex sums, In: General Inequalities, 5, Oberwolfach, May 4–10, 1986, Internat. Schriftenreihe Numer. Math., 80, Birkhäuser, Basel, 1987, 433–438 http://dx.doi.org/10.1007/978-3-0348-7192-1_35[Crossref] 

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