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2012 | 10 | 6 | 2215-2228
Tytuł artykułu

Fixed points and iterations of mean-type mappings

Treść / Zawartość
Warianty tytułu
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.
Kategorie tematyczne
Wydawca
Czasopismo
Rocznik
Tom
10
Numer
6
Strony
2215-2228
Opis fizyczny
Daty
wydano
2012-12-01
online
2012-10-12
Twórcy
  • Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, Szafrana 4a, 65-516, Zielona Góra, Poland, J.Matkowski@wmie.uz.zgora.pl
Bibliografia
  • [1] Bajraktarević M., Sur une équation fonctionnelle aux valeurs moyennes, Glasnik Mat.-Fiz. Astronom. Društvo Mat. Fiz. Hrvatske. Ser. II, 1958, 13, 243–248
  • [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] Bullen P.S., Handbook of Means and their Inequalities, Math. Appl., 560, Kluwer, Dordrecht-Boston-London, 2003
  • [4] Gauss C.F., Bestimmung der Anziehung eines Elliptischen Ringen, Ostwalds Klassiker Exakt. Wiss., 225, Akademische Verlagsgesellschaft, Leipzig, 1927
  • [5] Matkowski J., Iterations of mean-type mappings and invariant means, Ann. Math. Sil., 1999, 13, 211–226
  • [6] Matkowski J., Invariant and complementary quasi-arithmetic means, Aequationes Math., 1999, 57(1), 87–107 http://dx.doi.org/10.1007/s000100050072[Crossref]
  • [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] 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]
  • [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]
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-012-0106-7
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