Fixed points and iterations of mean-type mappings

Abstract

Let (X ) be a metric space and T : X → X a continuous map. If the sequence (T ) ∈N of iterates of T is pointwise convergent in X , then for any ∈ X , the limitis a fixed point of T . The problem of determining the form of µ T leads to the invariance equation µ T • T = µ T , 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 , where I is a real interval, ≥ 2 a fixed positive integer and T is the mean-type mapping M =(M 1 M ) of I . In this paper we give the explicit forms of µ M 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. MSC:26E60, 26A18

Invariant means and iterates of mean-type mappings

“…For two means M, N : I 2 → I, a mean K : I 2 → I is called invariant with respect to the mean-type mapping (M, N ) : The following result is a consequence of bivariable version of Corollary 1 in [5] (see also [1,2,4]). Theorem 2.3.…”

“…For the results of this type, with more restrictive assumptions, see [1]. In particular, instead of (1) it was assumed that both means are strict; in [2] it was assumed that at most one mean is not strict, and condition (1) appeared first in [4] (see also [6]). Moreover, in all these papers the uniqueness of the invariant mean was obtained under the condition that it is continuous.…”

Contractibility of Fixed Point Sets of Mean-Type Mappings

“…We are able to establish a convergence theorem for certain continuous quasi-nonexpansive mean-type mappings in general normed linear spaces (which immediately covers the result in [3]) and conclude the contractibility of their fixed point sets. This also gives a new result on the structure of fixed point sets of quasinonexpansive mappings outside the strict-convexity setting.…”

Invariance of means

“…The reader looking for the proof in Borweins' book [20] should compile it combining some particular results, viz. [ The Generalized Gaussian Algorithm, in its consolidated form, was rediscovered by Matkowski [111]: firstly in 1999 for p = 2, under slightly stronger assumptions (see Remark 1.2 below), and in the form presented in Theorem 2.1 in 2009 (see [118]). Notice that an important argument for the equality of some basic limits in the proof presented in [111] was missing.…”