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A proof of the Mazur-Orlicz theorem via the Markov-Kakutani common fixed point theorem, and vice versa
- Barbara Przebieracz^{1}Email author
https://doi.org/10.1186/s13663-014-0257-2
© Przebieracz; licensee Springer 2015
- Received: 29 September 2014
- Accepted: 30 December 2014
- Published: 1 February 2015
Abstract
In this paper, we present a new proof of the Mazur-Orlicz theorem, which uses the Markov-Kakutani common fixed point theorem, and a new proof of the Markov-Kakutani common fixed point theorem, which uses the Mazur-Orlicz theorem.
Keywords
- Mazur-Orlicz theorem
- Markov-Kakutani theorem
MSC
- 46A22
- 47H10
1 Introduction
The aim of this paper is to present new proofs of the well-known Mazur-Orlicz theorem and Markov-Kakutani theorem. The proof of the former is based on the latter and vice versa.
We present here the two main theorems under consideration.
Theorem 1.1
(Markov-Kakutani common fixed point theorem; see [1] and [2])
Let X be a (locally convex) linear topological space, \(\mathcal{C}\) a nonempty convex compact subset of X, ℱ a commuting family of continuous affine selfmappings of \(\mathcal{C}\). Then there is an \(x\in\mathcal{C}\) such that \(f(x)=x\) for every \(f\in\mathcal{F}\).
Theorem 1.2
(Mazur-Orlicz theorem; see [3])
- (i)there is a linear functional \(a\colon X\to\mathbb{R}\) such that$$\begin{aligned}& a(x)\leq p(x),\quad x\in X, \\& \beta_{t}\leq a(x_{t}),\quad t\in T; \end{aligned}$$
- (ii)for every \(n\in\mathbb{N}\), \(t_{1},\ldots,t_{n}\in T\) and \(\lambda _{1},\ldots,\lambda_{n}\in(0,\infty)\),$$ \sum_{i=1}^{n} \lambda_{i}\beta_{t_{i}}\leq p\Biggl(\sum _{i=1}^{n} \lambda_{i} x_{t_{i}} \Biggr). $$
We also recall a counterpart of the Mazur-Orlicz theorem for abelian groups.
Theorem 1.3
- (i)there is an additive function \(a\colon G\to\mathbb{R}\) such that$$\begin{aligned}& a(x)\leq p(x),\quad x\in G, \\& \beta_{t}\leq a(x_{t}),\quad t\in T; \end{aligned}$$
- (ii)for every \(n\in\mathbb{N}\) and \(t_{1},\ldots,t_{n}\in T\),$$ \sum_{i=1}^{n} \beta_{t_{i}}\leq p\Biggl(\sum_{i=1}^{n} x_{t_{i}}\Biggr). $$
In both versions the implication (i) ⇒ (ii) is obvious. Moreover, in the condition (ii) of Theorem 1.2, we can as well demand that \(\sum\lambda_{i}=1\).
There are many different proofs of the Mazur-Orlicz theorem and its generalizations in the literature. See for example [4–12] (from which the most elementary and elegant is [10]). It seems to us that the approach via the Markov-Kakutani fixed point theorem is new and interesting. Moreover, this approach enables us to prove Theorems 1.2 and 1.3 analogously. Let us emphasize that already in [2], a corollary of the Markov-Kakutani fixed point theorem was used to prove the Hahn-Banach theorem. This approach was simplified in [13, Lemma 4.5.1]. However, to prove the Mazur-Orlicz theorem, the authors of [13] employ a lemma on supporting at a point of sublinear functionals by functionals, and an important result in the theory of infinite systems of inequalities [13, Theorem 3.2.2].
It is well known that from the Mazur-Orlicz theorem follows immediately the Hahn-Banach theorem. Furthermore, in [14], the Markov-Kakutani common fixed point theorem was proven via the separation theorem (in locally convex spaces compact convex nonvoid disjoint sets can be strictly separated) which is a consequence of the Hahn-Banach theorem (not directly from the Hahn-Banach theorem, as the title of that paper suggests). Also from the separation theorem (a locally convex space separates points) as well as the already mentioned [13, Theorem 3.2.2], the Markov-Kakutani theorem is proved in [13]. Another proof of the Markov-Kakutani theorem, based also on the separation theorem, can be found in [15].
Our proof of the Markov-Kakutani common fixed point theorem uses directly the Mazur-Orlicz theorem and is valid, as in [1, 13–15], for locally convex spaces.
Let us mention that the proof of the Markov-Kakutani theorem from [2] can be found in [16], and that probably the most elementary and elegant proof of this theorem can be found in [17].
2 Proof of the Mazur-Orlicz theorem
In the proofs of Theorems 1.2 and 1.3 we use some standard argumentation, which we formulate here as lemmas. We are unable to indicate where these reasonings appeared for the first time.
Lemma 2.1
Let G be a group. Suppose that \(a\colon G\to \mathbb{R}\) is an additive function, \(p\colon G\to\mathbb{R}\) subadditive, \(s\in\mathbb{R}\). Moreover, assume that \(a(x)\leq p(x)+s\) for \(x\in G\). Then \(a(x)\leq p(x)\), \(x\in G\).
Proof
Lemma 2.2
Let X be a linear topological space. Suppose that \(a\colon X\to\mathbb {R}\) is an additive function, \(p\colon X\to\mathbb{R}\) a sublinear functional such that \(a(x)\leq p(x)\), \(x\in X\). Then a is linear.
Proof
As every additive function, a is ℚ-linear. To see that a is linear fix \(x\in X\). We may proceed as follows. The mapping \(\mathbb {R}\ni t\stackrel{\alpha}{\mapsto} a(tx)\in\mathbb{R}\) is additive and dominated from above by \(\mathbb{R}_{+}\ni t\mapsto tp(x)\in\mathbb{R}\) on \(\mathbb{R}_{+}\). Hence α is bounded from above on some interval. Therefore, α is continuous (see for example [18, Theorem 12.1.2]). This, together with ℚ-linearity gives \(a(tx)=ta(x)\) for every \(t\in\mathbb{R}\).
Proof of the implication (ii) ⇒ (i) of Theorem 1.3
3 Proof of the Markov-Kakutani common fixed point theorem
Theorem 3.1
Let X be a locally convex linear topological space, \(\mathcal{C}\subset X\) nonempty convex and compact, \(F\colon\mathcal{C}\to\mathcal{C}\) continuous and affine. Then F has a fixed point.
In the proof we will use Theorem 1.2, however, we could also use Theorem 1.3, as well.
Proof
The Markov-Kakutani theorem follows easily from Theorem 3.1. For convenience of the reader we repeat the argumentation from [13, 15, 17] or [14].
Proof of Theorem 1.1
Declarations
Acknowledgements
This research was supported by the Institute of Mathematics, University of Silesia, Katowice, Poland (Real Analysis and Iterative Functional Equations program). I gratefully acknowledge the referees’ helpful comments and suggestions.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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