The mean ergodic theorem for nonexpansive mappings in multi-Banach spaces
© Kenari et al.; licensee Springer. 2014
Received: 11 April 2014
Accepted: 13 June 2014
Published: 22 July 2014
In this paper, we prove a mean ergodic theorem for nonexpansive mappings in multi-Banach spaces.
MSC:39A10, 39B72, 47H10, 46B03.
for all . For each , let be the set of fixed points of . If X is a strictly convex Banach space, then is closed and convex.
converges weakly to a fixed point of . It was also shown by Pazy  that, if X is a real Hilbert space and converges weakly to , then . These results were extended by Baillon , Bruck  and Reich [5, 6] and .
2 Multi-Banach spaces
The notion of a multi-normed space was introduced by Dales and Polyakov in . This concept is somewhat similar to an operator sequence space and has some connections with the operator spaces and Banach lattices. Observations on multi-normed spaces and examples are given in [8–10].
Let be a complex normed space and let . We denote by the linear space consisting of k-tuples , where . The linear operations on are defined coordinate-wise. The zero element of either E or is denoted by 0. We denote by the set and by the group of permutations on k symbols.
Definition 2.1 A multi-norm on is a sequence such that is a norm on for each with satisfying the following conditions:
(A1) (, );
(A2) (, );
In this case, we say that is a multi-normed space.
Lemma 2.2 ()
It follows from (2) that, if is a Banach space, then is a Banach space for each . In this case is a multi-Banach space.
Now, we give two important examples of multi-norms for an arbitrary normed space E .
is a multi-norm, which is called the minimum multi-norm. The terminology ‘minimum’ is justified by the property (2).
Then is a multi-norm on , which is called the maximum multi-norm.
We need the following observation, which can easily be deduced from the triangle inequality for the norm and the property (2) of multi-norms.
if is a multi-null sequence.
3 Main results
To prove the main results in this paper, first, we introduce some lemmas.
Lemma 3.1 ()
for all , where .
To proceed, let denote a uniformly convex multi-Banach space with modulus of the convexity δ.
for all and .
which is a contradiction. This completes the proof. □
Lemma 3.3 (Browder )
Let C be a closed convex subset of X and be a nonexpansive mapping. If is a weakly convergent sequence in C with the weak limit and , then is a fixed point of .
uniformly for each .
for each .
which contradicts .
This completes the proof of the case .
for all and .
This completes the proof. □
Now, assume that the norm of X is Frechet differentiable and then we have the following.
Let C be a closed convex subset of X and, for each , be a nonexpansive mapping. If we put for all , then is at most one point.
In this paper, we give a new proof of the following theorem, which is due to Reich .
is bounded for all .
For all , converges weakly to a point uniformly for each .
Proof (1) ⟺ (2) is well known in .
(3) ⟺ (2) Suppose that, for some , there exists an unbounded subsequence of . For each , since is a nonexpansive mapping, it follows that, for each , the sequence is also unbounded, which contradicts the condition (3).
for all . Therefore, uniformly for each .
where , . Since multi-converges to uniformly for each , it follows that converges weakly to uniformly for each . This completes the proof. □
The third author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant Number: 2011-0021821).
- Baillon JB: Un theoreme de type ergodique pour les contractions non lineaires dans un espace de Hilbert. C. R. Acad. Sci. Paris Sér. A-B 1975, 280: 1511–1514.MathSciNetMATHGoogle Scholar
- Pazy A: On the asymptotic behavior of iterates of nonexpansive mappings in Hilbert space. Isr. J. Math. 1977, 26: 197–204. 10.1007/BF03007669MathSciNetView ArticleMATHGoogle Scholar
- Baillon JB:Comportement asymptotique des itérés de contractions non linéaires dans les espaces . C. R. Acad. Sci. Paris Sér. A-B 1978, 286: 157–159.MathSciNetMATHGoogle Scholar
- Bruk RE: A simple proof of the mean ergodic theorem for nonlinear contractions in Banach space. Isr. J. Math. 1979, 32: 107–116. 10.1007/BF02764907View ArticleGoogle Scholar
- Reich S: Almost convergence and nonlinear ergodic theorems. J. Approx. Theory 1978, 24: 269–272. 10.1016/0021-9045(78)90012-6MathSciNetView ArticleMATHGoogle Scholar
- Reich S: Weak convergence theorems for nonexpansive mappings in Banach space. J. Math. Anal. Appl. 1979, 67: 274–276. 10.1016/0022-247X(79)90024-6MathSciNetView ArticleMATHGoogle Scholar
- Reich, S: Nonlinear ergodic theory in Banach space. Argonne National Laboratory Report # 79–69 (1979)Google Scholar
- Dales, HG, Polyakov, ME: Multi-normed spaces and multi-Banach algebras. PreprintGoogle Scholar
- Dales HG, Moslehian MS: Stability of mappings on multi-normed spaces. Glasg. Math. J. 2007, 49: 321–332. 10.1017/S0017089507003552MathSciNetView ArticleMATHGoogle Scholar
- Moslehian MS, Nikodem K, Popa D: Asymptotic aspect of the quadratic functional equation in multi-normed spaces. J. Math. Anal. Appl. 2009, 355: 717–724. 10.1016/j.jmaa.2009.02.017MathSciNetView ArticleMATHGoogle Scholar
- Groetsch CW: A note on segmention Mann iterates. J. Math. Anal. Appl. 1972, 40: 369–372. 10.1016/0022-247X(72)90056-XMathSciNetView ArticleMATHGoogle Scholar
- Browder FE Pure Math. 18. In Nonlinear Operators and Nonlinear Equations of Evolution in Banach Space. Am. Math. Soc., Providence; 1976.View ArticleGoogle Scholar
- Hirano N: A proof of the mean ergodic theorem for nonexpansive mappings in Banach space. Proc. Am. Math. Soc. 1980, 78: 361–365. 10.1090/S0002-9939-1980-0553377-8MathSciNetView ArticleMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. 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.