Abstract
In this paper, we prove a mean ergodic theorem for nonexpansive mappings in multi-Banach spaces.
MSC:39A10, 39B72, 47H10, 46B03.
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1 Introduction
Let X be a Banach space and C be a closed convex subset of X. For each , a mapping is said to be nonexpansive on C if
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.
In [1], Baillon proved the first nonlinear ergodic theorem such that, if X is a real Hilbert space and for each , then, for each , the sequence defined by
converges weakly to a fixed point of . It was also shown by Pazy [2] that, if X is a real Hilbert space and converges weakly to , then . These results were extended by Baillon [3], Bruck [4] and Reich [5, 6] and [7].
2 Multi-Banach spaces
The notion of a multi-normed space was introduced by Dales and Polyakov in [8]. 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) (, );
(A3) ();
(A4) ().
In this case, we say that is a multi-normed space.
Lemma 2.2 ([10])
Suppose that is a multi-normed space and take . Then we have the following:
-
(1)
();
-
(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 [8].
Example 2.3 The sequence on defined by
is a multi-norm, which is called the minimum multi-norm. The terminology ‘minimum’ is justified by the property (2).
Example 2.4 Let be the (nonempty) family of all multi-norms on . For each , set
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.
Lemma 2.5 Suppose that and . For each , let be a sequence in E such that . Then, for each , we have
Definition 2.6 Let be a multi-normed space. A sequence in E is called a multi-null sequence if, for any , there exists such that
Let . We say that the sequence is multi-convergent to a point and write
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 ([11])
Let be a uniformly convex multi-Banach space with modulus of the convexity δ. Let . If , , and , then
for all , where .
To proceed, let denote a uniformly convex multi-Banach space with modulus of the convexity δ.
Lemma 3.2 Let C be a closed convex subset of X and for each , be a nonexpansive mapping. Let , for each and . Then, for any , there exists such that, for all ,
for all and .
Proof Put
For given , choose such that . Then there exists such that, for all ,
For each , and , we put
where . Then we have
and
Suppose that
Then, by Lemma 3.1, we have
Hence we have
which is a contradiction. This completes the proof. □
Lemma 3.3 (Browder [12])
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 .
Lemma 3.4 Let C be a closed convex subset of X and, for each , be a nonexpansive mapping. Then, for all and ,
uniformly for each .
Proof By induction on n, we prove this lemma. First, we prove the conclusion in the case . Put
for each .
If , then, for any , choose such that . Then there exists such that, for all ,
for each . If we put
where , and , then we have
Similarly, we have . Suppose that
Then, by Lemma 3.1, we have
which contradicts .
If , then, for any , choose so large that . Hence we have
This completes the proof of the case .
Now, suppose that
uniformly for each . We claim that
exists. Put
For any , choose such that
and
Then we have
for all . Therefore, we have
Since is arbitrary, we have
i.e., exists.
Now, we put
If , then, for any ϵ, choose such that
Then there exists such that, if, for all , we put
so
and
Hence, by the method in the proof of the case , we have
for all and .
If , then, as in the proof of the case , there exists such that, for each ,
Therefore, we have
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 [6].
Theorem 3.6 Let be a uniformly convex multi-Banach space which has the Fréchet differentiable norm. Let C be a closed convex subset of X and, for each , be a nonexpansive mapping. Then the following statements are equivalent:
-
(1)
.
-
(2)
is bounded for all .
-
(3)
For all , converges weakly to a point uniformly for each .
Proof (1) ⟺ (2) is well known in [12].
(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).
(2) ⟺ (3) Since is bounded and
there exists a sequence such that
Then, by Lemma 3.3 and Proposition 3.5, it follows that any weakly multi-convergent subsequence of multi-converges weakly to a point , i.e., , where . Also, by Lemma 3.4, it follows that
for all . Therefore, uniformly for each .
On the other hand, for each with , we have
where , . Since multi-converges to uniformly for each , it follows that converges weakly to uniformly for each . This completes the proof. □
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Acknowledgements
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).
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All authors carried out the proof. All authors conceived of the study, and participated in its design and coordination. All authors read and approved the final manuscript.
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Kenari, H.M., Saadati, R. & Cho, Y.J. The mean ergodic theorem for nonexpansive mappings in multi-Banach spaces. J Inequal Appl 2014, 259 (2014). https://doi.org/10.1186/1029-242X-2014-259
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DOI: https://doi.org/10.1186/1029-242X-2014-259