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Abstract and Applied Analysis
Volume 2017, Article ID 3689069, 7 pages
https://doi.org/10.1155/2017/3689069
Research Article

Contractibility of Fixed Point Sets of Mean-Type Mappings

Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand

Correspondence should be addressed to P. Chaoha; ht.ca.aluhc@c.tehcihp

Received 23 July 2017; Accepted 2 November 2017; Published 31 December 2017

Academic Editor: Ngai-Ching Wong

Copyright © 2017 S. Iampiboonvatana and P. Chaoha. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We establish a convergence theorem and explore fixed point sets of certain continuous quasi-nonexpansive mean-type mappings in general normed linear spaces. We not only extend previous works by Matkowski to general normed linear spaces, but also obtain a new result on the structure of fixed point sets of quasi-nonexpansive mappings in a nonstrictly convex setting.

1. Introduction

The theory of mean iteration has been studied long before the 19th century [1]. Johann Carl Friedrich Gauss observed the connection between the arithmetic-geometric mean iteration and an elliptic integral. Indeed, if we recursively define the following sequences of positive real numbers we know that both and converge to the same limit, say, . He found that which is later generalised to However, the convergence above is not coincidental as we will see in the next section.

In 1999, Matkowski introduced the notion of mean-type mappings on a real interval and showed the convergence of its Picard iteration if at most one of its coordinate means is not strict. Later in 2009, he showed the same result for mappings with a weaker condition. The fixed point set of such a mapping is exactly the diagonal; however, the fixed point set of a general (continuous) mean-type mapping only covers the diagonal and may not be contractible. On the other hand, in 2012, Chaoha and Chanthorn introduced the concept of virtually stable (fixed point iteration) schemes to connect topological structures of the convergence set of a scheme to those of the fixed point set via a retraction. Many schemes for nonexpansive-type mappings have been proved to be virtually stable.

With those results in mind, in this work, we first extend the concept of mean-type mappings to vector spaces and then explore their fixed point sets using the notion of virtually stable schemes developed in [2]. 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 quasi-nonexpansive mappings outside the strict-convexity setting.

2. Preliminaries

We begin this section by recalling the notion of means and mean-type mappings from [3].

Definition 1. Let be an interval in and an integer. A function is said to be a mean if for all .
A mean is strict if, in addition, for all , where denotes the diagonal:

Definition 2. A mapping is said to be a mean-type mapping if each coordinate function is a mean; i.e., is a mean for all , where Moreover, is strict if all coordinate means are strict.

Definition 3. Let be means. We say that and are comparable if one of the following holds: (i) for all .(ii) for all .

We are ready to recall the classical and well-known convergence theorem for a -dimensional mean iteration.

Theorem 4 (see [1]). Let be comparable continuous means. Suppose that or is strict. Define by Then there exists a continuous mean such that for all .

Remark 5. The convergence of the arithmetic-geometric mean iteration follows directly from Theorem 4 by letting for all . In this case,

In , Matkowski showed that the comparability between means is not necessary. He also extended the convergence theorem to a -dimensional mean iteration as follows.

Theorem 6 (see [4]). Let be a continuous mean-type mapping such that at most one of the coordinate means is not strict. Then there exists a continuous mean such that for all .

Again, in , he improved his earlier result to include the larger class of nonstrict mean-type mappings.

Theorem 7 (see [3]). Let be a continuous mean-type mapping such that for all . Then there exists a continuous mean such that for all .

Remark 8. If a mean-type mapping satisfies the condition in Theorem 7, then

Next, we recall the concept of virtual stability of fixed point iteration schemes [2]. We use this to conclude the contractibility of the fixed point sets of nonexpansive-type mappings.

Let be a sequence of self-maps on a Hausdorff space . Define the fixed point set of and the convergence set of as follows: Define a function by

Definition 9 (see [2]). Let be a sequence of self-maps on and The sequence is called a (fixed point iteration) scheme if .

Definition 10 (see [2]). Let be a scheme. A fixed point is called virtually stable if, for each neighbourhood of , there exist a neighbourhood of and a strictly increasing sequence such that for all and .
The scheme is called virtually stable if all its common fixed points are virtually stable.

Theorem 11 (see [2]). If is a regular space and is a virtually stable scheme having a subsequence consisting of continuous mappings, then the function defined above is continuous and hence is a retract of .

Theorem 12 (see [5]). A retract subspace of a contractible space is contractible.

Lastly, we recall a very well-known fact about the structure of the fixed point sets of quasi-nonexpansive mappings.

Definition 13. A normed linear space is called strictly convex if for all such that and ; equivalently the boundary of the unit ball does not contain any line segment.

Definition 14. Let be a subset of a normed linear space. A mapping is said to be (i)nonexpansive if for all ;(ii)quasi-nonexpansive if for all and .

It is easy to see that nonexpansive mappings are continuous and quasi-nonexpansive while continuous quasi-nonexpansive mappings may not be nonexpansive.

Theorem 15 (see [6]). The fixed point set of a quasi-nonexpansive mapping defined on a convex subset of a strictly convex space is convex.

3. Main Results

In this section, we extend the notions of means and mean-type mappings to general vector spaces. Then we prove a convergence theorem as well as the contractibility of fixed point sets for certain continuous quasi-nonexpansive mean-type mappings.

Let be a convex subset of a vector space (over ) and an integer with . As usual, the diagonal in is simply .

Definition 16. A function is said to be a mean if, for each , there is a function such that with for each . For simplicity, we usually write . We also call strict if, in addition, for all and .

We note that Definition 16 is equivalent to Definition 1 when is an interval in .

Definition 17. A mapping is said to be a mean-type mapping if each coordinate function is a mean; that is where is a mean for all . Moreover, is strict if each is strict.

Remark 18. For any mean , we clearly have , for each . Consequently, for any mean-type mapping , we have

Example 19. Define by Clearly, is a nonlinear mean-type mapping and it is straightforward to verify that is strict.

Theorem 20. If is a strict mean-type mapping, then

Proof. Let . We can form the following system of linear equations: which is equivalent to Since is strict, we can apply Gauss-Jordan elimination to the coefficient matrix to obtain which implies

When is nonstrict, may still be the diagonal , or even the whole space .

Example 21. Consider defined by Clearly, we have and .

When is also a metric space, the next theorem surprisingly gives an explicit construction of a continuous mean-type mapping, whose fixed point set is any closed subset of containing the diagonal.

Recall that the distance between a point in the metric space and is defined to be When is closed, we also have iff .

Theorem 22. Suppose further that is a metric space. For any closed subset of such that , there exists a continuous mean-type mapping such that

Proof. Define and by It is easy to verify that iff , and is continuous with .

From now on, let be a convex subset of a normed linear space , and we will always use the maximum norm on . Notice that together with maximum norm may not be strictly convex. This prevents us from using Theorem 15 to conclude the convexity (and hence contractibility) of fixed point sets of quasi-nonexpansive mean-type mappings.

The following theorem shows that, under a simple condition, a mean-type mapping is always quasi-nonexpansive.

Theorem 23. Let be a mean-type mapping. If , then is quasi-nonexpansive. In particular, strict mean-type mappings are always quasi-nonexpansive.

Proof. Let , , and write . Consider, for each , and hence,

Unfortunately, the converse of the previous theorem is not true (see Example 36 below). Moreover, the last line in the proof of the previous theorem shows that any mean-type mapping is continuous on the diagonal. However, the continuity may not hold at other points as in the next example.

Example 24. Define by It is easy to see that is a strict mean-type mapping. However, is not continuous at each .

Definition 25. For each , with , and , we define the -combination of as follows: where denotes the identity mapping (Id). The -combination of a mean-type mapping is defined similarly.

From the above definition, it is easy to verify that the -combination of a mean-type mapping is also a mean-type mapping, and it is continuous if is continuous. Before we establish a convergence theorem, let us recall some basic facts about quasi-nonexpansive mappings and the distance between points in a convex hull.

Lemma 26. Let be a convex subset of a normed linear space, a quasi-nonexpansive mapping, , and . For a given with , define a sequence by (1)If and are limit points of , then .(2)If is a limit point of , then .

Proof. Since is quasi-nonexpansive, we have for each ,(1)If and are limit points of , then there exist subsequences of such that and . From (33), for each , there exist such that , and hence This implies, as , that .(2)If is a limit point of , then there exists a subsequence of such that . Again, by (33), we must have .

For each and , recall that the convex hull of , the algebraic interior of the convex hull of , and the diameter of are, respectively, defined as follows (see [7] for details):

Note that always exists when is nonempty and bounded. Otherwise, let .

Lemma 27. Let be a subset of a normed linear space. is compact for each .

Proof. Let . Define by It is easy to verify that is the image of the compact set under the continuous function .

Lemma 28. Let be a subset of a normed linear space and . (1).(2)If , and , then .

Proof. (1) It is clear that . Conversely, let ; say and . Then (2) From the proof of (1), also notice that If , we have for all , which implies , and hence

Lemma 28(1) naturally leads us to the notion of diameter of : which will replace until the end of this paper.

Remark 29. From the above notation, we clearly have for some , and iff .

Definition 30. A mean-type mapping is called diametrically contractive if for all .

Example 31. It is easy to see that the mean-type mapping defined by is not diametrically contractive.

Theorem 32. Suppose is a convex subset of a normed linear spaces, and is a continuous quasi-nonexpansive mean-type mapping such that is diametrically contractive for some with and .
Let and define a sequence by Then converges to a fixed point of .

Proof. Let . Then and for all because is a mean-type mapping. Since is compact by Lemma 27, has a subsequence converging to some . It follows that , and hence both and are limit points of . By Lemma 26(1), we have for all . By writing and , the above equation becomesfor all .
By Remark 29, there exists such that .
By letting in (46), we have and hence for some .
Again, by letting in (46), we have which contradicts the diametrical contractivity unless .
Therefore, by Lemma 26(2), we have .

We now combine the above convergence theorem with virtual stability to obtain the contractibility of fixed point sets of certain continuous quasi-nonexpansive mean-type mappings.

Corollary 33. If is a continuous quasi-nonexpansive mean-type mapping such that is diametrically contractive for some , then there is a continuous mean-type mapping such that and hence is contractible.

Proof. For each and , let , where . Notice that and , where . Then, by Theorem 32, , and hence is a scheme consisting of continuous mappings, where and , which is contractible. It is easy to see that is a mean-type mapping. So is . Moreover, for each and , we have and hence Therefore, is virtually stable. By Theorem 11, is continuous and is a retract of . Hence, by Theorem 12, is contractible.

Remark 34. The previous corollary immediately extends Theorem 7 to general normed linear spaces because when is an interval , the condition for all , implies . Hence, is quasi-nonexpansive and diametrically contractive, and Theorem 7 follows (with ). Notice that, in this case, the existence of in Theorem 7 is the consequence of the fact that for each .
Moreover, it should be pointed out here that, for any mean-type mapping and , the combination is always diametrically contractive. If such that , we have Since , we must have Then, by Lemma 28(2),

The next example shows that, without quasi-nonexpansiveness, the diametrical contractivity alone may not be sufficient to obtain the contractibility of fixed point sets.

Example 35. Define by Clearly, is a continuous mean-type mapping and it is diametrically contractive by the previous remark. However, is not quasi-nonexpansive because and is not contractible.

We now end this paper by giving an example showing that, under the condition of Theorem 32, we only have the contractibility of fixed point sets but not the convexity.

Example 36. Let be equipped with the maximum norm. Define by It is easy to see that is a nonexpansive, hence quasi-nonexpansive, mean-type mapping. The previous remark assures that is diametrically contractive for any . We note that which is not convex but still contractible.
Moreover, with a slight modification of , we obtain Then is a continuous quasi-nonexpansive mean-type mapping and . We note that is not nonexpansive because

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work has been presented as the first author’s Ph.D. thesis and in a session of the 10th International Conference on Nonlinear Analysis and Convex Analysis (NACA2017).

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