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Fixed Point Theory and Applications

, 2010:471781 | Cite as

Halpern's Iteration in CAT(0) Spaces

  • Satit Saejung
Open Access
Research Article
Part of the following topical collections:
  1. Impact of Kirk's Results on the Development of Fixed Point Theory

Abstract

Motivated by Halpern's result, we prove strong convergence theorem of an iterative sequence in CAT(0) spaces. We apply our result to find a common fixed point of a family of nonexpansive mappings. A convergence theorem for nonself mappings is also discussed.

Keywords

Nonexpansive Mapping Common Fixed Point Iterative Sequence Geodesic Segment Nonnegative Real Number 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1. Introduction

Let Open image in new window be a metric space and Open image in new window with Open image in new window . A geodesic path from Open image in new window to Open image in new window is an isometry Open image in new window such that Open image in new window and Open image in new window . The image of a geodesic path is called a geodesic segment. A metric space Open image in new window is a (uniquely) geodesic space if every two points of Open image in new window are joined by only one geodesic segment. A geodesic triangle Open image in new window in a geodesic space Open image in new window consists of three points Open image in new window of Open image in new window and three geodesic segments joining each pair of vertices. A comparison triangle of a geodesic triangle Open image in new window is the triangle Open image in new window in the Euclidean space Open image in new window such that Open image in new window for all Open image in new window .

A geodesic space Open image in new window is a CAT(0) space if for each geodesic triangle Open image in new window in Open image in new window and its comparison triangle Open image in new window in Open image in new window , the CAT(0) inequality

is satisfied by all Open image in new window and Open image in new window . The meaning of the CAT(0) inequality is that a geodesic triangle in Open image in new window is at least thin as its comparison triangle in the Euclidean plane. A thorough discussion of these spaces and their important role in various branches of mathematics are given in [1, 2]. The complex Hilbert ball with the hyperbolic metric is an example of a CAT(0) space (see [3]).

The concept of Open image in new window -convergence introduced by Lim in 1976 was shown by Kirk and Panyanak [4] in CAT(0) spaces to be very similar to the weak convergence in Banach space setting. Several convergence theorems for finding a fixed point of a nonexpansive mapping have been established with respect to this type of convergence (e.g., see [5, 6, 7]). The purpose of this paper is to prove strong convergence of iterative schemes introduced by Halpern [8] in CAT(0) spaces. Our results are proved under weaker assumptions as were the case in previous papers and we do not use Open image in new window -convergence. We apply our result to find a common fixed point of a countable family of nonexpansive mappings. A convergence theorem for nonself mappings is also discussed.

In this paper, we write Open image in new window for the the unique point Open image in new window in the geodesic segment joining from Open image in new window to Open image in new window such that

We also denote by Open image in new window the geodesic segment joining from Open image in new window to Open image in new window , that is, Open image in new window . A subset Open image in new window of a CAT(0) space is convex if Open image in new window for all Open image in new window . For elementary facts about CAT(0) spaces, we refer the readers to [1] (or, briefly in [5]).

The following lemma plays an important role in our paper.

Lemma 1.1.

A geodesic space Open image in new window is a CAT(0) space if and only if the following inequality
is satisfied by all Open image in new window and all Open image in new window . In particular, if Open image in new window are points in a CAT(0) space and Open image in new window , then

Recall that a continuous linear functional Open image in new window on Open image in new window , the Banach space of bounded real sequences, is called a Banach limit if Open image in new window and Open image in new window for all Open image in new window .

Lemma 1.2 (see [9, Proposition Open image in new window ]).

Let Open image in new window be such that Open image in new window for all Banach limits Open image in new window and Open image in new window . Then Open image in new window .

Lemma 1.3 (see [10, Lemma Open image in new window ]).

Let Open image in new window be a sequence of nonnegative real numbers, Open image in new window a sequence of real numbers in Open image in new window with Open image in new window , Open image in new window a sequence of nonnegative real numbers with Open image in new window , and Open image in new window a sequence of real numbers with Open image in new window . Suppose that

Then Open image in new window .

2. Halpern's Iteration for a Single Mapping

Lemma 2.1.

Let Open image in new window be a closed convex subset of a complete CAT(0) space Open image in new window and let Open image in new window be a nonexpansive mapping. Let Open image in new window be fixed. For each Open image in new window , the mapping Open image in new window defined by
has a unique fixed point Open image in new window , that is,

Proof.

For Open image in new window , we consider the triangle Open image in new window and its comparison triangle and we have the following:

This implies that Open image in new window is a contraction mapping and hence the conclusion follows.

The following result is proved by Kirk in [11, Theorem Open image in new window ] under the boundedness assumption on Open image in new window . We present here a new proof which is modified from Kirk's proof.

Lemma 2.2.

Let Open image in new window , Open image in new window be as the preceding lemma. Then Open image in new window if and only if Open image in new window given by the formula (2.2) remains bounded as Open image in new window . In this case, the following statements hold:

(1) Open image in new window converges to the unique fixed point Open image in new window of Open image in new window which is nearest Open image in new window ;

(2) Open image in new window for all Banach limits Open image in new window and all bounded sequences Open image in new window with Open image in new window .

Proof.

If Open image in new window , then it is clear that Open image in new window is bounded. Conversely, suppose that Open image in new window is bounded. Let Open image in new window be any sequence in Open image in new window such that Open image in new window and define Open image in new window by
for all Open image in new window . By the boundedness of Open image in new window , we have Open image in new window . We choose a sequence Open image in new window in Open image in new window such that Open image in new window . It follows from Lemma 1.1 that
Then, by the convexity of Open image in new window ,
This implies that Open image in new window is a Cauchy sequence in Open image in new window and hence it converges to a point Open image in new window . Suppose that Open image in new window is a point in Open image in new window satisfying Open image in new window . It follows then that
and hence Open image in new window . Moreover, Open image in new window is a fixed point of Open image in new window . To see this, we consider
This implies that Open image in new window and hence Open image in new window .
  1. (1)

    is proved in [12, Theorem Open image in new window ]. In fact, it is shown that Open image in new window is the nearest point of Open image in new window to Open image in new window . Finally, we prove (2). Suppose that Open image in new window is a sequence given by the formula (2.2), where Open image in new window is a sequence in Open image in new window such that Open image in new window . We also assume that Open image in new window is the nearest point of Open image in new window to Open image in new window . By the first inequality in Lemma 1.1, we have

     
Let Open image in new window be a Banach limit. Then
This implies that
In particular,

Inspired by the results of Wittmann [13] and of Shioji and Takahashi [9], we use the iterative scheme introduced by Halpern to obtain a strong convergence theorem for a nonexpansive mapping in CAT(0) space setting. A part of the following theorem is proved in [14].

Theorem 2.3.

Let Open image in new window be a closed convex subset of a complete CAT(0) space Open image in new window and let Open image in new window be a nonexpansive mapping with a nonempty fixed point set Open image in new window . Suppose that Open image in new window are arbitrarily chosen and Open image in new window is iteratively generated by

where Open image in new window is a sequence in Open image in new window satisfying

(C1) Open image in new window ;

(C2) Open image in new window ;

(C3) Open image in new window or Open image in new window .

Then Open image in new window converges to Open image in new window which is the nearest point of Open image in new window to Open image in new window .

Proof.

We first show that the sequence Open image in new window is bounded. Let Open image in new window . Then
By induction, we have

for all Open image in new window . This implies that Open image in new window is bounded and so is the sequence Open image in new window .

Next, we show that Open image in new window . To see this, we consider the following:

By the conditions (C2) and (C3), we have
Consequently, by the condition (C1),
From Lemma 2.2, let Open image in new window where Open image in new window is given by the formula (2.2). Then Open image in new window is the nearest point of Open image in new window to Open image in new window . We next consider the following:
By Lemma 2.2, we have Open image in new window for all Banach limits Open image in new window . Moreover, since Open image in new window ,
It follows from Open image in new window and Lemma 1.2 that

Hence the conclusion follows by Lemma 1.3.

3. Halpern's Iteration for a Family of Mappings

3.1. Finitely Many Mappings

We use the "cyclic method" [15] and Bauschke's condition [16] to obtain the following strong convergence theorem for a finite family of nonexpansive mappings.

Theorem 3.1.

Let Open image in new window be a complete CAT(0) space and Open image in new window a closed convex subset of Open image in new window . Let Open image in new window be nonexpansive mappings with Open image in new window and let Open image in new window be arbitrarily chosen. Define an iterative sequence Open image in new window by

where Open image in new window is a sequence in Open image in new window satisfying

(C1) Open image in new window ;

(C2) Open image in new window ;

(C3) Open image in new window or Open image in new window .

Suppose, in addition, that

Then Open image in new window converges to Open image in new window which is nearest Open image in new window .

Here the Open image in new window function takes values in Open image in new window .

Proof.

The proof line now follows from the proofs of Theorem 2.3 and [15, Theorem Open image in new window ].

3.2. Countable Mappings

The following concept is introduced by Aoyama et al. [10]. Let Open image in new window be a complete CAT(0) space and Open image in new window a subset of Open image in new window . Let Open image in new window be a countable family of mappings from Open image in new window into itself. We say that a family Open image in new window satisfies AKTT-condition if

for each bounded subset of Open image in new window of Open image in new window .

If Open image in new window is a closed subset and Open image in new window satisfies AKTT-condition, then we can define Open image in new window such that

In this case, we also say that Open image in new window satisfies AKTT-condition.

Theorem 3.2.

Let Open image in new window be a complete CAT(0) space and Open image in new window a closed convex subset of Open image in new window . Let Open image in new window be a countable family of nonexpansive mappings with Open image in new window . Suppose that Open image in new window are arbitrarily chosen and Open image in new window is defined by

where Open image in new window is a sequence in Open image in new window satisfying

(C1) Open image in new window ;

(C2) Open image in new window ;

(C3) Open image in new window or Open image in new window .

Suppose, in addition, that

(M1) Open image in new window satisfies AKTT-condition;

(M2) Open image in new window .

Then Open image in new window converges to Open image in new window which is nearest Open image in new window .

Proof.

Since the proof of this theorem is very similar to that of Theorem 2.3, we present here only the sketch proof. First, we notice that both sequences Open image in new window and Open image in new window are bounded and
By conditions (C2), (C3), AKTT-condition, and Lemma 1.3, we have
Let Open image in new window be the nearest point of Open image in new window to Open image in new window . As in the proof of Theorem 2.3, we have Open image in new window for all Banach limits Open image in new window and Open image in new window . We observe that
and this implies that

Therefore, Open image in new window and hence Open image in new window converges to Open image in new window .

We next show how to generate a family of mappings from a given family of mappings to satisfy conditions (M1) and (M2) of the preceding theorem. The following is an analogue of Bruck's result [17] in CAT(0) space setting. The idea using here is from [10].

Theorem 3.3.

Let Open image in new window be a complete CAT(0) space and Open image in new window a closed convex subset of Open image in new window . Suppose that Open image in new window is a countable family of nonexpansive mappings with Open image in new window . Then there exist a family of nonexpansive mappings Open image in new window and a nonexpansive mapping Open image in new window such that

(M1) Open image in new window satisfies AKTT-condition;

(M2) Open image in new window .

Lemma 3.4.

Let Open image in new window and Open image in new window be as above. Suppose that Open image in new window are nonexpansive mappings and Open image in new window . Then, for any Open image in new window , the mapping Open image in new window is nonexpansive and Open image in new window .

Proof.

To see that Open image in new window is nonexpansive, we only apply the triangle inequality and two applications of the second inequality in Lemma 1.1. We next prove the latter. It is clear that Open image in new window . To see the reverse inclusion, let Open image in new window and Open image in new window . Then, by the first inequality of Lemma 1.1,

This implies Open image in new window . As Open image in new window , we have Open image in new window , as desired.

Proof of Theorem 3.3.

We first define a family of mappings Open image in new window by
By Lemma 3.4, each Open image in new window is a nonexpansive mapping satisfying Open image in new window . Notice that, for fixed Open image in new window ,
From the estimation above, we have
for each bounded subset Open image in new window of Open image in new window . In particular, Open image in new window is a Cauchy sequence for each Open image in new window . We now define the nonexpansive mapping Open image in new window by
Finally, we prove that
The latter equality is clearly verified and Open image in new window holds. On the other hand, let Open image in new window and Open image in new window . We consider the following:

Because Open image in new window , we have Open image in new window . Continuing this procedure we obtain that Open image in new window and hence Open image in new window . This completes the proof.

4. Nonself Mappings

From Bridson and Haefliger's book (page 176), the following result is proved.

Theorem 4.1.

Let Open image in new window be a complete CAT(0) space and Open image in new window a closed convex subset of Open image in new window . Then the followings hold true.

(i)For each Open image in new window , there exists an element Open image in new window such that

(ii) Open image in new window for all Open image in new window .

(iii)The mapping Open image in new window is nonexpansive.

The mapping Open image in new window in the preceding theorem is called the metric projection from Open image in new window onto Open image in new window . From this, we have the following result.

Theorem 4.2.

Let Open image in new window be a complete CAT(0) space and Open image in new window a closed convex subset of Open image in new window . Let Open image in new window be a nonself nonexpansive mapping with Open image in new window and Open image in new window the metric projection from Open image in new window onto Open image in new window . Then the mapping Open image in new window is nonexpansive and Open image in new window .

Proof.

It follows from Theorem 4.1 that Open image in new window is nonexpansive. To see the latter, it suffices to show that Open image in new window . Let Open image in new window and Open image in new window . Since

we have Open image in new window and this finishes the proof.

By the preceding theorem and Theorem 2.3, we obtain the following result.

Theorem 4.3.

Let Open image in new window , Open image in new window , Open image in new window , and Open image in new window be as the same as Theorem 4.2. Suppose that Open image in new window are arbitrarily chosen and the sequence Open image in new window is defined by

where Open image in new window is a sequence in Open image in new window satisfying

(C1) Open image in new window ;

(C2) Open image in new window ;

(C3) Open image in new window or Open image in new window .

Then Open image in new window converges to Open image in new window which is nearest Open image in new window .

Notes

Acknowledgments

The author would like to thank the referee for the information that a part of Theorem 2.3 was proved in [14]. This work was supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand.

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Copyright information

© Satit Saejung. 2010

This article is published under license to BioMed Central Ltd. 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.

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceKhon Kaen UniversityKhon KaenThailand
  2. 2.Centre of Excellence in MathematicsCHEBangkokThailand

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