- Research Article
- Open Access

# Halpern's Iteration in CAT(0) Spaces

- Satit Saejung
^{1, 2}Email author

**2010**:471781

https://doi.org/10.1155/2010/471781

© Satit Saejung. 2010

**Received:**26 September 2009**Accepted:**24 November 2009**Published:**7 December 2009

## 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

## 1. Introduction

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

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

is satisfied by all and . The meaning of the CAT(0) inequality is that a geodesic triangle in 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 -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 [5–7]). 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 -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.

We also denote by
the geodesic segment joining from
to
, that is,
. A subset
of a CAT(0) space is *convex* if
for all
. 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.

Recall that a continuous linear functional
on
, the Banach space of bounded real sequences, is called a *Banach limit* if
and
for all
.

Lemma 1.2 (see [9, Proposition ]).

Let be such that for all Banach limits and . Then .

Lemma 1.3 (see [10, Lemma ]).

Then .

## 2. Halpern's Iteration for a Single Mapping

Lemma 2.1.

Proof.

This implies that is a contraction mapping and hence the conclusion follows.

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

Lemma 2.2.

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

(1) converges to the unique fixed point of which is nearest ;

(2) for all Banach limits and all bounded sequences with .

Proof.

- (1)
is proved in [12, Theorem ]. In fact, it is shown that is the nearest point of to . Finally, we prove (2). Suppose that is a sequence given by the formula (2.2), where is a sequence in such that . We also assume that is the nearest point of to . By the first inequality in Lemma 1.1, we have

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.

where is a sequence in satisfying

(C1) ;

(C2) ;

(C3) or .

Then converges to which is the nearest point of to .

Proof.

for all . This implies that is bounded and so is the sequence .

Next, we show that . To see this, we consider the following:

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.

where is a sequence in satisfying

(C1) ;

(C2) ;

(C3) or .

Then converges to which is nearest .

Here the function takes values in .

Proof.

The proof line now follows from the proofs of Theorem 2.3 and [15, Theorem ].

### 3.2. Countable Mappings

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

for each bounded subset of of .

If is a closed subset and satisfies AKTT-condition, then we can define such that

In this case, we also say that satisfies AKTT-condition.

Theorem 3.2.

where is a sequence in satisfying

(C1) ;

(C2) ;

(C3) or .

Suppose, in addition, that

(M1) satisfies AKTT-condition;

(M2) .

Then converges to which is nearest .

Proof.

Therefore, and hence converges to .

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

(M1) satisfies AKTT-condition;

(M2) .

Lemma 3.4.

Let and be as above. Suppose that are nonexpansive mappings and . Then, for any , the mapping is nonexpansive and .

Proof.

This implies . As , we have , as desired.

Proof of Theorem 3.3.

Because , we have . Continuing this procedure we obtain that and hence . This completes the proof.

## 4. Nonself Mappings

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

Theorem 4.1.

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

(ii) for all .

(iii)The mapping is nonexpansive.

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

Theorem 4.2.

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

Proof.

we have and this finishes the proof.

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

Theorem 4.3.

where is a sequence in satisfying

(C1) ;

(C2) ;

(C3) or .

Then converges to which is nearest .

## Declarations

### 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.

## Authors’ Affiliations

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