# 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.

## 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 ]).

## 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

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

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

Suppose, in addition, that

(M1) satisfies AKTT-condition;

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;

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.

(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.

## 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

## References

- Bridson MR, Haefliger A:
*Metric Spaces of Non-Positive Curvature, Grundlehren der Mathematischen Wissenschaften*.*Volume 319*. Springer, Berlin, Germany; 1999:xxii+643.View ArticleMATHGoogle Scholar - Burago D, Burago Y, Ivanov S:
*A Course in Metric Geometry, Graduate Studies in Mathematics*.*Volume 33*. American Mathematical Society, Providence, RI, USA; 2001:xiv+415.MATHGoogle Scholar - Goebel K, Reich S:
*Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Monographs and Textbooks in Pure and Applied Mathematics*.*Volume 83*. Marcel Dekker, New York, NY, USA; 1984:ix+170.Google Scholar - Kirk WA, Panyanak B:
**A concept of convergence in geodesic spaces.***Nonlinear Analysis: Theory, Methods & Applications*2008,**68**(12):3689–3696. 10.1016/j.na.2007.04.011MathSciNetView ArticleMATHGoogle Scholar - Dhompongsa S, Panyanak B:
**On -convergence theorems in CAT(0) spaces.***Computers & Mathematics with Applications*2008,**56**(10):2572–2579. 10.1016/j.camwa.2008.05.036MathSciNetView ArticleMATHGoogle Scholar - Dhompongsa S, Fupinwong W, Kaewkhao A:
**Common fixed points of a nonexpansive semigroup and a convergence theorem for Mann iterations in geodesic metric spaces.***Nonlinear Analysis: Theory, Methods & Applications*2009,**70**(12):4268–4273. 10.1016/j.na.2008.09.012MathSciNetView ArticleMATHGoogle Scholar - Laokul T, Panyanak B:
**Approximating fixed points of nonexpansive mappings in CAT(0) spaces.***International Journal of Mathematical Analysis*2009,**3:**1305–1315.MathSciNetMATHGoogle Scholar - Halpern B:
**Fixed points of nonexpanding maps.***Bulletin of the American Mathematical Society*1967,**73:**957–961. 10.1090/S0002-9904-1967-11864-0MathSciNetView ArticleMATHGoogle Scholar - Shioji N, Takahashi W:
**Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces.***Proceedings of the American Mathematical Society*1997,**125**(12):3641–3645. 10.1090/S0002-9939-97-04033-1MathSciNetView ArticleMATHGoogle Scholar - Aoyama K, Kimura Y, Takahashi W, Toyoda M:
**Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space.***Nonlinear Analysis: Theory, Methods & Applications*2007,**67**(8):2350–2360. 10.1016/j.na.2006.08.032MathSciNetView ArticleMATHGoogle Scholar - Kirk WA:
**Geodesic geometry and fixed point theory.**In*Seminar of Mathematical Analysis (Malaga/Seville, 2002/2003), Colección Abierta*.*Volume 64*. University of Sevilla Secretary, Seville, Spain; 2003:195–225.Google Scholar - Kirk WA:
**Fixed point theorems in CAT(0) spaces and -trees.***Fixed Point Theory and Applications*2004,**2004**(4):309–316.MathSciNetView ArticleMATHGoogle Scholar - Wittmann R:
**Approximation of fixed points of nonexpansive mappings.***Archiv der Mathematik*1992,**58**(5):486–491. 10.1007/BF01190119MathSciNetView ArticleMATHGoogle Scholar - Aoyama K, Eshita K, Takahashi W:
**Iteration processes for nonexpansive mappings in convex metric spaces.**In*Nonlinear Analysis and Convex Analysis*. Yokohama, Yokohama, Japan; 2007:31–39.Google Scholar - Bauschke HH:
**The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space.***Journal of Mathematical Analysis and Applications*1996,**202**(1):150–159. 10.1006/jmaa.1996.0308MathSciNetView ArticleMATHGoogle Scholar - Suzuki T:
**Some notes on Bauschke's condition.***Nonlinear Analysis: Theory, Methods & Applications*2007,**67**(7):2224–2231. 10.1016/j.na.2006.09.014MathSciNetView ArticleMATHGoogle Scholar - Bruck RE Jr.:
**Properties of fixed-point sets of nonexpansive mappings in Banach spaces.***Transactions of the American Mathematical Society*1973,**179:**251–262.MathSciNetView ArticleMATHGoogle Scholar

## Copyright

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.