- Research Article
- Open Access

# Approximating Fixed Points of Nonexpansive Nonself Mappings in CAT(0) Spaces

- W Laowang
^{1}and - B Panyanak
^{1}Email author

**2010**:367274

https://doi.org/10.1155/2010/367274

© W. Laowang and B. Panyanak. 2010

**Received:**23 July 2009**Accepted:**30 November 2009**Published:**2 December 2009

## Abstract

Suppose that is a nonempty closed convex subset of a complete CAT(0) space with the nearest point projection from onto . Let be a nonexpansive nonself mapping with . Suppose that is generated iteratively by , , , where and are real sequences in for some . Then -converges to some point in . This is an analog of a result in Banach spaces of Shahzad (2005) and extends a result of Dhompongsa and Panyanak (2008) to the case of nonself mappings.

## Keywords

- Nonexpansive Mapping
- Hyperbolic Space
- Common Fixed Point
- Nonempty Closed Convex Subset
- Uniform Convexity

## 1. Introduction

A metric space is a CAT(0) space if it is geodesically connected and if every geodesic triangle in is at least as "thin" as its comparison triangle in the Euclidean plane. It is well known that any complete, simply connected Riemannian manifold having nonpositive sectional curvature is a CAT space. Other examples include Pre-Hilbert spaces, -trees (see [1]), Euclidean buildings (see [2]), the complex Hilbert ball with a hyperbolic metric (see [3]), and many others. For a thorough discussion of these spaces and of the fundamental role they play in geometry see Bridson and Haefliger [1]. The work by Burago et al. [4] contains a somewhat more elementary treatment, and by Gromov [5] a deeper study.

Fixed point theory in a CAT(0) space was first studied by Kirk (see [6, 7]). He showed that every nonexpansive (single-valued) mapping defined on a bounded closed convex subset of a complete CAT(0) space always has a fixed point. Since then the fixed point theory for single-valued and multivalued mappings in CAT(0) spaces has been rapidly developed and much papers have appeared (see, e.g., [8–19]).

In 2008, Kirk and Panyanak [20] used the concept of -convergence introduced by Lim [21] to prove the CAT(0) space analogs of some Banach space results which involve weak convergence, and Dhompongsa and Panyanak [22] obtained -convergence theorems for the Picard, Mann and Ishikawa iterations in the CAT(0) space setting.

The purpose of this paper is to study the iterative scheme defined as follows. Let is a nonempty closed convex subset of a complete CAT(0) space with the nearest point projection from onto . If is a nonexpansive mapping with nonempty fixed point set, and if is generated iteratively by

where and are real sequences in for some we show that the sequence defined by (1.1) -converges to a fixed point of This is an analog of a result in Banach spaces of Shahzad [23] and also extends a result of Dhompongsa and Panyanak [22] to the case of nonself mappings. It is worth mentioning that our result immediately applies to any CAT( ) space with since any CAT( ) space is a CAT( ) space for every (see [1, page 165]).

## 2. Preliminaries and Lemmas

Let
be a metric space. A *geodesic path* joining
to
(or, more briefly, a *geodesic* from
to
) is a map
from a closed interval
to
such that
and
for all
In particular,
is an isometry and
The image
of
is called a *geodesic* (or *metric*) *segment* joining
and
. When it is unique this geodesic segment is denoted by
. The space
is said to be a *geodesic space* if every two points of
are joined by a geodesic, and
is said to be *uniquely geodesic* if there is exactly one geodesic joining
and
for each
A subset
is said to be *convex* if
includes every geodesic segment joining any two of its points.

A *geodesic triangle*
in a geodesic metric space
consists of three points
in
(the*vertices* of
) and a geodesic segment between each pair of vertices (the *edges* of
). A *comparison triangle* for the geodesic triangle
in
is a triangle
in the Euclidean plane
such that
for

A geodesic space is said to be a CAT(0) space if all geodesic triangles of appropriate size satisfy the following comparison axiom.

CAT(0):Let
be a geodesic triangle in
and let
be a comparison triangle for
. Then
is said to satisfy the CAT(0) *inequality* if for all
and all comparison points

This is the (CN) inequality of Bruhat and Tits [24]. In fact (cf. [1, page 163]), a geodesic space is a CAT(0) space if and only if it satisfies the (CN) inequality.

We now collect some elementary facts about CAT(0) spaces which will be used frequently in the proofs of our main results.

Lemma 2.1.

- (i)
[1, Proposition 2.4] Let be a convex subset of which is complete in the induced metric. Then, for every there exists a unique point such that Moreover, the map is a nonexpansive retract from onto

- (ii)

- (iii)

- (iv)

Let
be a nonempty subset of a CAT(0) space
and let
be a mapping.
is called *nonexpansive* if for each

A point is called a fixed point of if . We shall denote by the set of fixed points of The existence of fixed points for nonexpansive nonself mappings in a CAT(0) space was proved by Kirk [6] as follows.

Theorem 2.2.

Then has a fixed point in

Let be a bounded sequence in a CAT(0) space . For we set

The *asymptotic radius*
of
is given by

and the *asymptotic center*
of
is the set

It is known (see, e.g., [12, Proposition ]) that in a CAT(0) space, consists of exactly one point.

We now give the definition of -convergence.

Definition 2.3 (see [20, 21]).

A sequence in a CAT(0) space is said to -converge to if is the unique asymptotic center of for every subsequence of . In this case one writes - and call the -limit of

The following lemma was proved by Dhompongsa and Panyanak (see [22, Lemma ]).

Lemma 2.4.

Let be a closed convex subset of a complete CAT(0) space and let be a nonexpansive mapping. Suppose is a bounded sequence in such that and converges for all , then Here where the union is taken over all subsequences of Moreover, consists of exactly one point.

We now turn to a wider class of spaces, namely, the class of hyperbolic spaces, which contains the class of CAT(0) spaces (see Lemma 2.8).

Definition 2.5 (see [16]).

A hyperbolic space is a triple where is a metric space and is such that

(W1)

(W2)

(W3)

(W4)

for all

It follows from (W1) that for each and

In fact, we have

since if

we get

which is a contradiction. By comparing between (2.2) and (2.11), we can also use the notation for in a hyperbolic space

Definition 2.6 (see [16]).

A mapping providing such a for given and is called a modulus of uniform convexity.

Lemma 2.7 (see [16, Lemma ]).

Lemma 2.8 (see [16, Proposition ]).

is a modulus of uniform convexity.

The following result is a characterization of uniformly convex hyperbolic spaces which is an analog of Lemma of Schu [25]. It can be applied to a CAT(0) space as well.

Lemma 2.9.

Proof.

which contradicts to the hypothesis.

## 3. Main Results

In this section, we prove our main theorems.

Theorem 3.1.

Let be a nonempty closed convex subset of a complete CAT(0) space and let be a nonexpansive mapping with Let and be sequences in for some Starting from arbitrary define the sequence by the recursion (1.1). Then exists.

Proof.

This implies that is bounded and decreasing. Hence exists.

Theorem 3.2.

Proof.

where Since

By (3.8), we have

This implies

Since is nonexpansive, we get that and hence

This completes the proof.

The following theorem is an analog of [23, Theorem ] and extends [22, Theorem ] to nonself mappings.

Theorem 3.3.

Let be a nonempty closed convex subset of a complete CAT(0) space and let be a nonexpansive mapping with Let and be sequences in for some From arbitrary define the sequence by the recursion (1.1). Then -converges to a fixed point of

Proof.

By Theorem 3.2, It follows from (3.2) that is bounded and decreasing for each and so it is convergent. By Lemma 2.4, consists of exactly one point and is contained in . This shows that the sequence -converges to an element of

We now state two strong convergence theorems. Recall that a mapping
is said to satisfy *Condition I* ([26]) if there exists a nondecreasing function
with
and
for all
such that

Theorem 3.4.

Let be a nonempty closed convex subset of a complete CAT(0) space and let be a nonexpansive mapping with Let and be sequences in for some From arbitrary define the sequence by the recursion (1.1). Suppose that satisfies condition I. Then converges strongly to a fixed point of

Theorem 3.5.

Let be a nonempty compact convex subset of a complete CAT(0) space and let be a nonexpansive mapping with Let and be sequences in for some From arbitrary define the sequence by the recursion (1.1). Then converges strongly to a fixed point of

Another result in [23] is that the author obtains a common fixed point theorem of two nonexpansive self-mappings. The proof is metric in nature and carries over to the present setting. Therefore, we can state the following result.

Theorem 3.6.

Then -converges to a common fixed point of and

## Declarations

### Acknowledgments

The authors are grateful to Professor Sompong Dhompongsa for his suggestions and advices during the preparation of the article. The second author was supported by the Commission on Higher Education and Thailand Research Fund under Grant MRG5280025. This work is dedicated to Professor Wataru Takahashi.

## Authors’ Affiliations

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