Open Access

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

Fixed Point Theory and Applications20092010:367274

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

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.

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., [819]).

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

(1.1)

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 (thevertices 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

(2.1)
If are points in a CAT(0) space and if is the midpoint of the segment then the CAT(0) inequality implies
(CN)

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.

Let be a CAT(0) space.
  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

     
  2. (ii)
    [22, Lemma  2.1(iv)] For and there exists a unique point such that
    (2.2)
     
one uses the notation for the unique point satisfying (2.2).
  1. (iii)
    [22, Lemma  2.4] For and one has
    (2.3)
     
  1. (iv)
    [22, Lemma  2.5] For and one has
    (2.4)
     

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

(2.5)

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.

Let be a bounded closed convex subset of a complete CAT(0) space . Suppose that is a nonexpansive mapping for which
(2.6)

Then has a fixed point in

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

(2.7)

The asymptotic radius of is given by

(2.8)

and the asymptotic center of is the set

(2.9)

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

(2.10)

In fact, we have

(2.11)

since if

(2.12)

we get

(2.13)

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]).

The hyperbolic space is called uniformly convex if for any and there exists a such that for all
(2.14)

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

Lemma 2.7 (see [16, Lemma ]).

Let be a uniformly convex hyperbolic with modulus of uniform convexity For any , and
(2.15)

Lemma 2.8 (see [16, Proposition ]).

Assume that is a CAT(0) space. Then is uniformly convex, and
(2.16)

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.

Let be a uniformly convex hyperbolic space with modulus of convexity , and let . Suppose that increases with (for a fixed ) and suppose that is a sequence in for some and , are sequences in such that , and for some Then
(2.17)

Proof.

The case is trivial. Now suppose . If it is not the case that as then there are subsequences, denoted by and , such that
(2.18)
Choose such that
(2.19)
Since and This implies Choose such that
(2.20)
Since
(2.21)
there are further subsequences again denoted by and , such that
(2.22)
Then by Lemma 2.7 and (2.20),
(2.23)
for all Taking we obtain
(2.24)

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.

By Lemma 2.1(i) the nearest point projection is nonexpansive. Then
(3.1)
Consequently, we have
(3.2)

This implies that is bounded and decreasing. Hence exists.

Theorem 3.2.

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
(3.3)

Proof.

Let Then, by Theorem 3.1, exists. Let
(3.4)
If then by the nonexpansiveness of the conclusion follows. If , we let By Lemma 2.1(iv) we have
(3.5)
Therefore
(3.6)
It follows from (3.6) and Lemma 2.1(iv) that
(3.7)
Therefore
(3.8)

where Since

By (3.8), we have

(3.9)

This implies

Since is nonexpansive, we get that and hence

(3.10)
On the other hand, we can get from (3.6) that
(3.11)
Thus . This fact and (3.6) imply
(3.12)
Since is nonexpansive,
(3.13)
It follows from (3.4), (3.12), (3.13), and Lemma 2.9 that
(3.14)

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

(3.15)

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.

Let be a nonempty closed convex subset of a complete CAT(0) space and let be two nonexpansive mappings with Let and be sequences in for some From arbitrary define the sequence by the recursion
(3.16)

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

(1)
Department of Mathematics, Faculty of Science, Chaing Mai University

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Copyright

© W. Laowang and B. Panyanak. 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.