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Abstract and Applied Analysis
Volume 2011 (2011), Article ID 824718, 14 pages
http://dx.doi.org/10.1155/2011/824718
Research Article

A Generalization of Suzuki's Lemma

Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand

Received 4 February 2011; Accepted 27 April 2011

Academic Editor: D. Anderson

Copyright © 2011 B. Panyanak and A. Cuntavepanit. 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.

Abstract

Let , , and be bounded sequences in a metric space of hyperbolic type , and let be a sequence in with . If for all , , and , then . This is a generalization of Lemma 2.2 in (T. Suzuki, 2005). As a consequence, we obtain strong convergence theorems for the modified Halpern iterations of nonexpansive mappings in CAT(0) spaces.

1. Introduction

Suppose that is a metric space which contains a family of metric segments (isometric images of real line segments) such that distinct points lie on exactly one member of . Let , we use the notation to denote the point of the segment with distance from , that is, We will say that is of hyperbolic type if for each and , It is proved in [1] that (1.2) implies It is well-known that Banach spaces are of hyperbolic type. Notice also that CAT(0) spaces and hyperconvex metric spaces are of hyperbolic type (see [2, 3]).

In 1983, Goebel and Kirk [4] proved that if and are sequences in a metric space of hyperbolic type and which satisfy for all , (i) , (ii) , (iii) , (iv) , and (v) , then . It was proved by Suzuki [5] that one obtains the same conclusion if the conditions (i)–(v) are replaced by the conditions (S1)–(S4) as follows:(S1), (S2), (S3) and are bounded sequences,(S4).

Both Goebel-Kirk’s and Suzuki’s results have been used to prove weak and strong convergence theorems for approximating fixed points of various types of mappings. The purpose of this paper is to generalize Suzuki's result by relaxing the condition (S1), namely, we can define in terms of and such that . Precisely, we are going to prove the following lemma.

Lemma 1.1. Let , , and be bounded sequences in a metric space of hyperbolic type , and let be a sequence in with . Suppose that for all , and , then .

2. Proof of Lemma 1.1

We begin by proving a crucial lemma.

Lemma 2.1. Let , , and be sequences in a metric space of hyperbolic type , and let be a sequence in with . Put Suppose that for all , and then holds for all .

Proof. (This proof is patterned after the proof of [5, Lemma  1.1]). For each , let , then by (1.2), we have This implies Since , then This fact and (2.4) yield Since , we have By using this fact, we have, for , Put . We note that . Fix and , then there exists such that , and , for all and . In the case of , we choose satisfying and for all . We note that for . In the case of , we choose satisfying and for all . We note that for . In both cases, such satisfies that , for all , and for . We next show that for . Since and , we have Hence, (2.15) holds for . We assume that (2.15) holds for some . Then, since we obtain Hence, (2.15) holds for . Therefore, (2.15) holds for all . Specially, we have On the other hand, we have This fact and (2.20) imply Since and are arbitrary, we obtain the desired result.

By using Lemma 2.1 together with the argument in the proof of [5, Lemma  2.2], simply replacing by , we can obtain Lemma 1.1 as desired.

3. Applications

In this section, we apply Lemma 1.1 to prove two strong convergence theorems for the modified Halpern iterations of nonexpansive mappings in CAT(0) spaces. The results we obtain are analogs of the Banach space results of Song and Li [6].

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(0) space. Other examples include pre-Hilbert spaces (see [7]), –trees (see [8]), Euclidean buildings (see [9]), the complex Hilbert ball with a hyperbolic metric (see [10]), and many others. For a thorough discussion of these spaces and of the fundamental role they play in geometry, we refer the reader to Bridson and Haefliger [7].

Fixed-point theory in CAT(0) spaces was first studied by Kirk (see [2, 11]). 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 many papers have appeared (see, e.g., [1224] and the references therein). It is worth mentioning that fixed-point theorems in CAT(0) spaces (specially in –trees) can be applied to graph theory, biology, and computer science (see, e.g., [8, 2528]).

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 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 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 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 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 , We now collect some elementary facts about CAT(0) spaces.

Lemma 3.1. Let be a CAT(0) space. (i)(see [7, Proposition  2.4]) Let be a closed-convex subset of , then, for every , there exists a unique point such that . The mapping is called the nearest point (or metric) projection from onto .(ii)(see [15, Lemma  2.5]) For and , one has

Recall that a mapping on a CAT(0) space is called nonexpansive if A point is called a fixed point of if . We will denote by the set of fixed points of . The following result can be found in [13] (see also [2, Theorem  12]).

Theorem 3.2. Let be a convex subset of a CAT(0) space, and let be a nonexpansive mapping whose fixed-point set is nonempty, then is closed and convex.

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

Lemma 3.3 (see [29], Proposition  2). Let be such that for all Banach limits and , then .

Lemma 3.4 (see [21], Lemma  2.1). Let be a closed-convex subset of a complete CAT(0) space , and let be a nonexpansive mapping. Let be fixed. For each , the mapping defined by has a unique fixed-point , that is,

Lemma 3.5 (see [21], Lemma  2.2). Let and be as the preceding lemma, then if and only if given by the formula (3.5) remains bounded as . In this case, the following statements hold: (1) converges to the unique fixed-point of which is the nearest ,(2) for all Banach limits and all bounded sequences with .

Lemma 3.6 (see [30], Lemma 2.1). Let be a sequence of nonnegative real numbers satisfying the condition where and are sequences of real numbers, such that(i) and , (ii)either or , then .

The following result is an analog of [6, Theorem  3.1].

Theorem 3.7. Let be a nonempty closed-convex subset of a complete CAT(0) space , and let be a nonexpansive mapping such that . Given a point and sequences and in , the following conditions are satisfied: (C1), (C2), (C3). Define a sequence in by arbitrarily, and then converges to a fixed-point of , where is the nearest point projection from onto .

Proof. For each , we let . We divide the proof into 3 steps. (i) We show that and are bounded sequences. (ii) We show that . (iii) We show that converges to a fixed-point which is the nearest .
(i) Let , then we have Now, an induction yields Hence, is bounded and so is .
(ii) First, we show that . Consider This implies By the condition (C1), we have It follows from Lemma 1.1 that . Now,
(iii) From Lemma 3.4, let where is given by the formula (3.5). Then is the point of which is the nearest . By applying Lemma 3.1, we have By Lemma 3.5, we have for all Banach limit . Moreover, since It follows from condition (C1) and Lemma 3.3 that Hence, the conclusion follows from Lemma 3.6.

Remark 3.8. In the proof of Theorem 3.7, one may observe that it is not necessary to use Lemma 1.1 because Suzuki's original lemma is sufficient. However, in [6], there is a strong convergence theorem for another type of modified Halpern iteration (see [6, Theorem  3.2]). We show that the proof is quite easy when we use Lemma 1.1.

Theorem 3.9. Let be a nonempty closed-convex subset of a complete CAT(0) space , and let be a nonexpansive mapping such that . Given a point and sequences and in , the following conditions are satisfied: (C1), (C2), (C3). Define a sequence in by arbitrarily, and Then converges to a fixed-point of , where is the nearest point projection from onto .

Proof. Using the same technique as in the proof of Theorem 3.7, we easily obtain that both and are bounded. Let , then . By the condition (C1), we have It follows from the nonexpansiveness of that By Lemma 1.1, we have From (3.18) and (3.20), we get that Let where is given by (3.5), then is the point of which is the nearest . Consider By Lemma 3.5, we have for all Banach limit . Moreover, since , then It follows from condition (C1) and Lemma 3.3 that Hence, the conclusion follows from Lemma 3.6.

Acknowledgment

This research was supported by the Faculty of Science, Chiang Mai University, Chiang Mai, Thailand.

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