- Research Article
- Open Access
Strong Convergence of Modified Halpern Iterations in CAT(0) Spaces
© A. Cuntavepanit and B. Panyanak. 2011
- Received: 28 November 2010
- Accepted: 10 January 2011
- Published: 11 January 2011
Strong convergence theorems are established for the modified Halpern iterations of nonexpansive mappings in CAT(0) spaces. Our results extend and improve the recent ones announced by Kim and Xu (2005), Hu (2008), Song and Chen (2008), Saejung (2010), and many others.
- Nonexpansive Mapping
- Strong Convergence Theorem
- Smooth Banach Space
- Complete Space
- Geodesic Triangle
that the sequence converges strongly to a fixed point of .
The purpose of this paper is to extend Kim-Xu's result to a special kind of metric spaces, namely, CAT(0) spaces. We also prove a strong convergence theorem for another kind of modified Halpern iteration defined by Hu  in this setting.
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. The precise definition is given below. 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 ), -trees (see ), Euclidean buildings (see ), the complex Hilbert ball with a hyperbolic metric (see ), 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 .
Fixed point theory in CAT(0) spaces was first studied by Kirk (see [18, 19]). 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., [20–31] 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., [15, 32–35]).
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 satisfy the following comparison axiom.
From now on, we will use the notation for the unique point satisfying (2.2). We now collect some elementary facts about CAT(0) spaces which will be used in the proofs of our main results.
- (i)(see [23, Lemma 2.4]) for each and , one has
- (ii)(see ) for each and , one has
Recall that a continuous linear functional on , the Banach space of bounded real sequences, is called a Banach limit if and for all .
Lemma 2.2 (see [8, Proposition 2]).
Let be such that for all Banach limits and . Then, .
Lemma 2.3 (see [28, Lemma 2.1]).
Lemma 2.4 (see [28, Lemma 2.2]).
Let and be as the preceding lemma. Then, if and only if given by (2.8) 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 .
Lemma 2.5 (see [10, Lemma 2.1]).
where and are sequences of real numbers such that
either or .
The following result is an analog of Theorem 1 of Kim and Xu . They prove the theorem by using the concept of duality mapping, while we use the concept of Banach limit. We also observe that the condition in [12, Theorem 1] is superfluous.
Let be a nonempty closed convex subset of a complete space , and let be a nonexpansive mapping such that . Given a point and sequences and in , the following conditions are satisfied:
(A1) and ,
(A2) , and .
Then, converges to a fixed point which is nearest .
- (i)As in the first part of the proof of [12, Theorem 1], we can show that is bounded and so is and . Notice also that
Hence, the conclusion follows from Lemma 2.5.
Suppose that both and are sequences in satisfying
Then, converges to a fixed point which is nearest .
Let . We divide the proof into 3 steps.
Hence, is bounded and so are and .
Hence, the conclusion follows by Lemma 2.5.
The authors are grateful to Professor Sompong Dhompongsa for his suggestions and advices during the preparation of the paper. This research was supported by the National Research University Project under Thailand's Office of the Higher Education Commission.
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