- Research Article
- Open Access
Halpern's Iteration in CAT(0) Spaces
© Satit Saejung. 2010
- Received: 26 September 2009
- Accepted: 24 November 2009
- Published: 7 December 2009
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.
- Nonexpansive Mapping
- Common Fixed Point
- Iterative Sequence
- Geodesic Segment
- Nonnegative Real Number
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 .
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 ).
The concept of -convergence introduced by Lim in 1976 was shown by Kirk and Panyanak  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  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  (or, briefly in ).
The following lemma plays an important role in our paper.
Lemma 1.2 (see [9, Proposition ]).
Lemma 1.3 (see [10, Lemma ]).
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.
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  and of Shioji and Takahashi , 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 .
Hence the conclusion follows by Lemma 1.3.
3.1. Finitely Many Mappings
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. . 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
Suppose, in addition, that
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  in CAT(0) space setting. The idea using here is from .
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
Proof of Theorem 3.3.
From Bridson and Haefliger's book (page 176), the following result is proved.
By the preceding theorem and Theorem 2.3, we obtain the following result.
The author would like to thank the referee for the information that a part of Theorem 2.3 was proved in . This work was supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand.
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