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.
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 ]).
2. Halpern's Iteration for a Single Mapping
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. Halpern's Iteration for a Family of Mappings
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.
4. Nonself Mappings
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.
- Bridson MR, Haefliger A: Metric Spaces of Non-Positive Curvature, Grundlehren der Mathematischen Wissenschaften. Volume 319. Springer, Berlin, Germany; 1999:xxii+643.View ArticleMATHGoogle Scholar
- Burago D, Burago Y, Ivanov S: A Course in Metric Geometry, Graduate Studies in Mathematics. Volume 33. American Mathematical Society, Providence, RI, USA; 2001:xiv+415.MATHGoogle Scholar
- Goebel K, Reich S: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Monographs and Textbooks in Pure and Applied Mathematics. Volume 83. Marcel Dekker, New York, NY, USA; 1984:ix+170.Google Scholar
- Kirk WA, Panyanak B: A concept of convergence in geodesic spaces. Nonlinear Analysis: Theory, Methods & Applications 2008,68(12):3689–3696. 10.1016/j.na.2007.04.011MathSciNetView ArticleMATHGoogle Scholar
- Dhompongsa S, Panyanak B: On -convergence theorems in CAT(0) spaces. Computers & Mathematics with Applications 2008,56(10):2572–2579. 10.1016/j.camwa.2008.05.036MathSciNetView ArticleMATHGoogle Scholar
- Dhompongsa S, Fupinwong W, Kaewkhao A: Common fixed points of a nonexpansive semigroup and a convergence theorem for Mann iterations in geodesic metric spaces. Nonlinear Analysis: Theory, Methods & Applications 2009,70(12):4268–4273. 10.1016/j.na.2008.09.012MathSciNetView ArticleMATHGoogle Scholar
- Laokul T, Panyanak B: Approximating fixed points of nonexpansive mappings in CAT(0) spaces. International Journal of Mathematical Analysis 2009, 3: 1305–1315.MathSciNetMATHGoogle Scholar
- Halpern B: Fixed points of nonexpanding maps. Bulletin of the American Mathematical Society 1967, 73: 957–961. 10.1090/S0002-9904-1967-11864-0MathSciNetView ArticleMATHGoogle Scholar
- Shioji N, Takahashi W: Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces. Proceedings of the American Mathematical Society 1997,125(12):3641–3645. 10.1090/S0002-9939-97-04033-1MathSciNetView ArticleMATHGoogle Scholar
- Aoyama K, Kimura Y, Takahashi W, Toyoda M: Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space. Nonlinear Analysis: Theory, Methods & Applications 2007,67(8):2350–2360. 10.1016/j.na.2006.08.032MathSciNetView ArticleMATHGoogle Scholar
- Kirk WA: Geodesic geometry and fixed point theory. In Seminar of Mathematical Analysis (Malaga/Seville, 2002/2003), Colección Abierta. Volume 64. University of Sevilla Secretary, Seville, Spain; 2003:195–225.Google Scholar
- Kirk WA: Fixed point theorems in CAT(0) spaces and -trees. Fixed Point Theory and Applications 2004,2004(4):309–316.MathSciNetView ArticleMATHGoogle Scholar
- Wittmann R: Approximation of fixed points of nonexpansive mappings. Archiv der Mathematik 1992,58(5):486–491. 10.1007/BF01190119MathSciNetView ArticleMATHGoogle Scholar
- Aoyama K, Eshita K, Takahashi W: Iteration processes for nonexpansive mappings in convex metric spaces. In Nonlinear Analysis and Convex Analysis. Yokohama, Yokohama, Japan; 2007:31–39.Google Scholar
- Bauschke HH: The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space. Journal of Mathematical Analysis and Applications 1996,202(1):150–159. 10.1006/jmaa.1996.0308MathSciNetView ArticleMATHGoogle Scholar
- Suzuki T: Some notes on Bauschke's condition. Nonlinear Analysis: Theory, Methods & Applications 2007,67(7):2224–2231. 10.1016/j.na.2006.09.014MathSciNetView ArticleMATHGoogle Scholar
- Bruck RE Jr.: Properties of fixed-point sets of nonexpansive mappings in Banach spaces. Transactions of the American Mathematical Society 1973, 179: 251–262.MathSciNetView ArticleMATHGoogle Scholar
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.