- Research Article
- Open Access
Approximating Fixed Points of Nonexpansive Nonself Mappings in CAT(0) Spaces
© W. Laowang and B. Panyanak. 2010
- Received: 23 July 2009
- Accepted: 30 November 2009
- Published: 2 December 2009
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.
- Nonexpansive Mapping
- Hyperbolic Space
- Common Fixed Point
- Nonempty Closed Convex Subset
- Uniform Convexity
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 ), 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 see Bridson and Haefliger . The work by Burago et al.  contains a somewhat more elementary treatment, and by Gromov  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., [8–19]).
In 2008, Kirk and Panyanak  used the concept of -convergence introduced by Lim  to prove the CAT(0) space analogs of some Banach space results which involve weak convergence, and Dhompongsa and Panyanak  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
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  and also extends a result of Dhompongsa and Panyanak  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]).
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.
We now collect some elementary facts about CAT(0) spaces which will be used frequently in the proofs of our main results.
- (iii)[22, Lemma 2.4] For and one has
- (iv)[22, Lemma 2.5] For and one has
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  as follows.
It is known (see, e.g., [12, Proposition ]) that in a CAT(0) space, consists of exactly one point.
The following lemma was proved by Dhompongsa and Panyanak (see [22, Lemma ]).
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 ).
In fact, we have
Definition 2.6 (see ).
Lemma 2.7 (see [16, Lemma ]).
Lemma 2.8 (see [16, Proposition ]).
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 . It can be applied to a CAT(0) space as well.
which contradicts to the hypothesis.
In this section, we prove our main theorems.
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.
By (3.8), we have
This completes the proof.
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
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 () if there exists a nondecreasing function with and for all such that
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
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  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.
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.
- Bridson M, Haefliger A: Metric Spaces of Non-Positive Curvature, Fundamental Principles of Mathematical Sciences. Volume 319. Springer, Berlin, Germany; 1999:xxii+643.View ArticleMATHGoogle Scholar
- Brown KS: Buildings. Springer, New York, NY, USA; 1989:viii+215.View ArticleMATHGoogle 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
- 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
- Gromov M: Metric Structures for Riemannian and Non-Riemannian Spaces, Progress in Mathematics. Volume 152. Birkhäuser, Boston, Mass, USA; 1999:xx+585.MATHGoogle Scholar
- Kirk WA: Geodesic geometry and fixed point theory. In Seminar of Mathematical Analysis (Malaga/Seville, 2002/2003), Colecc. Abierta. Volume 64. Seville University Publications, Seville, Spain; 2003:195–225.Google Scholar
- Kirk WA: Geodesic geometry and fixed point theory II. In International Conference on Fixed Point Theory and Applications. Yokohama Publications, Yokohama, Japan; 2004:113–142.Google Scholar
- Chaoha P, Phon-on A: A note on fixed point sets in CAT(0) spaces. Journal of Mathematical Analysis and Applications 2006,320(2):983–987. 10.1016/j.jmaa.2005.08.006MathSciNetView 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
- Dhompongsa S, Kaewkhao A, Panyanak B: Lim's theorems for multivalued mappings in CAT(0) spaces. Journal of Mathematical Analysis and Applications 2005,312(2):478–487. 10.1016/j.jmaa.2005.03.055MathSciNetView ArticleMATHGoogle Scholar
- Dhompongsa S, Kirk WA, Panyanak B: Nonexpansive set-valued mappings in metric and Banach spaces. Journal of Nonlinear and Convex Analysis 2007,8(1):35–45.MathSciNetMATHGoogle Scholar
- Dhompongsa S, Kirk WA, Sims B: Fixed points of uniformly Lipschitzian mappings. Nonlinear Analysis: Theory, Methods & Applications 2006,65(4):762–772. 10.1016/j.na.2005.09.044MathSciNetView ArticleMATHGoogle Scholar
- Hussain N, Khamsi MA: On asymptotic pointwise contractions in metric spaces. Nonlinear Analysis: Theory, Methods & Applications 2009,71(10):4423–4429. 10.1016/j.na.2009.02.126MathSciNetView ArticleMATHGoogle Scholar
- Kaewcharoen A, Kirk WA: Proximinality in geodesic spaces. Abstract and Applied Analysis 2006, Article ID 43591 2006:-10.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
- Leustean L: A quadratic rate of asymptotic regularity for CAT(0)-spaces. Journal of Mathematical Analysis and Applications 2007,325(1):386–399. 10.1016/j.jmaa.2006.01.081MathSciNetView ArticleMATHGoogle Scholar
- Shahzad N: Fixed point results for multimaps in CAT(0) spaces. Topology and Its Applications 2009,156(5):997–1001. 10.1016/j.topol.2008.11.016MathSciNetView ArticleMATHGoogle Scholar
- Shahzad N: Invariant approximations in CAT(0) spaces. Nonlinear Analysis: Theory, Methods & Applications 2009,70(12):4338–4340. 10.1016/j.na.2008.10.002MathSciNetView ArticleMATHGoogle Scholar
- Shahzad N, Markin J: Invariant approximations for commuting mappings in CAT(0) and hyperconvex spaces. Journal of Mathematical Analysis and Applications 2008,337(2):1457–1464. 10.1016/j.jmaa.2007.04.041MathSciNetView ArticleMATHGoogle 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
- Lim TC: Remarks on some fixed point theorems. Proceedings of the American Mathematical Society 1976, 60: 179–182. 10.1090/S0002-9939-1976-0423139-XMathSciNetView 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
- Shahzad N: Approximating fixed points of non-self nonexpansive mappings in Banach spaces. Nonlinear Analysis: Theory, Methods & Applications 2005,61(6):1031–1039. 10.1016/j.na.2005.01.092MathSciNetView ArticleMATHGoogle Scholar
- Bruhat F, Tits J: Groupes réductifs sur un corps local. Publications Mathématiques de l'Institut des Hautes Études Scientifiques 1972, (41):5–251.View ArticleGoogle Scholar
- Schu J: Weak and strong convergence to fixed points of asymptotically nonexpansive mappings. Bulletin of the Australian Mathematical Society 1991,43(1):153–159. 10.1017/S0004972700028884MathSciNetView ArticleMATHGoogle Scholar
- Senter HF, Dotson, WG Jr.: Approximating fixed points of nonexpansive mappings. Proceedings of the American Mathematical Society 1974, 44: 375–380. 10.1090/S0002-9939-1974-0346608-8MathSciNetView ArticleMATHGoogle Scholar
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