△-Convergence analysis of improved Kuhfittig iterative for asymptotically nonexpansive nonself-mappings in W-hyperbolic spaces
© Yi and Bo; licensee Springer. 2014
Received: 15 February 2014
Accepted: 19 July 2014
Published: 19 August 2014
Throughout this paper, we introduce a class of asymptotically nonexpansive nonself-mapping and modify the classical Kuhfittig iteration algorithm in the general setup of hyperbolic space. Also, △-convergence results are obtained under a limit condition. The results presented in the paper extend various results in the existing literature.
As we know, fixed point theory proposed in the setting of normed linear spaces or Banach spaces mainly depends on the linear structure of the underlying space. However, a nonlinear framework for fixed point theory is a metric space embedded with a ‘convex structure’. The class of hyperbolic spaces, nonlinear in nature, is a general abstract theoretic setting with rich geometrical structures for metric fixed point theory.
In fact, a few important results of the problems in various disciplines of science being nonlinear in nature were studied only in CAT(0) space. In 1976, the concept of △-convergence in general metric spaces was coined by Lim . Since then, Kirk and Panyanak  specialized this concept to CAT(0) spaces and proved that it is very similar to the weak convergence in the Banach space setting. Dhompongsa and Panyanak  and Abbas et al.  obtained △-convergence theorems for the Mann and Ishikawa iterations in the CAT(0) space setting. Moreover, Yang and Zhao  studied the strong and Δ-convergence theorems for total asymptotically nonexpansive nonself-mappings in CAT(0) spaces. As for more details of this work, one can refer to the aforementioned papers and references therein.
In recent years, hyperbolic space has attracted much attention of many authors. The study of hyperbolic spaces has been largely motivated and dominated by questions about hyperbolic groups. It should be noted that one of the main object of study is in geometric group theory. For example, Wan  proved some Δ-convergence theorems in a hyperbolic space, in which a mixed Agarwal-O’Regan-Sahu type iterative scheme for approximating a common fixed point of totally asymptotically nonexpansive mappings was constructed.
In this paper, following the work of Yang and Wan, by introducing a class of asymptotically nonexpansive nonself-mapping, we modify a classical Kuhfittig iteration algorithm in the general setup of hyperbolic space. Under a limit condition, we also establish some △-convergence results, which extend various results in the existing literature.
for all and . A nonempty subset C of a hyperbolic space E is convex if () and . The class of hyperbolic spaces contains normed spaces and convex subsets thereof, the Hilbert ball equipped with the hyperbolic metric , Hadamard manifolds as well as CAT(0) spaces in the sense of Gromov .
provided that , and .
Recall that C is said to be a retraction of E if there exists a continuous map such that , for all . A map is said to be a retraction if . Consequently, if P is a retraction, then for all y in the range of P.
Definition 2.1 ()
- (1)asymptotically nonexpansive nonself-mapping if there exists a sequence with such that(2.6)
- (2)totally asymptotically nonexpansive nonself-mapping if there exist nonnegative sequences , with , and a strictly continuous function with such that(2.7)
- (3)uniformly L-Lipschitzian if there exists a constant such that(2.8)
Remark 2.1 From the definitions above, we know that each nonexpansive mapping is an asymptotically nonexpansive nonself-mapping, and each asymptotically nonexpansive nonself-mapping is a totally asymptotically nonexpansive nonself-mapping.
A sequence in hyperbolic space E is said to △-converge to , if p is the unique asymptotic center of for every subsequence of . In this case, we call p the △-limit of .
The following lemmas are important in our paper.
Lemma 2.1 (see )
Let be a complete uniformly convex hyperbolic space with monotone modulus of uniform convexity, and let C be a nonempty, closed, convex subset of E. Then every bounded sequence in E has a unique asymptotic center with respect to C.
Let be a uniformly convex hyperbolic space with monotone modulus of uniform convexity η. Let and be a sequence in for some . If and are sequences in E such that , , and for some , then .
Lemma 2.3 (see )
Let C be a nonempty, closed, convex subset of a uniformly convex hyperbolic space, and let be a bounded sequence in C such that and . If is another sequence in C such that , then .
If and , then exists. If there exists a subsequence such that , then .
3 Main results
where C is a nonempty closed and convex subset of a complete uniformly hyperbolic space E, is a uniformly L-Lipschitzian and -asymptotically nonexpansive nonself-mapping with and , and such that . P is nonexpansive retraction of E onto C.
is valid under the condition .
for all ;
there exist constants with such that .
and . Then the sequence △-converges to a point .
Proof (I) First, we prove that () and exist, respectively.
where . By condition (i), we get . Therefore, from Lemma 2.4, () and exist.
(II) Next, we prove that (as ).
(III) Now we prove that △-converges to a point .
Consequently, . By the uniqueness of asymptotic centers, we get . It implies that is the unique asymptotic of for each subsequence , that is, △-converges to a point . The proof of Theorem 3.1 is completed. □
From Remark 2.1, we have the following result.
If there exist constants with such that , and , then the sequence △-converges to a point .
The authors are very grateful to both reviewers for carefully reading this paper and their comments.
- Lim TC: Remarks on some fixed point theorems. Proc. Am. Math. Soc. 1976, 60: 179–182. 10.1090/S0002-9939-1976-0423139-XView ArticleMathSciNetMATHGoogle Scholar
- Kirk WA, Panyanak B: A concept of convergence in geodesic spaces. Nonlinear Anal. 2008, 68: 3689–3696. 10.1016/j.na.2007.04.011MathSciNetView ArticleMATHGoogle Scholar
- Dhompongsa S, Panyanak B: On △-convergence theorem in CAT(0) spaces. Comput. Math. Appl. 2008,56(10):2572–2579. 10.1016/j.camwa.2008.05.036MathSciNetView ArticleMATHGoogle Scholar
- Abbas M, Thakur BS, Thakur D: Fixed points of asymptotically nonexpansive mapping in the intermediate sense in CAT(0) spaces. Commun. Korean Math. Soc. 2013,28(1):107–121. 10.4134/CKMS.2013.28.1.107MathSciNetView ArticleMATHGoogle Scholar
- Yang L, Zhao FH: Srong and Δ-convergence theorems for total asymptotically nonexpansive nonself mappings in CAT(0) spaces. J. Inequal. Appl. 2013., 2013: Article ID 557Google Scholar
- Wan L: Δ-Convergence for mixed-type total asymptotically nonexpansive mappings in hyperbolic spaces. J. Inequal. Appl. 2013., 2013: Article ID 553Google Scholar
- Kohlenbach U: Some logical metatheorems with applications in functional analysis. Trans. Am. Math. Soc. 2005, 357: 89–128. 10.1090/S0002-9947-04-03515-9MathSciNetView ArticleMATHGoogle Scholar
- Goebel K, Kirk WA: Iteration processes for nonexpansive mappings. Contemp. Math. 21. In Topological Methods in Nonlinear Functional Analysis. Edited by: Singh SP, Thomeier S, Watson B. Am. Math. Soc., Providence; 1983:115–123.View ArticleGoogle Scholar
- Reich S, Shafrir I: Nonexpansive iterations in hyperbolic spaces. Nonlinear Anal. 1990, 15: 537–558. 10.1016/0362-546X(90)90058-OMathSciNetView ArticleMATHGoogle Scholar
- Goebel K, Reich S: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Dekker, New York; 1984.MATHGoogle Scholar
- Bridson M, Haefliger A: Metric Spaces of Non-Positive Curvature. Springer, Berlin; 1999.View ArticleMATHGoogle Scholar
- Wang L, Chang SS, Ma Z: Convergence theorem for total asymptotically nonexpansive nonself mappings in CAT(0) space. J. Inequal. Appl. 2013., 2013: Article ID 135Google Scholar
- Khan AR, Fukharuddin H, Khan MAA: An implicit algorithm for two finite families of nonexpansive maps in hyperbolic space. Fixed Point Theory Appl. 2012., 2012: Article ID 54Google Scholar
- Leustean L: Nonexpansive iterations uniformly covex W -hyperbolic spaces. Contemporary Math. 513. In Nonlinear Analysis and Optimization I. Nonlinear Analysis. Edited by: Leizarowitz A, Mordukhovich BS, Shafrir I, Zaslavski A. Am. Math. Soc, Providence; 2010:193–209.Google Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.