- Open Access
△-Convergence analysis of improved Kuhfittig iterative for asymptotically nonexpansive nonself-mappings in W-hyperbolic spaces
Journal of Inequalities and Applications volume 2014, Article number: 303 (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.
In this paper, we work in the setting of hyperbolic spaces introduced by Kohlenbach , which is more restrictive than the hyperbolic space introduced in Goebel and Kirk  and more general than the hyperbolic space in Reich and Shafrir . Concretely, is called a hyperbolic space if is a metric space and is a function satisfying
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 .
A hyperbolic space E is uniformly convex if for , and , there exists such that
provided that , and .
A map is called modulus of uniform convexity if for given . Besides, η is monotone if it decreases with r, that is,
Let C be a nonempty subset of a metric space . Recall that a mapping is said to be nonexpansive if
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 ()
Let C be a nonempty and closed subset of a metric space , a map is a retraction, a mapping is said to be
asymptotically nonexpansive nonself-mapping if there exists a sequence with such that(2.6)
totally asymptotically nonexpansive nonself-mapping if there exist nonnegative sequences , with , and a strictly continuous function with such that(2.7)
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.
To study our results in the general setup of hyperbolic spaces, we first collect some basic concepts. Let be a bounded sequence in hyperbolic space E. For , define a continuous functional by
The asymptotic radius of is given by
The asymptotic radius of with respect to is given by
The asymptotic center of is the set
The asymptotic center of with respect to is the set
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 .
Lemma 2.4 Let , , and be sequences of nonnegative numbers such that
If and , then exists. If there exists a subsequence such that , then .
3 Main results
Now we modify a classical Kuhfittig iteration algorithm in the general setup of hyperbolic space, and prove a △-convergence theorem for the following implicit iterative scheme:
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.
Remark 3.1 For and a fixed , define the mapping by
It can be seen obviously that is contraction on C. Indeed, for , we have
Therefore, is a contraction mapping, that is,
is valid under the condition .
Theorem 3.1 Let E be a complete hyperbolic space, C be a nonempty, bounded, closed, convex subset of E and be the nonexpansive retraction. Let be -asymptotically nonexpansive nonself-mapping with sequence and such that T is uniformly L-Lipschitz continuous, satisfying the following conditions:
for all ;
there exist constants with such that .
Define as follows: ,
and . Then the sequence △-converges to a point .
Proof (I) First, we prove that () and exist, respectively.
Since is -asymptotically nonexpansive nonself-mapping with sequence and , from Definition 2.1(1), for any , we have
For each , it follows from (3.1) and (2.1) that
From condition (iii), we have
Since , there exists an integer such that for all . Hence we have
where . By condition (i), we get . Therefore, from Lemma 2.4, () and exist.
(II) Next, we prove that (as ).
For , according to the proof of (I), we know that exists. Assume that
From (3.4) and (3.6), we get
which implies that
In addition, since
from (3.6), we have
It follows from (3.6)-(3.8) and Lemma 2.2 that
Hence, from (3.9) and (3.10), we get
Since T is uniformly L-Lipschitzian, we have
It follows from (3.11) and (3.12) that
(III) Now we prove that △-converges to a point .
Since is bounded, by Lemma 2.1, it has a unique asymptotic center . If is any subsequence of with , then, from (3.12), we have
We claim that . In fact, for any ,
From (3.13), we get
By the definition of the asymptotic center of a bounded sequence with respect C, we have
This implies that
Therefore, we have
By Lemma 2.3, one shows that . Because T is uniformly continuous, we have
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.
Corollary 3.1 Let E be a complete hyperbolic space, C be a nonempty, bounded, closed, convex subset of E, and be the nonexpansive retraction. Let be nonexpansive nonself-mapping such that T be uniformly L-Lipschitz continuous. Define as follows: ,
If there exist constants with such that , and , then the sequence △-converges to a point .
Lim TC: Remarks on some fixed point theorems. Proc. Am. Math. Soc. 1976, 60: 179–182. 10.1090/S0002-9939-1976-0423139-X
Kirk WA, Panyanak B: A concept of convergence in geodesic spaces. Nonlinear Anal. 2008, 68: 3689–3696. 10.1016/j.na.2007.04.011
Dhompongsa S, Panyanak B: On △-convergence theorem in CAT(0) spaces. Comput. Math. Appl. 2008,56(10):2572–2579. 10.1016/j.camwa.2008.05.036
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.2013.28.1.107
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 557
Wan L: Δ-Convergence for mixed-type total asymptotically nonexpansive mappings in hyperbolic spaces. J. Inequal. Appl. 2013., 2013: Article ID 553
Kohlenbach U: Some logical metatheorems with applications in functional analysis. Trans. Am. Math. Soc. 2005, 357: 89–128. 10.1090/S0002-9947-04-03515-9
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.
Reich S, Shafrir I: Nonexpansive iterations in hyperbolic spaces. Nonlinear Anal. 1990, 15: 537–558. 10.1016/0362-546X(90)90058-O
Goebel K, Reich S: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Dekker, New York; 1984.
Bridson M, Haefliger A: Metric Spaces of Non-Positive Curvature. Springer, Berlin; 1999.
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 135
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 54
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.
The authors are very grateful to both reviewers for carefully reading this paper and their comments.
The authors declare that they have no competing interests.
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
About this article
Cite this article
Yi, L., Bo, L.H. △-Convergence analysis of improved Kuhfittig iterative for asymptotically nonexpansive nonself-mappings in W-hyperbolic spaces. J Inequal Appl 2014, 303 (2014). https://doi.org/10.1186/1029-242X-2014-303
- asymptotically nonexpansive nonself-mapping
- improved Kuhfittig iterative
- hyperbolic space
- nonexpansive retraction