- Research Article
- Open access
- Published:
Strong Convergence of Modified Halpern Iterations in CAT(0) Spaces
Fixed Point Theory and Applications volume 2011, Article number: 869458 (2011)
Abstract
Strong convergence theorems are established for the modified Halpern iterations of nonexpansive mappings in CAT(0) spaces. Our results extend and improve the recent ones announced by Kim and Xu (2005), Hu (2008), Song and Chen (2008), Saejung (2010), and many others.
1. Introduction
Let be a nonempty subset of a metric space
. A mapping
is said to be nonexpansive if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F869458/MediaObjects/13663_2010_Article_1439_Equ1_HTML.gif)
A point is called a fixed point of
if
. We will denote by
the set of fixed points of
. In 1967, Halpern [1] introduced an explicit iterative scheme for a nonexpansive mapping
on a subset
of a Hilbert space by taking any points
and defined the iterative sequence
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F869458/MediaObjects/13663_2010_Article_1439_Equ2_HTML.gif)
where . He pointed out that the control conditions: (C1)
and (C2)
are necessary for the convergence of
to a fixed point of
. Subsequently, many mathematicians worked on the Halpern iterations both in Hilbert and Banach spaces (see, e.g., [2–11] and the references therein). Among other things, Wittmann [7] proved strong convergence of the Halpern iteration under the control conditions (C1), (C2), and (C4)
in a Hilbert space. In 2005, Kim and Xu [12] generalized Wittmann's result by introducing a modified Halpern iteration in a Banach space as follows. Let
be a closed convex subset of a uniformly smooth Banach space
, and let
be a nonexpansive mapping. For any points
, the sequence
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F869458/MediaObjects/13663_2010_Article_1439_Equ3_HTML.gif)
where and
are sequences in
. They proved under the following control conditions:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F869458/MediaObjects/13663_2010_Article_1439_Equ4_HTML.gif)
that the sequence converges strongly to a fixed point of
.
The purpose of this paper is to extend Kim-Xu's result to a special kind of metric spaces, namely, CAT(0) spaces. We also prove a strong convergence theorem for another kind of modified Halpern iteration defined by Hu [13] in this setting.
2. CAT(0) Spaces
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. The precise definition is given below. It is well known that any complete, simply connected Riemannian manifold having nonpositive sectional curvature is a CAT(0) space. Other examples include Pre-Hilbert spaces (see [14]),
-trees (see [15]), Euclidean buildings (see [16]), the complex Hilbert ball with a hyperbolic metric (see [17]), and many others. For a thorough discussion of these spaces and of the fundamental role they play in geometry, we refer the reader to Bridson and Haefliger [14].
Fixed point theory in CAT(0) spaces was first studied by Kirk (see [18, 19]). 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 many papers have appeared (see, e.g., [20–31] and the references therein). It is worth mentioning that fixed point theorems in CAT(0) spaces (specially in -trees) can be applied to graph theory, biology, and computer science (see, e.g., [15, 32–35]).
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 satisfy the following comparison axiom.
CAT(0): let be a geodesic triangle in
, and let
be a comparison triangle for
. Then,
is said to satisfy the CAT(0) inequality if for all
and all comparison points
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F869458/MediaObjects/13663_2010_Article_1439_Equ5_HTML.gif)
Let , and by Lemma  2.1 (iv) of [23] for each
, there exists a unique point
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F869458/MediaObjects/13663_2010_Article_1439_Equ6_HTML.gif)
From now on, we will use the notation for the unique point
satisfying (2.2). We now collect some elementary facts about CAT(0) spaces which will be used in the proofs of our main results.
Lemma 2.1.
Let be a
space. Then,
-
(i)
(see [23, Lemma  2.4]) for each
and
, one has
(23)
-
(ii)
(see [21]) for each
and
, one has
(24)
 (iii) (see [19, Lemma  3]) for each and
, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F869458/MediaObjects/13663_2010_Article_1439_Equ9_HTML.gif)
 (iv) (see [23, Lemma  2.5]) for each and
, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F869458/MediaObjects/13663_2010_Article_1439_Equ10_HTML.gif)
Recall that a continuous linear functional on
, the Banach space of bounded real sequences, is called a Banach limit if
and
for all
.
Lemma 2.2 (see [8, Proposition  2]).
Let be such that
for all Banach limits
and
. Then,
.
Lemma 2.3 (see [28, Lemma  2.1]).
Let be a closed convex subset of a complete
space
, and let
be a nonexpansive mapping. Let
be fixed. For each
, the mapping
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F869458/MediaObjects/13663_2010_Article_1439_Equ11_HTML.gif)
has a unique fixed point , that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F869458/MediaObjects/13663_2010_Article_1439_Equ12_HTML.gif)
Lemma 2.4 (see [28, Lemma  2.2]).
Let and
be as the preceding lemma. Then,
if and only if
given by (2.8) remains bounded as
. In this case, the following statements hold:
(1) converges to the unique fixed point
of
which is nearest
,
(2) for all Banach limits
and all bounded sequences
with
.
Lemma 2.5 (see [10, Lemma  2.1]).
Let be a sequence of nonnegative real numbers satisfying the condition
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F869458/MediaObjects/13663_2010_Article_1439_Equ13_HTML.gif)
where and
are sequences of real numbers such that
(1) and
,
-
(2)
either
or
.
Then, .
Let and
be bounded sequences in a
space
, and let
be a sequence in
with
. Suppose that
for all
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F869458/MediaObjects/13663_2010_Article_1439_Equ14_HTML.gif)
Then, .
3. Main Results
The following result is an analog of Theorem  1 of Kim and Xu [12]. They prove the theorem by using the concept of duality mapping, while we use the concept of Banach limit. We also observe that the condition in [12, Theorem  1] is superfluous.
Theorem 3.1.
Let be a nonempty closed convex subset of a complete
space
, and let
be a nonexpansive mapping such that
. Given a point
and sequences
and
in
, the following conditions are satisfied:
(A1) and
,
(A2) ,
and
.
Define a sequence in
by
arbitrarily, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F869458/MediaObjects/13663_2010_Article_1439_Equ15_HTML.gif)
Then, converges to a fixed point
which is nearest
.
Proof.
For each , we let
. We divide the proof into 3 steps. (i) We will show that
,
, and
are bounded sequences. (ii) We show that
. Finally, we show that (iii)
converges to a fixed point
which is nearest
.
-
(i)
As in the first part of the proof of [12, Theorem  1], we can show that
is bounded and so is
and
. Notice also that
(32)
 (ii) It suffices to show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F869458/MediaObjects/13663_2010_Article_1439_Equ17_HTML.gif)
Indeed, if (3.3) holds, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F869458/MediaObjects/13663_2010_Article_1439_Equ18_HTML.gif)
By using Lemma 2.1, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F869458/MediaObjects/13663_2010_Article_1439_Equ19_HTML.gif)
Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F869458/MediaObjects/13663_2010_Article_1439_Equ20_HTML.gif)
where is a constant such that
for all
. By assumptions, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F869458/MediaObjects/13663_2010_Article_1439_Equ21_HTML.gif)
Hence, Lemma 2.5 is applicable to (3.6), and we obtain .
-
(iii)
From Lemma 2.3, let
, where
is given by (2.8). Then,
is the point of
which is nearest
. We observe that
(38)
By Lemma 2.4, we have for all Banach limit
. Moreover, since
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F869458/MediaObjects/13663_2010_Article_1439_Equ23_HTML.gif)
It follows from and Lemma 2.2 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F869458/MediaObjects/13663_2010_Article_1439_Equ24_HTML.gif)
Hence, the conclusion follows from Lemma 2.5.
By using the similar technique as in the proof of Theorem 3.1, we can obtain a strong convergence theorem which is an analog of [13, Theorem  3.1] (see also [37, 38] for subsequence comments).
Theorem 3.2.
Let C be a nonempty closed and convex subset of a complete space
, and let
be a nonexpansive mapping such that
. Given a point
and an initial value
. The sequence
is defined iteratively by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F869458/MediaObjects/13663_2010_Article_1439_Equ25_HTML.gif)
Suppose that both and
are sequences in
satisfying
(B1) ,
(B2) ,
(B3) .
Then, converges to a fixed point
which is nearest
.
Proof.
Let . We divide the proof into 3 steps.
Step 1.
We show that ,
, and
are bounded sequences. Let
, then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F869458/MediaObjects/13663_2010_Article_1439_Equ26_HTML.gif)
Now, an induction yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F869458/MediaObjects/13663_2010_Article_1439_Equ27_HTML.gif)
Hence, is bounded and so are
and
.
Step 2.
We show that . By using Lemma 2.1, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F869458/MediaObjects/13663_2010_Article_1439_Equ28_HTML.gif)
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F869458/MediaObjects/13663_2010_Article_1439_Equ29_HTML.gif)
Since and
are bounded and
, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F869458/MediaObjects/13663_2010_Article_1439_Equ30_HTML.gif)
Hence, by Lemma 2.6, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F869458/MediaObjects/13663_2010_Article_1439_Equ31_HTML.gif)
On the other hand,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F869458/MediaObjects/13663_2010_Article_1439_Equ32_HTML.gif)
Using (3.17) and (3.18), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F869458/MediaObjects/13663_2010_Article_1439_Equ33_HTML.gif)
Step 3.
We show that converges to a fixed point of
. Let
, where
is given by (2.8), then
. Finally, we show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F869458/MediaObjects/13663_2010_Article_1439_Equ34_HTML.gif)
By Lemma 2.4, we have for all Banach limit
. Moreover, since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F869458/MediaObjects/13663_2010_Article_1439_Equ35_HTML.gif)
it follows from condition , 
and Lemma 2.2 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F869458/MediaObjects/13663_2010_Article_1439_Equ36_HTML.gif)
Hence, the conclusion follows by Lemma 2.5.
References
Halpern B: Fixed points of nonexpanding maps. Bulletin of the American Mathematical Society 1967, 73: 957–961. 10.1090/S0002-9904-1967-11864-0
Reich S: Some fixed point problems. Atti della Accademia Nazionale dei Lincei 1974,57(3–4):194–198.
Lions P-L: Approximation de points fixes de contractions. Comptes Rendus de l'Académie des Sciences de Paris A-B 1977,284(21):A1357-A1359.
Reich S: Strong convergence theorems for resolvents of accretive operators in Banach spaces. Journal of Mathematical Analysis and Applications 1980,75(1):287–292. 10.1016/0022-247X(80)90323-6
Reich S: Some problems and results in fixed point theory. In Topological Methods in Nonlinear Functional Analysis (Toronto, Ont., 1982), Contemporary Mathematics. Volume 21. American Mathematical Society, Providence, RI, USA; 1983:179–187.
Takahashi W, Ueda Y: On Reich's strong convergence theorems for resolvents of accretive operators. Journal of Mathematical Analysis and Applications 1984,104(2):546–553. 10.1016/0022-247X(84)90019-2
Wittmann R: Approximation of fixed points of nonexpansive mappings. Archiv der Mathematik 1992,58(5):486–491. 10.1007/BF01190119
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/S0002-9939-97-04033-1
Xu H-K: Iterative algorithms for nonlinear operators. Journal of the London Mathematical Society. Second Series 2002,66(1):240–256. 10.1112/S0024610702003332
Xu H-K: An iterative approach to quadratic optimization. Journal of Optimization Theory and Applications 2003,116(3):659–678. 10.1023/A:1023073621589
Xu H-K: A strong convergence theorem for contraction semigroups in Banach spaces. Bulletin of the Australian Mathematical Society 2005,72(3):371–379. 10.1017/S000497270003519X
Kim T-H, Xu H-K: Strong convergence of modified Mann iterations. Nonlinear Analysis: Theory, Methods & Applications 2005,61(1–2):51–60. 10.1016/j.na.2004.11.011
Hu L-G: Strong convergence of a modified Halpern's iteration for nonexpansive mappings. Fixed Point Theory and Applications 2008, 2008:-9.
Bridson MR, Haefliger A: Metric Spaces of Non-Positive Curvature, Grundlehren der Mathematischen Wissenschaften. Volume 319. Springer, Berlin, Germany; 1999:xxii+643.
Kirk WA: Fixed point theorems in CAT(0) spaces and -trees. Fixed Point Theory and Applications 2004,2004(4):309–316.
Brown KS: Buildings. Springer, New York, NY, USA; 1989:viii+215.
Goebel K, Reich S: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Monographs and Textbooks in Pure and Applied Mathematics. Volume 83. Marcel Dekker Inc., New York, NY, USA; 1984:ix+170.
Kirk WA: Geodesic geometry and fixed point theory. In Seminar of Mathematical Analysis (Malaga/Seville, 2002/2003), Colección Abierta. Volume 64. University of Seville, Secretary of Publications, Seville, Spain; 2003:195–225.
Kirk WA: Geodesic geometry and fixed point theory. II. In International Conference on Fixed Point Theory and Applications. Yokohama Publishers, Yokohama, Japan; 2004:113–142.
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.2005.03.055
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.2005.08.006
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.2006.01.081
Dhompongsa S, Panyanak B: On -convergence theorems in CAT(0) spaces. Computers & Mathematics with Applications 2008,56(10):2572–2579. 10.1016/j.camwa.2008.05.036
Shahzad N: Fixed point results for multimaps in CAT(0) spaces. Topology and Its Applications 2009,156(5):997–1001. 10.1016/j.topol.2008.11.016
Espinola R, Fernandez-Leon A: CAT()-spaces, weak convergence and fixed points. Journal of Mathematical Analysis and Applications 2009,353(1):410–427. 10.1016/j.jmaa.2008.12.015
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.2009.02.126
Laowang W, Panyanak B: Strong and convergence theorems for multivalued mappings in CAT(0) spaces. Journal of Inequalities and Applications 2009, 2009:-16.
Saejung S: Halpern's iteration in CAT(0) spaces. Fixed Point Theory and Applications 2010, 2010:-13.
Khan AR, Khamsi MA, Fukhar-Ud-Din H: Strong convergence of a general iteration scheme in CAT(0) spaces. Nonlinear Analysis: Theory, Methods and Applications 2011,74(3):783–791. 10.1016/j.na.2010.09.029
Khan SH, Abbas M: Strong and -convergence of some iterative schemes in CAT(0) spaces. Computers and Mathematics with Applications 2011,61(1):109–116. 10.1016/j.camwa.2010.10.037
Abkar A, Eslamian M: Common fixed point results in CAT(0) spaces. Nonlinear Analysis: Theory, Methods and Applications 2011,74(5):1835–1840. 10.1016/j.na.2010.10.056
Bestvina M: -trees in topology, geometry, and group theory. In Handbook of Geometric Topology. North-Holland, Amsterdam, The Netherlands; 2002:55–91.
Semple C, Steel M: Phylogenetics, Oxford Lecture Series in Mathematics and Its Applications. Volume 24. Oxford University Press, Oxford, UK; 2003:xiv+239.
Espinola R, Kirk WA: Fixed point theorems in -trees with applications to graph theory. Topology and Its Applications 2006,153(7):1046–1055. 10.1016/j.topol.2005.03.001
Kirk WA: Some recent results in metric fixed point theory. Journal of Fixed Point Theory and Applications 2007,2(2):195–207. 10.1007/s11784-007-0031-8
Suzuki T: Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces. Fixed Point Theory and Applications 2005,2005(1):103–123.
Song Y, Chen R: Strong convergence of an iterative method for non-expansive mappings. Mathematische Nachrichten 2008,281(8):1196–1204. 10.1002/mana.200510670
Wang S: A note on strong convergence of a modified Halpern's iteration for nonexpansive mappings. Fixed Point Theory and Applications 2010, 2010:-2.
Acknowledgments
The authors are grateful to Professor Sompong Dhompongsa for his suggestions and advices during the preparation of the paper. This research was supported by the National Research University Project under Thailand's Office of the Higher Education Commission.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Cuntavepanit, A., Panyanak, B. Strong Convergence of Modified Halpern Iterations in CAT(0) Spaces. Fixed Point Theory Appl 2011, 869458 (2011). https://doi.org/10.1155/2011/869458
Received:
Accepted:
Published:
DOI: https://doi.org/10.1155/2011/869458