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Approximating Fixed Points of Nonexpansive Nonself Mappings in CAT(0) Spaces
Fixed Point Theory and Applications volume 2010, Article number: 367274 (2009)
Abstract
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.
1. Introduction
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 [1]), Euclidean buildings (see [2]), the complex Hilbert ball with a hyperbolic metric (see [3]), and many others. For a thorough discussion of these spaces and of the fundamental role they play in geometry see Bridson and Haefliger [1]. The work by Burago et al. [4] contains a somewhat more elementary treatment, and by Gromov [5] 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 [20] used the concept of -convergence introduced by Lim [21] to prove the CAT(0) space analogs of some Banach space results which involve weak convergence, and Dhompongsa and Panyanak [22] 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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F367274/MediaObjects/13663_2009_Article_1268_Equ1_HTML.gif)
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 [23] and also extends a result of Dhompongsa and Panyanak [22] 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]).
2. Preliminaries and Lemmas
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.
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%2F2010%2F367274/MediaObjects/13663_2009_Article_1268_Equ2_HTML.gif)
If are points in a CAT(0) space and if
is the midpoint of the segment
then the CAT(0) inequality implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F367274/MediaObjects/13663_2009_Article_1268_Equ3_HTML.gif)
This is the (CN) inequality of Bruhat and Tits [24]. In fact (cf. [1, page 163]), a geodesic space is a CAT(0) space if and only if it satisfies the (CN) inequality.
We now collect some elementary facts about CAT(0) spaces which will be used frequently in the proofs of our main results.
Lemma 2.1.
Let be a CAT(0) space.
-
(i)
[1, Proposition  2.4] Let
be a convex subset of
which is complete in the induced metric. Then, for every
there exists a unique point
such that
Moreover, the map
is a nonexpansive retract from
onto
-
(ii)
[22, Lemma  2.1(iv)] For
and
there exists a unique point
such that
(2.2)
one uses the notation for the unique point
satisfying (2.2).
-
(iii)
[22, Lemma  2.4] For
and
one has
(2.3)
-
(iv)
[22, Lemma  2.5] For
and
one has
(2.4)
Let be a nonempty subset of a CAT(0) space
and let
be a mapping.
is called nonexpansive if for each
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F367274/MediaObjects/13663_2009_Article_1268_Equ7_HTML.gif)
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 [6] as follows.
Theorem 2.2.
Let be a bounded closed convex subset of a complete CAT(0) space
. Suppose that
is a nonexpansive mapping for which
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F367274/MediaObjects/13663_2009_Article_1268_Equ8_HTML.gif)
Then has a fixed point in
Let be a bounded sequence in a CAT(0) space
. For
we set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F367274/MediaObjects/13663_2009_Article_1268_Equ9_HTML.gif)
The asymptotic radius of
is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F367274/MediaObjects/13663_2009_Article_1268_Equ10_HTML.gif)
and the asymptotic center of
is the set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F367274/MediaObjects/13663_2009_Article_1268_Equ11_HTML.gif)
It is known (see, e.g., [12, Proposition ]) that in a CAT(0) space,
consists of exactly one point.
We now give the definition of -convergence.
Definition 2.3 (see [20, 21]).
A sequence in a CAT(0) space
is said to
-converge to
if
is the unique asymptotic center of
for every subsequence
of
. In this case one writes
-
and call
the
-limit of
The following lemma was proved by Dhompongsa and Panyanak (see [22, Lemma ]).
Lemma 2.4.
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 [16]).
A hyperbolic space is a triple where
is a metric space and
is such that
(W1)
(W2)
(W3)
(W4)
for all
It follows from (W1) that for each and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F367274/MediaObjects/13663_2009_Article_1268_Equ12_HTML.gif)
In fact, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F367274/MediaObjects/13663_2009_Article_1268_Equ13_HTML.gif)
since if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F367274/MediaObjects/13663_2009_Article_1268_Equ14_HTML.gif)
we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F367274/MediaObjects/13663_2009_Article_1268_Equ15_HTML.gif)
which is a contradiction. By comparing between (2.2) and (2.11), we can also use the notation for
in a hyperbolic space
Definition 2.6 (see [16]).
The hyperbolic space is called uniformly convex if for any
and
there exists a
such that for all
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F367274/MediaObjects/13663_2009_Article_1268_Equ16_HTML.gif)
A mapping providing such a
for given
and
is called a modulus of uniform convexity.
Lemma 2.7 (see [16, Lemma ]).
Let be a uniformly convex hyperbolic with modulus of uniform convexity
For any
,
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F367274/MediaObjects/13663_2009_Article_1268_Equ17_HTML.gif)
Lemma 2.8 (see [16, Proposition ]).
Assume that is a CAT(0) space. Then
is uniformly convex, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F367274/MediaObjects/13663_2009_Article_1268_Equ18_HTML.gif)
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 [25]. It can be applied to a CAT(0) space as well.
Lemma 2.9.
Let be a uniformly convex hyperbolic space with modulus of convexity
, and let
. Suppose that
increases with
(for a fixed
) and suppose that
is a sequence in
for some
and
,
are sequences in
such that
,
and
for some
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F367274/MediaObjects/13663_2009_Article_1268_Equ19_HTML.gif)
Proof.
The case is trivial. Now suppose
. If it is not the case that
as
then there are subsequences, denoted by
and
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F367274/MediaObjects/13663_2009_Article_1268_Equ20_HTML.gif)
Choose such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F367274/MediaObjects/13663_2009_Article_1268_Equ21_HTML.gif)
Since and
This implies
Choose
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F367274/MediaObjects/13663_2009_Article_1268_Equ22_HTML.gif)
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F367274/MediaObjects/13663_2009_Article_1268_Equ23_HTML.gif)
there are further subsequences again denoted by and
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F367274/MediaObjects/13663_2009_Article_1268_Equ24_HTML.gif)
Then by Lemma 2.7 and (2.20),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F367274/MediaObjects/13663_2009_Article_1268_Equ25_HTML.gif)
for all Taking
we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F367274/MediaObjects/13663_2009_Article_1268_Equ26_HTML.gif)
which contradicts to the hypothesis.
3. Main Results
In this section, we prove our main theorems.
Theorem 3.1.
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.
Proof.
By Lemma 2.1(i) the nearest point projection is nonexpansive. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F367274/MediaObjects/13663_2009_Article_1268_Equ27_HTML.gif)
Consequently, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F367274/MediaObjects/13663_2009_Article_1268_Equ28_HTML.gif)
This implies that is bounded and decreasing. Hence
exists.
Theorem 3.2.
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F367274/MediaObjects/13663_2009_Article_1268_Equ29_HTML.gif)
Proof.
Let Then, by Theorem 3.1,
exists. Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F367274/MediaObjects/13663_2009_Article_1268_Equ30_HTML.gif)
If then by the nonexpansiveness of
the conclusion follows. If
, we let
By Lemma 2.1(iv) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F367274/MediaObjects/13663_2009_Article_1268_Equ31_HTML.gif)
Therefore
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F367274/MediaObjects/13663_2009_Article_1268_Equ32_HTML.gif)
It follows from (3.6) and Lemma 2.1(iv) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F367274/MediaObjects/13663_2009_Article_1268_Equ33_HTML.gif)
Therefore
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F367274/MediaObjects/13663_2009_Article_1268_Equ34_HTML.gif)
where Since
By (3.8), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F367274/MediaObjects/13663_2009_Article_1268_Equ35_HTML.gif)
This implies
Since is nonexpansive, we get that
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F367274/MediaObjects/13663_2009_Article_1268_Equ36_HTML.gif)
On the other hand, we can get from (3.6) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F367274/MediaObjects/13663_2009_Article_1268_Equ37_HTML.gif)
Thus . This fact and (3.6) imply
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F367274/MediaObjects/13663_2009_Article_1268_Equ38_HTML.gif)
Since is nonexpansive,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F367274/MediaObjects/13663_2009_Article_1268_Equ39_HTML.gif)
It follows from (3.4), (3.12), (3.13), and Lemma 2.9 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F367274/MediaObjects/13663_2009_Article_1268_Equ40_HTML.gif)
This completes the proof.
The following theorem is an analog of [23, Theorem ] and extends [22, Theorem
] to nonself mappings.
Theorem 3.3.
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
Proof.
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 ([26]) if there exists a nondecreasing function
with
and
for all
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F367274/MediaObjects/13663_2009_Article_1268_Equ41_HTML.gif)
Theorem 3.4.
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
Theorem 3.5.
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 [23] 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.
Theorem 3.6.
Let be a nonempty closed convex subset of a complete CAT(0) space
and let
be two nonexpansive mappings with
Let
and
be sequences in
for some
From arbitrary
define the sequence
by the recursion
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F367274/MediaObjects/13663_2009_Article_1268_Equ42_HTML.gif)
Then -converges to a common fixed point of
and
References
Bridson M, Haefliger A: Metric Spaces of Non-Positive Curvature, Fundamental Principles of Mathematical Sciences. Volume 319. Springer, Berlin, Germany; 1999:xxii+643.
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, New York, NY, USA; 1984:ix+170.
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.
Gromov M: Metric Structures for Riemannian and Non-Riemannian Spaces, Progress in Mathematics. Volume 152. Birkhäuser, Boston, Mass, USA; 1999:xx+585.
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.
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.
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
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.2008.09.012
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
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.
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.2005.09.044
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
Kaewcharoen A, Kirk WA: Proximinality in geodesic spaces. Abstract and Applied Analysis 2006, Article ID 43591 2006:-10.
Kirk WA: Fixed point theorems in CAT(0) spaces and -trees. Fixed Point Theory and Applications 2004,2004(4):309–316.
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
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
Shahzad N: Invariant approximations in CAT(0) spaces. Nonlinear Analysis: Theory, Methods & Applications 2009,70(12):4338–4340. 10.1016/j.na.2008.10.002
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.2007.04.041
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.2007.04.011
Lim TC: Remarks on some fixed point theorems. Proceedings of the American Mathematical Society 1976, 60: 179–182. 10.1090/S0002-9939-1976-0423139-X
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: 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.2005.01.092
Bruhat F, Tits J: Groupes réductifs sur un corps local. Publications Mathématiques de l'Institut des Hautes Études Scientifiques 1972, (41):5–251.
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/S0004972700028884
Senter HF, Dotson, WG Jr.: Approximating fixed points of nonexpansive mappings. Proceedings of the American Mathematical Society 1974, 44: 375–380. 10.1090/S0002-9939-1974-0346608-8
Acknowledgments
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.
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Laowang, W., Panyanak, B. Approximating Fixed Points of Nonexpansive Nonself Mappings in CAT(0) Spaces. Fixed Point Theory Appl 2010, 367274 (2009). https://doi.org/10.1155/2010/367274
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DOI: https://doi.org/10.1155/2010/367274