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Strong Convergence of an Implicit Algorithm in CAT(0) Spaces
Fixed Point Theory and Applications volume 2011, Article number: 173621 (2011)
Abstract
We establish strong convergence of an implicit algorithm to a common fixed point of a finite family of generalized asymptotically quasi-nonexpansive maps in CAT spaces. Our work improves and extends several recent results from the current literature.
1. Introduction
A metric space is said to be a length space if any two points of
are joined by a rectifiable path (i.e., a path of finite length), and the distance between any two points of
is taken to be the infimum of the lengths of all rectifiable paths joining them. In this case,
is said to be a length metric (otherwise known as an inner metric or intrinsic metric). In case no rectifiable path joins two points of the space, the distance between them is taken to be
.
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 α of
is called a geodesic (or metric) segment joining
and
. We say
is (i) a geodesic space if any two points of
are joined by a geodesic and (ii) uniquely geodesic if there is exactly one geodesic joining
and
for each
, which we will denote by
, called the segment joining
to
.
A geodesic triangle in a geodesic metric space
consists of three points in
(the vertices of
) and a geodesic segment between each pair of vertices (the edges of
). A comparison triangle for geodesic triangle
in
is a triangle
in
such that
for
. Such a triangle always exists (see [1]).
A geodesic metric space is said to be a CAT space if all geodesic triangles of appropriate size satisfy the following CAT
comparison axiom.
Let be a geodesic triangle in
, and let
be a comparison triangle for
. Then
is said to satisfy the CAT
  inequality if for all
and all comparison points
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173621/MediaObjects/13663_2010_Article_1379_Equ1_HTML.gif)
Complete CAT spaces are often called Hadamard spaces (see [2]). If
are points of a CAT
space and
is the midpoint of the segment
, which we will denote by
, then the CAT
inequality implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173621/MediaObjects/13663_2010_Article_1379_Equ2_HTML.gif)
The inequality (1.2) is the () inequality of Bruhat and Titz [3]. The above inequality has been extended in [4] as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173621/MediaObjects/13663_2010_Article_1379_Equ3_HTML.gif)
for any and
.
Let us recall that a geodesic metric space is a CAT  space if and only if it satisfies the (CN) inequality (see [1, page 163]). Moreover, if
is a CAT
metric space and
, then for any
, there exists a unique point
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173621/MediaObjects/13663_2010_Article_1379_Equ4_HTML.gif)
for any and
.
A subset of a CAT
space
is convex if for any
, we have
.
Let be a selfmap on a nonempty subset
of
. Denote the set of fixed points of
by
. We say
is: (i) asymptotically nonexpansive if there is a sequence
with
such that
for all
and
, (ii) asymptotically quasi-nonexpansive if
and there is a sequence
with
such that
for all
,
and
, (iii) generalized asymptotically quasi-nonexpansive [5] if
and there exist two sequences of real numbers
and
with
such that
  for all
,
  and
, (iv) uniformly
-Lipschitzian if for some
,
for all
  and
, and (v) semicompact if for any bounded sequence
in
with
as
, there is a convergent subsequence of
.
Denote the indexing set by
. Let
be the set of
selfmaps of
. Throughout the paper, it is supposed that
. We say condition
is satisfied if there exists a nondecreasing function
with
,
for all
and at least one
such that
for all
where
If in definition (iii), for all
, then
becomes asymptotically quasi-nonexpansive, and hence the class of generalized asymptotically quasi-nonexpansive maps includes the class of asymptotically quasi-nonexpansive maps.
Let be a sequence in a metric space
, and let
be a subset of
. We say that
is: (vi) of monotone type(A) with respect to
if for each
, there exist two sequences
and
of nonnegative real numbers such that
,
and
, (vii) of monotone type(B) with respect to
if there exist sequences
and
of nonnegative real numbers such that
,
and
(also see [6]).
From the above definitions, it is clear that sequence of monotone type(A) is a sequence of monotone type(B) but the converse is not true, in general.
Recently, numerous papers have appeared on the iterative approximation of fixed points of asymptotically nonexpansive (asymptotically quasi-nonexpansive) maps through Mann, Ishikawa, and implicit iterates in uniformly convex Banach spaces, convex metric spaces and CAT spaces (see, e.g., [5, 7–16]).
Using the concept of convexity in CAT spaces, a generalization of Sun's implicit algorithm [15] is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173621/MediaObjects/13663_2010_Article_1379_Equ5_HTML.gif)
where .
Starting from arbitrary , the above process in the compact form is written as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173621/MediaObjects/13663_2010_Article_1379_Equ6_HTML.gif)
where ,  
and
is a positive integer such that
as
.
In a normed space, algorithm (1.6) can be written as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173621/MediaObjects/13663_2010_Article_1379_Equ7_HTML.gif)
where ,  
and
is a positive integer such that
as
.
The algorithms (1.6)-(1.7) exist as follows.
Let be a CAT
space. Then, the following inequality holds:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173621/MediaObjects/13663_2010_Article_1379_Equ8_HTML.gif)
for all (see [17]).
Let be the set of
uniformly
-Lipschitzian selfmaps of
. We show that (1.6) exists. Let  
and
. Define
by:
for all
. The existence of
is guaranteed if
has a fixed point. For any
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173621/MediaObjects/13663_2010_Article_1379_Equ9_HTML.gif)
Now, is a contraction if
or
. As
, therefore
is a contraction even if
. By the Banach contraction principle,
has a unique fixed point. Thus, the existence of
is established. Similarly, we can establish the existence of
. Thus, the implicit algorithm (1.6) is well defined. Similarly, we can prove that (1.7) exists.
For implicit iterates, Xu and Ori [16] proved the following theorem.
Theorem XO (see [16, Theorem  2]).
Let be nonexpansive selfmaps on a closed convex subset
of a Hilbert space with
, let
, and let
be a sequence in
such that
. Then, the sequence
, where
and
, converges weakly to a point in
.
They posed the question: what conditions on the maps and (or) the parameters
are sufficient to guarantee strong convergence of the sequence in Theorem XO?
The aim of this paper is to study strong convergence of iterative algorithm (1.6) for the class of uniformly -Lipschitzian and generalized asymptotically quasi-nonexpansive selfmaps on a CAT
space. Thus, we provide a positive answer to Xu and Ori's question for the general class of maps which contains asymptotically quasi-nonexpansive, asymptotically nonexpansive, quasi-nonexpansive, and nonexpansive maps in the setup of CAT
spaces. It is worth mentioning that if an implicit iteration algorithm without an error term converges, then the method of proof generally carries over easily to algorithm with bounded error terms. Thus, our results also hold if we add bounded error terms to the implicit iteration scheme considered. Our results constitute generalizations of several important known results.
We need the following useful lemma for the development of our convergence results.
Lemma 1.1 (see [14, Lemma  1.1]).
Let and
be two nonnegative sequences of real numbers, satisfying the following condition:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173621/MediaObjects/13663_2010_Article_1379_Equ10_HTML.gif)
If , then
exists.
2. Convergence in CAT(0) Spaces
We establish some convergence results for the algorithm (1.6) to a common fixed point of a finite family of uniformly -Lipschitzian and generalized asymptotically quasi-nonexpansive selfmaps in the general class of CAT
spaces. The following result extends Theorem XO; our methods of proofs are based on the ideas developed in [15].
Theorem 2.1.
Let be a complete CAT
space, and let
be a nonempty closed convex subset of
. Let
be
uniformly
-Lipschitzian and generalized asymptotically quasi-nonexpansive selfmaps of
with
,
such that
and
for all
. Suppose that
is closed. Starting from arbitrary
, define the sequence
by the algorithm (1.6), where
for some
. Then,
is of monotone type(A) and monotone type(B) with respect to
. Moreover,
converges strongly to a common fixed point of the maps
if and only if
.
Proof.
First, we show that   is of monotone type(A) and monotone type(B) with respect to
. Let
. Then, from (1.6), we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173621/MediaObjects/13663_2010_Article_1379_Equ11_HTML.gif)
Since , the above inequlaity gives that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173621/MediaObjects/13663_2010_Article_1379_Equ12_HTML.gif)
On simplification, we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173621/MediaObjects/13663_2010_Article_1379_Equ13_HTML.gif)
Let and
. Since
for all
, therefore
, and hence, there exists a natural number
such that
for
or
. Then, we have that
. Similarly,
.
Now, from (2.3), for , we get that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173621/MediaObjects/13663_2010_Article_1379_Equ14_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173621/MediaObjects/13663_2010_Article_1379_Equ15_HTML.gif)
These inequalities, respectively, prove that is a sequence of monotone type(A) and monotone type(B) with respect to
.
Next, we prove that converges strongly to a common fixed point of the maps
if and only if
.
If , then
. Since
, we have
.
Conversely, suppose that . Applying Lemma 1.1 to (2.5), we have that
exists. Further, by assumption
, we conclude that
. Next, we show that
is a Cauchy sequence.
Since for
, therefore from (2.4), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173621/MediaObjects/13663_2010_Article_1379_Equ16_HTML.gif)
for the natural numbers , where
. Since
, therefore for any
, there exists a natural number
such that
and
for all
. So, we can find
such that
. Hence, for all
and
, we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173621/MediaObjects/13663_2010_Article_1379_Equ17_HTML.gif)
This proves that is a Cauchy sequence. Let
. Since
is closed, therefore
. Next, we show that
. Now, the following two inequalities:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173621/MediaObjects/13663_2010_Article_1379_Equ18_HTML.gif)
give that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173621/MediaObjects/13663_2010_Article_1379_Equ19_HTML.gif)
That is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173621/MediaObjects/13663_2010_Article_1379_Equ20_HTML.gif)
As and
, we conclude that
.
We deduce some results from Theorem 2.1 as follows.
Corollary 2.2.
Let be a complete CAT
space, and let
be a nonempty closed convex subset of
. Let
be
uniformly
-Lipschitzian and generalized asymptotically quasi-nonexpansive selfmaps of
with
,
such that
and
for all
. Suppose that
is closed. Starting from arbitaray
, define the sequence
by the algorithm (1.6), where
for some
. Then,
converges strongly to a common fixed point of the maps
if and only if there exists some subsequence
of
which converges to
.
Corollary 2.3.
Let be a complete CAT
space, and let
be a nonempty closed convex subset of
. Let
be
uniformly
-Lipschitzian and asymptotically quasi-nonexpansive selfmaps of
with
such that
for all
. Starting from arbitaray
, define the sequence
by the algorithm (1.6), where
for some
. Then,
is of monotone type(A) and monotone type(B) with respect to
. Moreover,
converges strongly to a common fixed point of the maps
if and only if
.
Proof.
Follows from Theorem 2.1 with for all
.
Corollary 2.4.
Let be a Banach space, and let
be a nonempty closed convex subset of
. Let
be
asymptotically quasi-nonexpansive self-maps of
with
such that
for all
. Starting from arbitaray
, define the sequence
by the algorithm (1.7), where
for some
. Then,
is of monotone type(A) and monotone type(B) with respect to
. Moreover,
converges strongly to a common fixed point of the maps
if and only if
.
Proof.
Take in Corollary 2.3.
The lemma to follow establishes an approximate sequence, and as a consequence of that, we find another strong convergence theorem for (1.6).
Lemma 2.5.
Let be a complete CAT
space, and let
be a nonempty closed convex subset of
. Let
be
uniformly
-Lipschitzian and generalized asymptotically quasi-nonexpansive selfmaps of
with
,
such that
and
for all
. Suppose that
is closed. Let
for some
. From arbitaray
, define the sequence
by (1.6). Then,
for all
.
Proof.
Note that is bounded as
exists (proved in Theorem 2.1). So, there exists
and
such that
for all
. Denote
by
.
We claim that .
For any , apply (1.3) to (1.6) and get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173621/MediaObjects/13663_2010_Article_1379_Equ21_HTML.gif)
further, using (2.4), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173621/MediaObjects/13663_2010_Article_1379_Equ22_HTML.gif)
which implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173621/MediaObjects/13663_2010_Article_1379_Equ23_HTML.gif)
for some consant . This gives that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173621/MediaObjects/13663_2010_Article_1379_Equ24_HTML.gif)
where .
For , we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173621/MediaObjects/13663_2010_Article_1379_Equ25_HTML.gif)
When , we have that
as
.
Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173621/MediaObjects/13663_2010_Article_1379_Equ26_HTML.gif)
Further,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173621/MediaObjects/13663_2010_Article_1379_Equ27_HTML.gif)
implies that .
For a fixed , we have
, and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173621/MediaObjects/13663_2010_Article_1379_Equ28_HTML.gif)
For ,
. Also,
. Hence,
.
That is, and
.
Therefore, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173621/MediaObjects/13663_2010_Article_1379_Equ29_HTML.gif)
which together with (2.16) and (2.18) yields that .
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173621/MediaObjects/13663_2010_Article_1379_Equ30_HTML.gif)
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173621/MediaObjects/13663_2010_Article_1379_Equ31_HTML.gif)
Hence, for all ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173621/MediaObjects/13663_2010_Article_1379_Equ32_HTML.gif)
together with (2.18) and (2.21) implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F173621/MediaObjects/13663_2010_Article_1379_Equ33_HTML.gif)
Thus, for all
.
Theorem 2.6.
Let be a complete CAT
space, and let
be a nonempty closed convex subset of
. Let
be
-uniformly
-Lipschitzian and generalized asymptotically quasi-nonexpansive selfmaps of
with
,
such that
and
for all
. Suppose that
is closed, and there exists one member
in
which is either semicompact or satisfies condition (
). Let
for some
. From arbitaray
, define the sequence
by algorithm (1.6). Then,
converges strongly to a common fixed point of the maps in
.
Proof.
Without loss of generality, we may assume that is either semicompact or satisfies condition (
). If
is semicompact, then there exists a subsequence
of
such that
as
. Now, Lemma 2.5 guarantees that
for all
and so
for all
. This implies that
. Therefore,
. If
satisfies condition (
), then we also have
. Now, Theorem 2.1 gaurantees that
converges strongly to a point in
.
Finally, we state two corollaries to the above theorem.
Corollary 2.7.
Let be a complete CAT
space and let
be a nonempty closed convex subset of
. Let
be
uniformly
-Lipschizian and asymptotically quasi-nonexpansive selfmaps of
with
such that
for all
. Suppose that there exists one member
in
which is either semicompact or satisfies condition (
). From arbitaray
, define the sequence
by algorithm (1.6), where
for some
. Then,
converges strongly to a common fixed point of the maps in
.
Corollary 2.8.
Let be a complete CAT
space, and let
be a nonempty closed convex subset of
. Let
be
asymptotically nonexpansive selfmaps of
with
such that
for all
. Suppose that there exists one member
in
which is either semicompact or satisfies condition (
). From arbitrary
, define the sequence
by algorithm (1.6), where
for some
. Then,
converges strongly to a common fixed point of the maps in
.
Remark 2.9.
The corresponding approximation results for a finite family of asymptotically quasi-nonexpansive maps on: (i) uniformly convex Banach spaces [5, 14, 15], (ii) convex metric spaces [13], (iii) CAT spaces [12] are immediate consequences of our results.
Remark 2.10.
Various algorithms and their strong convergence play an important role in finding a common element of the set of fixed (common fixed) point for different classes of mapping(s) and the set of solutions of an equilibrium problem in the framework of Hilbert spaces and Banach spaces; for details we refer to [18–20].
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Acknowledgments
The author A. R. Khan gratefully acknowledges King Fahd University of Petroleum and Minerals and SABIC for supporting research project no. SB100012.
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Fukhar-ud-din, H., Domlo, A. & Khan, A. Strong Convergence of an Implicit Algorithm in CAT(0) Spaces. Fixed Point Theory Appl 2011, 173621 (2011). https://doi.org/10.1155/2011/173621
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DOI: https://doi.org/10.1155/2011/173621