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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 
,
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
The inequality (1.2) is the (
) inequality of Bruhat and Titz [3]. The above inequality has been extended in [4] as
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
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
where 
.
Starting from arbitrary 
, the above process in the compact form is written as
where 
,  
and 
is a positive integer such that 
as 
.
In a normed space, algorithm (1.6) can be written as
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:
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
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:
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
Since 
, the above inequlaity gives that
On simplification, we have that
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
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
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
This proves that 
is a Cauchy sequence. Let 
. Since 
is closed, therefore 
. Next, we show that 
. Now, the following two inequalities:
give that
That is,
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
further, using (2.4), we obtain
which implies that
for some consant 
. This gives that
where 
.
For 
, we have that
When 
, we have that 
as 
.
Hence,
Further,
implies that 
.
For a fixed 
, we have 
, and hence
For 
, 
. Also, 
. Hence, 
.
That is, 
and 
.
Therefore, we have
which together with (2.16) and (2.18) yields that 
.
Since
we have
Hence, for all 
,
together with (2.18) and (2.21) implies that
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].
References
Bridson MR, Haefliger A: Metric Spaces of Non-Positive Curvature, Grundlehren der Mathematischen Wissenschaften. Volume 319. Springer, Berlin, Germany; 1999:xxii+643.
Khamsi MA, Kirk WA: An Introduction to Metric Spaces and Fixed Point Theory, Pure and Applied Mathematics. Wiley-Interscience, New York, NY, USA; 2001:x+302.
Bruhat F, Tits J: Groupes réductifs sur un corps local. Institut des Hautes Études Scientifiques. Publications Mathématiques 1972, (41):5–251.
Dhompongsa S, Panyanak B: On
-convergence theorems in CAT(0) spaces. Computers & Mathematics with Applications 2008,56(10):2572–2579. 10.1016/j.camwa.2008.05.036Imnang S, Suantai S: Common fixed points of multistep Noor iterations with errors for a finite family of generalized asymptotically quasi-nonexpansive mappings. Abstract and Applied Analysis 2009, 2009:-14.
Zhou HY, Gao GL, Guo GT, Cho YJ: Some general convergence principles with applications. Bulletin of the Korean Mathematical Society 2003,40(3):351–363.
Fukhar-ud-din H, Khan AR: Approximating common fixed points of asymptotically nonexpansive maps in uniformly convex Banach spaces. Computers & Mathematics with Applications 2007,53(9):1349–1360. 10.1016/j.camwa.2007.01.008
Fukhar-ud-din H, Khan AR: Convergence of implicit iterates with errors for mappings with unbounded domain in Banach spaces. International Journal of Mathematics and Mathematical Sciences 2005, (10):1643–1653.
Fukhar-ud-din H, Khan AR, O'Regan D, Agarwal RP: An implicit iteration scheme with errors for a finite family of uniformly continuous mappings. Functional Differential Equations 2007,14(2–4):245–256.
Fukhar-ud-din H, Khan SH: Convergence of iterates with errors of asymptotically quasi-nonexpansive mappings and applications. Journal of Mathematical Analysis and Applications 2007,328(2):821–829. 10.1016/j.jmaa.2006.05.068
Guo W, Cho YJ: On the strong convergence of the implicit iterative processes with errors for a finite family of asymptotically nonexpansive mappings. Applied Mathematics Letters 2008,21(10):1046–1052. 10.1016/j.aml.2007.07.034
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 AR, Ahmed MA: Convergence of a general iterative scheme for a finite family of asymptotically quasi-nonexpansive mappings in convex metric spaces and applications. Computers & Mathematics with Applications 2010,59(8):2990–2995. 10.1016/j.camwa.2010.02.017
Khan AR, Domlo A-A, Fukhar-ud-din H: Common fixed points Noor iteration for a finite family of asymptotically quasi-nonexpansive mappings in Banach spaces. Journal of Mathematical Analysis and Applications 2008,341(1):1–11. 10.1016/j.jmaa.2007.06.051
Sun Z: Strong convergence of an implicit iteration process for a finite family of asymptotically quasi-nonexpansive mappings. Journal of Mathematical Analysis and Applications 2003,286(1):351–358. 10.1016/S0022-247X(03)00537-7
Xu H-K, Ori RG: An implicit iteration process for nonexpansive mappings. Numerical Functional Analysis and Optimization 2001,22(5–6):767–773. 10.1081/NFA-100105317
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
Tada A, Takahashi W: Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem. Journal of Optimization Theory and Applications 2007, 133: 359–370. 10.1007/s10957-007-9187-z
Ceng L-C, Al-Homidan S, Ansari QH, Yao J-C: An iterative scheme for equilibrium problems and fixed point problems of strict pseudo-contraction mappings. Journal of Computational and Applied Mathematics 2009, 223: 967–974. 10.1016/j.cam.2008.03.032
Qin X, Cho YJ, Kang SM: Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces. Journal of Computational and Applied Mathematics 2009, 225: 20–30. 10.1016/j.cam.2008.06.011
-convergence theorems in CAT(0) spaces. Computers & Mathematics with Applications 2008,56(10):2572–2579. 10.1016/j.camwa.2008.05.036Acknowledgments
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