Open Access

Strong Convergence of an Implicit Algorithm in CAT(0) Spaces

Fixed Point Theory and Applications20102011:173621

https://doi.org/10.1155/2011/173621

Received: 23 November 2010

Accepted: 23 December 2010

Published: 29 December 2010

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 ,
(1.1)
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
(1.2)
The inequality (1.2) is the ( ) inequality of Bruhat and Titz [3]. The above inequality has been extended in [4] as
(1.3)

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
(1.4)

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, 716]).

Using the concept of convexity in CAT spaces, a generalization of Sun's implicit algorithm [15] is given by
(1.5)

where .

Starting from arbitrary , the above process in the compact form is written as
(1.6)

where ,   and is a positive integer such that as .

In a normed space, algorithm (1.6) can be written as
(1.7)

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:
(1.8)

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
(1.9)

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:
(1.10)

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
(2.1)
Since , the above inequlaity gives that
(2.2)
On simplification, we have that
(2.3)

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
(2.4)
(2.5)

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
(2.6)
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
(2.7)
This proves that is a Cauchy sequence. Let . Since is closed, therefore . Next, we show that . Now, the following two inequalities:
(2.8)
give that
(2.9)
That is,
(2.10)

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
(2.11)
further, using (2.4), we obtain
(2.12)
which implies that
(2.13)
for some consant . This gives that
(2.14)

where .

For , we have that
(2.15)

When , we have that as .

Hence,
(2.16)
Further,
(2.17)

implies that .

For a fixed , we have , and hence
(2.18)

For , . Also, . Hence, .

That is, and .

Therefore, we have
(2.19)

which together with (2.16) and (2.18) yields that .

Since
(2.20)
we have
(2.21)
Hence, for all ,
(2.22)
together with (2.18) and (2.21) implies that
(2.23)

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 [1820].

Declarations

Acknowledgments

The author A. R. Khan gratefully acknowledges King Fahd University of Petroleum and Minerals and SABIC for supporting research project no. SB100012.

Authors’ Affiliations

(1)
Department of Mathematics, The Islamia University of Bahawalpur
(2)
Department of Mathematics, Taibah University
(3)
Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals

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© Hafiz Fukhar-ud-din et al. 2011

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