Abstract and Applied Analysis

Volume 2013 (2013), Article ID 503198, 8 pages

http://dx.doi.org/10.1155/2013/503198

Research Article

Implicit Ishikawa Approximation Methods for Nonexpansive Semigroups in CAT(0) Spaces

Zhi-bin Liu,1,2 Yi-shen Chen,1,3 Xue-song Li,⁴ and Yi-bin Xiao⁵

¹Southwest Petroleum University, Chengdu 610500, China

²State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Chengdu 610500, China

³China Petroleum Engineering and Construct Corporation, Beijing 100120, China

⁴Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China

⁵School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan 610054, China

Received 29 November 2012; Accepted 12 June 2013

Academic Editor: Mohamed Amine Khamsi

Copyright © 2013 Zhi-bin Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper is devoted to the convergence of the implicit Ishikawa iteration processes for approximating a common fixed point of nonexpansive semigroup in CAT(0) spaces. We obtain the -convergence results of the implicit Ishikawa iteration sequences for a family of nonexpansive mappings in CAT(0) spaces. Under certain and different conditions, we also get the strong convergence theorems of implicit Ishikawa iteration sequences for nonexpansive semigroups in the CAT(0) spaces. The results presented in this paper extend and generalize some previous results.

1. Introduction

Let be a metric space and be a subset of . A mapping is said to be nonexpansive if for all . We denote the set of all nonnegative elements in by and denote the set of all fixed points of by , that is,

For each , let be nonexpansive mappings and denote the common fixed points set of the family by . A family of mappings is said to be uniformly asymptotically regular if, for any bounded subset of ,

for all .

A nonexpansive semigroup is a family,

of mappings on such that

(1) for all and ; (2) is nonexpansive for each ; (3)for each , the mapping from to is continuous.

We denote by the common fixed points set of nonexpansive semigroup , that is,

Note that, if is compact, then is nonempty (see [1, 2, 28]).

A geodesic from to in is a mapping 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 . The space is said to be a geodesic space if any two points of are joined by a geodesic, and is said to be uniquely geodesic if there is exactly one geodesic joining and for any , which is denoted by and is called the segment joining and . A subset of a geodesic space is said to be convex if for any , .

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 the geodesic triangle in is a triangle in such that for all . It is known that such a triangle always exists (see [3]). A geodesic space is said to be a CAT(0) space if all geodesic triangles of appropriate size satisfy the following comparison axiom (CA).

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 ,

The complete CAT(0) spaces are often called, Hadamard spaces (see [4]). For any , we denote by the unique point which satisfies

It is known that if is a CAT(0) space and , then for any , there exists a unique point . For any , the following inequality holds:

where (for metric spaces of hyperbolic type, see [5]).

Recently, Cho et al. [6] proved the strong convergence of an explicit iterative sequence in a CAT(0) space, where is generated by the following iterative scheme for a nonexpansive semigroup :

where and . The existence of fixed points, an invariant approximation, and convergence theorems for several mappings in CAT(0) spaces have been studied by many authors (see [7–18]).

On the other hand, Thong [19] considered an implicit Mann iteration process for a nonexpansive semigroup on a closed convex subset of a Banach space as follows:

Under different conditions, Thong [19] proved the weak convergence and strong convergence results of the implicit Mann iteration scheme (9) for nonexpansive semigroups in certain Banach spaces. Many authors have studied the convergence of implicit iteration sequences for nonexpansive mappings, nonexpansive semigroups and pseudocontractive semigroups in the Banach spaces (see [20–23]). Readers may consult [24, 25] for the convergence of the Ishikawa iteration sequences for nonexpansive mappings and nonexpansive semigroups in certain Banach spaces. However, to our best knowledge, there is no paper to study the convergence of the implicit Ishikawa type iteration processes for nonexpansive semigroups in CAT(0) spaces. Therefore, it is of interest to investigate the convergence of implicit Ishikawa type iteration processes for nonexpansive semigroups in CAT(0) spaces under some suitable conditions.

Motivated and inspired by the work mentioned previously, we consider the following implicit Ishikawa iteration scheme for a family of nonexpansive mappings in a CAT(0) space:

where and are given sequences. We prove that generated by (10) is -convergent to some point in under appropriate conditions. We also consider the following implicit Ishikawa iteration process for a nonexpansive semigroup in a CAT(0) space:

where and are given sequences. Under various and appropriate conditions, we obtain that generated by (11) converges strongly to a common fixed point of . The results presented in this paper extend and generalize some previous results in [6, 19].

2. Definitions and Lemmas

Let be a bounded sequence in a CAT(0) space . For any , denote

Consider the following:

(i) is called the asymptotic radius of ;(ii) is called the asymptotic radius of with respect to ;(iii)the set is called the asymptotic center of ;(iv)the set is called the asymptotic center of with respect to .

Definition 1 (see [12, 26]). A sequence in a CAT(0) space is said to be -convergent to a point in , if is the unique asymptotic center of for all subsequences . In this case, we write -, and is called the -limit of .

For the sake of convenience, we restate the following lemmas that shall be used.

Lemma 2 (see [10]). Let be a CAT(0) space. Then,

for all and .

Lemma 3 (see [10]). Let be a CAT(0) space. Then,

for all and .

Lemma 4 (see [10]). Let be a closed convex subset of a complete CAT(0) space and be a nonexpansive mapping. Suppose that is a bounded sequence in such that and converges for all . Then, , where the union is taken over all subsequences of . Moreover, consists of exactly one point.

Lemma 5 (see [6]). Let and be bounded sequences in a CAT(0) space . Let be a sequence in such that . Define for all and suppose that

Then, .

3. Main Results

It is necessary for us to show that the implicit Ishikawa iteration sequences generated by schemes (10) and (11) are well defined, before providing the main results of this present paper.

Lemma 6. Let be a nonempty, closed, and convex subset of a complete CAT(0) space and be nonexpansive mappings. Suppose that and are given parameter sequences. Then, the sequence generated by the implicit Ishikawa iteration process (10) is well defined.

Proof. For each and any given , define a mapping by

It can be verified that for any fixed , is a contractive mapping. Indeed, if setting and , then we have and . It follows from Lemmas 3 and 2 that

Consequently, , and thus,

which shows that for each , is a contractive mapping. By induction, Banach’s fixed theorem yields that the sequence generated by (10) is well defined. This completes the proof.

We need the following lemma for our main results. The analogs of [6, Lemma 3.1] and [27, Lemma 2.2] are given in what follows. We sketch the proof here for the convenience of the reader.

Lemma 7. Let be a nonempty, closed, and convex subset of a complete CAT(0) space and be nonexpansive mappings. Let and be given sequences such that . Suppose that generated by (10) is bounded and either

holds. If , then .

Proof. First, we show that the boundedness of implies the boundedness of . If is bounded, then set

for some given point and , . With , there exists such that for all ,

It follows from Lemma 2 that

Hence, for all , from (21), we have

which means that is bounded.

Next, we prove the conclusion of Lemma 7. If , then we have

Similarly, if , then we have

It follows from Lemma 5 that . Since

we obtain that . This completes the proof.

As a direct consequence of Lemma 7, the following lemma is immediate.

Lemma 8. Let be a nonempty, closed, and convex subset of a complete CAT(0) space and be nonexpansive mappings. Let and be given sequences such that . Suppose that generated by (10) is bounded and

holds. If , then .

We now present our main results in this paper. The following theorem discusses the -convergence of the implicit Ishikawa iteration sequence (10) for a family of nonexpansive mappings in CAT(0) spaces.

Theorem 9. Let be a nonempty, closed, and convex subset of a complete CAT(0) space and be uniformly asymptotically regular and nonexpansive mappings such that . Let and be given sequences such that . Then, the sequence generated by (10) is well defined. Suppose that either

holds. If , then -converges to some point in .

Proof. By Lemma 6, we know that the sequence generated by (10) is well defined. For any , from (10) and Lemma 2, we have

Since , it follows that

Consequently, converges, and is thus bounded.

It follows from (10) and (21) that for sufficiently large ,

Applying Lemma 7, we have . Hence,

and thus,

We prove that for each , . Since

we know that . Because the family of nonexpansive mappings is uniformly asymptotically regular, we have

Since converges for any , an application of Lemma 4 yields that consists of exactly one point and is contained in , for all . This shows that -converges to some point in . This completes the proof.

In the special case where , from Theorem 9, we have the following corollary.

Corollary 10 (see [6, Theorem 3.4]). Let be a nonempty, closed, and convex subset of a complete CAT(0) space and be uniformly asymptotically regular and nonexpansive mappings such that . Let be a given sequence of real numbers such that . Let be a sequence defined by

Suppose that either

holds. Then, -converges to some point in .

Remark 11. Theorem 9 extends and improves [6, Theorem 3.4] from the explicit Mann iteration schemes to the implicit Ishikawa iteration schemes.

By Lemma 8 and Theorem 9, the following theorem holds trivially.

Theorem 12. Let be a nonempty, closed, and convex subset of a complete CAT(0) space and be uniformly asymptotically regular and nonexpansive mappings such that . Let and be given sequences such that . Then, the sequence generated by (10) is well defined. Suppose that

holds. If , then -converges to some point in .

Finally, we study the strong convergence of the implicit Ishikawa iteration sequence (11) for nonexpansive semigroups in CAT(0) spaces, under various and appropriate conditions.

Theorem 13. Let be a compact convex subset of a complete CAT(0) space and be a nonexpansive semigroup on . Let and be given sequences of real numbers such that . Then, the sequence generated by the implicit Ishikawa iteration process (11) is well defined. Suppose that is a sequence in such that

If , then converges strongly to some point in .

Proof. It is known that is nonempty (see [1, 2, 28]). From Lemma 6, we know that the sequence generated by (11) is well defined. Then, we show that

Assume for the contrary that (40) does not hold. There exist a subsequence , a sequence , and an such that for all ,

Since is compact, there exists a convergent subsequence contained in . Without loss of generality, we assume that with . Consequently,

which is a contradiction. Formula (40) follows readily. Now, Lemma 8 yields that

Similar to the proof of [6, Theorem 3.5], it is easy to see that there exists a subsequence which converges to , where is a common fixed point in . Since is a cluster of , we have . It follows from (11) and (30) that exists. Hence, we obtain , which completes the proof.

Remark 14. The proof of Theorem 13 is an analog of [6, Theorem 3.5]. If and , then Theorem 13 reduces to [6, Theorem 3.5]. Therefore, Theorem 13 extends and generalizes [6, Theorem 3.5] from the explicit Mann iteration processes to the implicit Ishikawa iteration processes.

We prove another strong convergence theorem which differs from Theorem 13.

Theorem 15. Let be a compact convex subset of a complete CAT(0) space and be a nonexpansive semigroup on . Let and be given sequences. Then, the sequence generated by the implicit Ishikawa iteration process (11) is well defined. Moreover, if

then converges strongly to a common fixed point of .

Proof. It is known that is nonempty (see [1, 2, 28]). From Lemma 6, we know that generated by (11) is well defined.

Claim 1. If is a sequence of nonnegative real numbers such that , then

Assume for the contrary that (45) does not hold. There exist a subsequence , a sequence , and an such that for all ,

Since is compact, there exists a convergent subsequence of . Without loss of generality, we assume that with . Consequently,

which is a contradiction. Formula (45) follows readily.

Claim 2. Consider that . Since is a compact convex subset of , there exists a subsequence such that as . It follows from (11) and Lemma 2 that

Hence, we have

For any given , let . Since is compact and , we know that and for sufficiently large . Consequently,

Thus, from (45) we get

For any given , it follows from (50) that

where is the integer part of . Since , it follows from (45) and (51) that

Therefore, from the previous formula, we know that . Iteration process (11) and inequality (30) imply that converges strongly to . This completes the proof.

Remark 16. If is a Banach space, the notation “” with is replaced by “” and , then Theorem 15 reduces to [19, Theorem 2.3]. Therefore, Theorem 15 extends and generalizes [19, Theorem 2.3] from the implicit Mann iteration processes in the Banach spaces to the implicit Ishikawa iteration processes in CAT(0) spaces.

Remark 17. The results presented in this paper can be immediately applied to any CAT() space with , because any CAT() space is a CAT() space for any (see [3, 6]).

Acknowledgments

This work was supported by the National Natural Science Foundation of China (11101069 and 11026063) and the Open Fund (PLN1104 and PLN1102) of State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation (Southwest Petroleum University).

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