# Strong Convergence to Common Fixed Points for Countable Families of Asymptotically Nonexpansive Mappings and Semigroups

- Kriengsak Wattanawitoon
^{1, 2}and - Poom Kumam
^{2, 3}Email author

**2010**:301868

https://doi.org/10.1155/2010/301868

© Kriengsak Wattanawitoon and Poom Kumam. 2010

**Received: **15 April 2010

**Accepted: **11 October 2010

**Published: **12 October 2010

## Abstract

We prove strong convergence theorems for countable families of asymptotically nonexpansive mappings and semigroups in Hilbert spaces. Our results extend and improve the recent results of Nakajo and Takahashi (2003) and of Zegeye and Shahzad (2008) from the class of nonexpansive mappings to asymptotically nonexpansive mappings.

## 1. Introduction

*nonexpansive*if , for all . We denote the set of fixed points of by , that is, . A mapping is said to be

*asymptotically nonexpansive*if there exists a sequence with for all , , and

where the initial guess is taken in arbitrarily and the sequence is in the interval .

for all , where and is a sequence in . This iteration process is called a Halpern-type iteration.

Recall also that a one-parameter family of self-mappings of a nonempty closed convex subset of a Hilbert space is said to be a (continuous) Lipschitzian semigroup on if the following conditions are satisfied:

(c)for each , the map is continuous on

(d)there exists a bounded measurable function such that, for each , , for all

A Lipschitzian semigroup is called nonexpansive if for all , and asymptotically nonexpansive if . We denote by the set of fixed points of the semigroup , that is, .

Recently, Su and Qin [6] modified the hybrid iteration method of Nakajo and Takahashi through the monotone hybrid method, and to prove strong convergence theorems.

In this paper, motivated and inspired by the above results, we modify iteration process (1.4)–(1.11) by the new hybrid methods for countable families of asymptotically nonexpansive mappings and semigroups in a Hilbert space, and to prove strong convergence theorems. Our results presented are improvement and extension of the corresponding results in [3, 5–8] and many authors.

## 2. Preliminaries

This section collects some lemmas which will be used in the proofs for the main results in the next section.

Lemma 2.1.

we get that a Hilbert space has the Kadec-Klee property.

Let be a nonempty closed convex subset of a Hilbert space . Motivated by Nakajo et al. [9], we give the following definitions: Let and be families of nonexpansive mappings of into itself such that , where is the set of all fixed points of and is the set of all common fixed points of . We consider the following conditions of and (see [9]):

(i)NST-condition (I). For each bounded sequence , implies that for all .

(ii)NST-condition (II). For each bounded sequence , implies that for all .

(iii)NST-condition (III). There exists with such that for every bounded subset of , there exists such that holds for all and

Lemma 2.2.

Let be a nonempty closed convex subset of and let be a nonexpansive mapping of into itself with . Then, the following hold:

(i) with and satisfy the condition (I) with .

(ii) with and satisfy the condition (I) with

Lemma 2.3 (Opial [10]).

Let be a closed convex subset of a real Hilbert space and let be a nonexpansive mapping such that . If is a sequence in such that and , then .

Lemma 2.4 (Lin et al. [11]).

Let be an asymptotically nonexpansive mapping defined on a bounded closed convex subset of a bounded closed convex subset of a Hilbert space . If is a sequence in such that and , then .

Lemma 2.5 (Nakajo and Takahashi [3]).

Let be a real Hilbert space. Given a closed convex subset and points . Given also a real number . The set is convex and closed.

Lemma 2.6 (Kim and Xu [4]).

Let be a nonempty bounded closed convex subset of and be an asymptotically nonexpansive semigroup on . If is a sequence in satisfying the properties

Lemma 2.7 (Kim and Xu [4]).

## 3. Strong Convergence for a Family of Asymptotically Nonexpansive Mappings

Theorem 3.1.

where as . Then converges in norm to .

Proof.

We first show that is closed and convex for all . From the Lemma 2.5, it is observed that is closed and convex for each .

Thus and hence for all . Thus is well defined.

for all Therefore is nondecreasing.

Since exists, we conclude that .

and hence . This completes the proof.

Corollary 3.2.

where as . Then converges in norm to .

Proof.

Setting for all from Lemma 2.2(i) and Theorem 3.1, we immediately obtain the corollary.

Since every family's nonexpansive mapping is family's asymptotically nonexpansive mapping we obtain the following result.

Corollary 3.3.

Assume that if for each bounded sequence , for all implies that . Then converges in norm to .

We have the following corollary for nonexpansive mappings by Lemma 2.2(i) and Theorem 3.1.

Corollary 3.4 (Takahashi et al. [7, Theorem ]).

## 4. Strong Convergence for a Family of Asymptotically Nonexpansive Semigroups

Theorem 4.1.

where as with . Then converges in norm to .

Proof.

Furthermore, from (4.9) and Lemma 2.6 and the boundedness of we obtain that . By the fact that for any , where and the weak lower semi-continuity of the norm, we have for all . However, since , we must have for all . Thus and then converges weakly to . Moreover, following the method of Theorem 3.1, . This completes the proof.

Corollary 4.2.

converges in norm to , where as .

Proof.

By Theorem 4.1, if the semigroup , then for all and for all . Hence for all and then, (4.1) reduces to (4.11).

Corollary 4.3 (Takahashi et al. [7, Theorem ]).

## Declarations

### Acknowledgments

The authors would like to thank *professor Somyot Plubtieng* for drawing my attention to the subject and for many useful discussions and the referees for helpful suggestions that improved the contents of the paper. This research is supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand.

## Authors’ Affiliations

## References

- Mann WR:
**Mean value methods in iteration.***Proceedings of the American Mathematical Society*1953,**4:**506–510. 10.1090/S0002-9939-1953-0054846-3MathSciNetView ArticleMATHGoogle Scholar - Halpern B:
**Fixed points of nonexpanding maps.***Bulletin of the American Mathematical Society*1967,**73:**957–961. 10.1090/S0002-9904-1967-11864-0MathSciNetView ArticleMATHGoogle Scholar - Nakajo K, Takahashi W:
**Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups.***Journal of Mathematical Analysis and Applications*2003,**279**(2):372–379. 10.1016/S0022-247X(02)00458-4MathSciNetView ArticleMATHGoogle Scholar - Kim T-H, Xu H-K:
**Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups.***Nonlinear Analysis: Theory, Methods & Applications*2006,**64**(5):1140–1152. 10.1016/j.na.2005.05.059MathSciNetView ArticleMATHGoogle Scholar - Zegeye H, Shahzad N:
**Strong convergence theorems for a finite family of asymptotically nonexpansive mappings and semigroups.***Nonlinear Analysis: Theory, Methods & Applications*2008,**69**(12):4496–4503. 10.1016/j.na.2007.11.005MathSciNetView ArticleMATHGoogle Scholar - Su Y, Qin X:
**Strong convergence of monotone hybrid method for fixed point iteration processes.***Journal of Systems Science & Complexity*2008,**21**(3):474–482. 10.1007/s11424-008-9129-3MathSciNetView ArticleMATHGoogle Scholar - Takahashi W, Takeuchi Y, Kubota R:
**Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces.***Journal of Mathematical Analysis and Applications*2008,**341**(1):276–286. 10.1016/j.jmaa.2007.09.062MathSciNetView ArticleMATHGoogle Scholar - Su Y, Qin X:
**Strong convergence theorems for asymptotically nonexpansive mappings and asymptotically nonexpansive semigroups.***Fixed Point Theory and Applications*2006,**2006:**-11.Google Scholar - Nakajo K, Shimoji K, Takahashi W:
**Strong convergence to common fixed points of families of nonexpansive mappings in Banach spaces.***Journal of Nonlinear and Convex Analysis*2007,**8**(1):11–34.MathSciNetMATHGoogle Scholar - Opial Z:
**Weak convergence of the sequence of successive approximations for nonexpansive mappings.***Bulletin of the American Mathematical Society*1967,**73:**591–597. 10.1090/S0002-9904-1967-11761-0MathSciNetView ArticleMATHGoogle Scholar - Lin P-K, Tan K-K, Xu HK:
**Demiclosedness principle and asymptotic behavior for asymptotically nonexpansive mappings.***Nonlinear Analysis: Theory, Methods & Applications*1995,**24**(6):929–946. 10.1016/0362-546X(94)00128-5MathSciNetView ArticleMATHGoogle Scholar

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