# Strong Convergence Theorems for Nonexpansive Semigroups without Bochner Integrals

- Satit Saejung
^{1}Email author

**2008**:745010

**DOI: **10.1155/2008/745010

© Satit Saejung. 2008

**Received: **28 November 2007

**Accepted: **30 January 2008

**Published: **5 February 2008

## Abstract

We prove a convergence theorem by the new iterative method introduced by Takahashi et al. (2007). Our result does not use Bochner integrals so it is different from that by Takahashi et al. We also correct the strong convergence theorem recently proved by He and Chen (2007).

## 1. Introduction

- (1)
for all ;

- (2)
for all ;

- (3)
for each the mapping is continuous;

- (4)
for all and .

needs to be proved precisely. So, the aim of this short paper is to correct He-Chen's result and also to give a new result by using the method recently introduced by Takahashi et al.

We need the following lemma proved by Suzuki [4, Lemma 1].

Lemma 1.1.

- (i)
, or

- (ii)
.

Then is a cluster point of . Moreover, for , , there exists such that for every integer with .

## 2. Results

### 2.1. The Shrinking Projection Method

The following method is introduced by Takahashi et al. in [5]. We use this method to approximate a common fixed point of a nonexpansive semigroup without Bochner integrals as was the case in [5, Theorem 4.4].

Theorem 2.1.

where , , , and . Then

Proof.

That is, as required.

This implies that each subset is convex. It is also clear that is closed. Hence the first claim is proved.

In particular, for for all , the sequence is bounded and hence so is .

Consequently, (2.14) is satisfied.

As and (2.14), we have and so .

Hence as required. This completes the proof.

### 2.2. The Hybrid Method

We consider the iterative scheme computing by the hybrid method (some authors call the CQ-method). The following result is proved by He and Chen [3]. However, the important part of the proof seems to be overlooked. Here we present the correction under some additional restriction on the parameter .

Theorem 2.2.

where , , , and . Then .

Proof.

As in the proof of the preceding theorem, we have for all . Clearly, . Suppose that for some , we have . In particular, , that is, . It follows from the induction that for all . This proves the claim.

Hence the whole sequence must converge to , as required.

## Declarations

### Acknowledgments

The author would like to thank the referee(s) for his comments and suggestions on the manuscript. This work is supported by the Commission on Higher Education and the Thailand Research Fund (Grant MRG4980022).

## Authors’ Affiliations

## References

- Suzuki T: On strong convergence to common fixed points of nonexpansive semigroups in Hilbert spaces.
*Proceedings of the American Mathematical Society*2003, 131(7):2133-2136. 10.1090/S0002-9939-02-06844-2MATHMathSciNetView ArticleGoogle 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)00458-4MATHMathSciNetView ArticleGoogle Scholar - He H, Chen R: Strong convergence theorems of the CQ method for nonexpansive semigroups.
*Fixed Point Theory and Applications*2007, 2007:-8.Google Scholar - Suzuki T: Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals.
*Journal of Mathematical Analysis and Applications*2005, 305(1):227-239. 10.1016/j.jmaa.2004.11.017MATHMathSciNetView ArticleGoogle 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*2007, 341(1):276-286.MathSciNetView ArticleGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. 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.