# Strong Convergence Theorems for Nonexpansive Semigroups without Bochner Integrals

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## 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).

### Keywords

Hilbert Space Mathematical Programming Convergence Theorem Hybrid Method Positive Real Number## 1. Introduction

- (1)
- (2)
- (3)
for each Open image in new window the mapping Open image in new window is continuous;

- (4)

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)
- (ii)

Then Open image in new window is a cluster point of Open image in new window . Moreover, for Open image in new window , Open image in new window , there exists Open image in new window such that Open image in new window for every integer Open image in new window with Open image in new window .

## 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 Open image in new window , Open image in new window , Open image in new window , and Open image in new window . Then Open image in new window

Proof.

That is, Open image in new window as required.

This implies that each subset Open image in new window is convex. It is also clear that Open image in new window is closed. Hence the first claim is proved.

In particular, for Open image in new window for all Open image in new window , the sequence Open image in new window is bounded and hence so is Open image in new window .

Consequently, (2.14) is satisfied.

As Open image in new window and (2.14), we have Open image in new window and so Open image in new window .

Hence Open image in new window 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 Open image in new window .

Theorem 2.2.

where Open image in new window , Open image in new window , Open image in new window , and Open image in new window . Then Open image in new window .

Proof.

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

Hence the whole sequence must converge to Open image in new window , as required.

## Notes

### 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).

### References

- 1.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/S0002-9939-02-06844-2MATHMathSciNetCrossRefGoogle Scholar - 2.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)00458-4MATHMathSciNetCrossRefGoogle Scholar - 3.He H, Chen R: Strong convergence theorems of the CQ method for nonexpansive semigroups.
*Fixed Point Theory and Applications*2007, 2007:-8.Google Scholar - 4.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.2004.11.017MATHMathSciNetCrossRefGoogle Scholar - 5.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.MathSciNetCrossRefGoogle Scholar

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