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Fixed Point Theory and Applications

, 2008:745010 | Cite as

Strong Convergence Theorems for Nonexpansive Semigroups without Bochner Integrals

  • Satit Saejung
Open Access
Research Article

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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1. Introduction

Let Open image in new window be a real Hilbert space with the inner product Open image in new window and the norm Open image in new window . Let Open image in new window be a family of mappings from a subset Open image in new window of Open image in new window into itself. We call it a nonexpansive semigroup on Open image in new window if the following conditions are satisfied:
Motivated by Suzuki's result [1] and Nakajo-Takahashi's results [2], He and Chen [3] recently proved a strong convergence theorem for nonexpansive semigroups in Hilbert spaces by hybrid method in the mathematical programming. However, their proof of the main result ([3, Theorem 2.3]) is very questionable. Indeed, the existence of the subsequence Open image in new window such that (2.16) of [3] are satisfied, that is,

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.

Let Open image in new window be a real sequence and let Open image in new window be a real number such that Open image in new window . Suppose that either of the following holds:

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.

Let Open image in new window be a closed convex subset of a real Hilbert space Open image in new window . Let Open image in new window be a nonexpansive semigroup on Open image in new window with a nonempty common fixed point Open image in new window , that is, Open image in new window . Suppose that Open image in new window is a sequence iteratively generated by the following scheme:

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.

It is well known that Open image in new window is closed and convex. We first show that the iterative scheme is well defined. To see that each Open image in new window is nonempty, it suffices to show that Open image in new window . The proof is by induction. Clearly, Open image in new window . Suppose that Open image in new window . Then, for Open image in new window ,

That is, Open image in new window as required.

Notice that
is convex since

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 .

Moreover, since the sequence Open image in new window is bounded,
Note that
It then follows from the existence of Open image in new window that Open image in new window is a Cauchy sequence. In fact, for Open image in new window , there exists a natural number Open image in new window such that, for all Open image in new window ,
Moreover,
The last convergence follows from (2.12). We choose a sequence Open image in new window of positive real number such that
We now show that how such a special subsequence can be constructed. First we fix Open image in new window such that
From (2.13), there exists Open image in new window such that Open image in new window for all Open image in new window . By Lemma 1.1, Open image in new window is a cluster point of Open image in new window . In particular, there exists Open image in new window such that Open image in new window . Next, we choose Open image in new window such that Open image in new window for all Open image in new window . Again, by Lemma 1.1, Open image in new window is a cluster point of Open image in new window and this implies that there exists Open image in new window such that Open image in new window . Continuing in this way, we obtain a subsequence Open image in new window of Open image in new window satisfying

Consequently, (2.14) is satisfied.

We next show that Open image in new window . To see this, we fix Open image in new window ,

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.

Let Open image in new window be a closed convex subset of a real Hilbert space Open image in new window . Let Open image in new window be a nonexpansive semigroup on Open image in new window with a nonempty common fixed point Open image in new window , that is, Open image in new window . Suppose that Open image in new window is a sequence iteratively generated by the following scheme:

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.

For the sake of clarity, we give the whole sketch proof even though some parts of the proof are the same as [3]. To see that the scheme is well defined, it suffices to show that both Open image in new window and Open image in new window are closed and convex, and Open image in new window for all Open image in new window . It follows easily from the definition that Open image in new window and Open image in new window are just the intersection of Open image in new window and the half-spaces, respectively,

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.

We next show that Open image in new window . To see this, we first prove that
This implies that sequence Open image in new window is bounded and
Notice that
This implies that
As in Theorem 2.1, we can choose a subsequence Open image in new window of Open image in new window such that
Consequently, for any Open image in new window ,
This implies that
In virtue of Opial's condition of Open image in new window , we have Open image in new window for all Open image in new window , that is, Open image in new window . Next, we observe that
This implies that
Consequently,

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. 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
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Copyright information

© Satit Saejung. 2008

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.

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceKhon Kaen UniversityKhon KaenThailand

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