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Strong Convergence Theorems for Nonexpansive Semigroups without Bochner Integrals
Fixed Point Theory and Applications volumeÂ 2008, ArticleÂ number:Â 745010 (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
Let be a real Hilbert space with the inner product and the norm . Let be a family of mappings from a subset of into itself. We call it a nonexpansive semigroup on if the following conditions are satisfied:

(1)
for all ;

(2)
for all ;

(3)
for each the mapping is continuous;

(4)
for all and .
Motivated by Suzuki's result [1] and NakajoTakahashi'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 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 HeChen'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 be a real sequence and let be a real number such that . Suppose that either of the following holds:

(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.
Let be a closed convex subset of a real Hilbert space . Let be a nonexpansive semigroup on with a nonempty common fixed point , that is, . Suppose that is a sequence iteratively generated by the following scheme:
where , , , and . Then
Proof.
It is well known that is closed and convex. We first show that the iterative scheme is well defined. To see that each is nonempty, it suffices to show that . The proof is by induction. Clearly, . Suppose that . Then, for ,
That is, as required.
Notice that
is convex since
This implies that each subset is convex. It is also clear that is closed. Hence the first claim is proved.
Next, we prove that is bounded. As ,
In particular, for for all , the sequence is bounded and hence so is .
Next, we show that is a Cauchy sequence. As and ,
Moreover, since the sequence is bounded,
Note that
In particular, since for all ,
It then follows from the existence of that is a Cauchy sequence. In fact, for , there exists a natural number such that, for all ,
where . In particular, if and , then
Moreover,
We now assume that for some . Now since for all and ,
The last convergence follows from (2.12). We choose a sequence of positive real number such that
We now show that how such a special subsequence can be constructed. First we fix such that
From (2.13), there exists such that for all . By Lemma 1.1, is a cluster point of . In particular, there exists such that . Next, we choose such that for all . Again, by Lemma 1.1, is a cluster point of and this implies that there exists such that . Continuing in this way, we obtain a subsequence of satisfying
Consequently, (2.14) is satisfied.
We next show that . To see this, we fix ,
As and (2.14), we have and so .
Finally, we show that . Since and ,
But ; we have
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 CQmethod). 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.
Let be a closed convex subset of a real Hilbert space . Let be a nonexpansive semigroup on with a nonempty common fixed point , that is, . Suppose that is a sequence iteratively generated by the following scheme:
where , , , and . Then .
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 and are closed and convex, and for all . It follows easily from the definition that and are just the intersection of and the halfspaces, respectively,
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.
We next show that . To see this, we first prove that
As and ,
For fixed . It follows from for all that
This implies that sequence is bounded and
Notice that
This implies that
It then follows from that and hence
As in Theorem 2.1, we can choose a subsequence of such that
Consequently, for any ,
This implies that
In virtue of Opial's condition of , we have for all , that is, . Next, we observe that
This implies that
Consequently,
Hence the whole sequence must converge to , as required.
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):21332136. 10.1090/S0002993902068442
Nakajo K, Takahashi W: Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. Journal of Mathematical Analysis and Applications 2003, 279(2):372379. 10.1016/S0022247X(02)004584
He H, Chen R: Strong convergence theorems of the CQ method for nonexpansive semigroups. Fixed Point Theory and Applications 2007, 2007:8.
Suzuki T: Strong convergence of Krasnoselskii and Mann's type sequences for oneparameter nonexpansive semigroups without Bochner integrals. Journal of Mathematical Analysis and Applications 2005, 305(1):227239. 10.1016/j.jmaa.2004.11.017
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):276286.
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).
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Saejung, S. Strong Convergence Theorems for Nonexpansive Semigroups without Bochner Integrals. Fixed Point Theory Appl 2008, 745010 (2008). https://doi.org/10.1155/2008/745010
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DOI: https://doi.org/10.1155/2008/745010