Strong Convergence Theorems for Nonexpansive Semigroups without Bochner Integrals
- Satit Saejung1Email author
https://doi.org/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).
Keywords
1. Introduction







- (1)
- (2)
- (3)
- (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
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.







Proof.






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.
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.







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.
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.