© Qiao-Li Dong et al. 2011
Received: 9 December 2010
Accepted: 2 February 2011
Published: 17 February 2011
Let be a real Hilbert space with inner product and norm , and let be a nonempty closed convex subset of . Recall that a mapping is said to be nonexpansive if for all . The set of fixed points of is Fix .
The class is fundamental because it contains several types of operators commonly found in various areas of applied mathematics and in particular in approximation and optimization theory (see  for details).
Using Lemma 1.2 given below and the fact that a nonexpansive mapping is quasi-nonexpansive, one can easily obtain hybrid methods introduced by Nakajo and Takahashi  for a nonexpansive mapping.
Inspired by Bauschke and Combettes  and Takahashi et al. , we present a shrinking projection method for the class of mappings. Furthermore, we obtain a shrinking projection method for a nonexpansive mapping on Hilbert spaces, which is the same as presented by Takahashi et al. .
We will use the following notations:
Lemma 1.2 (see ).
Next lemma shows that nonexpansive mappings are demeiclosed at 0.
Lemma 1.5 (Goebel and Kirk ).
Lemma 1.6 (see ).
Lemma 1.7 (Goebel and Kirk ).
2. Main Results
Set . By Lemma 1.2 (ii), we have that . From , it follows that which implies that the sequence is coherent. It is obvious that Fix( ) = Fix( ), for all . Hence . Using Theorem 2.1, we get the desired result.
3. Deduced Results
Let in (2.11) for all . Following the proof of Theorem 2.1, we can easily get (2.5) and . By (2.5), we obtain that is bounded and is nonempty. For any , there exists a subsequence of the sequence such that . From , it follows that . Since is demiclosed at 0, we get Fix . Thus Fix which together with Lemma 1.6 and (2.5) implies that .
Since a nonexpansive mapping is quasi-nonexpansive, using Lemma 1.5 and Theorem 3.2, we have following corollary.
Corollary 3.3 is a special case of Theorem 4.1 in  when .
The authors would like to express their thanks to the referee for the valuable comments and suggestions for improving this paper. This paper is supported by Research Funds of Civil Aviation University of China Grant (08QD10X) and Fundamental Research Funds for the Central Universities Grant (ZXH2009D021).
- Bauschke HH, Combettes PL: A weak-to-strong convergence principle for Fejér-monotone methods in Hilbert spaces. Mathematics of Operations Research 2001,26(2):248–264. 10.1287/moor.188.8.131.5258MATHMathSciNetView ArticleGoogle Scholar
- Combettes PL: Quasi-Fejérian analysis of some optimization algorithms. In Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications. Volume 8. North-Holland, Amsterdam, The Netherlands; 2001:115–152.View 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
- 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 2008,341(1):276–286. 10.1016/j.jmaa.2007.09.062MATHMathSciNetView ArticleGoogle Scholar
- Goebel K, Kirk WA: Topics in Metric Fixed Point Theory, Cambridge Studies in Advanced Mathematics. Volume 28. Cambridge University Press, Cambridge, UK; 1990:viii+244.View ArticleMATHGoogle Scholar
- Martinez-Yanes C, Xu H-K: Strong convergence of the CQ method for fixed point iteration processes. Nonlinear Analysis. Theory, Methods & Applications 2006,64(11):2400–2411. 10.1016/j.na.2005.08.018MATHMathSciNetView ArticleGoogle Scholar
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.