Strong Convergence of Monotone Hybrid Algorithm for Hemi-Relatively Nonexpansive Mappings
© Yongfu Su et al. 2008
Received: 1 June 2007
Accepted: 16 October 2007
Published: 27 November 2007
The purpose of this article is to prove strong convergence theorems for fixed points of closed hemi-relatively nonexpansive mappings. In order to get these convergence theorems, the monotone hybrid iteration method is presented and is used to approximate those fixed points. Note that the hybrid iteration method presented by S. Matsushita and W. Takahashi can be used for relatively nonexpansive mapping, but it cannot be used for hemi-relatively nonexpansive mapping. The results of this paper modify and improve the results of S. Matsushita and W. Takahashi (2005), and some others.
They proved the following convergence theorem.
Theorem 1.1 (MT).
Let be a uniformly convex and uniformly smooth real Banach space, let be a nonempty, closed, and convex subset of , let be a relatively nonexpansive mapping from into itself, and let be a sequence of real numbers such that and . Suppose that is given by (1.1), where is the duality mapping on . If the set of fixed points of is nonempty, then converges strongly to , where is the generalized projection from onto .
The purpose of this article is to prove strong convergence theorems for fixed points of closed hemi-relatively nonexpansive mappings. In order to get these convergence theorems, the monotone hybrid iteration method is presented and is used to approximate those fixed points. Note that the hybrid iteration method presented by S.Matsushita and W. Takahashi can be used for relatively nonexpansive mapping, but it cannot be used for hemi-relatively nonexpansive mapping. The results of this paper modify and improve the results of S.Matsushita and W. Takahashi , and some others.
Recall that if is a nonempty, closed, and convex subset of a Hilbert space and is the metric projection of onto , then is nonexpansive. This is true only when is a real Hilbert space. In this connection, Alber  has recently introduced a generalized projection operator in a Banach space which is an analogue of the metric projection in Hilbert spaces.
If is a reflexive strict convex and smooth Banach space, then for , if and only if . It is sufficient to show that if , then . From (2.4), we have . This implies From the definition of we have , that is, see  for more details.
We refer the interested reader to the , where additional information on the duality mapping may be found.
A point in is said to be an asymptotic fixed point of  if contains a sequence which converges weakly to such that the strong The set of asymptotic fixed points of will be denoted by . A hemi-relatively nonexpansive mapping from into itself is called relatively nonexpansive [1, 7, 8] if .
We need the following lemmas for the proof of our main results.
By using the similar method as [1, Proposition 2.4], the following lemma is not hard to prove.
3. Strong Convergence for Hemi-Relatively Nonexpansive Mappings
On the other hand, from (3.8) and (3.9) we have
Since every relatively nonexpansive mapping is a hemi-relatively one, Theorem 3.2 is implied by Theorem 3.1.
In recent years, the hybrid iteration methods for approximating fixed points of nonlinear mappings have been introduced and studied by various authors [1, 8–11]. In fact, all hybrid iteration methods can be replaced (or modified) by monotone hybrid iteration methods, respectively. In addition, by using the monotone hybrid method we can easily show that the iteration sequence is a Cauchy sequence, without the use of the Kadec-Klee property, demiclosedness principle, and Opial's condition or other methods which make use of the weak topology.
The authors would like to thank the referee for valuable suggestions which helped to improve this manuscript. This project is supported by the National Natural Science Foundation of China under Grant no. 10771050.
- Matsushita S-Y, Takahashi W: A strong convergence theorem for relatively nonexpansive mappings in a banach space. Journal of Approximation Theory 2005, 134(2):257–266. 10.1016/j.jat.2005.02.007MATHMathSciNetView ArticleGoogle Scholar
- Alber YaI: Metric and generalized projection operators in Banach spaces: properties and applications. In Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, Lecture Notes in Pure and Appl. Math.. Volume 178. Edited by: Kartsatos AG. Marcel Dekker, New York, NY, USA; 1996:15–50.Google Scholar
- Alber YaI, Reich S: An iterative method for solving a class of nonlinear operator equations in Banach spaces. PanAmerican Mathematical Journal 1994, 4(2):39–54.MATHMathSciNetGoogle Scholar
- Kamimura S, Takahashi W: Strong convergence of a proximal-type algorithm in a Banach space. SIAM Journal on Optimization 2002, 13(3):938–945. 10.1137/S105262340139611XMathSciNetView ArticleGoogle Scholar
- Cioranescu I: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Mathematics and Its Applications. Volume 62. Kluwer Academic, Dordrecht, The Netherlands; 1990:xiv+260.View ArticleGoogle Scholar
- Reich S: Review of geometry of Banach spaces, duality mappings and nonlinear problems by Ioana Cioranescu. Bulletin of the American Mathematical Society 1992, 26: 367–370. 10.1090/S0273-0979-1992-00287-2View ArticleGoogle Scholar
- Butnariu D, Reich S, Zaslavski AJ: Asymptotic behavior of relatively nonexpansive operators in Banach spaces. Journal of Applied Analysis 2001, 7(2):151–174. 10.1515/JAA.2001.151MATHMathSciNetView ArticleGoogle 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
- Kim T-H, Xu H-K: Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups. Nonlinear Analysis: Theory, Methods & Applications 2006, 64(5):1140–1152. 10.1016/j.na.2005.05.059MATHMathSciNetView 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
- Qin X, Su Y: Strong convergence theorems for relatively nonexpansive mappings in a banach space. Nonlinear Analysis: Theory, Methods & Applications 2007, 67(6):1958–1965. 10.1016/j.na.2006.08.021MATHMathSciNetView 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.