- Research Article
- Open Access

# Strong Convergence of Monotone Hybrid Algorithm for Hemi-Relatively Nonexpansive Mappings

- Yongfu Su
^{1}Email author, - Dongxing Wang
^{1}and - Meijuan Shang
^{1, 2}

**2008**:284613

https://doi.org/10.1155/2008/284613

© Yongfu Su et al. 2008

**Received:**1 June 2007**Accepted:**16 October 2007**Published:**27 November 2007

## Abstract

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.

## Keywords

- Banach Space
- Convergence Theorem
- Nonexpansive Mapping
- Strong Convergence
- Duality Mapping

## 1. Introduction

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 [1], and some others.

## 2. Preliminaries

where denotes the generalized duality pairing. It is well known that if is uniformly convex, then is uniformly continuous on bounded subsets of . In this case, is singe valued and also one to one.

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 [2] has recently introduced a generalized projection operator in a Banach space which is an analogue of the metric projection in Hilbert spaces.

Observe that, in a Hilbert space , (2.2) reduces to ,

Remark 2.1.

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 [5] for more details.

We refer the interested reader to the [6], where additional information on the duality mapping may be found.

Let be a closed convex subset of , and Let be a mapping from into itself. We denote by the set of fixed points of . is called hemi-relatively nonexpansive if for all and .

A point in is said to be an asymptotic fixed point of [7] 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.

Lemma 2.2 ({Kamimura and Takahashi [4], [1, Proposition 2.1]}).

Let be a uniformly convex and smooth real Banach space and let be two sequences of . If and either or is bounded, then

Lemma 2.3 ({Alber [2], [1, Proposition 2.2]}).

Lemma 2.4 ({Alber [2], [1, Proposition 2.3]}).

By using the similar method as [1, Proposition 2.4], the following lemma is not hard to prove.

Lemma 2.5.

Let be a strictly convex and smooth real Banach space, let be a closed convex subset of , and let be a hemi-relatively nonexpansive mapping from into itself. Then is closed and convex.

Recall that an operator in a Banach space is called closed, if , then .

## 3. Strong Convergence for Hemi-Relatively Nonexpansive Mappings

Theorem 3.1.

where is the duality mapping on . Then converges strongly to , where is the generalized projection from onto .

Proof.

As by the induction assumptions, the last inequality holds, in particular, for all . This together with the definition of implies that .

On the other hand, from (3.8) and (3.9) we have

This is a contradiction, so that is a Cauchy sequence, therefore there exists a point such that converges strongly to .

Since is a closed operator and , then is a fixed point of .

By the definition of , it follows that both and , whence . Therefore, it follows from the uniqueness of that . This completes the proof.

Theorem 3.2.

where is the duality mapping on . Then converges strongly to , where is the generalized projection from onto .

Proof.

Since every relatively nonexpansive mapping is a hemi-relatively one, Theorem 3.2 is implied by Theorem 3.1.

Remaek 3.3.

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.

## Declarations

### Acknowledgments

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.

## Authors’ Affiliations

## References

- 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

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