- Haiyun Zhou
^{1}Email author and - Xinghui Gao
^{2}

**2009**:351265

**DOI: **10.1155/2009/351265

© H. Zhou and X. Gao. 2009

**Received: **26 May 2009

**Accepted: **14 September 2009

**Published: **29 September 2009

## Abstract

The purpose of this paper is to propose a modified hybrid projection algorithm and prove a strong convergence theorem for a family of quasi- -nonexpansive mappings. The strong convergence theorem is proven in the more general reflexive, strictly convex, and smooth Banach spaces with the property (K). The results of this paper improve and extend the results of S. Matsushita and W. Takahashi (2005), X. L. Qin and Y. F. Su (2007), and others.

## 1. Introduction

It is well known that, in an infinite-dimensional Hilbert space, the normal Mann's iterative algorithm has only weak convergence, in general, even for nonexpansive mappings. Consequently, in order to obtain strong convergence, one has to modify the normal Mann's iteration algorithm, the so-called hybrid projection iteration method is such a modification.

The hybrid projection iteration algorithm (HPIA) was introduced initially by Haugazeau [1] in 1968. For 40 years, HPIA has received rapid developments. For details, the readers are referred to papers [2–7] and the references therein.

They proved the following convergence theorem.

Theorem 1 MT.

Let be a uniformly convex and uniformly smooth Banach space, let be a nonempty closed 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 normalized duality mapping on . If is nonempty, then converges strongly to , where is the generalized projection from onto .

They proved the following convergence theorem.

Theorem 1 QS.

Let be a uniformly convex and uniformly smooth Banach space, let be a nonempty closed convex subset of , let be a relatively nonexpansive mapping from into itself such that . Assume that and are sequences in such that and . Suppose that is given by (1.2). If is uniformly continuous, then converges strongly to .

Question 1.

Can both Theorems MT and QS be extended to more general reflexive, strictly convex, and smooth Banach spaces with the property (K)?

Question 2.

Can both Theorems MT and QS be extended to more general class of quasi- -nonexpansive mappings?

The purpose of this paper is to give some affirmative answers to the questions mentioned previously, by introducing a modified hybrid projection iteration algorithm and by proving a strong convergence theorem for a family of closed and quasi- -nonexpansive mappings by using new analysis techniques in the setting of reflexive, strictly convex, and smooth Banach spaces with the property (K). The results of this paper improve and extend the results of Matsushita and Takahashi [5], Qin and Su [2], and others.

## 2. Preliminaries

where denotes the generalized duality pairing between and . It is well known that if is reflexive, strictly convex, and smooth, then is single-valued, demi-continuous and strictly monotone (see, e.g., [8, 9]).

It is also very well known that if is a nonempty closed convex subset of a Hilbert space and is the metric projection of onto , then is nonexpansive. This fact actually characterizes Hilbert spaces and consequently, it is not available in more general Banach spaces. In this connection, Alber [10] 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.

Remark 2.2.

If is a reflexive, strictly 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 in turn implies that From the smoothness of , we know that is single valued, and hence we have . Since is strictly convex, is strictly monotone, in particular, is one to one, which implies that one may consult [8, 9] for the details.

Let be a closed convex subset of , and a mapping from into itself. A point in is said to be asymptotic fixed point of [13] if contains a sequence which converges weakly to such that . The set of asymptotic fixed point of will be denoted by . A mapping from into itself is said to be relatively nonexpansive [5, 14–16] if and for all and . The asymptotic behavior of a relatively nonexpansive mapping was studied in [14–16].

is said to be quasi- -nonexpansive if and for all and .

Remark 2.3.

The class of quasi- -nonexpansive mappings is more general than the class of relatively nonexpansive mappings [5, 14–16] which requires the strong restriction: .

We present two examples which are closed and quasi- -nonexpansive.

Example 2.4.

Let be the generalized projection from a smooth, strictly convex, and reflexive Banach space onto a nonempty closed convex subset of . Then, is a closed and quasi- -nonexpansive mapping from onto with .

Example 2.5.

Let be a reflexive, strictly convex, and smooth Banach space, and is a maximal monotone mapping such that its zero set is nonempty. Then, is a closed and quasi- -nonexpansive mapping from onto and .

Recall that a Banach space has the property (K) if for any sequence and , if weakly and , then . For more information concerning property (K) the reader is referred to [17] and references cited therein.

In order to prove our main result of this paper, we need to the following facts.

Lemma 2.6 (see, e.g., [10–12]).

Lemma 2.7 (see, e.g., [10–12]).

Now we are in a proposition to prove the main results of this paper.

## 3. Main Results

Theorem 3.1.

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

Proof.

We split the proof into six steps.

Step 1.

Show that is well defined for every .

This implies that . Hence is closed and convex for all and consequently is closed and convex. By our assumption that , we have is well defined for every .

Step 2.

Show that is closed and convex for each .

It is easy to see that is closed and convex. Then, for all , is closed and convex. Consequently, is closed and convex for all .

Step 3.

which implies that and consequently . So . Hence is well defined for each . Therefore, the iterative algorithm (3.1) is well defined.

Step 4.

and hence , since is strictly monotone.

which shows that . By the property (K) of , we have , where .

Step 5.

Step 6.

which implies that . Hence, . Then converges strongly to . This completes the proof.

From Theorem 3.1, we can obtain the following corollary.

Corollary 3.2.

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

Remark 3.3.

Theorem 3.1 and its corollary improve and extend Theorems MT and QS at several aspects.

(i)From uniformly convex and uniformly smooth Banach spaces extend to reflexive, strictly convex and smooth Banach spaces with the property (K). In Theorem 3.1 and its corollary the hypotheses on are weaker than the usual assumptions of uniform convexity and uniform smoothness. For example, any strictly convex, reflexive and smooth Musielak-Orlicz space satisfies our assumptions [17] while, in general, these spaces need not to be uniformly convex or uniformly smooth.

(ii)From relatively nonexpansive mappings extend to closed and quasi- -non-expansive mappings.

(iii)The continuity assumption on mapping in Theorem QS is removed.

(iv)Relax the restriction on from to .

Remark 3.4.

Corollary 3.2 presents some affirmative answers to Questions 1 and 2.

## 4. Applications

In this section, we present some applications of the main results in Section 3.

Theorem 4.1.

where satisfies the restriction: and . Then defined by (4.1) converges strongly to a minimizer of the family .

Proof.

Now the desired conclusion follows from Theorem 3.1. This completes the proof.

## Declarations

### Acknowledgment

This work was supported by the National Natural Science Foundation of China under Grant (10771050).

## Authors’ Affiliations

## References

- Haugazeau Y: Sur les inéquations variationnelles et la minimisation de fonctionnelles convexes, Thése. Université de Paris, Paris, France;Google Scholar
- Qin XL, Su YF:
**Strong convergence theorems for relatively nonexpansive mappings in a Banach space.***Nonlinear Analysis*2007,**67**(6):1958–1965. 10.1016/j.na.2006.08.021MathSciNetView ArticleMATHGoogle 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)00458-4MathSciNetView ArticleMATHGoogle Scholar - Carlos MY, Xu HK:
**Strong convergence of the CQ method for fixed point iteration processes.***Nonlinear Analysis*2006,**64**(11):2400–2411. 10.1016/j.na.2005.08.018MathSciNetView ArticleMATHGoogle Scholar - Matsushita S, 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.2005.02.007MathSciNetView ArticleMATHGoogle Scholar - Su YF, Qin XL:
**Strong convergence of modified Ishikawa iterations for nonlinear mappings.***Proceedings of the Indian Academy of Sciences, Mathematical Sciences*2007,**117**(1):97–107. 10.1007/s12044-007-0008-yMathSciNetView ArticleMATHGoogle Scholar - Su YF, Wang DX, Shang MJ:
**Strong convergence of monotone hybrid algorithm for hemi-relatively nonexpansive mappings.***Fixed Point Theory and Applications*2008, Article ID 284613,**2008:**-8.Google Scholar - Cioranescu I:
*Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems*.*Volume 62*. Kluwer Academic Publishers Group, Dordrecht, The Netherlands; 1990.View ArticleMATHGoogle Scholar - Takahashi W:
*Nonlinear Functional Analysis*. Yokohama Publishers, Yokohama, Japan; 2000.MATHGoogle 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*.*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.MathSciNetMATHGoogle 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 ArticleMATHGoogle Scholar - Reich S:
**A weak convergence theorem for the alternating method with Bregman distances.**In*Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, Lecture Notes in Pure and Applied Mathmatics*.*Volume 178*. Edited by: Kartsatos AG. Marcel Dekker, New York, NY, USA; 1996:313–318.Google 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.2001.151MathSciNetView ArticleMATHGoogle Scholar - Butnariu D, Reich S, Zaslavski AJ:
**Weak convergence of orbits of nonlinear operators in reflexive Banach spaces.***Numerical Functional Analysis and Optimization*2003,**24**(5–6):489–508. 10.1081/NFA-120023869MathSciNetView ArticleMATHGoogle Scholar - Censor Y, Reich S:
**Iterations of paracontractions and firmly nonexpansive operators with applications to feasibility and optimization.***Optimization*1996,**37**(4):323–339. 10.1080/02331939608844225MathSciNetView ArticleMATHGoogle Scholar - Hudzik H, Kowalewski W, Lewicki G:
**Approximate compactness and full rotundity in Musielak-Orlicz spaces and Lorentz-Orlicz spaces.***Zeitschrift fuer Analysis und ihre Anwendungen*2006,**25**(2):163–192.MathSciNetView ArticleMATHGoogle Scholar - Rockafellar RT:
**On the maximal monotonicity of subdifferential mappings.***Pacific Journal of Mathematics*1970,**33:**209–216.MathSciNetView ArticleMATHGoogle 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.