- Research Article
- Open Access

# Strong Convergence Theorems by Shrinking Projection Methods for Class Mappings

- Qiao-Li Dong
^{1, 2}Email author, - Songnian He
^{1, 2}and - Fang Su
^{3}

**2011**:681214

https://doi.org/10.1155/2011/681214

© Qiao-Li Dong et al. 2011

**Received:**9 December 2010**Accepted:**2 February 2011**Published:**17 February 2011

## Abstract

We prove a strong convergence theorem by a shrinking projection method for the class of mappings. Using this theorem, we get a new result. We also describe a shrinking projection method for a nonexpansive mapping on Hilbert spaces, which is the same as that of Takahashi et al. (2008).

## Keywords

- Hilbert Space
- Applied Mathematic
- Differential Geometry
- Convex Subset
- Optimization Theory

## 1. Introduction

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 .

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

be the half-space generated by . A mapping is said to be the class (or a cutter) if .

Remark 1.1.

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 [1] 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 [3] for a nonexpansive mapping.

where , denotes the metric projection from onto a closed convex subset of .

Inspired by Bauschke and Combettes [1] and Takahashi et al. [4], 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. [4].

We will use the following notations:

(1) for weak convergence and for strong convergence;

(2) denotes the weak -limit of .

We need some facts and tools in a real Hilbert space which are listed below.

Lemma 1.2 (see [1]).

Let be a Hilbert space. Let be the identity operator of .

(i)If dom , then is quasi-nonexpansive if and only if .

(ii)If , then , for all .

Definition 1.3.

Definition 1.4.

is called demiclosed at if whenever , and .

Next lemma shows that nonexpansive mappings are demeiclosed at 0.

Lemma 1.5 (Goebel and Kirk [5]).

Let be a closed convex subset of a real Hilbert space , and let be a nonexpansive mapping such that . If a sequence in is such that and , then .

Lemma 1.6 (see [6]).

then .

Lemma 1.7 (Goebel and Kirk [5]).

## 2. Main Results

In this section, we will introduce a shrinking projection method for the class of mappings and prove strong convergence theorem.

Theorem 2.1.

Then, converges strongly to .

Proof.

It is obvious that is closed and convex. So, from the definition, is closed and convex for all . So we get that is well defined.

We therefore get . Since the sequence is coherent, we have . From Lemma 1.6 and (2.5), the result holds.

Remark 2.2.

We take so that is satisfied.

Theorem 2.3.

where . Then, converges strongly to .

Proof.

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

In this section, using Theorem 2.3, we obtain some new strong convergence results for the class of mappings, a quasi-nonexpansive mapping and a nonexpansive mapping in a Hilbert space.

Theorem 3.1.

where . Then, converges strongly to .

Proof.

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 .

Theorem 3.2.

where . Then, converges strongly to .

Proof.

Since is demiclosed at 0, is demiclosed at 0. So we can obtain the result by using Theorem 3.1.

Since a nonexpansive mapping is quasi-nonexpansive, using Lemma 1.5 and Theorem 3.2, we have following corollary.

Corollary 3.3.

where . Then, converges strongly to .

Remark 3.4.

Corollary 3.3 is a special case of Theorem 4.1 in [4] when .

## Declarations

### Acknowledgments

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).

## Authors’ Affiliations

## References

- 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.26.2.248.10558MATHMathSciNetView 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

## 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.