# Strong Convergence of Two Iterative Algorithms for Nonexpansive Mappings in Hilbert Spaces

- Yonghong Yao
^{1}Email author, - Yeong Cheng Liou
^{2}and - Giuseppe Marino
^{3}

**2009**:279058

**DOI: **10.1155/2009/279058

© Yonghong Yao et al. 2009

**Received: **6 April 2009

**Accepted: **12 September 2009

**Published: **12 October 2009

## Abstract

We introduce two iterative algorithms for nonexpansive mappings in Hilbert spaces. We prove that the proposed algorithms strongly converge to a fixed point of a nonexpansive mapping .

## 1. Introduction

Let be a nonempty closed convex subset of a real Hilbert space . Recall that a mapping is said to be nonexpansive if

for all . We use to denote the set of fixed points of .

Construction of fixed points of nonlinear mappings is an important and active research area. In particular, iterative algorithms for finding fixed points of nonexpansive mappings have received vast investigation (cf. [1, 2]) since these algorithms find applications in a variety of applied areas of inverse problem, partial differential equations, image recovery, and signal processing see; [3–8]. Iterative methods for nonexpansive mappings have been extensively investigated in the literature; see [1–7, 9–21].

It is our purpose in this paper to introduce two iterative algorithms for nonexpansive mappings in Hilbert spaces. We prove that the proposed algorithms strongly converge to a fixed point of nonexpansive mapping .

## 2. Preliminaries

Let be a nonempty closed convex subset of . For every point , there exists a unique nearest point in , denoted by such that

The mapping is called the metric projection of onto . It is well known that is a nonexpansive mapping.

In order to prove our main results, we need the following well-known lemmas.

Lemma 2.1 (see [22], Demiclosed principle).

Let be a nonempty closed convex of a real Hilbert space . Let be a nonexpansive mapping. Then is demiclosed at , that is, if and , then .

Lemma 2.2 (see [20]).

Let , be bounded sequences in a Banach space , and let be a sequence in which satisfies the following condition: . Suppose that for all and , then .

Lemma 2.3 (see [22]).

Assume, that is a sequence of nonnegative real numbers such that , where is a sequence in and is a sequence in such that

(i) ,

(ii) or ,

then .

## 3. Main Results

Let be a nonempty closed convex subset of a real Hilbert space . Let be a nonexpansive mapping. For each , we consider the following mapping given by

It is easy to check that which implies that is a contraction. Using the Banach contraction principle, there exists a unique fixed point of in , that is,

Theorem 3.1.

Let be a nonempty closed convex subset of a real Hilbert space . Let be a nonexpansive mapping with . For each , let the net be generated by (3.2). Then, as , the net converges strongly to a fixed point of .

Proof.

Hence, is bounded.

Hence, the weak convergence of to actually implies that strongly. This has proved the relative norm compactness of the net as .

which implies that . This completes the proof.

Theorem 3.2.

Suppose that the following conditions are satisfied:

(i) and ,

(ii) ,

then the sequence generated by (3.15) strongly converges to a fixed point of .

Proof.

Hence, is bounded and so is .

We can check that all assumptions of Lemma 2.3 are satisfied. Therefore, . This completes the proof.

## Declarations

### Acknowledgment

The second author was partially supposed by the Grant NSC 98-2622-E-230-006-CC3 and NSC 98-2923-E-110-003-MY3.

## Authors’ Affiliations

## References

- Goebel K, Reich S:
*Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Monographs and Textbooks in Pure and Applied Mathematics*.*Volume 83*. Marcel Dekker, New York, NY, USA; 1984.MATHGoogle Scholar - Reich S:
**Almost convergence and nonlinear ergodic theorems.***Journal of Approximation Theory*1978,**24**(4):269–272. 10.1016/0021-9045(78)90012-6MathSciNetView ArticleMATHGoogle Scholar - Byrne C:
**A unified treatment of some iterative algorithms in signal processing and image reconstruction.***Inverse Problems*2004,**20**(1):103–120. 10.1088/0266-5611/20/1/006MathSciNetView ArticleMATHGoogle Scholar - Combettes PL:
**On the numerical robustness of the parallel projection method in signal synthesis.***IEEE Signal Processing Letters*2001,**8**(2):45–47. 10.1109/97.895371View ArticleGoogle Scholar - Combettes PL:
**The convex feasibility problem in image recovery.**In*Advances in Imaging and Electron Physics*.*Volume 95*. Edited by: Hawkes P. Academic Press, New York, NY, USA; 1996:155–270.Google Scholar - Engl HW, Leitão A:
**A Mann iterative regularization method for elliptic Cauchy problems.***Numerical Functional Analysis and Optimization*2001,**22**(7–8):861–884. 10.1081/NFA-100108313MathSciNetView ArticleMATHGoogle Scholar - Podilchuk CI, Mammone RJ:
**Image recovery by convex projections using a least-squares constraint.***Journal of the Optical Society of America*1990,**7**(3):517–512. 10.1364/JOSAA.7.000517View ArticleGoogle Scholar - Youla D:
**Mathematical theory of image restoration by the method of convex projection.**In*Image Recovery Theory and Applications*. Edited by: Stark H. Academic Press, Orlando, Fla, USA; 1987:29–77.Google Scholar - Bauschke HH:
**The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space.***Journal of Mathematical Analysis and Applications*1996,**202**(1):150–159. 10.1006/jmaa.1996.0308MathSciNetView ArticleMATHGoogle Scholar - Browder FE, Petryshyn WV:
**Construction of fixed points of nonlinear mappings in Hilbert space.***Journal of Mathematical Analysis and Applications*1967,**20:**197–228. 10.1016/0022-247X(67)90085-6MathSciNetView ArticleMATHGoogle Scholar - Halpern B:
**Fixed points of nonexpanding maps.***Bulletin of the American Mathematical Society*1967,**73:**957–961. 10.1090/S0002-9904-1967-11864-0MathSciNetView ArticleMATHGoogle Scholar - Jung JS:
**Viscosity approximation methods for a family of finite nonexpansive mappings in Banach spaces.***Nonlinear Analysis: Theory, Methods & Applications*2006,**64**(11):2536–2552. 10.1016/j.na.2005.08.032MathSciNetView ArticleMATHGoogle Scholar - Kim T-H, Xu H-K:
**Robustness of Mann's algorithm for nonexpansive mappings.***Journal of Mathematical Analysis and Applications*2007,**327**(2):1105–1115. 10.1016/j.jmaa.2006.05.009MathSciNetView ArticleMATHGoogle Scholar - Lions P-L:
**Approximation de points fixes de contractions.***Comptes Rendus de l'Académie des Sciences. Série I. Mathématique*1977,**284**(21):A1357-A1359.MathSciNetGoogle Scholar - Moudafi A:
**Viscosity approximation methods for fixed-points problems.***Journal of Mathematical Analysis and Applications*2000,**241**(1):46–55. 10.1006/jmaa.1999.6615MathSciNetView ArticleMATHGoogle Scholar - Reich S:
**Weak convergence theorems for nonexpansive mappings in Banach spaces.***Journal of Mathematical Analysis and Applications*1979,**67**(2):274–276. 10.1016/0022-247X(79)90024-6MathSciNetView ArticleMATHGoogle Scholar - Reich S:
**Strong convergence theorems for resolvents of accretive operators in Banach spaces.***Journal of Mathematical Analysis and Applications*1980,**75**(1):287–292. 10.1016/0022-247X(80)90323-6MathSciNetView ArticleMATHGoogle Scholar - Shioji N, Takahashi W:
**Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces.***Proceedings of the American Mathematical Society*1997,**125**(12):3641–3645. 10.1090/S0002-9939-97-04033-1MathSciNetView ArticleMATHGoogle Scholar - Suzuki T:
**Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces.***Fixed Point Theory and Applications*2005,**2005**(1):103–123. 10.1155/FPTA.2005.103View ArticleMathSciNetMATHGoogle Scholar - Suzuki T:
**Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces.***Proceedings of the American Mathematical Society*2007,**135:**99–106.MathSciNetView ArticleGoogle Scholar - Wittmann R:
**Approximation of fixed points of nonexpansive mappings.***Archiv der Mathematik*1992,**58**(5):486–491. 10.1007/BF01190119MathSciNetView ArticleMATHGoogle Scholar - Xu H-K:
**Viscosity approximation methods for nonexpansive mappings.***Journal of Mathematical Analysis and Applications*2004,**298**(1):279–291. 10.1016/j.jmaa.2004.04.059MathSciNetView 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.