# 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

https://doi.org/10.1155/2009/279058

© Yonghong Yao et al. 2009

**Received: **6 April 2009

**Accepted: **12 September 2009

**Published: **12 October 2009

## Abstract

## 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

## 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, 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:

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

Proof.

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

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