- Research Article
- Open Access

# Strong Convergence of a Generalized Iterative Method for Semigroups of Nonexpansive Mappings in Hilbert Spaces

- Husain Piri
^{1}Email author and - Hamid Vaezi
^{1}

**2010**:907275

https://doi.org/10.1155/2010/907275

© H. Piri and H. Vaezi. 2010

**Received:**20 April 2010**Accepted:**18 June 2010**Published:**8 July 2010

## Abstract

Using -strongly accretive and -strictly pseudocontractive mapping, we introduce a general iterative method for finding a common fixed point of a semigroup of non-expansive mappings in a Hilbert space, with respect to a sequence of left regular means defined on an appropriate space of bounded real-valued functions of the semigroup. We prove the strong convergence of the proposed iterative algorithm to the unique solution of a variational inequality.

## Keywords

- Hilbert Space
- Banach Space
- Variational Inequality
- Iterative Algorithm
- Nonexpansive Mapping

## 1. Introduction

Let be a real Hilbert space. A mapping of into itself is called non-expansive if , for all . By , we denote the set of fixed points of (i.e., ).

where is a sequence in . See also [2].

where . They proved that if is a sequence in satisfying the following conditions:

where is a potential function for (i.e., , for all ).

Using the Hahn-Banach theorem, it is immediately clear that for each . The multivalued mapping from into is said to be the (normalized) duality mapping. A Banach space is said to be smooth if the duality mapping is single valued. As it is well known, the duality mapping is the identity when is a Hilbert space; see [7].

see [8].

Various applications to the additive semigroup of nonnegative real numbers and commuting pairs of non-expansive mappings are also presented. It is worth mentioning that we obtain our result without assuming condition .

## 2. Preliminaries

for each , where is the adjoint operator of .

The open ball of radius centered at is denoted by . For subset of , by , we denote the closed convex hull of . Weak convergence is denoted by , and strong convergence is denoted by .

Lemma 2.2 (see [13]).

Let be a closed convex subset of a Hilbert space , a semigroup from into such that , the mapping an element of for each and , and a mean on . If one writes instead of , then the following holds.

(i) is non-expansive mapping from into .

(iv)If is left invariant, then is a non-expansive retraction from onto .

This defines a mapping from into and is called metric (the nearest point) projection onto .

Lemma 2.3 (see [7]).

Let be a nonempty convex subset of a smooth Banach space and let and . Then, the following is equivalent.

(i) is the best approximation to .

Lemma 2.4 (see [14]).

Let be a nonempty closed convex subset of a Hilbert space and suppose that is non-expansive. Then, the mapping is demiclosed at zero.

The following lemma is well known.

Lemma 2.5.

Let be a real Hilbert space. Then, for all

Lemma 2.6 (see [11]).

where and are sequences of real numbers satisfying the following conditions:

The following lemma will be frequently used throughout the paper. For the sake of completeness, we include its proof.

Lemma 2.7.

Let be a real smooth Banach space and a mapping.

(i)If is -strongly accretive and -strictly pseudo-contractive with , then, is contractive with constant .

(ii)If is -strongly accretive and -strictly pseudo-contractive with , then, for any fixed number , is contractive with constant .

This shows that is contractive with constant .

Throughout this paper, will denote a -strongly accretive and -strictly pseudo-contractive mapping with , and is a contraction with coefficient on a Hilbert space . We will also always use to mean a number in .

## 3. Strong Convergence Theorem

The following is our main result.

Theorem 3.1.

Proof.

Therefore, is bounded and so is .

Since is arbitrary, we get (3.11).

and hence is a contraction due to

Consequently, applying Lemma 2.6, to (3.26), we conclude that .

Corollary 3.2.

Proof.

This shows that is -strictly pseudo-contractive. Now apply Theorem 3.1 to conclude the result.

Corollary 3.3.

Proof.

## 4. Some Application

Corollary 4.1.

Proof.

Therefore, applying Theorem 3.1, the result follows.

Corollary 4.2.

Proof.

For , we define for each , where denotes the space of all real-valued bounded continuous functions on with supremum norm. Then, is regular sequence of means [16]. Furthermore, for each , we have . Now, apply Theorem 3.1 to conclude the result.

Corollary 4.3.

Proof.

For , we define for each . Then is regular sequence of means [16]. Furthermore, for each , we have . Now, apply Theorem 3.1 to conclude the result.

Corollary 4.4.

Proof.

for each . Since is a strongly regular matrix, for each , we have , as ; see [17]. Then, it is easy to see that is regular sequence of means. Furthermore, for each , we have Now, apply Theorem 3.1 to conclude the result.

## Declarations

### Acknowledgments

The authors thank the referee(s) for the helpful comments, which improved the presentation of this paper. This paper is dedicated to Professor Anthony To Ming Lau. This paper is based on final report of the research project of the Ph.D. thesis which is done with financial support of research office of the University of Tabriz.

## Authors’ Affiliations

## References

- Mann WR:
**Mean value methods in iteration.***Proceedings of the American Mathematical Society*1953,**4:**506–510. 10.1090/S0002-9939-1953-0054846-3MathSciNetView 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 - 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 - Xu HK:
**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 - Yamada I:
**The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings.**In*Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, Studies in Computational Mathematics*.*Volume 8*. North-Holland, Amsterdam, The Netherlands; 2001:473–504.Google Scholar - Marino G, Xu HK:
**A general iterative method for nonexpansive mappings in Hilbert spaces.***Journal of Mathematical Analysis and Applications*2006,**318**(1):43–52. 10.1016/j.jmaa.2005.05.028MathSciNetView ArticleMATHGoogle Scholar - Agarwal RP, O'Regan D, Sahu DR:
*Fixed Point Theory for Lipschitzian-Type Mappings with Applications, Topological Fixed Point Theory and Its Applications*. Springer, New York, NY, USA; 2009:x+368.MATHGoogle Scholar - Zeidler E:
*Nonlinear Functional Analysis and Its Applications. III*. Springer, New York, NY, USA; 1985:xxii+662.View ArticleGoogle Scholar - Atsushiba S, Takahashi W:
**Approximating common fixed points of nonexpansive semigroups by the Mann iteration process.***Annales Universitatis Mariae Curie-Skłodowska A*1997,**51**(2):1–16.MathSciNetMATHGoogle Scholar - Lau AT, Miyake H, Takahashi W:
**Approximation of fixed points for amenable semigroups of nonexpansive mappings in Banach spaces.***Nonlinear Analysis. Theory, Methods & Applications*2007,**67**(4):1211–1225. 10.1016/j.na.2006.07.008MathSciNetView ArticleMATHGoogle Scholar - Xu HK:
**An iterative approach to quadratic optimization.***Journal of Optimization Theory and Applications*2003,**116**(3):659–678. 10.1023/A:1023073621589MathSciNetView ArticleMATHGoogle Scholar - Lau AT, Shioji N, Takahashi W:
**Existence of nonexpansive retractions for amenable semigroups of nonexpansive mappings and nonlinear ergodic theorems in Banach spaces.***Journal of Functional Analysis*1999,**161**(1):62–75. 10.1006/jfan.1998.3352MathSciNetView ArticleMATHGoogle Scholar - Takahashi W:
**A nonlinear ergodic theorem for an amenable semigroup of nonexpansive mappings in a Hilbert space.***Proceedings of the American Mathematical Society*1981,**81**(2):253–256. 10.1090/S0002-9939-1981-0593468-XMathSciNetView ArticleMATHGoogle Scholar - Jung JS:
**Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces.***Journal of Mathematical Analysis and Applications*2005,**302**(2):509–520. 10.1016/j.jmaa.2004.08.022MathSciNetView ArticleMATHGoogle Scholar - Bruck RE:
**On the convex approximation property and the asymptotic behavior of nonlinear contractions in Banach spaces.***Israel Journal of Mathematics*1981,**38**(4):304–314. 10.1007/BF02762776MathSciNetView ArticleMATHGoogle Scholar - Takahashi W:
*Nonlinear Functional Analysis*. Yokohama Publishers, Yokohama, Japan; 2000:iv+276. Fixed Point Theory and Its ApplicationMATHGoogle Scholar - Hirano N, Kido K, Takahashi W:
**Nonexpansive retractions and nonlinear ergodic theorems in Banach spaces.***Nonlinear Analysis. Theory, Methods & Applications*1988,**12**(11):1269–1281. 10.1016/0362-546X(88)90059-4MathSciNetView ArticleMATHGoogle Scholar

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