- Research Article
- Open Access

# New Iterative Approximation Methods for a Countable Family of Nonexpansive Mappings in Banach Spaces

- Kamonrat Nammanee
^{1, 2}and - Rabian Wangkeeree
^{2, 3}Email author

**2011**:671754

https://doi.org/10.1155/2011/671754

© Kamonrat Nammanee and Rabian Wangkeeree. 2011

**Received:**5 October 2010**Accepted:**13 November 2010**Published:**30 November 2010

## Abstract

We introduce new general iterative approximation methods for finding a common fixed point of a countable family of nonexpansive mappings. Strong convergence theorems are established in the framework of reflexive Banach spaces which admit a weakly continuous duality mapping. Finally, we apply our results to solve the the equilibrium problems and the problem of finding a zero of an accretive operator. The results presented in this paper mainly improve on the corresponding results reported by many others.

## Keywords

- Banach Space
- Variational Inequality
- Equilibrium Problem
- Nonexpansive Mapping
- Real Hilbert Space

## 1. Introduction

In recent years, the existence of common fixed points for a finite family of nonexpansive mappings has been considered by many authors (see [1–4] and the references therein). The well-known convex feasibility problem reduces to finding a point in the intersection of the fixed point sets of a family of nonexpansive mappings (see [5, 6]). The problem of finding an optimal point that minimizes a given cost function over the common set of fixed points of a family of nonexpansive mappings is of wide interdisciplinary interest and practical importance (see [2, 7]). A simple algorithmic solution to the problem of minimizing a quadratic function over the common set of fixed points of a family of nonexpansive mappings is of extreme value in many applications including set theoretic signal estimation (see [7, 8]).

*contraction*on if there exists a constant such that

where is a fixed point. Banach's contraction mapping principle guarantees that has a unique fixed point in . It is unclear, in general, what is the behavior of as , even if has a fixed point. However, in the case of having a fixed point, Browder [9] proved that if is a Hilbert space, then converges strongly to a fixed point of . Reich [10] extended Browder's result to the setting of Banach spaces and proved that if is a uniformly smooth Banach space, then converges strongly to a fixed point of and the limit defines the (unique) sunny nonexpansive retraction from onto . Xu [11] proved Reich's results hold in reflexive Banach spaces which have a weakly continuous duality mapping.

where is a strongly positive bounded linear operator on . They proved that if the sequence of parameters satisfies the following conditions:

which is the optimality condition for the minimization problem: , where is a potential function for for ).

^{'}) for all and . Very recently, Song and Zheng [19] also introduced the conception of the condition on a countable family of nonexpansive mappings and proved strong convergence theorems of the modified Halpern iteration (1.12) and the sequence defined by

in a reflexive Banach space with a weakly continuous duality mapping and in a reflexive strictly convex Banach space with a uniformly Gáteaux differentiable norm.

Other investigations of approximating common fixed points for a countable family of nonexpansive mappings can be found in [1, 20–24] and many results not cited here.

*strongly positive*[25] if there exists a constant with the property

where is the identity mapping. If is a real Hilbert space, then the inequality (1.14) reduces to (1.4).

Finally, we apply our results to solve the the equilibrium problems and the problem of finding a zero of an accretive operator.

## 2. Preliminaries

Throughout this paper, let
be a real Banach space, and
be its dual space. We write
(resp.,
) to indicate that the sequence
weakly (resp., weak*) converges to
; as usual
will symbolize strong convergence. Let
. A Banach space
is said to *uniformly convex* if, for any
, there exists
such that, for any
,
implies
. It is known that a uniformly convex Banach space is reflexive and strictly convex (see also [26]). A Banach space
is said to be *smooth* if the limit
exists for all
. It is also said to be *uniformly smooth* if the limit is attained uniformly for
.

*weakly*

*continuous*

*duality*

*mapping*if there exists a gauge for which the duality mapping is single valued and continuous from the weak topology to the weak* topology, that is, for any with , the sequence converges weakly* to . It is known that has a weakly continuous duality mapping with a gauge function for all . Set

where denotes the subdifferential in the sense of convex analysis.

Now, we collect some useful lemmas for proving the convergence result of this paper.

The first part of the next lemma is an immediate consequence of the subdifferential inequality and the proof of the second part can be found in [28].

Lemma 2.1 (see [28]).

Assume that a Banach space has a weakly continuous duality mapping with gauge .

(ii)Assume that a sequence in converges weakly to a point ,

Lemma 2.2 (see [1, Lemma 2.3]).

where satisfying the restrictions

Definition 2.3 (see [1]).

Remark 2.4.

The example of the sequence of mappings satisfying AKTT-condition is supported by Lemma 4.6.

Lemma 2.5 (see [1, Lemma 3.2]).

Then, for each bounded subset of , .

The next valuable lemma was proved by Wangkeeree et al. [25]. Here, we present the proof for the sake of completeness.

Lemma 2.6.

Assume that a Banach space has a weakly continuous duality mapping with gauge . Let be a strongly positive bounded linear operator on with coefficient and , then .

Proof.

Remark 2.7.

We note that space has a weakly continuous duality mapping with a gauge function for all . This shows that is invariant on .

Lemma 2.8 (see [25, Lemma 3.3]).

## 3. Main Results

We now state and prove the main theorems of this section.

Theorem 3.1.

Let be a reflexive Banach space which admits a weakly continuous duality mapping with gauge such that is invariant on . Let be a countable family of nonexpansive mappings satisfying . Let be an -contraction and a strongly positive bounded linear operator with coefficient and . Let the sequence be generated by (1.16), where is a sequence in satisfying the following conditions:

Proof.

Applying Lemma 2.2 to (3.20), we conclude that as ; that is, as . This completes the proof.

Setting , where is the identity mapping and for all in Theorem 3.1, we have the following result.

Corollary 3.2.

where is a sequence in satisfying the following conditions:

Applying Theorem 3.1, we can obtain the following two strong convergence theorems for the iterative sequences and defined by (1.17).

Theorem 3.3.

Let be a reflexive Banach space which admits a weakly continuous duality mapping with gauge such that is invariant on . Let be a countable family of nonexpansive mappings satisfying . Let be an -contraction and a strongly positive bounded linear operator with coefficient and . Let the sequence be generated by (1.17), where is a sequence in satisfying the following conditions:

Suppose that satisfies the AKTT-condition. Let be a mapping of into itself defined by for all , and suppose that , then converges strongly to which solves the variational inequality (3.1).

Proof.

It follows from , , and Lemma 2.2 that . Consequently, as required.

Theorem 3.4.

Let be a reflexive Banach space which admits a weakly continuous duality mapping with gauge such that is invariant on . Let be a countable family of nonexpansive mappings satisfying . Let be an -contraction and a strongly positive bounded linear operator with coefficient and . Let the sequence be generated by (1.17), where is sequence in satisfying the following conditions:

Suppose that satisfies the AKTT-condition. Let be a mapping of into itself defined by for all , and suppose that , then converges strongly to which solves the variational inequality (3.1).

Proof.

Hence, the sequence converges strongly to . This competes the proof.

Setting , where is the identity mapping and for all in Theorem 3.4, we have the following result.

Corollary 3.5.

## 4. Applications

### 4.1. -Mappings

Nonexpansivity of each ensures the nonexpansivity of . The mapping is called a -mapping generated by and .

Throughout this section, we will assume that . Concerning defined by (4.1), we have the following useful lemmas.

Lemma 4.1 (see [4]).

Let be a nonempty closed convex subset of a a strictly convex, reflexive Banach space , a family of infinitely nonexpansive mapping with , and a real sequence such that , , then:

(1) is nonexpansive and for each ;

(2)for each and for each positive integer , the limit exists;

is a nonexpansive mapping satisfying , and it is called the -mapping generated by and .

From Remark 3.1 of Peng and Yao [29], we obtain the following lemma.

Lemma 4.2.

Let be a strictly convex, reflexive Banach space, a family of infinitely nonexpansive mappings with , and a real sequence such that , . Then sequence satisfies the -condition.

Applying Lemma 4.2 and Theorem 3.1, we obtain the following result.

Theorem 4.3.

where is defined by (4.1) and is a sequence in satisfying the conditions (C1), (C2), and (C3). Then converges strongly to in .

Applying Lemma 4.2 and Theorem 3.3, we obtain the following result.

Theorem 4.4.

then converges strongly to in .

Applying Lemma 4.2 and Theorem 3.4, we obtain the following result.

Theorem 4.5.

### 4.2. Accretive Operators

From the Resolvent identity, we have the following lemma.

Lemma 4.6.

Let be a Banach space and a nonempty closed convex subset of . Let be an accretive operator such that and . Suppose that is a sequence of such that and , then

(i)the sequence satisfies AKTT-condition,

Proof.

By the proof of Theorem 4.3 in [1] and applying Lemma 4.6 and Theorem 3.1, we obtain the following result.

Theorem 4.7.

where is a sequence in satisfying the following conditions (C1), (C2), and (C3), then converges strongly to in .

Applying Lemma 4.6 and Theorem 3.3, we obtain the following result.

Theorem 4.8.

then converges strongly to in .

Applying Lemma 4.6 and Theorem 3.4, we obtain the following result.

Theorem 4.9.

### 4.3. The Equilibrium Problems

The set of solutions of (4.10) is denoted by . Given a mapping , let for all . Then, if and only if for all , that is, is a solution of the variational inequality. Numerous problems in physics, optimization, and economics reduce to find a solution of (4.10). Some methods have been proposed to solve the equilibrium problem; see, for instance, Blum and Oettli [31] and Combettes and Hirstoaga [32]. For the purpose of solving the equilibrium problem for a bifunction , let us assume that satisfies the following conditions:

(A2) is monotone, that is, for all ,

(A4) for each is convex and lower semicontinuous.

The following lemmas were also given in [31, 32], respectively.

Lemma 4.10 (see [31, Corollary 1]).

Let be a nonempty closed convex subset of , and let be a bifunction of satisfying . Let and , then there exists such that .

Lemma 4.11 (see [32, Lemma 2.12]).

then, the following hold:

(2) is firmly nonexpansive, that is, for any ,

Theorem 4.12.

for all , where is a sequence in and satisfying the following conditions:

then and converge strongly to .

Proof.

This completes the proof.

Applying Theorem 4.12, we can obtain the following result.

Corollary 4.13.

for all , where is a sequence in and satisfying the following conditions:

then and converge strongly to .

Proof.

It follows from , , and Lemma 2.2 that as . Consequently, as required.

## Declarations

### Acknowledgments

The authors would like to thank the Centre of Excellence in Mathematics, Thailand for financial support. Finally, They would like to thank the referees for reading this paper carefully, providing valuable suggestions and comments, and pointing out a major error in the original version of this paper.

## Authors’ Affiliations

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