- Research Article
- Open Access

# Two New Iterative Methods for a Countable Family of Nonexpansive Mappings in Hilbert Spaces

- Shuang Wang
^{1}Email author and - Changsong Hu
^{2}

**2010**:852030

https://doi.org/10.1155/2010/852030

© S.Wang and C. Hu. 2010

**Received:**6 August 2010**Accepted:**5 October 2010**Published:**11 October 2010

## Abstract

We consider two new iterative methods for a countable family of nonexpansive mappings in Hilbert spaces. We proved that the proposed algorithms strongly converge to a common fixed point of a countable family of nonexpansive mappings which solves the corresponding variational inequality. Our results improve and extend the corresponding ones announced by many others.

## Keywords

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

## 1. Introduction

Remark 1.1.

From the definition of , we note that a strongly positive bounded linear operator is a -Lipschitzian and -strongly monotone operator.

The strong convergence of the path has been studied by Browder [9] and Halpern [10] in a Hilbert space.

They proved that if and satisfying appropriate conditions, then the defined by (1.6) and defined by (1.7) converge strongly to a fixed point of .

where is a -Lipschitzian and -strongly monotone operator with , , . Then he proved that generated by (1.8) converges strongly to the unique solution of variational inequality , .

In this paper, motivated and inspired by the above results, we introduce two new algorithms (3.3) and (3.13) for a countable family of nonexpansive mappings in Hilbert spaces. We prove that the proposed algorithms strongly converge to which solves the variational inequality: , .

## 2. Preliminaries

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

Lemma 2.1 (see [13]).

Let be a Hilbert space, a closed convex subset of , and a nonexpansive mapping with , if is a sequence in weakly converging to and if converges strongly to , then .

Lemma 2.2 (see [14]).

where , , and satisfy the following conditions: (i) and , (ii) or , (iii) . Then .

Lemma 2.4 (see [17, Lemma ]).

Then, for each , converges strongly to some point of C. Moreover, let T be a mapping of C into itself defined by , for all . Then .

Lemma 2.5.

Let be a -Lipschitzian and -strongly monotone operator on a Hilbert space with and . Then is a contraction with contraction coefficient .

Proof.

where . Hence is a contraction with contraction coefficient .

## 3. Main Results

Theorem 3.1.

Proof.

The strong monotonicity of implies that and the uniqueness is proved. Below we use to denote the unique solution of (3.4).

From (3.8) and (3.10), we have that is bounded and so is .

From , we have and . Observe that, if , we have .

That is is a solution of (3.4); hence by uniqueness. In a summary, we have shown that each cluster point of (as ) equals . Therefore, as .

Setting in Theorem 3.1, we can obtain the following result.

Corollary 3.2.

Setting , the identity mapping, in Theorem 3.1, we can obtain the following result.

Corollary 3.3.

Remark 3.4.

The Corollary 3.3 complements the results of Theorem in Yao et al. [11], that is, is the solution of the variational inequality: .

Theorem 3.5.

Let be a nonempty closed convex subset of a real Hilbert space . Let be a sequence of nonexpansive mappings of into itself such that . Let be a -Lipschitzian and -strongly monotone operator on with . Let and be two real sequences in and satisfy the conditions:

Proof.

We proceed with the following steps.

Step 1.

where . Therefore, is bounded. We also obtain that , , and are bounded. Without loss of generality, we may assume that , , , and , where is a bounded set of .

Step 2.

Step 3.

Step 4.

Step 5.

Step 6.

where , , and . It is easy to see that , , and . Hence, by Lemma 2.3, the sequence converges strongly to . From and Theorem 3.1, we have that is the unique solution of the variational inequality:

Remark 3.6.

From Remark
of Peng and Yao [18], we obtain that
is a sequence of nonexpansive mappings satisfying condition
for any bounded subset *B* of *H*. Moreover, let
be the
mapping; we know that
for all
and that
. If we replace
by
in the recursion formula (3.19), we can obtain the corresponding results of the so-called
mapping.

Setting and in Theorem 3.5, we can obtain the following result.

Corollary 3.7.

Setting and in Theorem 3.5, we can obtain the following result.

Corollary 3.8.

Remark 3.9.

The Corollary 3.8 complements the results of Theorem in Yao et al. [11], that is, is the solution of the variational inequality: .

## Declarations

### Acknowledgment

This paper is supported by the National Science Foundation of China under Grant (10771175).

## Authors’ Affiliations

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