- Research Article
- Open Access

# An Iterative Scheme with a Countable Family of Nonexpansive Mappings for Variational Inequality Problems in Hilbert Spaces

- YeolJe Cho
^{1}and - Shenghua Wang
^{2, 3}Email author

**2010**:687374

https://doi.org/10.1155/2010/687374

© Y. J. Cho and S.Wang. 2010

**Received:**13 December 2009**Accepted:**8 February 2010**Published:**21 February 2010

## Abstract

We introduce a new iterative scheme with a countable family of nonexpansive mappings for the variational inequality problems in Hilbert spaces and prove some strong convergence theorems for the proposed schemes.

## Keywords

- Hilbert Space
- Unique Solution
- Variational Inequality
- Nonexpansive Mapping
- Lipschitzian Mapping

## 1. Introduction

Let be a Hilbert space and be a nonempty closed convex subset of . Let be a nonlinear mapping. The classical variational inequality problem (for short, ) is to find a point such that

This variational inequality was initially studied by Kinderlehrer and Stampacchia [1]. Since then, many authors have introduced and studied many kinds of the variational inequality problems (inclusions) and applied them to many fields.

It is well known that, if is a strongly monotone and Lipschitzian mapping on , then the has a unique solution (see [2]).

Let be a mapping. Recall that a mapping is nonexpansive if

The set of fixed points of is denoted by . Recently, the iterative methods for nonexpansive mappings and some kinds of nonlinear mappings have been applied to solve the convex minimization problems (see [3–7]).

A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on :

where is the fixed point set of a nonexpansive mapping on H, is a given point in and is a strongly positive operator, that is, there is a constant such that

Recently, for solving the variational inequality on , Marino and Xu [8] introduced the following general iterative scheme:

where is a strongly positive linear bounded operator on , is a contraction on and .

More precisely, they gave the following result.

Theorem 1 MX (see [8, Theorem 3.4]).

Let be generated by algorithm (1.5) with the sequence satisfying the following conditions:

(C1) ,

(C2) ,

(C3) either or .

Let be a contraction with coefficient and let be two strongly positive linear bounded operators with coefficients and , respectively.

Motivated and inspired by the iterative sheme (1.5), Ceng et al. [9] introduced the following so-called *hybrid viscosity-like approximation algorithms* with variable parameters for nonexpansive mappings in Hilbert spaces.

Theorem 1 CGYone (see [9, Theorem 3.1]).

Assume that

(i) ,

(ii) ,

(iii)either or ,

(iv) .

Theorem 1 CGYtwo (see [9, Theorem 3.2]).

Assume that

(i) ,

(ii) ,

(iii)either or ,

(iv) .

In this paper, motivated and inspired by the above research results, we introduce a new iterative process with a countable family of nonexpansive mappings for the variational inequality problem in Hilbert spaces.

More precisely, let be a Hilbert space and be a countable family of nonexpansive mappings from to such that . Let be a contraction with coefficient and be strongly positive linear bounded operators with coefficients and , respectively. Let and with . Take three fixed numbers , and such that , and . For any , generate the iterative scheme by

We prove that the iterative scheme defined by (1.12) strongly converges to an element which is the unique solution of the variational inequality for short, :

## 2. Preliminaries

Let be a Hilbert space and be a nonexpansive mapping of into itself such that . For all and , we have

and hence

Let be a sequence in a Hilbert space and let . Throughout this paper, and denote that strongly converges to and converges weakly to a point , respectively.

Lemma 2.1 (see [10]).

The following lemma is an immediate consequence of the equality:

Lemma 2.2.

Assume that . Then the following results hold.

(1)If , where , then is a bounded sequence.

then .

Lemma 2.4 (see [8]).

that is, is strongly monotone with coefficient .

Lemma 2.5.

Assume is a strongly monotone linear bounded operator on a Hilbert space with coefficient . Take a fixed number such that . Then .

Proof.

This completes the proof.

Remark 2.6.

Lemma 2.5 still holds if is a strongly positive linear bounded operator (see [8, Lemma 2.5]). That is, Lemma 2.5 in this section and Lemma 2.5 in [8] both hold when is a strongly monotone linear bounded operator or a strongly positive linear bounded one because an operator on a Hilbert space is strongly monotone linear if and only if it is strongly positive linear.

which shows that is strongly monotone and linear.

## 3. Main Results

Let be a Hilbert space and be a nonempty closed and convex subset of . Let be a contraction with coefficient . Let be strongly positive linear bounded operator with coefficient and , respectively. Take a fixed number such that . Then, from Lemma 2.4, it follows that is strongly monotone with coefficient . For any fixed numbers and , we have , which can be seen easily from the following:

Moreover, observe that

which implies that is Lipschitzian with coefficient .

On the other hand, from Lemma 2.4, it follows that

which implies that is strongly monotone with coefficient . Hence the variational inequality (for short, )

has the unique solution.

Let be a nonexpansive mapping. Take two fixed numbers and such that and and, for all , define a mapping by

Then we have the following results.

Lemma 3.1.

Proof.

This completes the proof.

Let be a countable family of nonexpansive mappings from into itself such that . Since each is closed and convex, then is closed and convex.

Throughout this paper, let be a contraction with coefficient . Let be strongly positive linear bounded mapping with coefficient and , respectively. Take a fixed number such that . Suppose that , (assuming that such that ( is nonempty), with and with .

Now, we can rewrite the iterative scheme (1.12) as follows:

where . Then, by Lemma 3.1, for all , we have

where .

Lemma 3.2.

If is strictly decreasing, then the scheme defined by (3.9) is bounded.

Proof.

Hence is bounded and so are and for each . This completes the proof.

Lemma 3.3.

then

Proof.

Put . Since is a strictly decreasing sequence and , we have . By Lemma 2.3, it follows that as . This completes the proof.

Lemma 3.4.

then

Proof.

This completes the proof.

Finally, we give the main result in this paper.

Theorem 3.5.

Proof.

First, we prove that .

where is a constant such that for all . Since and , by Lemma 2.3, we conclude that the scheme converges strongly to . This completes the proof.

Remark 3.6.

( ) For each , a simple example on control parameters is and , where is a constant in .

( ) We obtain the desired results without any assumptions on the family . Foe example, in Theorem CGY2, the authors gave the strong condition (1.10).

Remark 3.7.

## Declarations

### Acknowledgment

This work was supported by the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-313-C00050).

## Authors’ Affiliations

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