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

*Journal of Inequalities and Applications*
**volume 2010**, Article number: 687374 (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.

## 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 .

Then the scheme defined by (1.5) converges strongly to an element which is the unique solution of the variational inequality for short, :

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]).

Let and . Let be a sequence in and be a sequence in . Starting with an arbitrary initial guess , generate a sequence by the following iterative scheme:

Assume that

(i),

(ii),

(iii)either or ,

(iv).

Then the scheme defined by (1.7) converges strongly to an element which is the unique solution of the variational inequality for short, :

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

Let and . Let be a sequence in and be a sequence in . Starting with an arbitrary initial guess , generate a sequence by the following iterative scheme:

Assume that

(i),

(ii),

(iii)either or ,

(iv).

In addition, assume that

Then the scheme defined by (1.9) converges strongly to an element which is the unique solution of the variational inequality for short, :

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]).

Let be a closed convex subset of a Hilbert space and be a nonexpansive mapping from into itself. Then is demiclosed at zero, that is,

The following lemma is an immediate consequence of the equality:

Lemma 2.2.

Let be a real Hilbert space. Then the following identity holds:

Let , be the sequences of nonnegative real numbers and let . Suppose that is a sequence of real numbers such that

Assume that . Then the following results hold.

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

(2)If one has

then .

Lemma 2.4 (see [8]).

Let be a real Hilbert space, be a contraction with coefficient and be a strongly positive linear bounded operator with coefficient . Then, for any with ,

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.

The proof method is mainly from the idea of Marino and Xu [8, Lemma 2.5]. It is known that the norm of a linear bounded self-adjoint operator on is as follows:

Now, for all with , we see that (here denotes zero point in )

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.

In fact, if is a strongly monotone linear operator with coefficient on a Hilbert space , then, for all ,

which shows that is strongly positive linear. Assume that is a strongly positive linear operator with coefficient on . Then, for all ,

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.

If , then is a contraction with coefficient , where , that is,

Proof.

From Lemma 2.5 and Remark 2.6, it follows that, for all ,

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.

Since , it follows from (3.10) that, for all ,

By induction, we obtain

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

Lemma 3.3.

If is strictly decreasing and the following conditions hold:

then

Proof.

By the iterative scheme (3.9), we have

and hence

where is a constant. Since , there exists a constant such that for all . Therefore, we have

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.

If is strictly decreasing and the following conditions hold:

then

Proof.

By the iterative scheme (3.9), we have

that is,

Hence, for any , we get

Since each is nonexpansive, it follows from (2.2) that

Hence, combining (3.21) with (3.20), it follows that

which implies that

Since each and , then we have

that is,

Since and are both bounded, there exists a constant such that

By Lemma 3.3 and the assumption condition , it follows that

This completes the proof.

Finally, we give the main result in this paper.

Theorem 3.5.

If is strictly decreasing and the following conditions hold:

then the scheme defined by (3.9) converges strongly to an element which is the unique solution of the variational inequality :

Proof.

First, we prove that .

To prove this, we pick a subsequence of such that

Without loss of generality, we may, further, assume that for some . From Lemmas 2.1 and 3.3, it follows that for each and so . Since is the unique solution of the problem , we obtain

It follows from Lemma 2.2 and (3.10) 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.

() If in (3.9), then we have the following iterative scheme:

and the scheme defined by (3.33) converges strongly to an element which is the unique solution of the variational inequality ():

() If and in (3.33), then we have the following iterative scheme:

and the scheme defined by (3.35) converges strongly to an element which is the unique solution of the variational inequality ():

() Furthermore, if and in (3.35), then we have the following iterative scheme:

and the scheme defined by (3.37) converges strongly to an element which is the unique solution of the variational inequality (), which is Stampacchia's variational inequality:

## References

- 1.
Kinderlehrer D, Stampacchia G:

*An Introduction to Variational Inequalities and Their Applications, Pure and Applied Mathematics*.*Volume 88*. Academic Press, New York, NY, USA; 1980:xiv+313. - 2.
Xu H-K: Viscosity approximation methods for nonexpansive mappings.

*Journal of Mathematical Analysis and Applications*2004, 298(1):279–291. 10.1016/j.jmaa.2004.04.059 - 3.
Deutsch F, Yamada I: Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings.

*Numerical Functional Analysis and Optimization*1998, 19(1–2):33–56. - 4.
Xu H-K: Iterative algorithms for nonlinear operators.

*Journal of the London Mathematical Society*2002, 66(1):240–256. 10.1112/S0024610702003332 - 5.
Xu HK: An iterative approach to quadratic optimization.

*Journal of Optimization Theory and Applications*2003, 116(3):659–678. 10.1023/A:1023073621589 - 6.
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, Stud. Comput. Math.*.*Volume 8*. Edited by: Butnarium D, Censor Y, Reich S. North-Holland, Amsterdam, The Netherlands; 2001:473–504. - 7.
Yamada I, Ogura N, Yamashita Y, Sakaniwa K: Quadratic optimization of fixed points of nonexpansive mappings in Hilbert space.

*Numerical Functional Analysis and Optimization*1998, 19(1–2):165–190. 10.1080/01630569808816822 - 8.
Marino G, Xu H-K: 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.2005.05.028 - 9.
Ceng L-C, Guu S-M, Yao J-C: Hybrid viscosity-like approximation methods for nonexpansive mappings in Hilbert spaces.

*Computers & Mathematics with Applications*2009, 58(3):605–617. 10.1016/j.camwa.2009.02.035 - 10.
Goebel K, Kirk WA:

*Topics in Metric Fixed Point Theory, Cambridge Studies in Advanced Mathematics*.*Volume 28*. Cambridge University Press, Cambridge, UK; 1990:viii+244. - 11.
Maingé P-E: Approximation methods for common fixed points of nonexpansive mappings in Hilbert spaces.

*Journal of Mathematical Analysis and Applications*2007, 325(1):469–479. 10.1016/j.jmaa.2005.12.066

## Acknowledgment

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

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### Keywords

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