# A General Iterative Method for Solving the Variational Inequality Problem and Fixed Point Problem of an Infinite Family of Nonexpansive Mappings in Hilbert Spaces

- Rabian Wangkeeree
^{1}Email author and - Uthai Kamraksa
^{1}

**2009**:369215

https://doi.org/10.1155/2009/369215

© R.Wangkeeree and U. Kamraksa. 2009

**Received: **3 November 2008

**Accepted: **16 January 2009

**Published: **10 February 2009

## Abstract

We introduce an iterative scheme for finding a common element of the set of common fixed points of a family of infinitely nonexpansive mappings, and the set of solutions of the variational inequality for an inverse-strongly monotone mapping in a Hilbert space. Under suitable conditions, some strong convergence theorems for approximating a common element of the above two sets are obtained. As applications, at the end of the paper we utilize our results to study the problem of finding a common element of the set of fixed points of a family of infinitely nonexpansive mappings and the set of fixed points of a finite family of -strictly pseudocontractive mappings. The results presented in the paper improve some recent results of Qin and Cho (2008).

## 1. Introduction

We denote by the set of fixed points of . Recall that a mapping is said to be

(ii) -Lipschitz if there exists a constant such that , for all ;

Remark 1.1.

It is obvious that any -inverse-strongly monotone mapping is monotone and -Lipschitz continuous.

The set of solutions of variational inequality (1.3) is denoted by . The variational inequality has been extensively studied in the literature; see, for example, [3, 4] and the references therein.

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

where the sequence is in the interval .

where is a contraction, is a nonexpansive mapping, and is a strongly positive linear bounded self-adjoint operator, proved that, under certain appropriate assumptions on the parameters, defined by (1.12) converges strongly to a fixed point of , which solves some variational inequality and is also the optimality condition for the minimization problem (1.9).

where is an -inverse strongly monotone mapping, and satisfy some parameters controlling conditions. They showed that if is nonempty, then the sequence generated by (1.14) converges strongly to some .

The existence of common fixed points for a finite family of nonexpansive mappings has been considered by many authors (see [17–20] 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 [21, 22]). 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 [18, 23]). 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 [23, 24]).

where is a nonnegative real sequence with , for all , , form a family of infinitely nonexpansive mappings of into itself. Nonexpansivity of each ensures the nonexpansivity of . Such a is nonexpansive from to and it is called a -mapping generated by and .

where is a mapping defined by (1.15), is a contraction, is strongly positive linear bounded self-adjoint operator, is a -inverse strongly monotone, and we prove that under certain appropriate assumptions on the sequences , , , and , the sequences defined by (1.16) converge strongly to a common element of the set of common fixed points of a family of and the set of solutions of the variational inequality for an inverse-strongly monotone mapping, which solves some variational inequality and is also the optimality condition for the minimization problem (1.9).

## 2. Preliminaries

It is well known that each Hilbert space satisfies the Opial's condition.

Then is the maximal monotone and if and only if ; see [26].

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

Lemma 2.1.

Lemma 2.2 (see [27]).

Let and be bounded sequences in a Banach space and let be a sequence in with Suppose for all integers and Then,

Lemma 2.3 (see [28]).

where is a sequence in and is a sequence in such that

Lemma 2.4 (see [9]).

Assume that is a strongly positive linear bounded self-adjoint operator on a Hilbert space with coefficient and . Then .

Throughout this paper, we will assume that , for all . Concerning defined by (1.15), we have the following lemmas which are important to prove our main result.

Lemma 2.5 (see [29]).

Let be a nonempty closed convex subset of a Hilbert space , let be a family of infinitely nonexpansive mapping with , and let be a real sequence such that , for all . 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

Lemma 2.6 (see [30]).

## 3. Main Results

Now we are in a position to state and prove the main result in this paper.

Theorem 3.1.

Let be a closed convex subset of a real Hilbert space , let be a contraction of into itself, let be an -inverse strongly monotone mapping of into , and let be a family of infinitely nonexpansive mappings with . Let be a strongly positive linear bounded self-adjoint operator with the coefficient such that . Assume that . Let , , , and be sequences in satisfying the following conditions:

Proof.

which gives that the sequence is bounded, and so are and .

where is a constant such that . Similarly, there exists such that .

Banach's Contraction Mapping Principle guarantees that has a unique fixed point, say . That is, .

This is a contradiction, which shows that .

Since is maximal monotone, we have , and hence .

Using (C1), (3.54), and (3.55), we get . Now applying Lemma 2.3 to (3.58), we conclude that . This completes the proof.

Remark 3.2.

Theorem 3.1 mainly improve the results of Qin and Cho [14] from a single nonexpansive mapping to an infinite family of nonexpansive mappings.

## 4. Applications

In this section, we obtain two results by using a special case of the proposed method.

Theorem 4.1.

where , , , and are sequences in satisfying the following conditions:

Proof.

We have and . Applying Theorem 3.1, we obtain the desired result.

Next, we will apply the main results to the problem for finding a common element of the set of fixed points of a family of infinitely nonexpansive mappings and the set of fixed points of a finite family of -strictly pseudocontractive mappings.

Definition 4.2.

The following lemmas can be obtained from [31, Proposition 2.6] by Acedo and Xu, easily.

Lemma 4.3.

Let be a Hilbert space, let be a closed convex subset of . For any integer , assume that, for each is a -strictly pseudocontractive mapping for some . Assume that is a positive sequence such that . Then is a -strictly pseudocontractive mapping with .

Lemma 4.4.

Let and be as in Lemma 4.3. Suppose that has a common fixed point in . Then .

This shows that is -inverse-strongly monotone.

Theorem 4.5.

where , , , and are the sequences in satisfying the following conditions:

Proof.

The conclusion of Theorem 4.5 can be obtained from Theorem 3.1.

Remark 4.6.

Theorem 4.5 is a generalization and improvement of the theorems by Qin and Cho [14], Iiduka and Takahashi [16, Thorem 3.1], and Takahashi and Toyoda [15].

## Declarations

### Acknowledgments

The authors would like to thank the referee for the comments which improve the manuscript. R. Wangkeeree was supported for CHE-PhD-THA-SUP/2551 from the Commission on Higher Education and the Thailand Research Fund under Grant TRG5280011.

## Authors’ Affiliations

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