# A General Iterative Approach to Variational Inequality Problems and Optimization Problems

- JongSoo Jung
^{1}Email author

**2011**:284363

https://doi.org/10.1155/2011/284363

© Jong Soo Jung. 2011

**Received: **4 October 2010

**Accepted: **14 November 2010

**Published: **30 November 2010

## Abstract

We introduce a new general iterative scheme for finding a common element of the set of solutions of variational inequality problem for an inverse-strongly monotone mapping and the set of fixed points of a nonexpansive mapping in a Hilbert space and then establish strong convergence of the sequence generated by the proposed iterative scheme to a common element of the above two sets under suitable control conditions, which is a solution of a certain optimization problem. Applications of the main result are also given.

## 1. Introduction

Let be a real Hilbert space with inner product and induced norm . Let be a nonempty closed convex subset of and be self-mapping on . We denote by the set of fixed points of and by the metric projection of onto .

We denote the set of solutions of the variational inequality problem (1.1) by . The variational inequality problem has been extensively studied in the literature; see [1–5] and the references therein.

where is a contraction with constant , , , and . He proved that the sequence generated by (1.3) strongly converges strongly to a point in , which is the unique solution of a certain variational inequality.

where , are infinitely many closed convex subsets of such that , , is a real number, is a strongly positive bounded linear operator on (i.e., there is a constant such that , for all ), and is a potential function for (i.e., for all ). For this kind of optimization problems, see, for example, Deutsch and Yamada [11], Jung [10], and Xu [12, 13] when and for a given point in .

where is a potential function for . The result improved the corresponding results of Moudafi [15] and Xu [16].

In this paper, motivated by the above-mentioned results, we introduce a new general composite iterative scheme for finding a common point of the set of solutions of the variational inequality problem (1.1) for an inverse-strongly monotone mapping and the set of fixed points of a nonexapansive mapping and then prove that the sequence generated by the proposed iterative scheme converges strongly to a common point of the above two sets, which is a solution of a certain optimization problem. Applications of the main result are also discussed. Our results improve and complement the corresponding results of Chen et al. [6], Iiduka and Takahashi [8], Jung [10], and others.

## 2. Preliminaries and Lemmas

Let be a real Hilbert space and let be a nonempty closed convex subset of . We write to indicate that the sequence converges weakly to . implies that converges strongly to .

First we recall that a mapping
is a *contraction* on
if there exists a constant
such that
,
. A mapping
is called *nonexpansive* if
. We denote by
the set of fixed points of
.

*metric projection*of onto . It is well known that is nonexpansive and satisfies

*Opial condition*, that is, for any sequence with , the inequality

*inverse-strongly monotone*if there exists a positive real number such that

*strongly monotone*if there exists a positive real number such that

*Lipschitz continuous*, that is, for all , then is -inverse-strongly monotone. If is an -inverse-strongly monotone mapping of into , then it is obvious that is -Lipschitz continuous. We also have that for all and ,

So, if , then is a nonexpansive mapping of into . The following result for the existence of solutions of the variational inequality problem for inverse strongly-monotone mappings was given in Takahashi and Toyoda [9].

Proposition 2.1.

Let be a bounded closed convex subset of a real Hilbert space and let be an -inverse-strongly monotone mapping of into . Then, is nonempty.

*monotone*if for all , , and imply . A monotone mapping is

*maximal*if the graph of is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping is maximal if and only if for , for every implies . Let be an inverse-strongly monotone mapping of into and let be the

*normal cone*to at , that is, , and define

Then is maximal monotone and if and only if ; see [18, 19].

We need the following lemmas for the proof of our main results.

Lemma 2.2.

Lemma 2.3 (Xu [12]).

where and satisfy the following conditions:

Lemma 2.4 (Marino and Xu [14]).

Assume that is a strongly positive linear bounded operator on a Hilbert space with constant and . Then .

The following lemma can be found in [20, 21] (see also Lemma 2.2 in [22]).

Lemma 2.5.

## 3. Main Results

In this section, we present a new general composite iterative scheme for inverse-strongly monotone mappings and a nonexpansive mapping.

Theorem 3.1.

where , , and . Let , , and satisfy the following conditions:

where is a potential function for .

Proof.

Now we divide the proof into several steps.

Step 1.

Step 2.

Step 3.

Step 4.

Step 5.

from (3.9) and (3.29), we have .

Step 6.

Without loss of generality, we may assume that converges weakly to .

which is a contradiction. Thus we have .

Since in Step 4 and is -inverse-strongly monotone, we have as . Since is maximal monotone, we have and hence .

Step 7.

From (i), in Steps 3, and 6, it is easily seen that , , and . Hence, by Lemma 2.3, we conclude as . This completes the proof.

As a direct consequence of Theorem 3.1, we have the following results.

Corollary 3.2.

where is a potential function for .

Corollary 3.3.

## 4. Applications

In this section, as in [6, 8, 10], we prove two theorems by using Theorem 3.1. First of all, we recall the following definition.

Using Theorem 3.1, we found a strong convergence theorem for finding a common fixed point of a nonexpansive mapping and a strictly pseudo-contractive mapping.

Theorem 4.1.

where is a potential function for .

Proof.

Put . Then is -inverse-strongly monotone. We have and . Thus, the desired result follows from Theorem 3.1.

Using Theorem 3.1, we also obtain the following result.

Theorem 4.2.

where is a potential function for .

Proof.

We have . So, putting , by Theorem 3.1, we obtain the desired result.

## Declarations

### Acknowledgment

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2010-0017007).

## Authors’ Affiliations

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