- Research Article
- Open Access

# 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.

## Keywords

- Monotone Mapping
- Positive Real Number
- Nonexpansive Mapping
- Strong Convergence
- Iterative Scheme

## 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

holds for every with .

*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.

for all .

Lemma 2.3 (Xu [12]).

where and satisfy the following conditions:

(i) and or, equivalently, ;

(ii) or ;

(iii) .

Then .

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.

where is a potential function for .

## 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:

(i) ; ;

(ii) for all and for some ;

(iii) for some , with ;

(iv) , , .

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.

where is a potential function for .

## 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

## References

- Browder FE:
**Nonlinear monotone operators and convex sets in Banach spaces.***Bulletin of the American Mathematical Society*1965,**71:**780–785. 10.1090/S0002-9904-1965-11391-XMATHMathSciNetView ArticleGoogle Scholar - Bruck, RE Jr.:
**On the weak convergence of an ergodic iteration for the solution of variational inequalities for monotone operators in Hilbert space.***Journal of Mathematical Analysis and Applications*1977,**61**(1):159–164. 10.1016/0022-247X(77)90152-4MATHMathSciNetView ArticleGoogle Scholar - Lions J-L, Stampacchia G:
**Variational inequalities.***Communications on Pure and Applied Mathematics*1967,**20:**493–519. 10.1002/cpa.3160200302MATHMathSciNetView ArticleGoogle Scholar - Liu F, Nashed MZ:
**Regularization of nonlinear ill-posed variational inequalities and convergence rates.***Set-Valued Analysis*1998,**6**(4):313–344. 10.1023/A:1008643727926MATHMathSciNetView ArticleGoogle Scholar - 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 (Haifa, 2000), Studies in Computational Mathematics*.*Volume 8*. North-Holland, Amsterdam, The Netherlands; 2001:473–504.Google Scholar - Chen J, Zhang L, Fan T:
**Viscosity approximation methods for nonexpansive mappings and monotone mappings.***Journal of Mathematical Analysis and Applications*2007,**334**(2):1450–1461. 10.1016/j.jmaa.2006.12.088MATHMathSciNetView ArticleGoogle Scholar - Iiduka H, Takahashi W, Toyoda M:
**Approximation of solutions of variational inequalities for monotone mappings.***Panamerican Mathematical Journal*2004,**14**(2):49–61.MATHMathSciNetGoogle Scholar - Iiduka H, Takahashi W:
**Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings.***Nonlinear Analysis: Theory, Methods & Applications*2005,**61**(3):341–350. 10.1016/j.na.2003.07.023MATHMathSciNetView ArticleGoogle Scholar - Takahashi W, Toyoda M:
**Weak convergence theorems for nonexpansive mappings and monotone mappings.***Journal of Optimization Theory and Applications*2003,**118**(2):417–428. 10.1023/A:1025407607560MATHMathSciNetView ArticleGoogle Scholar - Jung JS:
**A new iteration method for nonexpansive mappings and monotone mappings in Hilbert spaces.***Journal of Inequalities and Applications*2010,**2010:**-16.Google Scholar - 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.MATHMathSciNetView ArticleGoogle Scholar - Xu H-K:
**Iterative algorithms for nonlinear operators.***Journal of the London Mathematical Society*2002,**66**(1):240–256. 10.1112/S0024610702003332MATHMathSciNetView ArticleGoogle Scholar - Xu H-K:
**An iterative approach to quadratic optimization.***Journal of Optimization Theory and Applications*2003,**116**(3):659–678. 10.1023/A:1023073621589MATHMathSciNetView ArticleGoogle Scholar - 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.2005.05.028MATHMathSciNetView ArticleGoogle Scholar - Moudafi A:
**Viscosity approximation methods for fixed-points problems.***Journal of Mathematical Analysis and Applications*2000,**241**(1):46–55. 10.1006/jmaa.1999.6615MATHMathSciNetView ArticleGoogle Scholar - Xu H-K:
**Viscosity approximation methods for nonexpansive mappings.***Journal of Mathematical Analysis and Applications*2004,**298**(1):279–291. 10.1016/j.jmaa.2004.04.059MATHMathSciNetView ArticleGoogle Scholar - Browder FE, Petryshyn WV:
**Construction of fixed points of nonlinear mappings in Hilbert space.***Journal of Mathematical Analysis and Applications*1967,**20:**197–228. 10.1016/0022-247X(67)90085-6MATHMathSciNetView ArticleGoogle Scholar - Rockafellar RT:
**On the maximality of sums of nonlinear monotone operators.***Transactions of the American Mathematical Society*1970,**149:**75–88. 10.1090/S0002-9947-1970-0282272-5MATHMathSciNetView ArticleGoogle Scholar - Rockafellar RT:
**Monotone operators and the proximal point algorithm.***SIAM Journal on Control and Optimization*1976,**14**(5):877–898. 10.1137/0314056MATHMathSciNetView ArticleGoogle Scholar - Oden JT:
*Qualitative Methods on Nonlinear Mechanics*. Prentice-Hall, Englewood Cliffs, NJ, USA; 1986.MATHGoogle Scholar - Yao Y, Noor MA, Zainab S, Liou Y-C:
**Mixed equilibrium problems and optimization problems.***Journal of Mathematical Analysis and Applications*2009,**354**(1):319–329. 10.1016/j.jmaa.2008.12.055MATHMathSciNetView ArticleGoogle Scholar - Jung JS:
**Iterative algorithms with some control conditions for quadratic optimizations.***Panamerican Mathematical Journal*2006,**16**(4):13–25.MATHMathSciNetGoogle Scholar - Jung JS, Cho YJ, Agarwal RP:
**Iterative schemes with some control conditions for a family of finite nonexpansive mappings in Banach spaces.***Fixed Point Theory and Applications*2005,**2005**(2):125–135.MATHMathSciNetView ArticleGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.