# A New Hybrid Iterative Algorithm for Fixed-Point Problems, Variational Inequality Problems, and Mixed Equilibrium Problems

- Yonghong Yao
^{1}, - Yeong-Cheng Liou
^{2}and - Jen-Chih Yao
^{3}Email author

**2008**:417089

**DOI: **10.1155/2008/417089

© Yonghong Yao et al. 2008

**Received: **29 August 2007

**Accepted: **6 February 2008

**Published: **14 February 2008

## Abstract

We introduce a new hybrid iterative algorithm for finding a common element of the set of fixed points of an infinite family of nonexpansive mappings, the set of solutions of the variational inequality of a monotone mapping, and the set of solutions of a mixed equilibrium problem. This study, proves a strong convergence theorem by the proposed hybrid iterative algorithm which solves fixed-point problems, variational inequality problems, and mixed equilibrium problems.

## 1. Introduction

Let be a nonempty closed convex subset of a real Hilbert space . Recall that a mapping is called contractive if there exists a constant such that for all . A mapping is said to be nonexpansive if for all . Denote the set of fixed points of by .

Denote the set of solutions of MEP by . The mixed equilibrium problems include fixed-point problems, optimization problems, variational inequality problems, Nash equilibrium problems, and the equilibrium problems as special cases (see, e.g., [1–5]). Some methods have been proposed to solve the MEP and EP (see, e.g., [5–14]). In 1997, Combettes and Hirstoaga [13] introduced an iterative method of finding the best approximation to the initial data and proved a strong convergence theorem. Subsequently, S. Takahashi and W. Takahashi [8] introduced another iterative scheme for finding a common element of the set of solutions of EP and the set of fixed-point points of a nonexpansive mapping. Yao et al. [12] considered an iterative scheme for finding a common element of the set of solutions of EP and the set of common fixed points of an infinite nonexpansive mappings. Very recently, Zeng and Yao [14] considered a new iterative scheme for finding a common element of the set of solutions of MEP and the set of common fixed points of finitely many nonexpansive mappings. Their results extend and improve many results in the literature.

Recently, some authors have proposed new iterative algorithms to approximate a common element of the set of fixed points of a nonxpansive mapping and the set of solutions of the variational inequality. For the details, see [15, 16] and the references therein.

Motivated by the recent works, in this paper we introduce a new hybrid iterative algorithm for finding a common element of the set of fixed points of an infinite family of nonexpansive mappings, the set of solutions of the variational inequality of a monotone mapping, and the set of solutions of a mixed equilibrium problem. We prove a strong convergence theorem by the proposed hybrid iterative algorithm which solves fixed-point problems, variational inequality problems, and mixed equilibrium problems.

## 2. Preliminaries

In this paper, for solving the mixed equilibrium problems for an equilibrium bifunction , we assume that satisfies the following conditions:

(H1) is monotone, that is, for all ;

(H2) for each fixed , is concave and upper semicontinuous;

We first need the following important and interesting result.

Lemma 2.1 (see [14]).

- (i)
- (a)
- (b)
- (c)

*Then there hold the following*:

- (i)
- (ii)

We also need the following lemmas for proving our main results.

Lemma 2.2 (see [17]).

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 [18]).

## 3. Iterative Algorithm and Strong Convergence Theorems

Such is called the -mapping generated by and . For the iterative algorithm for a finite family of nonexpansive mappings, we refer the reader to [19].

We have the following crucial Lemmas 3.1 and 3.2 concerning which can be found in [20]. Now we only need the following similar version in Hilbert spaces.

Lemma 3.1.

Let be a nonempty closed convex subset of a real Hilbert space . Let be nonexpansive mappings of into itself such that is nonempty, and let be real numbers such that for any . Then for every and , the limit exists.

Lemma 3.2.

Let be a nonempty closed convex subset of a real Hilbert space . Let be nonexpansive mappings of into itself such that is nonempty, and let be real numbers such that for any . Then .

The following remark [12] is important to prove our main results.

Remark 3.3.

Throughout this paper, we will assume that for every .

Now we introduce the following iteration algorithm.

Algorithm 3.4.

where , , and are three sequences in , and is a sequence in .

Now we study the strong convergence of the hybrid iterative algorithm (3.3).

Theorem 3.5.

- (i)
- (a)
- (b)
- (c)

Let be a contraction of into itself and given arbitrarily. Then the sequence generated by (3.3) converges strongly to , where provided that is firmly nonexpansive.

Proof.

We first note that is a contraction with coefficient . Then for all . Therefore is a contraction of into itself which implies that there exists a unique element such that .

Next we divide the following proofs into several steps.

Step 1 ( , , and are bounded).

which implies that is nonexpansive.

Therefore is bounded, so are and .

where is a constant such that .

Since is bounded, there exists a subsequence of which converges weakly to . Without loss of generality, we can assume that . From , we obtain .

which is a contradiction. Hence we get . By the same argument as that in the proof of [21, Theorem 3.1], we can prove that ; and by the same argument as that in the proof of [14, Theorem 4.1], we also can prove that . Hence .

Applying Lemma 2.3 and (3.34) to (3.36), we conclude that as . This completes the proof.

Concerning , we give the following remark.

Remark 3.6.

This indicates that is firmly nonexpansive.

Corollary 3.7.

- (i)
- (a)
- (b)
- (c)

Then the sequence generated by (3.46) converges strongly to , where provided that is firmly nonexpansive.

Proof.

Take for all , and for all in (3.1). Then for all . The conclusion follows immediately from Theorem 3.5. This completes the proof.

Corollary 3.8.

- (i)
- (ii)
- (iii)

converges strongly to , where .

Proof.

Set and for all and put . Take and for all . Then we have . Hence the conclusion follows. This completes the proof.

## Declarations

### Acknowledgments

The authors are extremely grateful to the anonymous referees and Professor T. Suzuki for their useful comments and suggestions. The first author was partially supposed by National Natural Science Foundation of China Grant no. 10771050. The second author was partially supposed by the Grant no. NSC 96-2221-E-230-003.

## Authors’ Affiliations

## References

- Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems.
*The Mathematics Student*1994, 63(1–4):123-145.MATHMathSciNetGoogle Scholar - Zeng L-C, Wu S-Y, Yao J-C: Generalized KKM theorem with applications to generalized minimax inequalities and generalized equilibrium problems.
*Taiwanese Journal of Mathematics*2006, 10(6):1497-1514.MATHMathSciNetGoogle Scholar - Chadli O, Wong NC, Yao J-C: Equilibrium problems with applications to eigenvalue problems.
*Journal of Optimization Theory and Applications*2003, 117(2):245-266. 10.1023/A:1023627606067MATHMathSciNetView ArticleGoogle Scholar - Chadli O, Schaible S, Yao J-C: Regularized equilibrium problems with application to noncoercive hemivariational inequalities.
*Journal of Optimization Theory and Applications*2004, 121(3):571-596.MATHMathSciNetView ArticleGoogle Scholar - Konnov IV, Schaible S, Yao J-C: Combined relaxation method for mixed equilibrium problems.
*Journal of Optimization Theory and Applications*2005, 126(2):309-322. 10.1007/s10957-005-4716-0MATHMathSciNetView ArticleGoogle Scholar - Combettes PL, Hirstoaga SA: Equilibrium programming in Hilbert spaces.
*Journal of Nonlinear and Convex Analysis*2005, 6(1):117-136.MATHMathSciNetGoogle Scholar - Flåm SD, Antipin AS: Equilibrium programming using proximal-like algorithms.
*Mathematical Programming*1996, 78(1):29-41. 10.1007/BF02614504View ArticleGoogle Scholar - Takahashi S, Takahashi W: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces.
*Journal of Mathematical Analysis and Applications*2007, 331(1):506-515. 10.1016/j.jmaa.2006.08.036MATHMathSciNetView ArticleGoogle Scholar - Chadli O, Konnov IV, Yao J-C: Descent methods for equilibrium problems in a Banach space.
*Computers & Mathematics with Applications*2004, 48(3-4):609-616. 10.1016/j.camwa.2003.05.011MATHMathSciNetView ArticleGoogle Scholar - Ding XP, Lin YC, Yao J-C: Predictor-corrector algorithms for solving generalized mixed implicit quasi-equilibrium problems.
*Applied Mathematics and Mechanics*2006, 27(9):1157-1164. 10.1007/s10483-006-0901-1MATHMathSciNetView ArticleGoogle Scholar - Tada A, Takahashi W: Strong convergence theorem for an equilibrium problem and a nonexpansive mapping. In
*Nonlinear Analysis and Convex Analysis*. Edited by: Takahashi W, Tanaka T. Yokohama Publishers, Yokohama, Japan; 2007:609-617.Google Scholar - Yao Y, Liou Y-C, Yao J-C: Convergence theorem for equilibrium problems and fixed point problems of infinite family of nonexpansive mappings.
*Fixed Point Theory and Applications*2007, 2007:-12.Google Scholar - Combettes PL, Hirstoaga AS: Equilibrium programming using proximal-like algorithms.
*Mathematical Programming*1997, 78(1):29-41.MathSciNetGoogle Scholar - Zeng L-C, Yao J-C: A hybrid iterative scheme for mixed equilibrium problems and fixed point problems.
*Journal of Computational and Applied Mathematics*2008, 214(1):186-201. 10.1016/j.cam.2007.02.022MathSciNetView ArticleGoogle Scholar - Zeng L-C, Yao J-C: Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems.
*Taiwanese Journal of Mathematics*2006, 10(5):1293-1303.MATHMathSciNetGoogle Scholar - Yao Y, Yao J-C: On modified iterative method for nonexpansive mappings and monotone mappings.
*Applied Mathematics and Computation*2007, 186(2):1551-1558. 10.1016/j.amc.2006.08.062MATHMathSciNetView ArticleGoogle Scholar - Suzuki T: Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces.
*Fixed Point Theory and Applications*2005, 2005(1):103-123. 10.1155/FPTA.2005.103MATHView 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 - Kuhfittig PKF: Common fixed points of nonexpansive mappings by iteration.
*Pacific Journal of Mathematics*1981, 97(1):137-139.MATHMathSciNetView ArticleGoogle Scholar - Shimoji K, Takahashi W: Strong convergence to common fixed points of infinite nonexpansive mappings and applications.
*Taiwanese Journal of Mathematics*2001, 5(2):387-404.MATHMathSciNetGoogle Scholar - Nadezhkina N, Takahashi W: Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings.
*Journal of Optimization Theory and Applications*2006, 128(1):191-201. 10.1007/s10957-005-7564-zMATHMathSciNetView 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.