# An Iterative Algorithm Combining Viscosity Method with Parallel Method for a Generalized Equilibrium Problem and Strict Pseudocontractions

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

**2009**:794178

**DOI: **10.1155/2009/794178

© Jian-Wen Peng et al. 2009

**Received: **5 August 2008

**Accepted: **4 January 2009

**Published: **10 February 2009

## Abstract

We introduce a new approximation scheme combining the viscosity method with parallel method for finding a common element of the set of solutions of a generalized equilibrium problem and the set of fixed points of a family of finitely strict pseudocontractions. We obtain a strong convergence theorem for the sequences generated by these processes in Hilbert spaces. Based on this result, we also get some new and interesting results. The results in this paper extend and improve some well-known results in the literature.

## 1. Introduction

The set of solutions of (1.1) is denoted by . Flores-Bazán [1] provided some characterizations of the nonemptiness of the solution set for problem (1.1) in reflexive Banach spaces in the quasiconvex case. Bigi et al. [2] studied a dual problem associated with the problem (1.1) with .

The set of solutions of (1.2) is denoted by . The problem (1.2) includes, as special cases, the optimization problem, the variational inequality problem, the fixed point problem, the nonlinear complementarity problem, the Nash equilibrium problem in noncooperative games, and the vector optimization problem. For more detail, please see [3–5] and the references therein.

The set of solutions of (1.3) is denoted by .

where denotes the identity operator on . When , is said to be nonexpansive. Note that the class of strict pseudocontraction mappings strictly includes the class of nonexpansive mappings. We denote the set of fixed points of by .

Ceng and Yao [6], Yao et al. [8], and Peng and Yao [9, 10] introduced some iterative schemes for finding a common element of the set of solutions of the mixed equilibrium problem (1.4) and the set of common fixed points of a family of finitely (infinitely) nonexpansive mappings (strict pseudocontractions) in a Hilbert space and obtained some strong convergence theorems(weak convergence theorems). Some methods have been proposed to solve the problem (1.2); see, for instance, [3–5, 11–18] and the references therein. Recently, S. Takahashi and W. Takahashi [12] introduced an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of problem (1.2) and the set of fixed points of a nonexpansive mapping in a Hilbert space and proved a strong convergence theorem. Su et al. [13] introduced an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of problem (1.2) and the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality problem for an -inverse strongly monotone mapping in a Hilbert space. Tada and Takahashi [14] introduced two iterative schemes for finding a common element of the set of solutions of problem (1.2) and the set of fixed points of a nonexpansive mapping in a Hilbert space and obtained both strong convergence theorem and weak convergence theorem. Ceng et al. [15] introduced an iterative algorithm for finding a common element of the set of solutions of problem (1.2) and the set of fixed points of a strict pseudocontraction mapping. Chang et al. [16] introduced some iterative processes based on the extragradient method for finding the common element of the set of fixed points of a family of infinitely nonexpansive mappings, the set of problem (1.2), and the set of solutions of a variational inequality problem for an -inverse strongly monotone mapping. Colao et al. [17] introduced an iterative method for finding a common element of the set of solutions of problem (1.2) and the set of fixed points of a finite family of nonexpansive mappings in a Hilbert space and proved the strong convergence of the proposed iterative algorithm to the unique solution of a variational inequality, which is the optimality condition for a minimization problem. To the best of our knowledge, there is not any algorithms for solving problem (1.1).

On the other hand, Marino and Xu [19] and Zhou [20] introduced and researched some iterative scheme for finding a fixed point of a strict pseudocontraction mapping. Acedo and Xu [21] introduced some parallel and cyclic algorithms for finding a common fixed point of a family of finite strict pseudocontraction mappings and obtained both weak and strong convergence theorems for the sequences generated by the iterative schemes.

In the present paper, we introduce a new approximation scheme combining the viscosity method with parallel method for finding a common element of the set of solutions of the generalized equilibrium problem and the set of fixed points of a family of finitely strict pseudocontractions. We obtain a strong convergence theorem for the sequences generated by these processes. Based on this result, we also get some new and interesting results. The results in this paper extend and improve some well-known results in the literature.

## 2. Preliminaries

for all and .

for all and .

For each , we denote by the convex hull of . A multivalued mapping is said to be a KKM map if, for every finite subset ,

We will use the following results in the sequel.

Lemma 2.1 (see [22]).

Let be a nonempty subset of a Hausdorff topological vector space and let be a KKM map. If is closed for all and is compact for at least one , then

For solving the generalized equilibrium problem, let us give the following assumptions for the bifunction , and the set :

(A1) for all ;

(A2) is monotone, that is, for any ;

(A3) for each , is weakly upper semicontinuous;

(A4)for each is convex;

(A5)for each is lower semicontinuous;

(B2) is a bounded set.

Lemma 2.2.

for all . Assume that either (B1) or (B2) holds. Then, the following conclusions hold:

(1)for each , ;

(2) is single-valued;

(4)

(5) is closed and convex.

Proof.

Hence, is weakly compact. Thus, in both cases, we can use Lemma 2.1 and have .

This implies that . Hence, is weakly closed. Hence, . Hence, from the arbitrariness of , we conclude that , .

At last, we claim that is a closed convex. Indeed, Since is firmly nonexpansive, is also nonexpansive. By [23, Proposition 5.3], we know that is closed and convex.

Remark 2.3.

It is easy to see that Lemma 2.2 is a generalization of [9, Lemma 2.3].

Then,

Lemma 2.5.

for all

## 3. Strong Convergence Theorems

In this section, we show a strong convergence of an iterative algorithm based on both viscosity approximation method and parallel method which solves the problem of finding a common element of the set of solutions of a generalized equilibrium problem and the set of fixed points of a family of finitely strict pseudocontractions in a Hilbert space.

We need the following assumptions for the parameters and :

(C1) and ;

(C2) ;

(C3) for some and ;

(C4) and ;

(C5) for all .

Theorem 3.1.

for every , where and are sequences of numbers satisfying the conditions (C1)–(C5). Then, , and converge strongly to .

Proof.

for all Since is complete, there exists a unique element such that .

for every Therefore, is bounded. From (3.3) and (3.5), we also obtain that and are bounded.

where

It follows from (C1) and (C2) that .

It follows from (C1), (C2), and that .

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

So by the demiclosedness principle [21, Proposition 2.6(ii)], it follows that .

Finally, we show that , where .

It follows from (C1), (3.43), (3.45), and Lemma 2.4 that . From and , we have and . The proof is now complete.

Theorem 3.2.

for every where and are sequences of numbers satisfying the conditions (C1)–(C5). Then, , and converge strongly to .

Proof.

Let for all , by Theorem 3.1, we obtain the desired result.

## 4. Applications

By Theorems 3.1 and 3.2, we can obtain many new and interesting strong convergence theorems. Now, give some examples as follows: for , let , by Theorems 3.1 and 3.2, respectively, we have the following results.

Theorem 4.1.

for every where , , and are sequences of numbers satisfying the conditions (C1)–(C4). Then, , and converge strongly to .

Theorem 4.2.

for every where , , and are sequences of numbers satisfying the conditions (C1)–(C4). Then, , and converge strongly to .

We need the following two assumptions.

Let , by Theorems 3.1 and 3.2, respectively, we obtain the following results.

Theorem 4.3.

for every where and are sequences of numbers satisfying the conditions (C1)–(C5). Then, , and converge strongly to .

Theorem 4.4.

for every where and are sequences of numbers satisfying the conditions (C1)–(C5). Then, , and converge strongly to .

Let for all , by Theorems 3.1 and 3.2, respectively, we obtain the following results.

Theorem 4.5.

for every , where and are sequences of numbers satisfying the conditions (C1)–(C5). Then, , and converge strongly to .

Theorem 4.6.

Let , and let for all . Then . By Theorems 3.1 and 3.2, we obtain the following results.

Theorem 4.7.

for every , where and are sequences of numbers satisfying the conditions (C1)–(C3) and (C5). Then, and converge strongly to .

Theorem 4.8.

for every , where and are sequences of numbers satisfying the conditions (C1)–(C3) and (C5). Then, and converge strongly to .

- (1)
Since the nonexpansive mappings have been replaced by the strict pseudocontractions, Theorems 3.1, 3.2, 4.1 and 4.2 extend and improve [6, Theorem 3.1], [8, Theorem 3.5], [9, Theorems 4.1 and 4.2], [18, Theorem 4.1], and the main results in [9–11, 13–16].

- (2)
Since the weak convergence has been replaced by strong convergence, Theorems 3.1, 3.2, 4.1−4.4 extend and improve [12, Theorem 3.1], [10, Corollary 4.1].

- (3)
Theorems 4.7 and 4.8 are strong convergence theorems for strict pseudocontractions without constraints and hence they improve the corresponding results in [19, 21]. Theorems 3.1 and 3.2 also improve [10, Corollary 3.1].

## Declarations

### Acknowledgments

The first author was supported by the National Natural Science Foundation of China (Grants No. 10771228 and No. 10831009), the Science and Technology Research Project of Chinese Ministry of Education (Grant no. 206123), the Education Committee project Research Foundation of Chongqing Normal University (Grant no. KJ070816); the second and third authors were partially supported by the Grants NSC97-2221-E-230-017 and NSC96-2628-E-110-014-MY3, respectively.

## Authors’ Affiliations

## References

- Flores-Bazán F:
**Existence theorems for generalized noncoercive equilibrium problems: the quasi-convex case.***SIAM Journal on Optimization*2000,**11**(3):675–690.MathSciNetView ArticleMATHGoogle Scholar - Bigi G, Castellani M, Kassay G:
**A dual view of equilibrium problems.***Journal of Mathematical Analysis and Applications*2008,**342**(1):17–26. 10.1016/j.jmaa.2007.11.034MathSciNetView ArticleMATHGoogle Scholar - Iusem AN, Sosa W:
**Iterative algorithms for equilibrium problems.***Optimization*2003,**52**(3):301–316. 10.1080/0233193031000120039MathSciNetView ArticleMATHGoogle Scholar - Flåm SD, Antipin AS:
**Equilibrium programming using proximal-like algorithms.***Mathematical Programming*1997,**78**(1):29–41.MathSciNetView ArticleMATHGoogle Scholar - Blum E, Oettli W:
**From optimization and variational inequalities to equilibrium problems.***The Mathematics Student*1994,**63**(1–4):123–145.MathSciNetMATHGoogle Scholar - Ceng 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 ArticleMATHGoogle 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)90085-6MathSciNetView ArticleMATHGoogle Scholar - Yao Y, Liou Y-C, Yao J-C:
**A new hybrid iterative algorithm for fixed-point problems, variational inequality problems, and mixed equilibrium problems.***Fixed Point Theory and Applications*2008,**2008:**-15.Google Scholar - Peng J-W, Yao J-C:
**A new hybrid-extragradient method for generalized mixed equilibrium problems, fixed point problems and variational inequality problems.***Taiwanese Journal of Mathematics*2008,**12**(6):1401–1432.MathSciNetMATHGoogle Scholar - Peng J-W, Yao J-C:
**Some new iterative algorithms for generalized mixed equilibrium problems with strict pseudo-contractions and monotone mappings.**to appear in*Taiwanese Journal of Mathematics* - Combettes PL, Hirstoaga SA:
**Equilibrium programming in Hilbert spaces.***Journal of Nonlinear and Convex Analysis*2005,**6**(1):117–136.MathSciNetMATHGoogle 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.2006.08.036MathSciNetView ArticleMATHGoogle Scholar - Su Y, Shang M, Qin X:
**An iterative method of solution for equilibrium and optimization problems.***Nonlinear Analysis: Theory, Methods & Applications*2008,**69**(8):2709–2719. 10.1016/j.na.2007.08.045MathSciNetView ArticleMATHGoogle Scholar - Tada A, Takahashi W:
**Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem.***Journal of Optimization Theory and Applications*2007,**133**(3):359–370. 10.1007/s10957-007-9187-zMathSciNetView ArticleMATHGoogle Scholar - Ceng L-C, AI-Homidan S, Ansari QH, Yao J-C:
**An iterative scheme for equilibrium problems and fixed point problems of strict pseudo-contraction mappings.***Journal of Computational and Applied Mathematics*2009,**223**(2):967–974. 10.1016/j.cam.2008.03.032MathSciNetView ArticleMATHGoogle Scholar - Chang SS, Joseph Lee HW, Chan CK:
**A new method for solving equilibrium problem, fixed point problem and variational inequality problem with application to optimization.***Nonlinear Analysis: Theory, Methods & Applications*. In press - Colao V, Marino G, Xu H-K:
**An iterative method for finding common solutions of equilibrium and fixed point problems.***Journal of Mathematical Analysis and Applications*2008,**344**(1):340–352. 10.1016/j.jmaa.2008.02.041MathSciNetView ArticleMATHGoogle Scholar - Takahashi S, Takahashi W:
**Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space.***Nonlinear Analysis: Theory, Methods & Applications*2008,**69**(3):1025–1033. 10.1016/j.na.2008.02.042MathSciNetView ArticleMATHGoogle Scholar - Marino G, Xu H-K:
**Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces.***Journal of Mathematical Analysis and Applications*2007,**329**(1):336–346. 10.1016/j.jmaa.2006.06.055MathSciNetView ArticleMATHGoogle Scholar - Zhou H:
**Convergence theorems of fixed points for -strict pseudo-contractions in Hilbert spaces.***Nonlinear Analysis: Theory, Methods & Applications*2008,**69**(2):456–462. 10.1016/j.na.2007.05.032MathSciNetView ArticleMATHGoogle Scholar - Acedo GL, Xu H-K:
**Iterative methods for strict pseudo-contractions in Hilbert spaces.***Nonlinear Analysis: Theory, Methods & Applications*2007,**67**(7):2258–2271. 10.1016/j.na.2006.08.036MathSciNetView ArticleMATHGoogle Scholar - Fan K:
**A generalization of Tychonoff's fixed point theorem.***Mathematische Annalen*1961,**142:**305–310. 10.1007/BF01353421MathSciNetView ArticleMATHGoogle Scholar - Goebel K, Reich S:
*Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Monographs and Textbooks in Pure and Applied Mathematics*.*Volume 83*. Marcel Dekker, New York, NY, USA; 1984:ix+170.Google Scholar - Xu H-K:
**Iterative algorithms for nonlinear operators.***Journal of the London Mathematical Society. Second Series*2002,**66**(1):240–256. 10.1112/S0024610702003332MathSciNetView ArticleMATHGoogle Scholar - Xu HK:
**An iterative approach to quadratic optimization.***Journal of Optimization Theory and Applications*2003,**116**(3):659–678. 10.1023/A:1023073621589MathSciNetView ArticleMATHGoogle Scholar - Marino G, Colao V, Qin X, Kang SM:
**Strong convergence of the modified Mann iterative method for strict pseudo-contractions.***Computers & Mathematics with Applications*2009,**57**(3):455–465. 10.1016/j.camwa.2008.10.073MathSciNetView ArticleMATHGoogle 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.2005.103View ArticleMATHMathSciNetGoogle 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.