- Research Article
- Open Access

# Iterative Schemes for Generalized Equilibrium Problem and Two Maximal Monotone Operators

- L. C. Zeng
^{1, 2}, - Y. C. Lin
^{3}and - J. C. Yao
^{4}Email author

**2009**:896252

https://doi.org/10.1155/2009/896252

© L. C. Zeng et al. 2009

**Received:**20 July 2009**Accepted:**27 October 2009**Published:**3 November 2009

## Abstract

The purpose of this paper is to introduce and study two new hybrid proximal-point algorithms for finding a common element of the set of solutions to a generalized equilibrium problem and the sets of zeros of two maximal monotone operators in a uniformly smooth and uniformly convex Banach space. We established strong and weak convergence theorems for these two modified hybrid proximal-point algorithms, respectively.

## Keywords

- Hilbert Space
- Banach Space
- Variational Inequality
- Nonexpansive Mapping
- Lower Semicontinuous

## 1. Introduction

Whenever a Hilbert space, problem (1.2) was very recently introduced and considered by Kamimura and Takahashi [12]. Problem (1.2) is very general in the sense that it includes, as spacial cases, optimization problems, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games, and others; see, for example, [13, 14]. A mapping is called nonexpansive if for all . Denote by the set of fixed points of , that is, . Iterative schemes for finding common elements of EP and fixed points set of nonexpansive mappings have been studied recently; see, for example, [12, 15–17] and the references therein.

where is a sequence in , for each is the resolvent operator for , and is the identity operator on . This algorithm was first introduced by Martinet [14] and generally studied by Rockafellar [18] in the framework of a Hilbert space . Later many authors studied (1.5) and its variants in a Hilbert space or in a Banach space ; see, for example, [13, 19–23] and the references therein.

Let be a uniformly smooth and uniformly convex Banach space and let be a nonempty closed convex subset of . Let be a bifunction from to satisfying the following conditions (A1)–(A4) which were imposed in [24]:

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

(A4)for all is convex and lower semicontinuous.

Let be a maximal monotone operator such that

The purpose of this paper is to introduce and study two new iterative algorithms for finding a common element of the set of solutions for the generalized equilibrium problem (1.2) and the set for maximal monotone operators in a uniformly smooth and uniformly convex Banach space . First, motivated by Kamimura and Takahashi [12, Theorem ], Ceng et al. [16, Theorem ], and Zhang [17, Theorem ], we introduce a sequence that, under some appropriate conditions, is strongly convergent to in Section 3. Second, inspired by Kamimura and Takahashi [12, Theorem ], Ceng et al. [16, Theorem ], and Zhang [17, Theorem ], we define a sequence weakly convergent to an element , where in Section 4. Our results represent a generalization of known results in the literature, including Takahashi and Zembayashi [15], Kamimura and Takahashi [12], Li and Song [22], Ceng and Yao [25], and Ceng et al. [16]. In particular, compared with Theorems and in [16], our results (i.e., Theorems 3.2 and 4.2 in this paper) extend the problem of finding an element of to the one of finding an element of . Meantime, the algorithms in this paper are very different from those in [16] (because of considering the complexity involving the problem of finding an element of ).

## 2. Preliminaries

In the sequel, we denote the strong convergence, weak convergence and weak* convergence of a sequence to a point by , and , respectively.

The proof of the main results of Sections 3 and 4 will be based on the following assumption.

Assumption A.

Let be a uniformly smooth and uniformly convex Banach space and let be a nonempty closed convex subset of . Let be a bifunction from to satisfying the same conditions (A1)–(A4) as in Section 1. Let be two maximal monotone operators such that

It is clear that in a Hilbert space , (2.2) reduces to .

Let be a mapping from into itself. A point in is called an asymptotically fixed point of if contains a sequence which converges weakly to such that [28]. The set of asymptotically fixed points of will be denoted by . A mapping from into itself is called relatively nonexpansive if and , for all and [15].

Observe that, if is a reflexive strictly convex and smooth Banach space, then for any if and only if . To this end, it is sufficient to show that if then . Actually, from (2.4), we have which implies that . From the definition of , we have and therefore, ; see [29] for more details.

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

Lemma 2.1 (Kamimura and Takahashi [12]).

Let be a smooth and uniformly convex Banach space and let and be two sequences of . If and either or is bounded, then .

Lemma 2.2 (Alber [26], Kamimura and Takahashi [12]).

Lemma 2.3 (Alber [26], Kamimura and Takahashi [12]).

Lemma 2.4 (Rockafellar [18]).

Let be a reflexive strictly convex and smooth Banach space and let be a multivalued operator. Then there hold the following hold:

(i) is closed and convex if is maximal monotone such that ;

(ii) is maximal monotone if and only if is monotone with for all .

Lemma 2.5 (Xu [30]).

Lemma 2.6 (Kamimura and Takahashi [12]).

The following result is due to Blum and Oettli [24].

Lemma 2.7 (Blum and Oettli [24]).

Motivated by Combettes and Hirstoaga [31] in a Hilbert space, Takahashi and Zembayashi [15] established the following lemma.

Lemma 2.8 (Takahashi and Zembayashi [15]).

for all . Then, the following hold:

Using Lemma 2.8, one has the following result.

Lemma 2.9 (Takahashi and Zembayashi [15]).

Utilizing Lemmas 2.7, 2.8 and 2.9 as previously mentioned, Zhang [17] derived the following result.

Proposition 2.10 (Zhang [21, Lemma ]).

Let be a smooth strictly convex and reflexive Banach space and let be a nonempty closed convex subset of . Let be an -inverse-strongly monotone mapping, let be a bifunction from to satisfying (A1)–(A4), and let . Then the following hold:

then the mapping has the following properties:

(iv) is a closed convex subset of ,

Lemma 2.11 (Kohsaka and Takahashi [13]).

Lemma 2.12 (Tan and Xu [32]).

Let and be two sequences of nonnegative real numbers satisfying the inequality: for all . If , then exists.

## 3. Strong Convergence Theorem

In this section, we prove a strong convergence theorem for finding a common element of the set of solutions for a generalized equilibrium problem and the set for two maximal monotone operators and .

Lemma 3.1.

where and is the duality mapping on . In particular, whenever a real Hilbert space, is a nonexpansive mapping on .

Proof.

Theorem 3.2.

and satisfies . Then, the sequence converges strongly to provided for any sequence with , where is the generalized projection of onto .

Remark 3.3.

In this case, we can remove the requirement that for any sequence with .

Proof of Theorem 3.2.

We divide the proof into several steps.

Step 1.

We claim that is closed and convex for each .

This implies that . Therefore, is closed and convex.

Step 2.

We claim that for each and that is well defined.

As by the induction assumption, the last inequality holds, in particular, for all . This, together with the definition of implies that . Hence (3.20) holds for all . So, for all . This implies that the sequence is well defined.

Step 3.

We claim that is bounded and that as .

Step 4.

and therefore, . Since is uniformly norm-to-norm continuous on bounded subsets of and , then .

Thus, from (3.35) it follows that . Since is uniformly norm-to-norm continuous on bounded subsets of , it follows that .

Step 5.

Thus . Therefore, we obtain that by the arbitrariness of .

Step 6.

We claim that converges strongly to .

which implies that . Since has the Kadec-Klee property, then . Therefore, converges strongly to .

Remark 3.4.

In this case, the previous Theorem 3.2 reduces to [20, Theorem ].

## 4. Weak Convergence Theorem

In this section, we present the following algorithm for finding a common element of the set of solutions for a generalized equilibrium problem and the set for two maximal monotone operators and .

where , and , is defined by (2.15).

Before proving a weak convergence theorem, we need the following proposition.

Proposition 4.1.

Then, converges strongly to , where is the generalized projection of onto .

Proof.

for all . So, from , and Lemma 2.12, we deduce that exists. This implies that is bounded. Thus, is bounded and so are , and .

Hence, from (4.12), it follows that .

Since is a convergent sequence, is bounded and is convergent, from the property of we have that is a Cauchy sequence. Since is closed, converges strongly to . This completes the proof.

Now, we are in a position to prove the following theorem.

Theorem 4.2.

and satisfies . If is weakly sequentially continuous, then converges weakly to , where .

Proof.

From Lemma 2.5 and as in the proof of Theorem 3.2, there exists a continuous, strictly increasing, and convex function with such that

Thus, from , and , it follows that . In terms of Lemma 2.1, we derive .

Next, let us show that , where .

Then the maximality of the operators implies that .

From the strict convexity of , it follows that . Therefore, , where . This completes the proof.

Remark 4.3.

Corollary 4.4.

Suppose that conditions (A1)–(A5) are fulfilled and let be a sequence defined by (4.65), where is defined in Lemma 2.8, satisfy the conditions and , and satisfies . If is weakly sequentially continuous, then converges weakly to , where .

## Declarations

### Acknowledgments

The first author (http://zenglc@hotmail.com) was partially supported by the National Science Foundation of China (10771141), Ph. D. Program Foundation of Ministry of Education of China (20070270004), and Science and Technology Commission of Shanghai Municipality grant (075105118), Leading Academic Discipline Project of Shanghai Normal University (DZL707), Shanghai Leading Academic Discipline Project (S30405) and Innovation Program of Shanghai Municipal Education Commission (09ZZ133). The third author (http://yaojc@math.nsysu.edu.tw) was partially supported by the Grant NSF 97-2115-M-110-001.

## Authors’ Affiliations

## References

- 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.MathSciNetMATHGoogle Scholar - Schaible S, Yao J-C, Zeng L-C:
**A proximal method for pseudomonotone type variational-like inequalities.***Taiwanese Journal of Mathematics*2006,**10**(2):497–513.MathSciNetMATHGoogle Scholar - Zeng LC, Lin LJ, Yao JC:
**Auxiliary problem method for mixed variational-like inequalities.***Taiwanese Journal of Mathematics*2006,**10**(2):515–529.MathSciNetMATHGoogle Scholar - Peng J-W, Yao J-C: Ishikawa iterative algorithms for a generalized equilibrium problem and fixed point problems of a pseudo-contraction mapping. to appear in Journal of Global Optimization to appear in Journal of Global OptimizationGoogle 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.MathSciNetMATHGoogle Scholar - Peng J-W, Yao J-C: Some new extragradient-like methods for generalized equilibrium problems, fixed points problems and variational inequality problems. to appear in Optimization Methods & Software to appear in Optimization Methods & SoftwareGoogle Scholar
- Ceng L-C, Lee C, Yao J-C:
**Strong weak convergence theorems of implicit hybrid steepest-descent methods for variational inequalities.***Taiwanese Journal of Mathematics*2008,**12**(1):227–244.MathSciNetMATHGoogle 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:
**Strong convergence theorems of iterative scheme based on the extragradient method for mixed equilibrium problems and fixed point problems.***Mathematical and Computer Modelling*2009,**49**(9–10):1816–1828. 10.1016/j.mcm.2008.11.014MathSciNetView ArticleMATHGoogle Scholar - Ceng L-C, Ansari QH, Yao J-C:
**Viscosity approximation methods for generalized equilibrium problems and fixed point problems.***Journal of Global Optimization*2009,**43**(4):487–502. 10.1007/s10898-008-9342-6MathSciNetView ArticleMATHGoogle Scholar - Peng JW, Yao JC: Some new iterative algorithms for generalized mixed equilibrium problems with strict pseudo-contractions and monotone mappings. to appear in Taiwanese Journal of Mathematics to appear in Taiwanese Journal of MathematicsGoogle Scholar
- Kamimura S, Takahashi W:
**Strong convergence of a proximal-type algorithm in a Banach space.***SIAM Journal on Optimization*2003,**13**(3):938–945.MathSciNetView ArticleMATHGoogle Scholar - Kohsaka F, Takahashi W:
**Strong convergence of an iterative sequence for maximal monotone operators in a Banach space.***Abstract and Applied Analysis*2004,**2004**(3):239–249. 10.1155/S1085337504309036MathSciNetView ArticleMATHGoogle Scholar - Martinet B:
**Régularisation d'inéquations variationnelles par approximations successives.***Revue Française d'Informatique et de Recherche Opérationnelle*1970,**4:**154–158.MathSciNetMATHGoogle Scholar - Takahashi W, Zembayashi K:
**Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces.***Nonlinear Analysis: Theory, Methods & Applications*2009,**70**(1):45–57. 10.1016/j.na.2007.11.031MathSciNetView ArticleMATHGoogle Scholar - Ceng LC, Mastroeni G, Yao JC:
**Hybrid proximal-point methods for common solutions of equilibrium problems and zeros of maximal monotone operators.***Journal of Optimization Theory and Applications*2009,**142**(3):431–449. 10.1007/s10957-009-9538-zMathSciNetView ArticleMATHGoogle Scholar - Zhang SS: Shrinking projection method for solving generalized equilibrium problem, variational inequality and common fixed point in Banach spaces with applications. to appear in Science in China Series A to appear in Science in China Series AGoogle Scholar
- Rockafellar RT:
**Monotone operators and the proximal point algorithm.***SIAM Journal on Control and Optimization*1976,**14**(5):877–898. 10.1137/0314056MathSciNetView ArticleMATHGoogle 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 - Zeng LC, Yao JC:
**An inexact proximal-type algorithm in Banach spaces.***Journal of Optimization Theory and Applications*2007,**135**(1):145–161. 10.1007/s10957-007-9261-6MathSciNetView ArticleMATHGoogle Scholar - Kamimura S, Kohsaka F, Takahashi W:
**Weak and strong convergence theorems for maximal monotone operators in a Banach space.***Set-Valued Analysis*2004,**12**(4):417–429. 10.1007/s11228-004-8196-4MathSciNetView ArticleMATHGoogle Scholar - Li L, Song W:
**Modified proximal-point algorithm for maximal monotone operators in Banach spaces.***Journal of Optimization Theory and Applications*2008,**138**(1):45–64. 10.1007/s10957-008-9370-xMathSciNetView ArticleMATHGoogle Scholar - Güler O:
**On the convergence of the proximal point algorithm for convex minimization.***SIAM Journal on Control and Optimization*1991,**29**(2):403–419. 10.1137/0329022MathSciNetView 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:
**Hybrid viscosity approximation schemes for equilibrium problems and fixed point problems of infinitely many nonexpansive mappings.***Applied Mathematics and Computation*2008,**198**(2):729–741. 10.1016/j.amc.2007.09.011MathSciNetView ArticleMATHGoogle Scholar - Alber YI:
**Metric and generalized projection operators in Banach spaces: properties and applications.**In*Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, Lecture Notes in Pure and Applied Mathematics*.*Volume 178*. Edited by: Kartsatos AG. Dekker, New York, NY, USA; 1996:15–50.Google Scholar - Alber YI, Guerre-Delabriere S:
**On the projection methods for fixed point problems.***Analysis*2001,**21**(1):17–39.MathSciNetView ArticleMATHGoogle Scholar - Reich S:
**A weak convergence theorem for the alternating method with Bregman distances.**In*Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, Lecture Notes in Pure and Applied Mathematics*.*Volume 178*. Edited by: Kartsatos AG. Dekker, New York, NY, USA; 1996:313–318.Google Scholar - Cioranescu I:
*Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Mathematics and Its Applications*.*Volume 62*. Kluwer Academic, Dordrecht, The Netherlands; 1990:xiv+260.View ArticleMATHGoogle Scholar - Xu HK:
**Inequalities in Banach spaces with applications.***Nonlinear Analysis: Theory, Methods & Applications*1991,**16**(12):1127–1138. 10.1016/0362-546X(91)90200-KMathSciNetView ArticleMATHGoogle Scholar - Combettes PL, Hirstoaga SA:
**Equilibrium programming in Hilbert spaces.***Journal of Nonlinear and Convex Analysis*2005,**6**(1):117–136.MathSciNetMATHGoogle Scholar - Tan K-K, Xu HK:
**Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process.***Journal of Mathematical Analysis and Applications*1993,**178**(2):301–308. 10.1006/jmaa.1993.1309MathSciNetView ArticleMATHGoogle Scholar

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