# Strong and Weak Convergence Theorems for Common Solutions of Generalized Equilibrium Problems and Zeros of Maximal Monotone Operators

- L-C Zeng
^{1, 2}, - QH Ansari
^{3}, - DavidS Shyu
^{4}and - J-C Yao
^{5}Email author

**2010**:590278

https://doi.org/10.1155/2010/590278

© L.-C. Zeng et al. 2010

**Received: **27 October 2009

**Accepted: **12 January 2010

**Published: **21 January 2010

## Abstract

The purpose of this paper is to introduce and study two modified hybrid proximal-point algorithms for finding a common element of the solution set *EP* of a generalized equilibrium problem and the set
for two maximal monotone operators
and
defined on a Banach space
. Strong and weak convergence theorems for these two modified hybrid proximal-point algorithms are established.

## 1. Introduction

Let be a real Banach space with its dual . The mapping defined by

is called the normalized duality mapping. From the Hahn-Banach theorem, it follows that for each .

A Banach space is said to be strictly convex, if for all with . is said to be uniformly convex if for each , there exists such that for all with . Recall that each uniformly convex Banach space has the Kadec-Klee property, that is,

It is well known that if is strictly convex, then is single-valued. In the sequel, we shall still denote the single-valued normalized duality mapping by . Let be a nonempty closed convex subset of , a bifunction, and a nonlinear mapping. Very recently, Zhang [1] considered and studied the generalized equilibrium problem of finding such that

The set of solutions of (1.3) is denoted by . Problem (1.3) and related problems have been studied and investigated extensively in the literature; See, for example, [2–12] and references therein. Whenever , problem (1.3) reduces to the equilibrium problem of finding such that

The set of solutions of (1.4) is denoted by . Whenever , problem (1.3) reduces to the variational inequality problem of finding such that

The set of solutions of (1.5) is denoted by .

Whenever a Hilbert space, problem (1.3) was very recently introduced and considered by S. Takahashi and W. Takahashi [13]. Problem (1.3) is very general in the sense that it includes, as spacial cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problem in noncooperative games, and others; See, for example, [1, 2, 4, 6–9, 14–17] which are references therein.

A mapping is called nonexpansive if for all . Denote by the set of fixed points of , that is, . Very recently, W. Takahashi and K. Zembayashi [18] proposed an iterative algorithm for finding a common element of the solution set of the equilibrium problem (1.4) and the set of fixed points of a relatively nonexpansive mapping in a Banach space . They also studied the strong and weak convergence of the sequences generated by their algorithm. In particular, they proposed the following iterative algorithm:

where for all , and for some . They proved that the sequence generated by the above algorithm converges strongly to , where is the generalized projection of onto . They have also studied the weak convergence of the sequence generated by the following algorithm:

Let be a nonempty closed convex subset of a uniformly smooth and uniformly convex Banach space . Let be an -inverse-strongly monotone mapping and a bifunction satisfying the following conditions:

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

(A4) for all , is convex and lower semicontinuous.

Let be two relatively nonexpansive mappings such that . Let be the sequence generated by

Zhang [1] proved the strong convergence of the sequence to under appropriate conditions.

On the other hand, a classic method of solving in a Hilbert space is the proximal point algorithm which generates, for any starting point , a sequence in by the iterative scheme

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 [19] and further studied by Rockafellar [20] in the framework of a Hilbert space . Later several authors studied (1.9) and its variants in the setting of a Hilbert space or in a Banach space ; See, for example, [15, 21–25] and references therein. Very recently, Li and Song [24] introduced and studied the following iterative scheme:

where and is the duality mapping on .

Algorithm (1.10) covers, as special cases, the algorithms introduced by Kohsaka and Takahashi [23] and Kamimura et al. [22] in a smooth and uniformly convex Banach space .

Let be a uniformly smooth and uniformly convex Banach space, and let be a nonempty closed convex subset of . Let be a maximal monotone operator such that:

In addition, for each , define a mapping as follows:

Very recently, utilizing the ideas of the above algorithms in [15, 16, 18, 21, 22, 24], we [17] introduced two iterative methods for finding an element of and established the following strong and weak convergence theorems.

Theorem 1.1 (see [17]).

is defined by (1.11), satisfy , , and satisfies . Then, the sequence converges strongly to , where is the generalized projection of onto .

Theorem 1.2 (see [17]).

where is defined by (1.11), satisfy the conditions and , and satisfies . If is weakly sequentially continuous, then converges weakly to an element , where .

The purpose of this paper is to introduce and study two new iterative methods for finding a common element of the solution set of generalized equilibrium problem (1.3) and the set for maximal monotone operators and in a uniformly smooth and uniformly convex Banach space . Firstly, motivated by Theorem 1.1 and a result of Zhang [1], we introduce a sequence that converges strongly to under some appropriate conditions.

Secondly, inspired by Theorem 1.2 and a result of Zhang [1], we define a sequence that converges weakly to an element , where (Section 4).

Our results represent a generalization of known results in the literature, including those in [16–18, 24]. Our Theorems 3.1 and 4.2 are the extension and improvements of Theorems 1.1 and 1.2 in the following way:

(i)the problem of finding an element of includes the one of finding an element of as a special case;

(ii)the algorithms in this paper are very different from those in [17] because of considering the complexity involving the problem of finding an element of .

## 2. Preliminaries

Throughout the paper, we denote the strong convergence, weak convergence, and weak convergence of a sequence to a point by , and , respectively.

Assumption 2.1.

Let be a uniformly smooth and uniformly convex Banach space and let be a nonempty closed convex subset of . Let be an -inverse-strongly monotone mapping and let be a bifunction satisfying the conditions (A1)–(A4). Let be two maximal monotone operators such that:

Recall that if is a nonempty closed convex subset of a Hilbert space , then the metric projection of onto is nonexpansive. This fact actually characterizes Hilbert spaces and hence, it is not available in more general Banach spaces. In this connection, Alber [26] recently introduced a generalized projection operator in a Banach space which is an analogue of the metric projection in Hilbert spaces.

Consider the functional defined as in [26] by

It is clear that in a Hilbert space , (2.1) reduces to .

The generalized projection is a mapping that assigns to an arbitrary point the minimum point of the functional , that is, , where is the solution to the minimization problem

The existence and uniqueness of the operator follow from the properties of the functional and strict monotonicity of the mapping ; See, for example, [27]. In a Hilbert space, . From [26], in a smooth, strictly convex and reflexive Banach space , we have

Moreover, by the property of subdifferential of convex functions, we easily get the following inequality:

Let be a mapping from into itself. A point in is called an asymptotic fixed point of [28] if contains a sequence which converges weakly to such that . The set of asymptotic fixed points of is denoted by . A mapping from into itself is called relatively nonexpansive [18, 29, 30] if and , for all and .

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 (2.3), we have , which implies that . From the definition of , we have and therefore, . For further details, we refer to [31].

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

Lemma 2.2 (see [32]).

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

Lemma 2.5 (see [33]).

Let be a reflexive, strictly convex and smooth Banach space and let be a multivalued operator. Then

(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 2.6 (see [34]).

Lemma 2.7 (see [32]).

The following result is due to Blum and Oettli [14].

Lemma 2.8 (see [14]).

Motivated by a result in [35] in a Hilbert space setting, Takahashi and Zembayashi [18] established the following lemma.

Lemma 2.9 (see [18]).

Using Lemma 2.9, we have the following result.

Lemma 2.10 (see [18]).

Utilizing Lemmas 2.8, 2.9, and 2.10, Zhang [1] derived the following result.

Proposition 2.11 (see [1]).

- (I)

then the mapping has the following properties:

(iv) is a closed convex subset of ,

Proof.

It is easy to verify that satisfies the conditions (A1)–(A4). Therefore, the conclusions (I) and (II) follow immediately from Lemmas 2.8, 2.9, and 2.10.

Let be two maximal monotone operators in a smooth Banach space . We denote the resolvent operators of and by and for each , respectively. Then and are two single-valued mappings. Also, and for each , where and are the sets of fixed points of and , respectively. For each , the Yosida approximations of and are defined by and , respectively. It is known that

Lemma 2.12 (see [23]).

Lemma 2.13 (see [36]).

Let and be two sequences of nonnegative real numbers such that 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 .

Theorem 3.1.

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

Proof.

We divide the proof into several steps.

Step 1.

We claim that is closed and convex for each .

Indeed, it is obvious that is closed and is closed and convex for each . Let us show that is convex. For and , put . It is sufficient to show that . We first write for each . Next, we prove that

This implies that . Therefore, is closed and convex.

Step 2.

We claim that for each and that is well defined.

Indeed, take arbitrarily. Note that is equivalent to

As by the induction assumption, the last inequality holds, in particular, for all . This, together with the definition of implies that . Hence (3.14) holds for all . So, for all . This implies that the sequence is well defined.

Step 3.

We claim that is bounded and that as .

Indeed, it follows from the definition of that . Since and , so for all , that is, is nondecreasing. It follows from and Lemma 2.4 that

Step 4.

Therefore, from and , it follows that .

Since and is uniformly convex and smooth, we have from Lemma 2.2 that

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

Let us set . Then, according to Lemma 2.5 and Proposition 2.11, we know that is a nonempty closed convex subset of such that . Fix arbitrarily. As in the proof of Step 2, we can show that ,

Observe first that

Since is uniformly norm-to-norm continuous on bounded subsets of , it follows from that . Thus from , , and the boundedness of both and , we deduce that . Utilizing the properties of , we have that . Since is uniformly norm-to-norm continuous on bounded subsets of , it follows that .

Step 5.

Indeed, since is bounded and is reflexive, we know that . Take arbitrarily. Then there exists a subsequence of such that . Hence it follows from , , and that and converge weakly to the same point . On the other hand, from (3.28a), (3.28b) and , we obtain that

Letting , we have that and . Then the maximality of the operators implies that and .

Next, let us show that . Since

So, from (3.39), , and , we have .

Since is uniformly convex and smooth, we conclude from Lemma 2.2 that

From , , and (3.42), we have and .

Since is uniformly norm-to-norm continuous on bounded subsets of , from (3.42) we derive

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

Step 6.

We claim that converges strongly to .

Indeed, from and , it follows that

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

Remark 3.2.

In this case, Theorem 3.1 reduces to [17, Theorem ].

## 4. Weak Convergence Theorem

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

Let be chosen arbitrarily and consider the sequence generated by

where , and is defined by (2.14).

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 2.13, we deduce that exists. This implies that is bounded. Thus, is bounded and so are , , , and .

Define for all . Let us show that is bounded. Indeed, observe that

Hence, from (4.8), it follows that .

Note that , , and is bounded, so that . Therefore, is a convergent sequence. On the other hand, from (4.6e) we derive, for all ,

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.

since is uniformly norm-to-norm continuous on bounded subsets of .

Now let us show that

Indeed, from (4.6e) we get

From (4.6d) it follows that

From (4.6c) it follows that

From (4.6c) it follows that

From (4.6a) it follows that

On the other hand, let us show that

Indeed, let . From Lemma 2.7, there exists a continuous, strictly increasing, and convex function with such that

Since , it follows from (4.24) and (4.41) that and .

Also, observe that

Thus, from , and , it follows that . In terms of Lemma 2.2, we derive .

Next, let us show that , where .

Indeed, since is bounded, there exists a subsequence of such that . Hence it follows from (4.24), (4.41), and that and converge weakly to the same point . Furthermore, from and (4.24), we have that

Then the maximality of the operators implies that .

Now, by the definition of , we have

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

Remark 4.3.

Compared with the algorithm of Theorem 1.2, the above algorithm (4.1) can be applied to find an element of . But, the algorithm of Theorem 1.2 cannot be applied. Therefore, algorithm (4.1) develops and improves the algorithm of Theorem 1.2.

## Declarations

### Acknowledgments

In this research, the first author was partially supported by the Leading Academic Discipline Project of Shanghai Normal University (DZL707), Innovation Program of Shanghai Municipal Education Commission Grant (09ZZ133), National Science Foundation of China (10771141), Ph.D. Program Foundation of Ministry of Education of China (20070270004), Science and Technology Commission of Shanghai Municipality Grant (075105118), and Shanghai Leading Academic Discipline Project (S30405). The Fourth author was partially supported by a grant NSC 98-2115-M-110-001.

## Authors’ Affiliations

## References

- Zhang S-S: 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 AGoogle Scholar
- Zeng 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 - Zeng 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 - 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:
**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 extragradient-like methods for generalized equilibrium problems, fixed points problems and variational inequality problems. to appear in Optimization Methods and SoftwareGoogle 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 - Peng J-W, Yao J-C:
**Some new iterative algorithms for generalized mixed equilibrium problems with strict pseudo-contractions and monotone mappings.***Taiwanese Journal of Mathematics*2009,**13**(5):1537–1582.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 OptimizationGoogle 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 - 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 - 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 - Blum E, Oettli W:
**From optimization and variational inequalities to equilibrium problems.***The Mathematics Student*1994,**63**(1–4):123–145.MathSciNetMATHGoogle 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 ArticleMATHGoogle Scholar - Zeng 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 - Zeng 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 - 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 - Martinet B:
**Régularisation d'inéquations variationnelles par approximations successives.***Revue Française d'Informatique et de Recherche Opérationnelle*1970,**4:**154–158.MathSciNetMATHGoogle 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 - 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 - 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 - Kohsaka F, Takahashi W: Strong convergence of an iterative sequence for maximal monotone operators in a Banach space. Abstract and Applied Analysis 2004, (3):239–249.Google 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 - Zeng L-C, Yao J-C:
**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 - 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. Marcel 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. Marcel Dekker, New York, NY, USA; 1996:313–318.Google Scholar - Matsushita S, Takahashi W:
**Weak and strong convergence theorems for relatively nonexpansive mappings in Banach spaces.***Fixed Point Theory and Applications*2004,**2004**(1):37–47.MathSciNetView ArticleMATHGoogle Scholar - Matsushita S, Takahashi W:
**A strong convergence theorem for relatively nonexpansive mappings in a Banach space.***Journal of Approximation Theory*2005,**134**(2):257–266. 10.1016/j.jat.2005.02.007MathSciNetView ArticleMATHGoogle Scholar - Cioranescu I:
*Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Mathematics and Its Applications*.*Volume 62*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1990:xiv+260.View ArticleMATHGoogle Scholar - Kamimura S, Takahashi W:
**Strong convergence of a proximal-type algorithm in a Banach space.***SIAM Journal on Optimization*2002,**13**(3):938–945. 10.1137/S105262340139611XMathSciNetView ArticleMATHGoogle Scholar - Rockafellar RT:
**On the maximality of sums of nonlinear monotone operators.***Transactions of the American Mathematical Society*1970,**149:**75–88. 10.1090/S0002-9947-1970-0282272-5MathSciNetView 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

## 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.