- Research Article
- Open Access

# A New Hybrid Algorithm for Variational Inclusions, Generalized Equilibrium Problems, and a Finite Family of Quasi-Nonexpansive Mappings

- Prasit Cholamjiak
^{1}and - Suthep Suantai
^{1}Email author

**2009**:350979

https://doi.org/10.1155/2009/350979

© P. Cholamjiak and S. Suantai. 2009

**Received:**12 June 2009**Accepted:**28 September 2009**Published:**13 October 2009

## Abstract

We proposed in this paper a new iterative scheme for finding common elements of the set of fixed points of a finite family of quasi-nonexpansive mappings, the set of solutions of variational inclusion, and the set of solutions of generalized equilibrium problems. Some strong convergence results were derived by using the concept of -mappings for a finite family of quasi-nonexpansive mappings. Strong convergence results are derived under suitable conditions in Hilbert spaces.

## Keywords

- Equilibrium Problem
- Monotone Mapping
- Nonexpansive Mapping
- Lipschitz Mapping
- Maximal Monotone

## 1. Introduction

Let be a real Hilbert space with inner product and inducted norm , and let be a nonempty closed and convex subset of . Then, a mapping is said to be

(1)*nonexpansive* if
, for all
;

(2)*quasi-nonexpansive* if
, for all
and
;

(3)
*-Lipschitzian* if there exists a constant
such that
, for all
. We denoted by
the set of fixed points of
.

where the initial point is taken in arbitrarily and is a sequence in .

However, we note that Mann's iteration process (1.1) has only weak convergence, in general; for instance, see [2, 3].

Many authors attempt to modify the process (1.1) so that strong convergence is guaranteed that has recently been made. Nakajo and Takahashi [4] proposed the following modification which is the so-called CQ method and proved the following strong convergence theorem for a nonexpansive mapping in a Hilbert space .

Theorem 1.1 (see [4]).

where . Then, converges strongly to .

The set of solutions of (1.3) is denoted by ; see also [5–7].

The set of solutions (1.5) is denoted by . Problem (1.5) 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; see [9–12] and the reference cited therein.

Recently, Tada and Takahashi [13] proposed a new iteration for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space and then obtain the following theorem.

Theorem 1.2 (see [13]).

where and . Then, converges strongly to .

*-inverse strongly monotone*if there exists a constant such that

*monotone*if for all , and imply . A monotone mapping is

*maximal*if its 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 all imply . We define the resolvent operator associated with and as follows:

It is known that the resolvent operator is single-valued, nonexpansive, and 1-inverse strongly monotone; see [14], and that a solution of problem (1.7) is a fixed point of the operator for all ; see also [15]. If , it is easy to see that is a nonexpansive mapping; consequently, is closed and convex.

The equilibrium problems, generalized equilibrium problems, variational inequality problems, and variational inclusions have been intensively studied by many authors; for instance, see [8, 16–43].

Motivated by Tada and Takahashi [13] and Peng et al. [7], we introduce a new approximation scheme for finding a common element of the set of fixed points of a finite family of quasi-nonexpansive and Lipschitz mappings, the set of solutions of a generalized equilibrium problem, and the set of solutions of a variational inclusion with set-valued maximal monotone and inverse strongly monotone mappings in the framework of Hilbert spaces.

## 2. Preliminaries and Lemmas

*metric projection*of on to . It is also known that for and , is equivalent to for all . Furthermore

Lemma 2.1 (see [45]).

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

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

(A3) for each is weakly upper semicontinuous;

(A5) for each , is lower semicontinuous;

Lemma 2.2 (see [7]).

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

Lemma 2.3 (see [14]).

Let be a maximal monotone mapping and let be a Lipshitz continuous mapping. Then the mapping is a maximal monotone mapping.

Lemma 2.4.

Let be a nonempty closed convex subset of a real Hilbert space . Let be a quasi-nonexpansive and -Lipschitz mapping of into itself. Then, is closed and convex.

Proof.

Since is -Lipschitz, it is easy to show that is closed.

which implies ; consequently, is convex. This completes the proof.

Lemma 2.5 (see [46]).

where is a finite mapping of into itself and for all with .

Such a mapping
is called the
*-mapping* generated by
and
; see also [48–50]. Throughout this paper, we denote
.

Next, we prove some useful lemmas concerning the -mapping.

Lemma 2.6.

Let be a nonempty closed convex subset of a strictly convex Banach space . Let be a finite family of quasi-nonexpansive and -Lipschitz mappings of into itself such that and let be real numbers such that for all , , and . Let be the -mapping generated by and . Then, the followings hold:

(i) is quasi-nonexpansive and Lipschitz;

This shows that is a quasi-nonexpansive mapping.

Applying Lemma 2.5 to (2.19), we get that and hence .

Applying Lemma 2.5 to (2.22), we get that and hence .

which yields that since . Hence .

Lemma 2.7.

Proof.

## 3. Strong Convergence Theorems

In this section, we prove a strong convergence theorem which solves the problem of finding a common element of the set of solutions of a generalized equilibrium problem and the set of solutions of a variational inclusion and the set of common fixed points of a finite family of quasi-nonexpansive and Lipschitz mappings.

Theorem 3.1.

where for some , for some and for some .

Then, , , , and converge strongly to .

Proof.

Since for all , we get that is nonexpansive for all . Hence, is closed and convex. By Lemma 2.2(5), we know that is closed and convex. By Lemma 2.4, we also know that is closed and convex. Hence, is a nonempty closed convex set; consequently, is well defined for every .

Next, we divide the proof into seven steps.

Step 1.

It follows that , and hence for all .

Step 2.

Hence is bounded; so are , , and .

From (3.3) and (3.4), we get that exists.

Step 3.

Show that is a Cauchy sequence.

as . Hence is a Cauchy sequence. By the completeness of and the closeness of , we can assume that .

Step 4.

By Lemma 2.7, we also get that . From Lemma 2.6(i), we know that is Lipschitz. Since as , it is easy to verify that . Moreover, by Lemma 2.6(ii), we can conclude that .

Step 5.

for all . Observe that if , then holds. Hence .

Step 6.

By the maximal monotonicity of , we have ; consequently, .

Step 7.

From Steps 1–7, we can conclude that , , , and converge strongly to . This completes the proof.

## 4. Applications

Theorem 4.1.

where for some , for some and for some .

Then, , , , and converge strongly to .

Proof.

In Theorem 3.1, take , where is the indicator function of . It is well known that the subdifferential is a maximal monotone operator. Then, problem (1.7) is equivalent to problem (4.1) and the resolvent operator for all . This completes the proof.

Next, we give a strong convergence theorem for finding a common element of the set of solutions of an equilibrium problem, the set of solutions of a variational inclusion and the set of common fixed points of a finite family of quasi-nonexpansive and Lipschitz mappings. In order to do this, let us assume that

Theorem 4.2.

where for some , for some , and for some .

Then, , , , and converge strongly to .

Proof.

In Theorem 3.1, take , for all . Then problem (1.3) reduces to the equilibrium problem (1.5).

Remark 4.3.

Theorem 3.1 improves and extends the main results in [4, 13] and the corresponding results.

## Declarations

### Acknowledgments

The authors would like to thank the referee for the valuable suggestions on the manuscript. The authors were supported by the Commission on Higher Education, the Thailand Research Fund, and the Graduate School of Chiang Mai University.

## Authors’ Affiliations

## References

- Mann WR:
**Mean value methods in iteration.***Proceedings of the American Mathematical Society*1953,**4:**506–510. 10.1090/S0002-9939-1953-0054846-3MathSciNetView ArticleMATHGoogle Scholar - Genel A, Lindenstrauss J:
**An example concerning fixed points.***Israel Journal of Mathematics*1975,**22**(1):81–86. 10.1007/BF02757276MathSciNetView ArticleMATHGoogle Scholar - Reich S:
**Weak convergence theorems for nonexpansive mappings in Banach spaces.***Journal of Mathematical Analysis and Applications*1979,**67**(2):274–276. 10.1016/0022-247X(79)90024-6MathSciNetView ArticleMATHGoogle Scholar - Nakajo K, Takahashi W:
**Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups.***Journal of Mathematical Analysis and Applications*2003,**279**(2):372–379. 10.1016/S0022-247X(02)00458-4MathSciNetView 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 - 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 - Peng J-W, Liou Y-C, Yao J-C:
**An iterative algorithm combining viscosity method with parallel method for a generalized equilibrium problem and strict pseudocontractions.***Fixed Point Theory and Applications*2009, Article ID 794178,**2009:**-21.Google 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 - Blum E, Oettli W:
**From optimization and variational inequalities to equilibrium problems.***The Mathematics Student*1994,**63**(1–4):123–145.MathSciNetMATHGoogle Scholar - Combettes PL, Hirstoaga SA:
**Equilibrium programming in Hilbert spaces.***Journal of Nonlinear and Convex Analysis*2005,**6**(1):117–136.MathSciNetMATHGoogle Scholar - Flåm SD, Antipin AS:
**Equilibrium programming using proximal-like algorithms.***Mathematical Programming*1997,**78**(1):29–41.MathSciNetView ArticleMATHGoogle Scholar - Iusem AN, Sosa W:
**Iterative algorithms for equilibrium problems.***Optimization*2003,**52**(3):301–316. 10.1080/0233193031000120039MathSciNetView 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 - Bŕezis H:
**Operateur maximaux monotones.**In*Mathematics Studies*.*Volume 5*. North-Holland, Amsterdam, The Netherlands; 1973.Google Scholar - Lemaire B:
**Which fixed point does the iteration method select?**In*Recent Advances in Optimization*.*Volume 452*. Springer, Berlin, Germany; 1997:154–167. 10.1007/978-3-642-59073-3_11View ArticleGoogle Scholar - Agarwal RP, Cho YJ, Huang N-J:
**Sensitivity analysis for strongly nonlinear quasi-variational inclusions.***Applied Mathematics Letters*2000,**13**(6):19–24. 10.1016/S0893-9659(00)00048-3MathSciNetView ArticleMATHGoogle Scholar - Ceng L-C, Ansari QH, Yao J-C:
**On relaxed viscosity iterative methods for variational inequalities in Banach spaces.***Journal of Computational and Applied Mathematics*2009,**230**(2):813–822. 10.1016/j.cam.2009.01.015MathSciNetView ArticleMATHGoogle Scholar - Ceng L-C, Ansari QH, Yao J-C:
**Mann-type steepest-descent and modified hybrid steepest-descent methods for variational inequalities in Banach spaces.***Numerical Functional Analysis and Optimization*2008,**29**(9–10):987–1033. 10.1080/01630560802418391MathSciNetView ArticleMATHGoogle Scholar - Ceng LC, Chen GY, Huang XX, Yao J-C:
**Existence theorems for generalized vector variational inequalities with pseudomonotonicity and their applications.***Taiwanese Journal of Mathematics*2008,**12**(1):151–172.MathSciNetMATHGoogle 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 - Ceng L-C, Petruşel A, Yao J-C:
**Weak convergence theorem by a modified extragradient method for nonexpansive mappings and monotone mappings.***Fixed Point Theory*2008,**9**(1):73–87.MathSciNetMATHGoogle Scholar - Ceng L-C, Xu H-K, Yao J-C:
**A hybrid steepest-descent method for variational inequalities in Hilbert spaces.***Applicable Analysis*2008,**87**(5):575–589. 10.1080/00036810802140608MathSciNetView ArticleMATHGoogle Scholar - Zeng LC, Schaible S, Yao J-C:
**Hybrid steepest descent methods for zeros of nonlinear operators with applications to variational inequalities.***Journal of Optimization Theory and Applications*2009,**141**(1):75–91. 10.1007/s10957-008-9501-4MathSciNetView ArticleMATHGoogle Scholar - Ceng L-C, Yao J-C:
**Relaxed viscosity approximation methods for fixed point problems and variational inequality problems.***Nonlinear Analysis: Theory, Methods & Applications*2008,**69**(10):3299–3309. 10.1016/j.na.2007.09.019MathSciNetView ArticleMATHGoogle Scholar - Chang SS:
**Set-valued variational inclusions in Banach spaces.***Journal of Mathematical Analysis and Applications*2000,**248**(2):438–454. 10.1006/jmaa.2000.6919MathSciNetView ArticleMATHGoogle Scholar - Cholamjiak P:
**A hybrid iterative scheme for equilibrium problems, variational inequality problems, and fixed point problems in Banach spaces.***Fixed Point Theory and Applications*2009, Article ID 719360,**2009:**-18.Google Scholar - Ding XP:
**Perturbed Ishikawa type iterative algorithm for generalized quasivariational inclusions.***Applied Mathematics and Computation*2003,**141**(2–3):359–373. 10.1016/S0096-3003(02)00261-8MathSciNetView ArticleMATHGoogle Scholar - Fang Y-P, Huang N-J:
**-monotone operator and resolvent operator technique for variational inclusions.***Applied Mathematics and Computation*2003,**145**(2–3):795–803. 10.1016/S0096-3003(03)00275-3MathSciNetView ArticleMATHGoogle Scholar - Kangtunyakarn A, Suantai S:
**A new mapping for finding common solutions of equilibrium problems and fixed point problems of finite family of nonexpansive mappings.***Nonlinear Analysis: Theory, Methods & Applications*2009,**71**(10):4448–4460. 10.1016/j.na.2009.03.003MathSciNetView ArticleMATHGoogle Scholar - Kangtunyakarn A, Suantai S:
**Hybrid iterative scheme for generalized equilibrium problems and fixed point problems of finite family of nonexpansive mappings.***Nonlinear Analysis: Hybrid Systems*2009,**3**(3):296–309. 10.1016/j.nahs.2009.01.012MathSciNetMATHGoogle Scholar - Kumam P:
**A hybrid approximation method for equilibrium and fixed point problems for a monotone mapping and a nonexpansive mapping.***Nonlinear Analysis: Hybrid Systems*2008,**2**(4):1245–1255. 10.1016/j.nahs.2008.09.017MathSciNetMATHGoogle Scholar - Nilsrakoo W, Saejung S:
**Strong convergence theorems for a countable family of quasi-Lipschitzian mappings and its applications.***Journal of Mathematical Analysis and Applications*2009,**356**(1):154–167. 10.1016/j.jmaa.2009.03.002MathSciNetView ArticleMATHGoogle Scholar - Peng J-W, Wang Y, Shyu DS, Yao J-C:
**Common solutions of an iterative scheme for variational inclusions, equilibrium problems, and fixed point problems.***Journal of Inequalities and Applications*2008, Article ID 720371,**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:
**A modified CQ method for equilibrium problems, fixed points and variational inequality.***Fixed Point Theory*2008,**9**(2):515–531.MathSciNetMATHGoogle Scholar - Petruşel A, Yao J-C:
**An extragradient iterative scheme by viscosity approximation methods for fixed point problems and variational inequality problems.***Central European Journal of Mathematics*2009,**7**(2):335–347. 10.2478/s11533-009-0003-xMathSciNetView ArticleMATHGoogle Scholar - Plubtieng S, Sriprad W:
**A viscosity approximation method for finding common solutions of variational inclusions, equilibrium problems, and fixed point problems in Hilbert spaces.***Fixed Point Theory and Applications*2009, Article ID 567147,**2009:**-20.Google 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 - Verma RU:
**-monotonicity and applications to nonlinear variational inclusion problems.***Journal of Applied Mathematics and Stochastic Analysis*2004,**2004**(2):193–195. 10.1155/S1048953304403013View ArticleMATHMathSciNetGoogle Scholar - Zeng L-C, Guu SM, Yao J-C:
**Hybrid approximate proximal point algorithms for variational inequalities in Banach spaces.***Journal of Inequalities and Applications*2009, Article ID 275208,**2009:**-17.Google Scholar - Zeng LC, Lin LJ, Yao J-C:
**Auxiliary problem method for mixed variational-like inequalities.***Taiwanese Journal of Mathematics*2006,**10**(2):515–529.MathSciNetMATHGoogle Scholar - Zeng LC, Yao J-C:
**A hybrid extragradient method for general variational inequalities.***Mathematical Methods of Operations Research*2009,**69**(1):141–158. 10.1007/s00186-008-0215-zMathSciNetView ArticleMATHGoogle Scholar - Zhang S-S, Lee JH, Chan CK:
**Algorithms of common solutions to quasi variational inclusion and fixed point problems.***Applied Mathematics and Mechanics*2008,**29**(5):571–581. 10.1007/s10483-008-0502-yMathSciNetView 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 - Martinez-Yanes C, Xu H-K:
**Strong convergence of the CQ method for fixed point iteration processes.***Nonlinear Analysis: Theory, Methods & Applications*2006,**64**(11):2400–2411. 10.1016/j.na.2005.08.018MathSciNetView ArticleMATHGoogle Scholar - Takahashi W:
*Nonlinear Functional Analysis*. Yokohama, Yokohama, Japan; 2000.MATHGoogle Scholar - Atsushiba S, Takahashi W:
**Strong convergence theorems for a finite family of nonexpansive mappings and applications.***Indian Journal of Mathematics*1999,**41**(3):435–453.MathSciNetMATHGoogle 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.MathSciNetMATHGoogle Scholar - Takahashi W:
**Weak and strong convergence theorems for families of nonexpansive mappings and their applications.***Annales Universitatis Mariae Curie-Skłodowska. Sectio A*1997,**51**(2):277–292.MATHMathSciNetGoogle Scholar - Takahashi W:
*Convex Analysis and Approximation of Fixed Points*.*Volume 2*. Yokohama, Yokohama, Japan; 2000.MATHGoogle 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.