- Research Article
- Open Access

# A Hybrid Iterative Scheme for Equilibrium Problems, Variational Inequality Problems, and Fixed Point Problems in Banach Spaces

- Prasit Cholamjiak
^{1}Email author

**2009**:719360

https://doi.org/10.1155/2009/719360

© Prasit Cholamjiak. 2009

**Received:**5 February 2009**Accepted:**10 April 2009**Published:**5 May 2009

## Abstract

The purpose of this paper is to introduce a new hybrid projection algorithm for finding a common element of the set of solutions of the equilibrium problem and the set of the variational inequality for an inverse-strongly monotone operator and the set of fixed points of relatively quasi-nonexpansive mappings in a Banach space. Then we show a strong convergence theorem. Using this result, we obtain some applications in a Banach space.

## Keywords

- Banach Space
- Variational Inequality
- Equilibrium Problem
- Nonexpansive Mapping
- Variational Inequality Problem

## 1. Introduction

The set of solutions of (1.1) is denoted by . Such a problem is connected with the convex minimization problem, the complementarity, the problem of finding a point satisfying , and so on. First, we recall that

Assume that

(C1) is -inverse-strongly monotone,

(C2) ,

(C3) for all and .

where is the duality mapping on , and is the generalized projection from onto . Assume that for some with where is the -uniformly convexity constant of . They proved that if is weakly sequentially continuous, then the sequence converges weakly to some element in where .

The problem of finding a common element of the set of the variational inequalities for monotone mappings in the framework of Hilbert spaces and Banach spaces has been intensively studied by many authors; see, for instance, [2–4] and the references cited therein.

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

For solving the equilibrium problem, let us assume that a bifunction satisfies the following conditions:

(A1) for all ;

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

(A4)for all is convex and lower semicontinuous.

Recently, Takahashi and Zembayashi [5], introduced the following iterative scheme which is called the shrinking projection method:

where is the duality mapping on and is the generalized projection from onto . They proved that the sequence converges strongly to under appropriate conditions.

Under suitable conditions over , and , they obtain that the sequence generated by (1.8) converges strongly to .

The problem of finding a common element of the set of fixed points and the set of solutions of an equilibrium problem in the framework of Hilbert spaces and Banach spaces has been studied by many authors; see [5, 7–16].

Motivated by Iiduka and Takahashi [1], Takahashi and Zembayashi [5], and Qin et al. [6], we introduce a new general process for finding common elements of the set of the equilibrium problem and the set of the variational inequality problem for an inverse-strongly monotone operator and the set of the fixed points for relatively quasi-nonexpansive mappings.

## 2. Preliminaries

*strictly convex*if for any ,

*modulus of convexity*of as follows:

*uniformly convex*if there exists a constant such that for all ; see [17–19] for more details. A Banach space is said to be

*smooth*if the limit

*uniformly smooth*if the limit (2.4) is attained uniformly for . One should note that no Banach space is -uniformly convex for ; see [19]. It is well known that a Hilbert space is -uniformly convex, uniformly smooth. For each , the

*generalized duality mapping*is defined by

for all
. In particular,
is called the *normalized duality mapping*. If
is a Hilbert space, then
, where
is the identity mapping. It is also known that if
is uniformly smooth, then
is uniformly norm-to-norm continuous on each bounded subset of
. See [20, 21] for more details.

where is the generalized duality mapping of and is the -uniformly convexity constant of .

for all . In a Hilbert space , we have for all .

Recall that a mapping is called nonexpansive if for all and relatively nonexpansive if satisfies the following conditions:

(1) , where is the set of fixed points of ;

(2) for all and ;

(3) , where is the set of all asymptotic fixed points of ;

see [10, 23, 24] for more details.

is said to be relatively quasi-nonexpansive if satisfies the conditions and . It is easy to see that the class of relatively quasi-nonexpansive mappings is more general than the class of relatively nonexpansive mappings [9, 25, 26].

We give some examples which are closed relatively quasi-nonexpansive; see [6].

Example 2.2.

Let be a uniformly smooth and strictly convex Banach space and be a maximal monotone mapping such that its zero set . Then, is a closed relatively quasi-nonexpansive mapping from onto and .

Example 2.3.

Let be the generalized projection from a smooth, strictly convex, and reflexive Banach space onto a nonempty closed convex subset of . Then, is a closed relatively quasi-nonexpansive mapping with .

Lemma 2.4 (Kamimura and Takahashi [27]).

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

Let be a nonempty closed convex subset of . If is reflexive, strictly convex and smooth, then there exists such that for and . The generalized projection defined by . The existence and uniqueness of the operator follows from the properties of the functional and strict monotonicity of the duality mapping ; for instance, see [20, 27–30]. In a Hilbert space, is coincident with the metric projection.

Lemma 2.5 (Alber [28]).

Let be a nonempty closed convex subset of a smooth Banach space and . Then if and only if for all .

Lemma 2.6 (Alber [28]).

Lemma 2.7 (Qin et al. [6]).

Let be a uniformly convex, smooth Banach space, let be a closed convex subset of , let be a closed and relatively quasi-nonexpansive mapping from into itself. Then is a closed convex subset of .

Lemma 2.8 (Cho et al. [31]).

for all , and with .

Lemma 2.9 (Blum and Oettli [7]).

Lemma 2.10 (Qin et al. [6]).

Then, the following hold:

(1) is single-valued;

(3) ;

(4) is closed and convex.

Lemma 2.11 (Takahashi and Zembayashi [14]).

for all and , that is, .

Lemma 2.12 (Alber [28]).

for all and .

*hemicontinuous*if for all , the mapping of into defined by is continuous with respect to the topology of . We define by the

*normal cone*for at a point , that is,

Theorem 2.13 (Rockafellar [33]).

Then is maximal monotone and .

## 3. Strong Convergence Theorems

Theorem 3.1.

where is the duality mapping on . Assume that , and are sequences in satisfying the restrictions:

(B1) ;

(B2) , ;

(B3) for some ;

(B4) for some with , where is the -uniformly convexity constant of .

Then, and converge strongly to .

Proof.

We divide the proof into eight steps.

Step 1.

Show that and are well defined.

It is obvious that is a closed convex subset of . By Lemma 2.7, we know that is closed and convex. From Lemma 2.10 , we also have is closed and convex. Hence is a nonempty, closed, and convex subset of ; consequently, is well defined.

So, is closed and convex. By induction, is closed and convex for all . This shows that is well-defined.

Step 2.

Show that for all .

This shows that ; consequently, . Hence for all .

Step 3.

Show that exists.

Combining (3.8) and (3.9), we obtain that exists.

Step 4.

Show that is a Cauchy sequence in .

Step 5.

Show that .

From (3.19), (3.26) and by the closedness of and , we get .

Step 6.

Show that .

From (A4) and , we get for all . For and . Define , then , which implies that . From (A1), we obtain that . Thus, . From (A3), we have for all . Hence .

Step 7.

Show that .

where . By taking the limit as and from (3.24) and (3.25), we obtain . By the maximality of , we have and hence .

Step 8.

Show that .

By Lemma 2.5, we can conclude that . Furthermore, it is easy to see that as . This completes the proof.

As a direct consequence of Theorem 3.1, we obtain the following results.

Corollary 3.2.

Let be a -uniformly convex and uniformly smooth Banach space, and let be a nonempty closed convex subset of . Let be a bifunction from to satisfying (A1)–(A4) and let be a closed relatively quasi-nonexpansive mapping from into itself such that . Assume that satisfies and for some . Then the sequence generated by (1.7) converges strongly to .

Proof.

Putting and in Theorem 3.1, we obtain the result.

Remark 3.3.

If in Theorem 3.1, then Theorem 3.1 reduces to Theorem of Qin et al. [6].

Remark 3.4.

Corollary 3.2 improves Theorem of Takahashi and Zembayashi [5] from the class of relatively nonexpansive mappings to the class of relatively quasi-nonexpansive mappings, that is, we relax the strong restriction: . Further, the algorithm in Corollary 3.2 is also simpler to compute than the one given in [14].

## 4. Applications

Next, we consider the problem of finding a zero point of an inverse-strongly monotone operator of into . Assume that satisfies the conditions:

(D1) is -inverse-strongly monotone,

(D2) .

Theorem 4.1.

where is the duality mapping on . Assume that , and are sequences in satisfying the conditions (B1)–(B4) of Theorem 3.1.

Then, and converge strongly to .

Proof.

Putting in Theorem 3.1, we have . We also have and then the condition (C3) of Theorem 3.1 holds for all and . So, we obtain the result.

*polar*in to be the set

The set of solutions of the complementarity problem is denoted by .

Assume that is an operator satisfying the conditions:

(E1) is -inverse-strongly monotone,

(E2) ,

(E3) for all and .

Theorem 4.2.

where is the duality mapping on . Assume that and are sequences in satisfying the conditions (B1)–(B4) of Theorem 3.1.

Then, and converge strongly to .

Proof.

From [20, Lemma ], we have . Hence, we obtain the result.

## Declarations

### Acknowledgments

The author would like to thank Professor Suthep Suantai and the referee for the valuable suggestions on the manuscript. The author was supported by the Commission on Higher Education and the Thailand Research Fund.

## Authors’ Affiliations

## References

- Iiduka H, Takahashi W:
**Weak convergence of a projection algorithm for variational inequalities in a Banach space.***Journal of Mathematical Analysis and Applications*2008,**339**(1):668–679. 10.1016/j.jmaa.2007.07.019MathSciNetView ArticleMATHGoogle Scholar - Browder FE, Petryshyn WV:
**Construction of fixed points of nonlinear mappings in Hilbert space.***Journal of Mathematical Analysis and Applications*1967,**20**(2):197–228. 10.1016/0022-247X(67)90085-6MathSciNetView ArticleMATHGoogle Scholar - Iiduka H, Takahashi W, Toyoda M:
**Approximation of solutions of variational inequalities for monotone mappings.***Panamerican Mathematical Journal*2004,**14**(2):49–61.MathSciNetMATHGoogle Scholar - Liu F, Nashed MZ:
**Regularization of nonlinear ill-posed variational inequalities and convergence rates.***Set-Valued Analysis*1998,**6**(4):313–344. 10.1023/A:1008643727926MathSciNetView ArticleMATHGoogle Scholar - Takahashi W, Zembayashi K:
**Strong convergence theorem by a new hybrid method for equilibrium problems and relatively nonexpansive mappings.***Fixed Point Theory and Applications*2008,**2008:**-11.Google Scholar - Qin X, Cho YJ, Kang SM:
**Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces.***Journal of Computational and Applied Mathematics*2009,**225**(1):20–30. 10.1016/j.cam.2008.06.011MathSciNetView 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 - 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 - 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. 10.1155/S1687182004310089MathSciNetView ArticleMATHGoogle Scholar - Plubtieng S, Ungchittrakool K:
**Strong convergence theorems for a common fixed point of two relatively nonexpansive mappings in a Banach space.***Journal of Approximation Theory*2007,**149**(2):103–115. 10.1016/j.jat.2007.04.014MathSciNetView 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 - 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 - 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 - Wattanawitoon K, Kumam P:
**Strong convergence theorems by a new hybrid projection algorithm for fixed point problems and equilibrium problems of two relatively quasi-nonexpansive mappings.***Nonlinear Analysis: Hybrid Systems*2009,**3**(1):11–20. 10.1016/j.nahs.2008.10.002MathSciNetMATHGoogle Scholar - Yao Y, Noor MA, Liou Y-C:
**On iterative methods for equilibrium problems.***Nonlinear Analysis: Theory, Methods & Applications*2009,**70**(1):497–509. 10.1016/j.na.2007.12.021MathSciNetView ArticleMATHGoogle Scholar - Ball K, Carlen EA, Lieb EH:
**Sharp uniform convexity and smoothness inequalities for trace norms.***Inventiones Mathematicae*1994,**115**(3):463–482.MathSciNetView ArticleMATHGoogle Scholar - Beauzamy B:
*Introduction to Banach Spaces and Their Geometry, North-Holland Mathematics Studies*.*Volume 68*. 2nd edition. North-Holland, Amsterdam, The Netherlands; 1985:xv+338.Google Scholar - Takahashi Y, Hashimoto K, Kato M:
**On sharp uniform convexity, smoothness, and strong type, cotype inequalities.***Journal of Nonlinear and Convex Analysis*2002,**3**(2):267–281.MathSciNetMATHGoogle Scholar - Takahashi W:
*Nonlinear Functional Analysis, Fixed Point Theory and Its Applications*. Yokohama, Yokohama, Japan; 2000:iv+276.MATHGoogle Scholar - Takahashi W:
*Convex Analysis and Approximation of Fixed Points, Mathematical Analysis Series*.*Volume 2*. Yokohama, Yokohama, Japan; 2000:iv+280.MATHGoogle Scholar - Zalinescu C:
**On uniformly convex functions.***Journal of Mathematical Analysis and Applications*1983,**95**(2):344–374. 10.1016/0022-247X(83)90112-9MathSciNetView 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 - 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 - Butnariu D, Reich S, Zaslavski AJ:
**Asymptotic behavior of relatively nonexpansive operators in Banach spaces.***Journal of Applied Analysis*2001,**7**(2):151–174. 10.1515/JAA.2001.151MathSciNetView ArticleMATHGoogle Scholar - Censor Y, Reich S:
**Iterations of paracontractions and firmly nonexpansive operators with applications to feasibility and optimization.***Optimization*1996,**37**(4):323–339. 10.1080/02331939608844225MathSciNetView 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 - Alber YaI:
**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 YaI, Reich S:
**An iterative method for solving a class of nonlinear operator equations in Banach spaces.***Panamerican Mathematical Journal*1994,**4**(2):39–54.MathSciNetMATHGoogle 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 - Cho YJ, Zhou H, Guo G:
**Weak and strong convergence theorems for three-step iterations with errors for asymptotically nonexpansive mappings.***Computers & Mathematics with Applications*2004,**47**(4–5):707–717. 10.1016/S0898-1221(04)90058-2MathSciNetView ArticleMATHGoogle Scholar - Kohsaka F, Takahashi W:
**Existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach spaces.***SIAM Journal on Optimization*2008,**19**(2):824–835. 10.1137/070688717MathSciNetView ArticleMATHGoogle Scholar - Rockafellar RT:
**On the maximality of sums of nonlinear monotone operators.***Transactions of the American Mathematical Society*1970,**149**(1):75–88. 10.1090/S0002-9947-1970-0282272-5MathSciNetView ArticleMATHGoogle Scholar

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