- Research Article
- Open Access

# New Iterative Scheme for Finite Families of Equilibrium, Variational Inequality, and Fixed Point Problems in Banach Spaces

- Shenghua Wang
^{1, 2}Email author and - Caili Zhou
^{3}

**2011**:372975

https://doi.org/10.1155/2011/372975

© Shenghua Wang and Caili Zhou. 2011

**Received:**6 December 2010**Accepted:**30 January 2011**Published:**24 February 2011

## Abstract

We introduced a new iterative scheme for finding a common element in the set of common fixed points of a finite family of quasi-*ϕ*-nonexpansive mappings, the set of common solutions of a finite family of equilibrium problems, and the set of common solutions of a finite family of variational inequality problems in Banach spaces. The proof method for the main result is simplified under some new assumptions on the bifunctions.

## Keywords

- Hilbert Space
- Banach Space
- Convex Subset
- Equilibrium Problem
- Nonexpansive Mapping

## 1. Introduction

We remark that if is a reflexive, strictly convex and smooth Banach space, then for , if and only if . For more details on and , the readers are referred to [1–4].

Let be a mapping from into itself. We denote the set of fixed points of by . is called to be nonexpansive if for all and quasi-nonexpansive if and for all and . A point is called to be an asymptotic fixed point of [5] if contains a sequence which converges weakly to such that . The set of asymptotic fixed points of is denoted by . The mapping is said to be relatively nonexpansive [6–8] if and for all and . The mapping is said to be -nonexpansive if for all . is called to be quasi- -nonexpansive [9] if and for all and .

where is the duality mapping on , is a relatively nonexpansive mapping from into itself, and is a sequence of real numbers such that and and proved that the sequence generated by (1.5) converges strongly to , where is the generalized projection from onto .

For studying the equilibrium problem, is usually assumed to satisfy the following conditions:

(A1) for all ;

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

(A3)for each , ;

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

where is a closed quasi- -nonexpansive mapping for each , are real sequences in satisfying for each and for each and is a real sequence in with . Then the authors proved that converges strongly to , where .

where , such that . Further, they proved that converges strongly to an element of , where .

where are the real numbers in satisfying and for each , are the real numbers in satisfying . We will prove that the sequence generated by (1.13) converges strongly to an element in . In this paper, in order to simplify the proof, we will replace the condition (A3) with (A3'): for each fixed , is continuous.

Obviously, the condition (A3') implies (A3). Under the condition (A3'), we will show that each (as well as , , ) is closed which is such that the proof for the main result of this paper is simplified.

## 2. Preliminaries

The space is said to be smooth if , , and is called uniformly smooth if and only if .

A Banach space is said to be strictly convex if for all with and . It is said to be uniformly convex if for any two sequences and in such that and . It is known that if a Banach space is uniformly smooth, then its dual space is uniformly convex.

A Banach space is called to have the Kadec-Klee property if for any sequence and with , where denotes the weak convergence, and , then as , where denotes the strong convergence. It is well known that every uniformly convex Banach space has the Kadec-Klee property. For more details on the Kadec-Klee property, the reader is referred to [3, 4].

Let be a nonempty closed and convex subset of a Banach space . A mapping is said to be closed if for any sequence such that and , .

The solution set of the above variational inequality problem is denoted by .

Next we state some lemmas which will be used later.

Lemma 2.1 (see [1]).

Lemma 2.2 (see [1]).

Lemma 2.3 (see [20]).

Let be a strictly convex and smooth Banach space, a nonempty closed and convex subset of , and a quasi- -nonexpansive mapping. Then is a closed and convex subset of .

Since the condition (A3') implies (A3), the following lemma is a natural result of [22, Lemmas 2.8 and 2.9].

Lemma 2.4.

- (a)
there exists such that

- (b)
define a mapping by

Then the following conclusions hold:

(1) is single-valued;

(2) is firmly nonexpansive, that is, for all ,

(3) ;

(4) is quasi- -nonexpansive;

(5) is closed and convex;

(6) .

Remark 2.5.

then satisfies all the conclusions in Lemma 2.4. See [25, Lemma 2.4].

Lemma 2.6 (see [26]).

for all and , where .

The following lemma can be obtained from Lemma 2.6 immediately; also see [20, Lemma 1.9].

Lemma 2.7 (see [20]).

for all and such that .

Lemma 2.8.

Let
be a closed and convex subset of a uniformly smooth and strictly convex Banach space
. Let
be a bifunction satisfying (A1), (A2), (A3'), and (A4). Let
and
be a mapping defined by (2.7). Then T_{r} is closed.

Proof.

which implies that . This completes the proof.

## 3. Main Results

Theorem 3.1.

where and are defined by (1.11) and (1.12), are the real numbers in satisfying and for each , are the real numbers in satisfying . Then the sequence converges strongly to , where is the generalized projection from onto .

Proof.

First we prove that is closed and convex for each . From the definition of , it is obvious that is closed. Moreover, since is equivalent to , it follows that is convex for each . By the definition of , we can conclude that is closed and convex for each .

for each . This shows that the sequence is bounded. It follows from (1.4) that the sequence is also bounded.

Since each is closed, by (3.30) and (3.31) we conclude that , that is, , . On the other hand, Lemma 2.4, Remark 2.5, and Lemma 2.8 show that and are closed. So, by (3.32) and (3.33) we have and . Now, it follows from Lemma 2.4 and Remark 2.5 that ( ) and ( ). Hence, ( ) and ( ). Therefore, .

In view of Lemma 2.1, we can obtain that . This completes the proof.

Remark 3.2.

Obviously, the proof process of is simple since we replace the condition (A3) with (A3') which is such that and are closed. In fact, although the condition (A3') is stronger than (A3), it is not easier to verify the condition (A3) than verify the condition (A3'). Hence, from this point, the condition (A3') is acceptable. On the other hand, the definition of is of some interest.

If for each , for each and for each , then Theorem 3.1 reduces to the following result.

Corollary 3.3.

where and are defined by (1.11) and (1.12) with and , are the real numbers in satisfying . Then the sequence converges strongly to , where is the generalized projection from onto .

Corollary 3.4.

where and are defined by (1.11) and (1.12) with ( is the identity mapping), are the real numbers in satisfying and for each , are the real numbers in satisfying . Then the sequence converges strongly to , where is the projection from onto .

Proof.

Since each is closed, we can conclude that , . Note that in a Hilbert space, a firmly-nonexpansive mapping is also nonexpansive. Hence, and are nonexpansive for each and . By demiclosed principle, we can conclude that and for each and . That is, . Then by the final part of proof of Theorem 3.1, we have . This completes the proof.

for all . Take . Then we have . That is, . This shows that , which implies that . So, . Based this, we have following result.

Corollary 3.5.

are the real numbers in satisfying and for each , are the real numbers in satisfying . Then the sequence converges strongly to , where is the projection from onto .

If , , and for each , , and , then Corollary 3.5 reduced the following result.

Corollary 3.6.

where is defined by (1.11) with and , is defined by (3.44) , and are the real numbers in satisfying . Then the sequence converges strongly to , where is the projection from onto .

## Declarations

### Acknowledgment

This work was supported by the Natural Science Foundation of Hebei Province (A2010001482).

## Authors’ Affiliations

## References

- 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 Appl. Math.*.*Volume 178*. 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.MATHMathSciNetGoogle 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 - Takahashi W:
*Nonlinear Functional Analysis*. Yokohama Publishers, Yokohama, Japan; 2000:iv+276. Fixed point theory and Its applicationMATHGoogle 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 Appl. Math.*.*Volume 178*. Edited by: Kartsatos AG. 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.2001.151MATHMathSciNetView ArticleGoogle Scholar - Butnariu D, Reich S, Zaslavski AJ:
**Weak convergence of orbits of nonlinear operators in reflexive Banach spaces.***Numerical Functional Analysis and Optimization*2003,**24**(5–6):489–508. 10.1081/NFA-120023869MATHMathSciNetView ArticleGoogle Scholar - Censor Y, Reich S:
**Iterations of paracontractions and firmly nonexpansive operators with applications to feasibility and optimization.***Optimization*1996,**37**(4):323–339. 10.1080/02331939608844225MATHMathSciNetView ArticleGoogle Scholar - Zhou H, Gao G, Tan B:
**Convergence theorems of a modified hybrid algorithm for a family of quasi-**ϕ**-asymptotically nonexpansive mappings.***Journal of Applied Mathematics and Computing*2010,**32**(2):453–464. 10.1007/s12190-009-0263-4MATHMathSciNetView ArticleGoogle Scholar - Matsushita S-y, 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.2005.02.007MATHMathSciNetView ArticleGoogle 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.2007.09.011MATHMathSciNetView ArticleGoogle Scholar - Ceng L-C, Al-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.2008.03.032MATHMathSciNetView ArticleGoogle Scholar - Ceng LC, Petruşel A, Yao JC:
**Iterative approaches to solving equilibrium problems and fixed point problems of infinitely many nonexpansive mappings.***Journal of Optimization Theory and Applications*2009,**143**(1):37–58. 10.1007/s10957-009-9549-9MATHMathSciNetView ArticleGoogle Scholar - Chang S-S, Cho YJ, Kim JK:
**Approximation methods of solutions for equilibrium problem in Hilbert spaces.***Dynamic Systems and Applications*2008,**17**(3–4):503–513.MATHMathSciNetGoogle Scholar - Chang S-S, 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*2009,**70**(9):3307–3319. 10.1016/j.na.2008.04.035MATHMathSciNetView ArticleGoogle Scholar - Kumam P, Katchang P:
**A viscosity of extragradient approximation method for finding equilibrium problems, variational inequalities and fixed point problems for nonexpansive mappings.***Nonlinear Analysis: Hybrid Systems*2009,**3**(4):475–486. 10.1016/j.nahs.2009.03.006MATHMathSciNetGoogle Scholar - Moudafi A:
**Weak convergence theorems for nonexpansive mappings and equilibrium problems.***Journal of Nonlinear and Convex Analysis*2008,**9**(1):37–43.MATHMathSciNetGoogle Scholar - Plubtieng S, Punpaeng R:
**A new iterative method for equilibrium problems and fixed point problems of nonexpansive mappings and monotone mappings.***Applied Mathematics and Computation*2008,**197**(2):548–558. 10.1016/j.amc.2007.07.075MATHMathSciNetView ArticleGoogle Scholar - Qin X, Cho YJ, Kang SM:
**Viscosity approximation methods for generalized equilibrium problems and fixed point problems with applications.***Nonlinear Analysis: Theory, Methods & Applications*2010,**72**(1):99–112. 10.1016/j.na.2009.06.042MATHMathSciNetView ArticleGoogle Scholar - Qin X, Cho SY, Kang SM:
**Strong convergence of shrinking projection methods for quasi-**ϕ**-nonexpansive mappings and equilibrium problems.***Journal of Computational and Applied Mathematics*2010,**234**(3):750–760. 10.1016/j.cam.2010.01.015MATHMathSciNetView ArticleGoogle 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.2006.08.036MATHMathSciNetView ArticleGoogle 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.2007.11.031MATHMathSciNetView ArticleGoogle Scholar - Wang S, Marino G, Wang F:
**Strong convergence theorems for a generalized equilibrium problem with a relaxed monotone mapping and a countable family of nonexpansive mappings in a Hilbert space.***Fixed Point Theory and Applications*2010,**2010:**-22.Google Scholar - Cho YJ, Wang S, Qin X:
**A new iterative method for solving equilibrium problems and fixed point problems for infinite family of nonexpansive mappings.***Fixed Point Theory and Applications*2010,**2010:**-18.Google Scholar - Zegeye H, Shahzad N:
**A hybrid scheme for finite families of equilibrium, variational inequality and fixed point problems.***Nonlinear Analysis: Theory, Methods & Applications*2011,**74**(1):263–272. 10.1016/j.na.2010.08.040MATHMathSciNetView ArticleGoogle 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-KMATHMathSciNetView ArticleGoogle 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.