# Strong Convergence of a New Iterative Method for Infinite Family of Generalized Equilibrium and Fixed-Point Problems of Nonexpansive Mappings in Hilbert Spaces

- Shenghua Wang
^{1, 2}Email author and - Baohua Guo
^{1, 2}

**2011**:392741

https://doi.org/10.1155/2011/392741

© S.Wang and B. Guo. 2011

**Received: **15 October 2010

**Accepted: **18 November 2010

**Published: **2 December 2010

## Abstract

We introduce an iterative algorithm for finding a common element of the set of solutions of an infinite family of equilibrium problems and the set of fixed points of a finite family of nonexpansive mappings in a Hilbert space. We prove some strong convergence theorems for the proposed iterative scheme to a fixed point of the family of nonexpansive mappings, which is the unique solution of a variational inequality. As an application, we use the result of this paper to solve a multiobjective optimization problem. Our result extends and improves the ones of Colao et al. (2008) and some others.

## Keywords

## 1. Introduction

If there exists a point such that , then the point is called a fixed point of . The set of fixed points of is denoted by . It is well known that is closed convex and also nonempty if has a bounded trajectory (see [1]).

Let be a fixed point, be a strongly positive linear bounded operator on and be a finite family of nonexpansive mappings of into itself such that .

where is a potential function for (i.e., for all ).

where is a potential function for (However, Colao et al. pointed out in [5] that there is a gap in Yao's proof).

In the case of , is deduced to . In the case of , is also denoted by .

and proved that, under certain appropriate conditions over and , the sequences and both converge strongly to .

where is a potential function for .

The proof method of the main result of this paper is different with ones of others in the literatures and our result extends and improves the ones of Colao et al. [5] and some others.

## 2. Preliminaries

*α*-inverse-strongly monotone mapping if there exists a positive real number

*α*such that

Hence, if , then is a nonexpansive mapping of into .

for all , then is called the solution of this variational inequality. The set of all solutions of the variational inequality is denoted by .

In this paper, we need the following lemmas.

Lemma 2.1 (see [21]).

Lemma 2.2 (see [22]).

where , , and satisfy the conditions:

Therefore, the following lemma naturally holds.

Lemma 2.3.

Lemma 2.4 (see [3]).

Assume that is a strongly positive linear bounded operator on a Hilbert space with coefficient and . Then .

Lemma 2.5 (see [2]).

where is a sequence in and is a sequence in such that

Lemma 2.6 (see [23]).

Let C be a nonempty closed convex subset of a Hilbert space H and let be a bifunction which satisfies the following:

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

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

Then is well defined and the following hold:

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

*α*-inverse strongly monotone mapping, then . In fact, since , one has

Such a mapping is called the - generated by and (see [5, 24, 25]).

Lemma 2.7 (see [26]).

Let be a nonempty closed convex subset of a Banach space. Let be nonexpansive mappings of into itself such that and let be real numbers such that for each and . Let be the -mapping of generated by and . Then .

Lemma 2.8 (see [5]).

## 3. Main Results

Now, we give our main results in this paper.

Theorem 3.1.

Let be a Hilbert space and be a nonempty closed convex subset of . Let be a contraction with coefficient , , be strongly positive linear bounded self-adjoint operators with coefficients and , respectively, be a finite family of nonexpansive mappings, be an infinite family of bifunctions satisfying be an infinite family of inverse-strongly monotone mappings with constants such that . Let and be two sequences in , be asequence in with , be a sequence in with and be a strictly decreasing sequence Set . Take a fixed number with . Assume that

Proof.

Next, we proceed the proof with following steps.

Step 1.

which shows that is bounded, so is .

Step 2.

By applying Lemma 2.2 to (3.23), we obtain as .

Step 3.

Step 4.

Step 5.

Without loss of generality, we may further assume that . Obviously, to prove Step 5, we only need to prove that .

which is a contradiction. Therefore, . Hence, .

Step 6.

The sequence strongly converges to some point .

Thus, applying Lemma 2.5 to (3.49), it follows that as . This completes the proof.

By Theorem 3.1, we have the following direct corollaries.

Corollary 3.2.

Let
be a Hilbert space and
be a nonempty closed convex subset of
. Let
be a contraction with coefficient
,
be strongly positive linear bounded self-adjoint operator with coefficient
,
be a finite family of nonexpansive mappings,
be a bifunction satisfying (A1)–(A4), and
be an *α*-inverse strongly monotone mapping such that
. Let
and
be two sequences in
,
be a sequence in
with
,
be a number in
, and
be a sequence
. Take a fixed number
with
. Assume that

Remark 3.3.

In the proof process of Theorem 3.1, we do not use Suzuki's Lemma (see [27]), which was used by many others for obtaining as (see [4, 5, 28]). The proof method of is simple and different with ones of others.

## 4. Applications for Multiobjective Optimization Problem

where and are both the convex and lower semicontinuous functions defined on a closed convex subset of of a Hilbert space .

Note that, if we find a solution , then one must have obviously.

By Theorem 3.1 with , , and for all , the sequence converges strongly to a solution , which is a solution of the multiobjective optimization problem (4.1).

## Authors’ Affiliations

## References

- Goebel K, Kirk WA:
*Topics in Metric Fixed Point Theory, Cambridge Studies in Advanced Mathematics*.*Volume 28*. Cambridge University Press, Cambridge, UK; 1990:viii+244.View ArticleMATHGoogle Scholar - Xu H-K:
**Viscosity approximation methods for nonexpansive mappings.***Journal of Mathematical Analysis and Applications*2004,**298**(1):279–291. 10.1016/j.jmaa.2004.04.059MathSciNetView ArticleMATHGoogle Scholar - Marino G, Xu H-K:
**A general iterative method for nonexpansive mappings in Hilbert spaces.***Journal of Mathematical Analysis and Applications*2006,**318**(1):43–52. 10.1016/j.jmaa.2005.05.028MathSciNetView ArticleMATHGoogle Scholar - Yao Y:
**A general iterative method for a finite family of nonexpansive mappings.***Nonlinear Analysis: Theory, Methods & Applications*2007,**66**(12):2676–2687. 10.1016/j.na.2006.03.047MathSciNetView ArticleMATHGoogle Scholar - Colao V, Marino G, Xu H-K:
**An iterative method for finding common solutions of equilibrium and fixed point problems.***Journal of Mathematical Analysis and Applications*2008,**344**(1):340–352. 10.1016/j.jmaa.2008.02.041MathSciNetView ArticleMATHGoogle 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 - Plubtieng S, Punpaeng R:
**A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces.***Journal of Mathematical Analysis and Applications*2007,**336**(1):455–469. 10.1016/j.jmaa.2007.02.044MathSciNetView ArticleMATHGoogle 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 - 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.2008.03.032MathSciNetView ArticleMATHGoogle 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/s10957-009-9549-9MathSciNetView ArticleMATHGoogle 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.MathSciNetMATHGoogle Scholar - Chang 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.2008.04.035MathSciNetView ArticleMATHGoogle 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.2009.03.006MathSciNetMATHGoogle Scholar - Moudafi A:
**Weak convergence theorems for nonexpansive mappings and equilibrium problems.***Journal of Nonlinear and Convex Analysis*2008,**9**(1):37–43.MathSciNetMATHGoogle 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.2007.07.075MathSciNetView ArticleMATHGoogle 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.2009.06.042MathSciNetView ArticleMATHGoogle Scholar - Qin X, Cho SY, Kang SM:
**Strong convergence of shrinking projection methods for quasi--nonexpansive mappings and equilibrium problems.***Journal of Computational and Applied Mathematics*2010,**234**(3):750–760. 10.1016/j.cam.2010.01.015MathSciNetView ArticleMATHGoogle Scholar - Wang SH, Marino G, Wang FH:
**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.MathSciNetMATHGoogle Scholar - Wang SH, Cho YJ, Qin XL:
**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.MathSciNetMATHGoogle Scholar - Iiduka H, Takahashi W:
**Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings.***Nonlinear Analysis: Theory, Methods & Applications*2005,**61**(3):341–350. 10.1016/j.na.2003.07.023MathSciNetView ArticleMATHGoogle Scholar - Takahashi W:
*Nonlinear Functional Analysis, Fixed Point Theory and Its Application*. Yokohama Publishers, Yokohama, Japan; 2000:iv+276.MATHGoogle Scholar - Xu H-K:
**Iterative algorithms for nonlinear operators.***Journal of the London Mathematical Society*2002,**66**(1):240–256. 10.1112/S0024610702003332MathSciNetView ArticleMATHGoogle Scholar - Combettes PL, Hirstoaga SA:
**Equilibrium programming in Hilbert spaces.***Journal of Nonlinear and Convex Analysis*2005,**6**(1):117–136.MathSciNetMATHGoogle Scholar - Takahashi W:
**Weak and strong convergence theorems for families of nonexpansive mappings and their applications.***Annales Universitatis Mariae Curie-Skłodowska*1997,**51**(2):277–292.MathSciNetMATHGoogle Scholar - Takahashi W, Shimoji K:
**Convergence theorems for nonexpansive mappings and feasibility problems.***Mathematical and Computer Modelling*2000,**32**(11–13):1463–1471.MathSciNetView ArticleMATHGoogle 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 - Suzuki T:
**Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals.***Journal of Mathematical Analysis and Applications*2005,**305**(1):227–239. 10.1016/j.jmaa.2004.11.017MathSciNetView ArticleMATHGoogle Scholar - Cho YJ, Qin X:
**Convergence of a general iterative method for nonexpansive mappings in Hilbert spaces.***Journal of Computational and Applied Mathematics*2009,**228**(1):458–465. 10.1016/j.cam.2008.10.004MathSciNetView ArticleMATHGoogle Scholar

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