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

- 783 Downloads

**Part of the following topical collections:**

## 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

Hilbert Space Variational Inequality Equilibrium Problem Nonexpansive Mapping Iterative Scheme## 1. Introduction

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

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

where Open image in new window is a potential function for Open image in new window (i.e., Open image in new window for all Open image in new window ).

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

In the case of Open image in new window , Open image in new window is deduced to Open image in new window . In the case of Open image in new window , Open image in new window is also denoted by Open image in new window .

and proved that, under certain appropriate conditions over Open image in new window and Open image in new window , the sequences Open image in new window and Open image in new window both converge strongly to Open image in new window .

where Open image in new window is a potential function for Open image in new window .

where Open image in new window .

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 Open image in new window , then Open image in new window is a nonexpansive mapping of Open image in new window into Open image in new window .

for all Open image in new window , then Open image in new window is called the solution of this variational inequality. The set of all solutions of the variational inequality is denoted by Open image in new window .

In this paper, we need the following lemmas.

Lemma 2.1 (see [21]).

Lemma 2.2 (see [22]).

where Open image in new window , Open image in new window , and Open image in new window satisfy the conditions:

(1) Open image in new window , Open image in new window or, equivalently, Open image in new window ;

(2) Open image in new window ;

(3) Open image in new window Open image in new window , Open image in new window .

Then Open image in new window .

Therefore, the following lemma naturally holds.

Lemma 2.3.

Lemma 2.4 (see [3]).

Assume that Open image in new window is a strongly positive linear bounded operator on a Hilbert space Open image in new window with coefficient Open image in new window and Open image in new window . Then Open image in new window .

Lemma 2.5 (see [2]).

where Open image in new window is a sequence in Open image in new window and Open image in new window is a sequence in Open image in new window such that

(1) Open image in new window ;

(2) Open image in new window or Open image in new window .

Then Open image in new window .

Lemma 2.6 (see [23]).

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

(A1) Open image in new window for all Open image in new window ;

(A2) Open image in new window is monotone, that is, Open image in new window for all Open image in new window ;

(A3) For each Open image in new window ,

(A4) For each Open image in new window , Open image in new window is convex and lower semicontinuous.

Then Open image in new window is well defined and the following hold:

(1) Open image in new window is single-valued;

(2) Open image in new window is firmly nonexpansive, that is, for any Open image in new window ,

(3) Open image in new window ;

(4) Open image in new window is closed and convex.

*α*-inverse strongly monotone mapping, then Open image in new window . In fact, since Open image in new window , one has

Hence, Open image in new window .

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

Lemma 2.7 (see [26]).

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

Lemma 2.8 (see [5]).

## 3. Main Results

Now, we give our main results in this paper.

Theorem 3.1.

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

(E2) Open image in new window ;

(E3) Open image in new window ;

(E4) Open image in new window with Open image in new window ;

(E5) Open image in new window .

Proof.

Next, we proceed the proof with following steps.

Step 1.

Open image in new window is bounded.

which shows that Open image in new window is bounded, so is Open image in new window .

Step 2.

Open image in new window as Open image in new window .

where Open image in new window .

where Open image in new window .

By applying Lemma 2.2 to (3.23), we obtain Open image in new window as Open image in new window .

Step 3.

Open image in new window as Open image in new window .

Step 4.

Open image in new window as Open image in new window .

Step 5.

Without loss of generality, we may further assume that Open image in new window . Obviously, to prove Step 5, we only need to prove that Open image in new window .

which is a contradiction. Therefore, Open image in new window . Hence, Open image in new window .

Step 6.

The sequence Open image in new window strongly converges to some point Open image in new window .

Thus, applying Lemma 2.5 to (3.49), it follows that Open image in new window as Open image in new window . This completes the proof.

By Theorem 3.1, we have the following direct corollaries.

Corollary 3.2.

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

(E1) Open image in new window and Open image in new window ;

(E2) Open image in new window for each Open image in new window ;

(E3) Open image in new window for all Open image in new window ;

(E4) Open image in new window with Open image in new window ;

(E5) Open image in new window , Open image in new window , Open image in new window .

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 Open image in new window as Open image in new window (see [4, 5, 28]). The proof method of Open image in new window is simple and different with ones of others.

## 4. Applications for Multiobjective Optimization Problem

where Open image in new window and Open image in new window are both the convex and lower semicontinuous functions defined on a closed convex subset of Open image in new window of a Hilbert space Open image in new window .

Note that, if we find a solution Open image in new window , then one must have Open image in new window obviously.

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

### References

- 1.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.CrossRefMATHGoogle Scholar - 2.Xu H-K:
**Viscosity approximation methods for nonexpansive mappings.***Journal of Mathematical Analysis and Applications*2004,**298**(1):279–291. 10.1016/j.jmaa.2004.04.059MathSciNetCrossRefMATHGoogle Scholar - 3.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.2005.05.028MathSciNetCrossRefMATHGoogle Scholar - 4.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.2006.03.047MathSciNetCrossRefMATHGoogle Scholar - 5.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.2008.02.041MathSciNetCrossRefMATHGoogle Scholar - 6.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.2006.08.036MathSciNetCrossRefMATHGoogle Scholar - 7.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.2007.02.044MathSciNetCrossRefMATHGoogle Scholar - 8.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.2007.09.011MathSciNetCrossRefMATHGoogle Scholar - 9.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.2008.03.032MathSciNetCrossRefMATHGoogle Scholar - 10.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/s10957-009-9549-9MathSciNetCrossRefMATHGoogle Scholar - 11.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 - 12.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.2008.04.035MathSciNetCrossRefMATHGoogle Scholar - 13.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 - 14.Moudafi A:
**Weak convergence theorems for nonexpansive mappings and equilibrium problems.***Journal of Nonlinear and Convex Analysis*2008,**9**(1):37–43.MathSciNetMATHGoogle Scholar - 15.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.2007.07.075MathSciNetCrossRefMATHGoogle Scholar - 16.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.2009.06.042MathSciNetCrossRefMATHGoogle Scholar - 17.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.2010.01.015MathSciNetCrossRefMATHGoogle Scholar - 18.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 - 19.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 - 20.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.2003.07.023MathSciNetCrossRefMATHGoogle Scholar - 21.Takahashi W:
*Nonlinear Functional Analysis, Fixed Point Theory and Its Application*. Yokohama Publishers, Yokohama, Japan; 2000:iv+276.MATHGoogle Scholar - 22.Xu H-K:
**Iterative algorithms for nonlinear operators.***Journal of the London Mathematical Society*2002,**66**(1):240–256. 10.1112/S0024610702003332MathSciNetCrossRefMATHGoogle Scholar - 23.Combettes PL, Hirstoaga SA:
**Equilibrium programming in Hilbert spaces.***Journal of Nonlinear and Convex Analysis*2005,**6**(1):117–136.MathSciNetMATHGoogle Scholar - 24.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 - 25.Takahashi W, Shimoji K:
**Convergence theorems for nonexpansive mappings and feasibility problems.***Mathematical and Computer Modelling*2000,**32**(11–13):1463–1471.MathSciNetCrossRefMATHGoogle Scholar - 26.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 - 27.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.2004.11.017MathSciNetCrossRefMATHGoogle Scholar - 28.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.2008.10.004MathSciNetCrossRefMATHGoogle Scholar

## Copyright information

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.