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

*Fixed Point Theory and Applications*
**volume 2011**, Article number: 392741 (2011)

## 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.

## 1. Introduction

Let be a real Hilbert space and be a mapping of into itself. is said to be nonexpansive if

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 mapping. If there exists a constant such that

then is called a contraction with the constant . Recall that an operator is called to be strongly positive with coefficient if

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 .

In 2003, Xu [2] introduced the following iterative scheme:

where is the identical mapping on and , and proved some strong convergence theorems for the iterative scheme to the solution of the quadratic minimization problem

under suitable hypotheses on and the additional hypothesis:

Recently, Marino and Xu [3] introduced a new iterative scheme from an arbitrary point by the viscosity approximation method as follows:

and prove that the scheme strongly converges to the unique solution of the variational inequality:

which is the optimality condition for the minimization problem:

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

Let be a finite family of nonexpansive mappings of into itself. In 2007, Yao [4] defined the mappings

and, by extending (1.10), proposed the iterative scheme:

Then he proved that the iterative scheme (1.10) strongly converges to the unique solution of the variational inequality:

where , which is the optimality condition for the minimization problem:

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

Let be a nonempty closed convex subset of and be a bifunction. The equilibrium problem for the function is to determine the equilibrium points, that is, the set

Let be a nonlinear mapping. Let denote the set of all solutions to the following equilibrium problem:

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

In 2007, S. Takahashi and W. Takahashi [6] introduced a viscosity approximation method for finding a common element of and from an arbitrary initial element

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

By combing the schemes (1.7) and (1.16), Plubtieg and Punpaeng [7] proposed the following algorithm:

and proved that the iterative schemes and converge strongly to the unique solution of the variational inequality:

which is the optimality condition for the minimization problem:

where is a potential function for .

Very recently, for finding a common element of the set of a finite family of nonexpansive mappings and the set of solutions of an equilibrium problem, by combining the schemes (1.11) and (1.17), Colao et al. [5] proposed the following explicit scheme:

and proved under some certain hypotheses that both sequences and converge strongly to a point which is an equilibrium point for and is the unique solution of the variational inequality:

where .

The equilibrium problems have been considered by many authors; see, for example, [6, 8–19] and the reference therein. But, in these references, the authors only considered at most finite family of equilibrium problems and few of authors investigate the infinite family of equilibrium problems in a Hilbert space or Banach space. In this paper, we consider a new iterative scheme for obtaining a common element in the solution set of an infinite family of generalized equilibrium problems and in the common fixed-point set of a finite family of nonexpansive mappings in a Hilbert space. Let () be a finite family of nonexpansive mappings of into itself, be be an infinite family of bifunctions, and be be an infinite family of -inverse-strongly monotone mappings. Let be a sequence such that with for each . Define the mapping by

Assume that . For an arbitrary initial point , we define the iterative scheme by

where , , and are three sequences in (0,1), and are both strongly positive linear bounded operators on , is defined by (1.10), and prove that, under some certain appropriate hypotheses on the control sequences, the sequence strongly converges to a point , which is the unique solution of the variational inequality:

If , and , then (1.23) is reduced to the iterative scheme:

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

Let be a closed convex subset of a Hilbert space . For any point , there exists a unique nearest point in , denoted by , such that

Then is called the metric projection of onto . It is well known that is a nonexpansive mapping of onto and satisfies the following:

Let be a mapping from into , then is called monotone if

for all . However, is called an *α*-inverse-strongly monotone mapping if there exists a positive real number *α* such that

for all . Let denote the identity mapping of , then for all and , one has [20]

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

If there exists such that

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]).

Given and . Then if and only if there holds the inequality

Lemma 2.2 (see [22]).

Let be a sequence of nonnegative real numbers satisfying

where , , and satisfy the conditions:

(1) , or, equivalently, ;

(2) ;

(3) , .

Then .

Let be a Hilbert space. For all , the following equality holds:

Therefore, the following lemma naturally holds.

Lemma 2.3.

Let be a real Hilbert space. The following identity holds:

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]).

Assume that is a sequence of nonnegative numbers such that

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

(1) ;

(2) or .

Then .

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:

(A1) for all ;

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

(A3) For each ,

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

For and , define a mapping by

Then is well defined and the following hold:

(1) is single-valued;

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

(3) ;

(4) is closed and convex.

It is easy to see that if there exists some point such that , where is an *α*-inverse strongly monotone mapping, then . In fact, since , one has

that is,

Hence, .

Let be a nonempty convex subset of a Banach space. Let be a finite family of nonexpansive mappings of into itself and be real numbers such that for each . Define a mapping of into itself as follows:

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]).

Let be a nonempty convex subset of a Banach space. Let be a finite family of nonexpansive mappings of into itself and let be sequences in such that for each . Moreover, for each , let and be the -mappings generated by and and and , respectively. Then, for all , it follows that

## 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

(E1)

(E2) ;

(E3) ;

(E4) with ;

(E5) .

Then the sequence defined by (1.23) converges strongly to , which is the unique solution of the variational inequality: (1.24), that is,

Proof.

Since as by the condition (E1), we may assume, without loss of generality, that for all . Noting that and are both the linear bounded self-adjoint operators, one has

Observing that

we obtain that is positive for all . It follows that

For each , define a quadratic function in as follows:

Note that

Hence, for each satisfying the condition (E4), one has

Moreover, it follows from (3.7), and (E4) that

Next, we proceed the proof with following steps.

Step 1.

is bounded.

Let . Lemma 2.6 shows that every is firmly nonexpansive and hence nonexpansive. Since , is nonexpansive for each . Therefore, is nonexpansive for each . Noting that is strictly decreasing, , we have

and hence

Then, from (3.4) and (3.11), it follows that (noting that is linear and )

It follows from and that . Therefore, by the simple induction, we have

which shows that is bounded, so is .

Step 2.

as .

First, we prove

Let and set

It follows from the definition of that

for each . Thus, using the above recursive inequalities repeatedly, we have

Also, we have

where .

Next, we prove . Observe (noting that is linear) that

Hence, by (3.4) and (3.18), we get

where .

Set . It follows from and (due to ) that . Thus we have

Set

Then it follows from (3.21) that

It follows from the assumption condition (E1), (E3), (E5), and (3.14) that

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

Step 3.

as .

For all , we have

and hence (noting (3.9))

It follows from the assumption conditions (E1), (E2), and Step 2 that

Step 4.

as .

Notice that, for any ,

Let and . By using (3.8), (3.9), (3.28), Lemmas 2.3, and 2.4, we have (noting that )

This shows that

and hence, for each ,

Since , and , we have

Now, for , we have, from Lemma 2.2,

and hence

Therefore,

By using (3.8), (3.9), (3.35), Lemmas 2.3 and 2.4, we have (noting that )

and hence

This shows that for, each ,

Since is strictly decreasing, , , and , we have, for each ,

Now, from we get

Since and for each , one has

Step 5.

.

To prove this, we pick a subsequence of such that

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

Indeed, for each , since , and is nonexpansive, by demiclosed principle of nonexpansive mapping we have

Assume that for each . Let be the -mapping generated by and . Then, by Lemma 2.8, we have

Moreover, it follows from Lemma 2.7 that . Assume that . Then . Since for each , by Step 3, (3.44) and Opial's property of the Hilbert space , we have

which is a contradiction. Therefore, . Hence, .

Step 6.

The sequence strongly converges to some point .

By using Lemmas 2.3 and 2.4, we have

which implies that

where is an appropriate constant such that . Put

Then we have

It follows from the assumption condition (E1) and (3.42) that

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

(E1) and ;

(E2) for each ;

(E3) for all ;

(E4) with ;

(E5) , , .

Then the sequence defined by (1.25) converges strongly to , which is the unique solution of the variational inequality:

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

In this section, we study a kind of multiobjective optimization problem by using the result of this paper. That is, we will give an iterative algorithm of solution for the following multiobjective optimization problem with the nonempty set of solutions:

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

We denote by the set of solutions of the problem (4.1) and assume that . Also, we denote the sets of solutions of the following two optimization problems by and , respectively,

and

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

Now, let and be two bifunctions from to defined by

respectively. It is easy to see that and , where denotes the set of solutions of the equilibrium problem:

respectively. In addition, it is easy to see that and satisfy the conditions (A1)–(A4). Let be a sequence in (0,1) and . Define a sequence by

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).

## 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. - 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.2004.04.059 - 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.2005.05.028 - 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.2006.03.047 - 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.2008.02.041 - 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.2006.08.036 - 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.2007.02.044 - 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.2007.09.011 - 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.2008.03.032 - 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/s10957-009-9549-9 - 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. - 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.2008.04.035 - 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.2009.03.006 - 14.
Moudafi A:

**Weak convergence theorems for nonexpansive mappings and equilibrium problems.***Journal of Nonlinear and Convex Analysis*2008,**9**(1):37–43. - 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.2007.07.075 - 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.2009.06.042 - 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.2010.01.015 - 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. - 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. - 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.2003.07.023 - 21.
Takahashi W:

*Nonlinear Functional Analysis, Fixed Point Theory and Its Application*. Yokohama Publishers, Yokohama, Japan; 2000:iv+276. - 22.
Xu H-K:

**Iterative algorithms for nonlinear operators.***Journal of the London Mathematical Society*2002,**66**(1):240–256. 10.1112/S0024610702003332 - 23.
Combettes PL, Hirstoaga SA:

**Equilibrium programming in Hilbert spaces.***Journal of Nonlinear and Convex Analysis*2005,**6**(1):117–136. - 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. - 25.
Takahashi W, Shimoji K:

**Convergence theorems for nonexpansive mappings and feasibility problems.***Mathematical and Computer Modelling*2000,**32**(11–13):1463–1471. - 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. - 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.2004.11.017 - 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.2008.10.004

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Wang, S., Guo, B. Strong Convergence of a New Iterative Method for Infinite Family of Generalized Equilibrium and Fixed-Point Problems of Nonexpansive Mappings in Hilbert Spaces.
*Fixed Point Theory Appl* **2011, **392741 (2011). https://doi.org/10.1155/2011/392741

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### Keywords

- Hilbert Space
- Variational Inequality
- Equilibrium Problem
- Nonexpansive Mapping
- Iterative Scheme