# A General Iterative Method for Variational Inequality Problems, Mixed Equilibrium Problems, and Fixed Point Problems of Strictly Pseudocontractive Mappings in Hilbert Spaces

## Abstract

We introduce an iterative scheme for finding a common element of the set of fixed points of a Open image in new window -strictly pseudocontractive mapping, the set of solutions of the variational inequality for an inverse-strongly monotone mapping, and the set of solutions of the mixed equilibrium problem in a real Hilbert space. Under suitable conditions, some strong convergence theorems for approximating a common element of the above three sets are obtained. As applications, at the end of the paper we first apply our results to study the optimization problem and we next utilize our results to study the problem of finding a common element of the set of fixed points of two families of finitely Open image in new window -strictly pseudocontractive mapping, the set of solutions of the variational inequality, and the set of solutions of the mixed equilibrium problem. The results presented in the paper improve some recent results of Kim and Xu (2005), Yao et al. (2008), Marino et al. (2009), Liu (2009), Plubtieng and Punpaeng (2007), and many others.

## Keywords

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

Throughout this paper, we always assume that Open image in new window is a real Hilbert space with inner product Open image in new window and norm Open image in new window , respectively, Open image in new window is a nonempty closed convex subset of Open image in new window . Let Open image in new window be a real-valued function and let Open image in new window be an equilibrium bifunction, that is, Open image in new window for each Open image in new window . Ceng and Yao [1] considered the following mixed equilibrium problem:

The set of solutions of (1.1) is denoted by Open image in new window . It is easy to see that Open image in new window is a solution of problem (1.1) implies that Open image in new window .

In particular, if Open image in new window , the mixed equilibrium problem (1.1) becomes the following equilibrium problem:

The set of solutions of (1.2) is denoted by Open image in new window .

If Open image in new window and Open image in new window for all Open image in new window , where Open image in new window is a mapping form Open image in new window into Open image in new window , then the mixed equilibrium problem (1.1) becomes the following variational inequality:

The set of solutions of (1.3) is denoted by Open image in new window . The variational inequality has been extensively studied in literature. See, for example, [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13] and the references therein.

The problem (1.1) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problem in noncooperative games and others; see for instance, [1, 2, 14, 15].

First we recall some relevant important results as follows.

In 1997, Combettes and Hirstoaga [14] introduced an iterative method of finding the best approximation to the initial data when Open image in new window is nonempty and proved a strong convergence theorem. Subsequently, S. Takahashi and W. Takahashi [16] introduced an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of Open image in new window and the set of fixed point points of a nonexpansive mapping. Using the idea of S. Takahashi and W. Takahashi [16], Plubtieng and Punpaeng [17] introduced an the general iterative method for finding a common element of the set of solutions of Open image in new window and the set of fixed points of a nonexpansive mapping which is the optimality condition for the minimization problem in a Hilbert space. Furthermore, Yao et al. [11] introduced some new iterative schemes for finding a common element of the set of solutions of Open image in new window and the set of common fixed points of finitely (infinitely) nonexpansive mappings. Very recently, Ceng and Yao [1] considered a new iterative scheme for finding a common element of the set of solutions of Open image in new window and the set of common fixed points of finitely many nonexpansive mappings in a Hilbert space and obtained a strong convergence theorem which used the following condition:

(E) Open image in new window is Open image in new window -strongly convex and its derivative Open image in new window is sequentially continuous from the weak topology to the strong topology.

Their results extend and improve the corresponding results in [6, 11, 14]. We note that the condition (E) for the function Open image in new window is a very strong condition. We also note that the condition (E) does not cover the case Open image in new window and Open image in new window . Motivated by Ceng and Yao [1], Peng and Yao [18] introduced a new iterative scheme based on only the extragradient method for finding a common element of the set of solutions of a mixed equilibrium problem, the set of fixed points of a family of finitely nonexpansive mappings and the set of the variational inequality for a monotone Lipschitz continuous mapping. They obtained a strong convergence theorem without the condition (E) for the sequences generated by these processes.

We recall that a mapping Open image in new window is said to be:

(i)monotone if Open image in new window

(ii) Open image in new window -Lipschitz if there exists a constant Open image in new window such that Open image in new window

(iii) Open image in new window -inverse-strongly monotone [19, 20] if there exists a positive real number Open image in new window such that

It is obvious that any Open image in new window -inverse-strongly monotone mapping Open image in new window is monotone and Lipschitz continuous. Recall that a mapping Open image in new window is called a Open image in new window -strictly pseudocontractive mapping if there exists a constant Open image in new window such that

Note that the class of Open image in new window -strictly pseudocontractive mappings strictly includes the class of nonexpansive mappings which are mappings Open image in new window on Open image in new window such that

That is, Open image in new window is nonexpansive if and only if Open image in new window is Open image in new window -strictly pseudocontractive. We denote by Open image in new window the set of fixed points of Open image in new window .

Iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems; see, for example, [21, 22, 23, 24] and the references therein. Convex minimization problems have a great impact and influence in the development of almost all branches of pure and applied sciences. A typical problem is to minimize a quadratic function over the set of the fixed points of nonexpansive mapping on a real Hilbert space:

where Open image in new window is a linear bounded operator, Open image in new window is the fixed point set of a nonexpansive mapping Open image in new window and Open image in new window is a given point in Open image in new window . Recall that a linear bounded operator Open image in new window is strongly positive if there is a constant Open image in new window with property

Recently, Marino and Xu [25] introduced the following general iterative scheme based on the viscosity approximation method introduced by Moudafi [26]:

where Open image in new window is a strongly positive bounded linear operator on Open image in new window . They proved that if the sequence Open image in new window of parameters satisfies appropriate conditions, then the sequence Open image in new window generated by (1.9) converges strongly to the unique solution of the variational inequality

which is the optimality condition for the minimization problem

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

Recall that the construction of fixed points of nonexpansive mappings via Manns algorithm [27] has extensively been investigated in literature; see, for example [27, 28, 29, 30, 31, 32] and references therein. If Open image in new window is a nonexpansive self-mapping of Open image in new window , then Mann's algorithm generates, initializing with an arbitrary Open image in new window , a sequence according to the recursive manner

where Open image in new window is a real control sequence in the interval Open image in new window .

If Open image in new window is a nonexpansive mapping with a fixed point and if the control sequence Open image in new window is chosen so that Open image in new window , then the sequence Open image in new window generated by Manns algorithm converges weakly to a fixed point of Open image in new window . Reich [33] showed that the conclusion also holds good in the setting of uniformly convex Banach spaces with a Fréhet differentiable norm. It is well known that Reich's result is one of the fundamental convergence results. However, this scheme has only weak convergence even in a Hilbert space [34]. Therefore, many authors try to modify normal Mann's iteration process to have strong convergence; see, for example, [35, 36, 37, 38, 39, 40] and the references therein.

Kim and Xu [36] introduced the following iteration process:

where Open image in new window is a nonexpansive mapping of Open image in new window into itself and Open image in new window is a given point. They proved the sequence Open image in new window defined by (1.13) strongly converges to a fixed point of Open image in new window provided the control sequences Open image in new window and Open image in new window satisfy appropriate conditions.

In [41], Yao et al. also modified iterative algorithm (1.13) to have strong convergence by using viscosity approximation method. To be more precisely, they considered the following iteration process:

where Open image in new window is a nonexpansive mapping of Open image in new window into itself and Open image in new window is an Open image in new window -contraction. They proved the sequence Open image in new window defined by (1.14) strongly converges to a fixed point of Open image in new window provided the control sequences Open image in new window and Open image in new window satisfy appropriate conditions.

Very recently, motivated by Acedo and Xu [35], Kim and Xu [36], Marino and Xu [42], and Yao et al. [41], Marino et al. [43] introduced a composite iteration scheme as follows:

where Open image in new window is a Open image in new window -strictly pseudocontractive mapping on Open image in new window Open image in new window is an Open image in new window -contraction, and Open image in new window is a linear bounded strongly positive operator. They proved that the iterative scheme Open image in new window defined by (1.15) converges to a fixed point of Open image in new window , which is a unique solution of the variational inequality (1.10) and is also the optimality condition for the minimization problem provided Open image in new window and Open image in new window are sequences in Open image in new window satifies the following control conditions:

(C2) Open image in new window for all Open image in new window and Open image in new window .

Moreover, for finding a common element of the set of fixed points of a Open image in new window -strictly pseudocontractive nonself mapping and the set of solutions of an equilibrium problem in a real Hilbert space, Liu [44] introduced the following iterative scheme:

where Open image in new window is a Open image in new window -strictly pseudocontractive mapping on Open image in new window Open image in new window is an Open image in new window -contraction and, Open image in new window is a linear bounded strongly positive operator. They proved that the iterative scheme Open image in new window defined by (1.16) converges to a common element of Open image in new window , which solves some variation inequality problems provided Open image in new window and Open image in new window are sequences in Open image in new window satifies the control conditions (C1) and the following conditions:

(2) Open image in new window for all Open image in new window , Open image in new window , and Open image in new window ;

(C3) Open image in new window .

All of the above bring us the following conjectures?

- (i)
Could we weaken or remove the control condition Open image in new window on parameter Open image in new window in (C1)?

- (ii)
Could we weaken or remove the control condition Open image in new window on parameter Open image in new window in (C2) and ( Open image in new window 2)?

- (iii)
Could we weaken or remove the control condition Open image in new window on the parameter Open image in new window in ( Open image in new window 2)?

- (iv)
Could we weaken the control condition (C3) on parameters Open image in new window ?

- (v)
Could we construct an iterative algorithm to approximate a common element of Open image in new window ?

It is our purpose in this paper that we suggest and analyze an iterative scheme for finding a common element of the set of fixed points of a Open image in new window -strictly pseudocontractive mapping, the set of solutions of a variational inequality and the set of solutions of a mixed equilibrium problem in the framework of a real Hilbert space. Then we modify our iterative scheme to finding a common element of the set of common fixed points of two finite families of Open image in new window -strictly pseudocontractive mappings, the set of solutions of a variational inequality and the set of solutions of a mixed equilibrium problem. Application to optimization problems which is one of the motivation in this paper is also given. The results in this paper generalize and improve some well-known results in [17, 36, 41, 43, 44].

## 2. Preliminaries

Let Open image in new window be a real Hilbert space with norm Open image in new window and inner product Open image in new window and let Open image in new window be a closed convex subset of Open image in new window . We denote weak convergence and strong convergence by notations Open image in new window Open image in new window and Open image in new window , respectively. It is well known that for any Open image in new window

For every point Open image in new window , there exists a unique nearest point in Open image in new window , denoted by Open image in new window , such that

Open image in new window is called the metric projection of Open image in new window onto Open image in new window It is well known that Open image in new window is a nonexpansive mapping of Open image in new window onto Open image in new window and satisfies

for every Open image in new window Moreover, Open image in new window is characterized by the following properties: Open image in new window and

for all Open image in new window . It is easy to see that the following is true:

A set-valued mapping Open image in new window is called monotone if for all Open image in new window , Open image in new window and Open image in new window imply Open image in new window . A monotone mapping Open image in new window is maximal if the graph of Open image in new window of Open image in new window is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping Open image in new window is maximal if and only if for Open image in new window , Open image in new window for every Open image in new window implies Open image in new window . Let Open image in new window be a monotone map of Open image in new window into Open image in new window and let Open image in new window be the normal cone to Open image in new window at Open image in new window , that is, Open image in new window and define

Then Open image in new window is the maximal monotone and Open image in new window if and only if Open image in new window ; see [45].

The following lemmas will be useful for proving the convergence result of this paper.

Lemma 2.1 ([46]).

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

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

Lemma 2.2 ([47]).

Let Open image in new window and Open image in new window be bounded sequences in a Banach space Open image in new window and let Open image in new window be a sequence in Open image in new window with Open image in new window Suppose Open image in new window for all integers Open image in new window and Open image in new window Then, Open image in new window

Lemma 2.3 ([42, Proposition 2.1]).

Assume that Open image in new window is a closed convex subset of Hilbert space Open image in new window , and let Open image in new window be a self-mapping of Open image in new window

(i)if Open image in new window is a Open image in new window -strictly pseudocontractive mapping, then Open image in new window satisfies the Lipscchitz condition

(ii)if Open image in new window is a Open image in new window -strictly pseudocontractive mapping, then the mapping Open image in new window is demiclosed(at Open image in new window ). That is, if Open image in new window is a sequence in Open image in new window such that Open image in new window and Open image in new window , Open image in new window

(iii)if Open image in new window is a Open image in new window -strictly pseudocontractive mapping, then the fixed point set Open image in new window of Open image in new window is closed and convex so that the projection Open image in new window is well defined.

Lemma 2.4 ([25]).

Assume 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

The following lemmas can be obtained from Acedo and Xu [35, Proposition 2.6] easily.

Lemma 2.5.

Let Open image in new window be a Hilbert space, Open image in new window be a closed convex subset of Open image in new window . For any integer Open image in new window , assume that, for each Open image in new window is a Open image in new window -strictly pseudocontractive mapping for some Open image in new window . Assume that Open image in new window is a positive sequence such that Open image in new window . Then Open image in new window is a Open image in new window -strictly pseudocontractive mapping with Open image in new window .

Lemma 2.6.

Let Open image in new window and Open image in new window be as in Lemma 2.5. Suppose that Open image in new window has a common fixed point in Open image in new window . Then Open image in new window .

For solving the mixed equilibrium problem, let us give the following assumptions for a bifunction Open image in new window and the set Open image in new window :

Open image in new window for all Open image in new window

Open image in new window is monotone, that is, Open image in new window for all Open image in new window

for each Open image in new window Open image in new window

for each Open image in new window is convex and lower semicontinuous;

Open image in new window is a bounded set.

By similar argument as in [48, proof of Lemma 2.3], we have the following result.

Lemma 2.7.

for all Open image in new window . Then, the following conditions hold:

(i)for each Open image in new window , Open image in new window ;

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

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

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

## 3. Main Results

In this section, we derive a strong convergence of an iterative algorithm which solves the problem of finding a common element of the set of solutions of a mixed equilibrium problem, the set of fixed points of a Open image in new window -strictly pseudocontractive mapping of Open image in new window into itself and the set of the variational inequality for an Open image in new window -inverse-strongly monotone mapping of Open image in new window into Open image in new window in a Hilbert space.

Theorem 3.1.

Let C be a nonempty closed convex subset of a Hilbert space H. Let Open image in new window be a bifunction from Open image in new window to Open image in new window satifies (A1)–(A4) and Open image in new window be a proper lower semicontinuous and convex function. Let Open image in new window be a Open image in new window -strictly pseudocontractive mapping of Open image in new window into itself. Let Open image in new window be a contraction of Open image in new window into itself with coefficient Open image in new window , Open image in new window an Open image in new window -inverse-strongly monotone mapping of Open image in new window into Open image in new window such that Open image in new window . Let Open image in new window be a strongly bounded linear self-adjoint operator with coefficient Open image in new window and Open image in new window . Assume that either (B1) or (B2) holds. Given the sequences Open image in new window and Open image in new window in Open image in new window satisfyies the following conditions

(D3) Open image in new window for all Open image in new window and Open image in new window ;

(D4) Open image in new window for some Open image in new window with Open image in new window and Open image in new window ;

(D5) Open image in new window .

Equivalently, one has Open image in new window

Proof.

Next, we divide the proof into six steps as follows.

Step 1.

which implies that Open image in new window is nonexpansive.

Step 2.

which gives that the sequence Open image in new window is bounded, so are Open image in new window and Open image in new window

Step 3.

As shown in [19], from the Open image in new window -strict pseudocontractivity of Open image in new window and the conditions (D4), it follows that Open image in new window is a nonexpansive maping for which Open image in new window .

Observing that

where Open image in new window .

Define a sequence Open image in new window such that

Hence (3.16) is proved.

Step 4.

- (a)
First we prove that Open image in new window . Observing that

- (b)
Next, we will show that Open image in new window for any Open image in new window Observe that

- (c)
Next we prove that Open image in new window . From (2.3), we have

- (d)
Next we prove that Open image in new window . Using Lemma 2.3 (i), we have

Hence (3.36) is proved.

Step 5.

Since Open image in new window is bounded, there exists a subsequence Open image in new window of Open image in new window which converges weakly to Open image in new window .

Next, we show that Open image in new window .

(a)We first show Open image in new window . In fact, using Lemma 2.3(ii) and (3.36), we obtain that Open image in new window .

(b)Next, we prove Open image in new window . For this purpose, let Open image in new window be the maximal monotone mapping defined by (2.6):

Since Open image in new window is maximal monotone, we have Open image in new window , and hence Open image in new window .

(c)We show Open image in new window . In fact, by Open image in new window , and we have,

Hence Open image in new window . Therefore, the conclusion Open image in new window is proved.

Consequently

Step 6.

Using (D1), and (3.79), we get Open image in new window . Now applying Lemma 2.1 to (3.82), we conclude that Open image in new window . From Open image in new window and Open image in new window , we obtain Open image in new window . The proof is now complete.

By Theorem 3.1, we can obtain some new and interesting strong convergence theorems. Now we give some examples as follows.

Setting Open image in new window in Theorem 3.1, we have the following result.

Corollary 3.2.

Let C be a nonempty closed convex subset of a Hilbert space H. Let Open image in new window be a bifunction from Open image in new window to Open image in new window satifies (A1)–(A4). Let Open image in new window be a Open image in new window -strictly pseudocontractive mapping of Open image in new window into itself. Let Open image in new window be a contraction of Open image in new window into itself with coefficient Open image in new window , Open image in new window an Open image in new window -inverse-strongly monotone mapping of Open image in new window into Open image in new window such that Open image in new window . Let Open image in new window be a strongly bounded linear self-adjoint operator with coefficient Open image in new window and Open image in new window . Given the sequences Open image in new window and Open image in new window in Open image in new window satisfies the following conditions

(D3) Open image in new window for all Open image in new window and Open image in new window

(D4) Open image in new window for some Open image in new window with Open image in new window and Open image in new window

(D5) Open image in new window .

Equivalently, one has Open image in new window

Setting Open image in new window and Open image in new window in Theorem 3.1, we have Open image in new window , then the following result is obtained.

Corollary 3.3.

Let C be a nonempty closed convex subset of a Hilbert space H. Let Open image in new window be a Open image in new window -strictly pseudocontractive mapping of Open image in new window into itself. Let Open image in new window be a contraction of Open image in new window into itself with coefficient Open image in new window , Open image in new window an Open image in new window -inverse-strongly monotone mapping of Open image in new window into Open image in new window such that Open image in new window . Let Open image in new window be a strongly bounded linear self-adjoint operator with coefficient Open image in new window and Open image in new window . Given the sequences Open image in new window and Open image in new window in Open image in new window satifies the following conditions

(D3) Open image in new window for all Open image in new window and Open image in new window

(D4) Open image in new window for some Open image in new window with Open image in new window and Open image in new window .

Equivalently, one has Open image in new window

- (i)
Since the conditions (C1) and (C2) have been weakened by the conditions (D1) and (D3) respectively. Theorem 3.1 and Corollary 3.2 generalize and improve [44, Theorem 3.2].

- (ii)
We can remove the control condition Open image in new window on the parameter Open image in new window in ( Open image in new window 2).

- (iii)
Since the conditions (C1) and (C2) have been weakened by the conditions (D1) and (D3) respectively. Theorem 3.1 and Corollary 3.3 generalize and improve [43, Theorem 2.1].

Setting Open image in new window and Open image in new window is nonexpansive in Theorem 3.1, we have the following result.

Corollary 3.5.

Let C be a nonempty closed convex subset of a Hilbert space H. Let Open image in new window be a bifunction from Open image in new window to Open image in new window satifies (A1)–(A4). Let Open image in new window be a nonexpansive mapping of Open image in new window into itself. Let Open image in new window be a contraction of Open image in new window into itself with coefficient Open image in new window such that Open image in new window . Let Open image in new window be a strongly bounded linear self-adjoint operator with coefficient Open image in new window and Open image in new window . Given the sequences Open image in new window and Open image in new window in Open image in new window satifies the following conditions

(D3) Open image in new window .

Equivalently, one has Open image in new window

Remark 3.6.

Since the conditions Open image in new window and Open image in new window have been weakened by the conditions Open image in new window and Open image in new window , respectively. Hence Corollary 3.5 generalize, extend and improve [17, Theorem 3.3].

## 4. Applications

First, we will utilize the results presented in this paper to study the following optimization problem:

where Open image in new window is a nonempty bounded closed convex subset of a Hilbert space and Open image in new window is a proper convex and lower semicontinuous function. We denote by Argmin Open image in new window the set of solutions in (4.1). Let Open image in new window for all Open image in new window , Open image in new window and Open image in new window in Theorem 3.1, then Open image in new window . It follows from Theorem 3.1 that the iterative sequence Open image in new window is defined by

where Open image in new window , Open image in new window satisfy the conditions (D1)–(D5) in Theorem 3.1. Then the sequence Open image in new window converges strongly to a solution Open image in new window .

Let Open image in new window for all Open image in new window , Open image in new window , Open image in new window , Open image in new window and Open image in new window in Theorem 3.1, then Open image in new window . It follows from Theorem 3.1 that the iterative sequence Open image in new window defined by

where Open image in new window , and Open image in new window satisfy the conditions (D1), (D2) and (D5), respectively in Theorem 3.1. Then the sequence Open image in new window converges strongly to a solution Open image in new window .

We remark that the algorithms (4.2) and (4.3) are variants of the proximal method for optimization problems introduced and studied by Martinet [49], Rockafellar [45], Ferris [50] and many others.

Next, we give the strong convergence theorem for finding a common element of the set of common fixed point of a finite family of strictly pseudocontractive mappings, the set of solutions of the variational inequality problem and the set of solutions of the mixed equilibrium problem in a Hilbert space.

Theorem 4.1.

Let C be a nonempty closed convex subset of a Hilbert space H. Let Open image in new window be a bifunction from Open image in new window to Open image in new window satifies (A1)–(A4) and Open image in new window be a proper lower semicontinuous and convex function. For each Open image in new window let Open image in new window be a Open image in new window -strictly pseudocontractive mapping of Open image in new window into itself for some Open image in new window . Let Open image in new window be a contraction of Open image in new window into itself with coefficient Open image in new window , Open image in new window an Open image in new window inverse-strongly monotone mapping of Open image in new window into Open image in new window such that Open image in new window . Let Open image in new window be a strongly bounded linear self-adjoint operator with coefficient Open image in new window and Open image in new window . Assume that either (B1) or (B2) holds. Given the sequences Open image in new window and Open image in new window in Open image in new window satifies the following conditions

(D3) Open image in new window for all Open image in new window and Open image in new window ;

(D4) Open image in new window for some Open image in new window with Open image in new window and Open image in new window ;

(D5) Open image in new window .

Equivalently, one has Open image in new window

Proof.

Let Open image in new window such that Open image in new window and define Open image in new window . By Lemmas 2.5 and 2.6, we conclude that Open image in new window is a Open image in new window -strictly pseudocontractive mapping with Open image in new window and Open image in new window . From Theorem 3.1, we can obtain the desired conclusion easily.

Finally, we will apply the main results to the problem for finding a common element of the set of fixed points of two finite families of Open image in new window -strictly pseudocontractive mappings, the set of solutions of the variational inequality and the set of solutions of the mixed equilibrium problem.

Let Open image in new window be a Open image in new window -strictly pseudocontractive mapping for some Open image in new window . We define a mapping Open image in new window where Open image in new window is a positive sequence such that Open image in new window , then Open image in new window is a Open image in new window -inverse-strongly monotone mapping with Open image in new window . In fact, from Lemma 2.5, we have

That is

On the other hand

Hence we have

This shows that Open image in new window is Open image in new window -inverse-strongly monotone.

Theorem 4.2.

Let Open image in new window be a nonempty closed convex subset of a Hilbert space Open image in new window . Let Open image in new window be a bifunction from Open image in new window to Open image in new window satifies (A1)–(A4) and Open image in new window be a proper lower semicontinuous and convex function. Let Open image in new window be a finite family of Open image in new window -strictly pseudocontractive mapping of Open image in new window into itself and Open image in new window be a finite family of Open image in new window -strictly pseudocontractive mapping of Open image in new window into Open image in new window for some Open image in new window such that Open image in new window . Let Open image in new window be a contraction of Open image in new window into itself with coefficient Open image in new window . Let Open image in new window be a strongly bounded linear self-adjoint operator with coefficient Open image in new window and Open image in new window . Assume that either (B1) or (B2) holds. Given the sequences Open image in new window and Open image in new window in Open image in new window satifies the following conditions

(D3) Open image in new window and Open image in new window for all Open image in new window and Open image in new window ;

(D4) Open image in new window for some Open image in new window with Open image in new window and Open image in new window ;

(D5) Open image in new window .

Equivalently, we have Open image in new window

Proof.

The conclusion can be obtained from Theorem 4.1.

## Notes

### Acknowledgments

R. Wangkeeree would like to thank The National Research Council of Thailand, Grant SC-AR-012/2552 for financial support. The authors would like to thank the referees for reading this paper carefully, providing valuable suggestions and comments, and pointing out a major error in the original version of this paper.

## References

- 1.Ceng L-C, Yao J-C:
**A hybrid iterative scheme for mixed equilibrium problems and fixed point problems.***Journal of Computational and Applied Mathematics*2008,**214**(1):186–201. 10.1016/j.cam.2007.02.022MathSciNetCrossRefMATHGoogle Scholar - 2.Blum E, Oettli W:
**From optimization and variational inequalities to equilibrium problems.***The Mathematics Student*1994,**63**(1–4):123–145.MathSciNetMATHGoogle Scholar - 3.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 - 4.Kumam P:
**A new hybrid iterative method for solution of equilibrium problems and fixed point problems for an inverse strongly monotone operator and a nonexpansive mapping.***Journal of Applied Mathematics and Computing*2009,**29**(1–2):263–280. 10.1007/s12190-008-0129-1MathSciNetCrossRefMATHGoogle Scholar - 5.Kumam W, Kumam P:
**Hybrid iterative scheme by a relaxed extragradient method for solutions of equilibrium problems and a general system of variational inequalities with application to optimization.***Nonlinear Analysis: Hybrid Systems*2009,**3**(4):640–656. 10.1016/j.nahs.2009.05.007MathSciNetMATHGoogle Scholar - 6.Takahashi W, Toyoda M:
**Weak convergence theorems for nonexpansive mappings and monotone mappings.***Journal of Optimization Theory and Applications*2003,**118**(2):417–428. 10.1023/A:1025407607560MathSciNetCrossRefMATHGoogle Scholar - 7.Kamraksa U, Wangkeeree R:
**A general iterative method for variational inequality problems and fixed point problems of an infinite family of nonexpansive mappings in Hilbert spaces.***Thai Journal of Mathematics*2008,**6**(1):147–170.MathSciNetMATHGoogle Scholar - 8.Wangkeeree R, Kamraksa U:
**A general iterative method for solving the variational inequality problem and fixed point problem of an infinite family of nonexpansive mappings in Hilbert spaces.***Fixed Point Theory and Applications*2009,**2009:**-23.Google Scholar - 9.Wangkeeree R:
**An extragradient approximation method for equilibrium problems and fixed point problems of a countable family of nonexpansive mappings.***Fixed Point Theory and Applications*2008,**2008:**-17.Google Scholar - 10.Wangkeeree R, Kamraksa U:
**An iterative approximation method for solving a general system of variational inequality problems and mixed equilibrium problems.***Nonlinear Analysis: Hybrid Systems*. In pressGoogle Scholar - 11.Yao Y, Liou Y-C, Yao J-C:
**An extragradient method for fixed point problems and variational inequality problems.***Journal of Inequalities and Applications*2007,**2007:**-12.Google Scholar - 12.Yao J-C, Chadli O:
**Pseudomonotone complementarity problems and variational inequalities.**In*Handbook of Generalized Convexity and Generalized Monotonicity, Nonconvex Optimization and Its Applications*.*Volume 76*. Edited by: Crouzeix JP, Haddjissas N, Schaible S. Springer, New York, NY, USA; 2005:501–558.CrossRefGoogle Scholar - 13.Zeng LC, Schaible S, Yao JC:
**Iterative algorithm for generalized set-valued strongly nonlinear mixed variational-like inequalities.***Journal of Optimization Theory and Applications*2005,**124**(3):725–738. 10.1007/s10957-004-1182-zMathSciNetCrossRefMATHGoogle Scholar - 14.Combettes PL, Hirstoaga SA:
**Equilibrium programming using proximal-like algorithms.***Mathematical Programming*1997,**78**(1):29–41.MathSciNetGoogle Scholar - 15.Flåm SD, Antipin AS:
**Equilibrium programming using proximal-like algorithms.***Mathematical Programming*1997,**78**(1):29–41.MathSciNetCrossRefMATHGoogle Scholar - 16.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 - 17.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 - 18.Peng J-W, Yao J-C:
**Strong convergence theorems of iterative scheme based on the extragradient method for mixed equilibrium problems and fixed point problems.***Mathematical and Computer Modelling*2009,**49**(9–10):1816–1828. 10.1016/j.mcm.2008.11.014MathSciNetCrossRefMATHGoogle Scholar - 19.Browder FE, Petryshyn WV:
**Construction of fixed points of nonlinear mappings in Hilbert space.***Journal of Mathematical Analysis and Applications*1967,**20:**197–228. 10.1016/0022-247X(67)90085-6MathSciNetCrossRefMATHGoogle Scholar - 20.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:1008643727926MathSciNetCrossRefMATHGoogle Scholar - 21.Deutsch F, Yamada I:
**Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings.***Numerical Functional Analysis and Optimization*1998,**19**(1–2):33–56.MathSciNetCrossRefMATHGoogle 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.Xu H-K:
**An iterative approach to quadratic optimization.***Journal of Optimization Theory and Applications*2003,**116**(3):659–678. 10.1023/A:1023073621589MathSciNetCrossRefMATHGoogle Scholar - 24.Yamada I:
**The hybrid steepest descent method for the variational inequality problem of the intersection of fixed point sets of nonexpansive mappings.**In*Inherently Parallel Algorithm for Feasibility and Optimization*. Edited by: Butnariu D, Censor Y, Reich S. Elsevier, London, UK; 2001:473–504.CrossRefGoogle Scholar - 25.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 - 26.Moudafi A:
**Viscosity approximation methods for fixed-points problems.***Journal of Mathematical Analysis and Applications*2000,**241**(1):46–55. 10.1006/jmaa.1999.6615MathSciNetCrossRefMATHGoogle Scholar - 27.Mann WR:
**Mean value methods in iteration.***Proceedings of the American Mathematical Society*1953,**4:**506–510. 10.1090/S0002-9939-1953-0054846-3MathSciNetCrossRefMATHGoogle Scholar - 28.Byrne C:
**A unified treatment of some iterative algorithms in signal processing and image reconstruction.***Inverse Problems*2004,**20**(1):103–120. 10.1088/0266-5611/20/1/006MathSciNetCrossRefMATHGoogle Scholar - 29.Tan K-K, Xu HK:
**Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process.***Journal of Mathematical Analysis and Applications*1993,**178**(2):301–308. 10.1006/jmaa.1993.1309MathSciNetCrossRefMATHGoogle Scholar - 30.Wittmann R:
**Approximation of fixed points of nonexpansive mappings.***Archiv der Mathematik*1992,**58**(5):486–491. 10.1007/BF01190119MathSciNetCrossRefMATHGoogle Scholar - 31.Xu H-K:
**Iterative algorithms for nonlinear operators.***Journal of the London Mathematical Society*2002,**66**(1):240–256. 10.1112/S0024610702003332MathSciNetCrossRefMATHGoogle Scholar - 32.Zeng L-C:
**A note on approximating fixed points of nonexpansive mappings by the Ishikawa iteration process.***Journal of Mathematical Analysis and Applications*1998,**226**(1):245–250. 10.1006/jmaa.1998.6053MathSciNetCrossRefMATHGoogle Scholar - 33.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)90024-6MathSciNetCrossRefMATHGoogle Scholar - 34.Genel A, Lindenstrauss J:
**An example concerning fixed points.***Israel Journal of Mathematics*1975,**22**(1):81–86. 10.1007/BF02757276MathSciNetCrossRefMATHGoogle Scholar - 35.Acedo GL, Xu H-K:
**Iterative methods for strict pseudo-contractions in Hilbert spaces.***Nonlinear Analysis: Theory, Methods & Applications*2007,**67**(7):2258–2271. 10.1016/j.na.2006.08.036MathSciNetCrossRefMATHGoogle Scholar - 36.Kim T-H, Xu H-K:
**Strong convergence of modified Mann iterations.***Nonlinear Analysis: Theory, Methods & Applications*2005,**61**(1–2):51–60. 10.1016/j.na.2004.11.011MathSciNetCrossRefMATHGoogle Scholar - 37.Martinez-Yanes C, Xu H-K:
**Strong convergence of the CQ method for fixed point iteration processes.***Nonlinear Analysis: Theory, Methods & Applications*2006,**64**(11):2400–2411. 10.1016/j.na.2005.08.018MathSciNetCrossRefMATHGoogle Scholar - 38.Nakajo K, Takahashi W:
**Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups.***Journal of Mathematical Analysis and Applications*2003,**279**(2):372–379. 10.1016/S0022-247X(02)00458-4MathSciNetCrossRefMATHGoogle Scholar - 39.Qin X, Su Y:
**Approximation of a zero point of accretive operator in Banach spaces.***Journal of Mathematical Analysis and Applications*2007,**329**(1):415–424. 10.1016/j.jmaa.2006.06.067MathSciNetCrossRefMATHGoogle Scholar - 40.Zhou H:
**Convergence theorems of fixed points for -strict pseudo-contractions in Hilbert spaces.***Nonlinear Analysis: Theory, Methods & Applications*2008,**69**(2):456–462. 10.1016/j.na.2007.05.032MathSciNetCrossRefMATHGoogle Scholar - 41.Yao Y, Chen R, Yao J-C:
**Strong convergence and certain control conditions for modified Mann iteration.***Nonlinear Analysis: Theory, Methods & Applications*2008,**68**(6):1687–1693. 10.1016/j.na.2007.01.009MathSciNetCrossRefMATHGoogle Scholar - 42.Marino G, Xu H-K:
**Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces.***Journal of Mathematical Analysis and Applications*2007,**329**(1):336–346. 10.1016/j.jmaa.2006.06.055MathSciNetCrossRefMATHGoogle Scholar - 43.Marino G, Colao V, Qin X, Kang SM:
**Strong convergence of the modified Mann iterative method for strict pseudo-contractions.***Computers & Mathematics with Applications*2009,**57**(3):455–465. 10.1016/j.camwa.2008.10.073MathSciNetCrossRefMATHGoogle Scholar - 44.Liu Y:
**A general iterative method for equilibrium problems and strict pseudo-contractions in Hilbert spaces.***Nonlinear Analysis: Theory, Methods & Applications*2009,**71**(10):4852–4861. 10.1016/j.na.2009.03.060MathSciNetCrossRefMATHGoogle Scholar - 45.Rockafellar RT:
**Monotone operators and the proximal point algorithm.***SIAM Journal on Control and Optimization*1976,**14**(5):877–898. 10.1137/0314056MathSciNetCrossRefMATHGoogle Scholar - 46.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 - 47.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 - 48.Peng J-W, Yao J-C:
**A new hybrid-extragradient method for generalized mixed equilibrium problems, fixed point problems and variational inequality problems.***Taiwanese Journal of Mathematics*2008,**12**(6):1401–1432.MathSciNetMATHGoogle Scholar - 49.Martinet B:
**Perturbation des méthodes d'optimisation. Applications.***RAIRO Analyse Numérique*1978,**12**(2):153–171.MathSciNetMATHGoogle Scholar - 50.Ferris MC:
**Finite termination of the proximal point algorithm.***Mathematical Programming*1991,**50**(3):359–366. 10.1007/BF01594944MathSciNetCrossRefMATHGoogle Scholar

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