Fixed Point Theory and Applications

, 2009:519065 | Cite as

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

Open Access
Research Article

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:

(C1) Open image in new window

(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?

Question 1.
  1. (i)

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

     
  2. (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)?

     
  3. (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)?

     
  4. (iv)

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

     
  5. (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]).

Assume Open image in new window is a sequence of nonnegative real numbers such that

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 2.2 ([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.

Let 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 bifunction satifies (A1)–(A4) and let Open image in new window be a proper lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. For Open image in new window and Open image in new window , define a mapping Open image in new window as follows:

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

(iv) 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

(D1) Open image in new window

(D2) Open image in new window

(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 .

Then Open image in new window and Open image in new window converge strongly to a point Open image in new window which is the unique solution of the variational inequality

Equivalently, one has Open image in new window

Proof.

Since Open image in new window , we may assume, without loss of generality, that Open image in new window for all Open image in new window . By Lemma 2.4, we have Open image in new window . We will assume that Open image in new window . Observe that Open image in new window is a contraction. Indeed, for all Open image in new window , we have
Since Open image in new window is complete, there exists a unique element Open image in new window such that Open image in new window On the other hand, since Open image in new window is a linear bounded self-adjoint operator, one has
Observing that
we obtain Open image in new window is positive. It follows that

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

Step 1.

which implies that Open image in new window is nonexpansive.

Step 2.

Next we prove that Open image in new window and Open image in new window are bounded. Indeed, pick any Open image in new window . From (2.5), we have Open image in new window Setting Open image in new window , we obtain from the nonexpansivity of Open image in new window that
From (2.1), we have
so, by (3.9) and the Open image in new window -strict pseudocontractivity of Open image in new window , it follows that
that is,
Observe that
From (3.8), (3.11) and the last inequality, we have
It follows that
By simple induction, we have

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.

Next we claim that
Notice that
Next, we define

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 is an appropriate constant such that Open image in new window . Substituting (3.20) into (3.17), we obtain
On the other hand, from Open image in new window and Open image in new window we note that
Putting Open image in new window in (3.22) and Open image in new window in (3.23), we have
So, from (A2) we have
and hence
Without loss of generality, let us assume that there exists a real number Open image in new window such that Open image in new window for all Open image in new window Then, we have
and hence
where Open image in new window . It follows from (3.21) and the last inequality that

where Open image in new window .

Define a sequence Open image in new window such that

Then, we have
It follows from (3.29) that
Observing the conditions (D1), (D3), (D4), (D5), and taking the superior limit as Open image in new window , we get
We can obtain Open image in new window easily by Lemma 2.2. Observing that
we obtain

Hence (3.16) is proved.

Step 4.

Next we prove that
  1. (a)

    First we prove that Open image in new window . Observing that

     
we arrive at
which implies that
Therefore, it follows from (3.16), (D1), and (D2) that
  1. (b)

    Next, we will show that Open image in new window for any Open image in new window Observe that

     
This implies that
It is easy to see that Open image in new window and then from (3.16), we obtain
  1. (c)

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

     
so, we obtain
It follows that
which implies that
Applying (3.16), (3.44), Open image in new window , and Open image in new window to the last inequality, we obtain that
It follows from (3.40) and (3.49) that
Then it follows from (D1), (3.49) and (3.50) that
For any Open image in new window , we have from Lemma 2.7,
From (3.41) we observe that
Using (D1), (D2) and (3.16), we obtain
  1. (d)

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

     
which implies that
By (3.16), (3.51), and (3.56), we have
Observing that
Using (3.40) and the last inequality, we obtain that
From Lemma 2.3(i), (3.59), and (3.61), we have

Hence (3.36) is proved.

Step 5.

We claim that

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):

On the other hand, from Open image in new window , we have
that is,
Therefore, we obtian
Noting that Open image in new window as Open image in new window and Open image in new window is Lipschitz continuous, hence from (3.69), we obtain

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,

From (A2), we also have
and hence
Letting Open image in new window , it follows from the weakly semicontinuity of Open image in new window that

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

Consequently

as required. This together with (3.40) implies that

Step 6.

Finally, we show that Open image in new window . Indeed, we note that
for all Open image in new window . It then follows that

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

(D1) Open image in new window

(D2) Open image in new window

(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 .

Then Open image in new window and Open image in new window converge strongly to a point Open image in new window which is the unique solution of the variational inequality

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

(D1) Open image in new window

(D2) Open image in new window

(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 .

Then Open image in new window and Open image in new window converge strongly to a point Open image in new window which is the unique solution of the variational inequality

Equivalently, one has Open image in new window

Remark 3.4.
  1. (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].

     
  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).

     
  3. (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

(D1) Open image in new window

(D2) Open image in new window

(D3) Open image in new window .

Then Open image in new window and Open image in new window converge strongly to a point Open image in new window which is the unique solution of the variational inequality

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

(D1) Open image in new window

(D2) Open image in new window

(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 .

where Open image in new window is a positive constant such that Open image in new window Then both Open image in new window and Open image in new window converge strongly to a point Open image in new window which is the unique solution of the variational inequality

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

(D1) Open image in new window

(D2) Open image in new window

(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 .

where Open image in new window and Open image in new window are positive constants such that Open image in new window and Open image in new window , respectively. Then Open image in new window and Open image in new window converge strongly to a point Open image in new window which is the unique solution of the variational inequality

Equivalently, we have Open image in new window

Proof.

Taking Open image in new window in Theorem 4.1, we know that Open image in new window is Open image in new window -inverse strongly monotone with Open image in new window . Hence, Open image in new window is a monotone Open image in new window -Lipschitz continuous mapping with Open image in new window . From Lemma 2.6, we know that Open image in new window is a Open image in new window -strictly pseudocontractive mapping with Open image in new window and then Open image in new window by Lemma 2.6. Observe that

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.

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Copyright information

© R.Wangkeeree and R.Wangkeeree. 2009

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.

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceNaresuan UniversityPhitsanulokThailand

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