A Hybrid Extragradient Viscosity Approximation Method for Solving Equilibrium Problems and Fixed Point Problems of Infinitely Many Nonexpansive Mappings

Abstract

We introduce a new hybrid extragradient viscosity approximation method for finding the common element of the set of equilibrium problems, the set of solutions of fixed points of an infinitely many nonexpansive mappings, and the set of solutions of the variational inequality problems for -inverse-strongly monotone mapping in Hilbert spaces. Then, we prove the strong convergence of the proposed iterative scheme to the unique solution of variational inequality, which is the optimality condition for a minimization problem. Results obtained in this paper improve the previously known results in this area.

1. Introduction

Let be a real Hilbert space, and let be a nonempty closed convex subset of Recall that a mapping of into itself is called nonexpansive (see [1]) if for all . We denote by the set of fixed points of . Recall also that a self-mapping is a contraction if there exists a constant such that In addition, let be a nonlinear mapping. Let be the projection of onto . The classical variational inequality which is denoted by is to find such that

(1.1)

For a given , satisfies the inequality

(1.2)

if and only if . It is well known that is a nonexpansive mapping of onto and satisfies

(1.3)

Moreover, is characterized by the following properties: and for all

(1.4)

(1.5)

It is easy to see that the following is true:

(1.6)

One can see that the variational inequality (1.1) is equivalent to a fixed point problem. The variational inequality has been extensively studied in literature; see, for instance, [2–6]. This alternative equivalent formulation has played a significant role in the studies of the variational inequalities and related optimization problems. Recall the following.

(1)A mapping of into is called monotone if

(1.7)

(2)A mapping is called -strongly monotone (see [7, 8]) if there exists a constant such that

(1.8)

(3)A mapping is called -Lipschitz continuous if there exists a positive real number such that

(1.9)

(4)A mapping is called -inverse-strongly monotone (see [7, 8]) if there exists a constant such that

(1.10)

Remark 1.1.

It is obvious that any -inverse-strongly monotone mapping is monotone and -Lipschitz continuous.

• (5) An operator is strongly positive on if there exists a constant with the property

(1.11)

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

(1.12)

Then is the maximal monotone and if and only if ; see [9].

• (7) Let be a bifunction of into , where is the set of real numbers. The equilibrium problem for is to find such that

(1.13)

The set of solutions of (1.13) is denoted by . Given a mapping let for all . Then, if and only if for all Numerous problems in physics, saddle point problem, fixed point problem, variational inequality problems, optimization, and economics are reduced to find a solution of (1.13). Some methods have been proposed to solve the equilibrium problem; see, for instance, [10–16]. Recently, Combettes and Hirstoaga [17] introduced an iterative scheme of finding the best approximation to the initial data when is nonempty and proved a strong convergence theorem.

In 1976, Korpelevich [18] introduced the following so-called extragradient method:

(1.14)

for all where is a closed convex subset of and is a monotone and -Lipschitz continuous mapping of into . He proved that if is nonempty, then the sequences and , generated by (1.14), converge to the same point . For finding a common element of the set of fixed points of a nonexpansive mapping and the set of solution of variational inequalities for -inverse-strongly monotone, Takahashi and Toyoda [19] introduced the following iterative scheme:

(1.15)

where is -inverse-strongly monotone, is a sequence in (0, 1), and is a sequence in . They showed that if is nonempty, then the sequence generated by (1.15) converges weakly to some . Recently, Iiduka and Takahashi [20] proposed a new iterative scheme as follows:

(1.16)

where is -inverse-strongly monotone, is a sequence in (0, 1), and is a sequence in . They showed that if is nonempty, then the sequence generated by (1.16) converges strongly to some .

Iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems; see, for example, [21–24] 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 a nonexpansive mapping on a real Hilbert space :

(1.17)

where is a linear bounded operator, is the fixed point set of a nonexpansive mapping on and is a given point in . Moreover, it is shown in [25] that the sequence defined by the scheme

(1.18)

converges strongly to Recently, Plubtieng and Punpaeng [26] proposed the following iterative algorithm:

(1.19)

They prove that if the sequences and of parameters satisfy appropriate condition, then the sequences and both converge to the unique solution of the variational inequality

(1.20)

which is the optimality condition for the minimization problem

(1.21)

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

Furthermore, for finding approximate common fixed points of an infinite countable family of nonexpansive mappings under very mild conditions on the parameters. Wangkeeree [27] introduced an iterative scheme for finding a common element of the set of solutions of the equilibrium problem (1.13) and the set of common fixed points of a countable family of nonexpansive mappings on . Starting with an arbitrary initial , define a sequence recursively by

(1.22)

where and are sequences in . It is proved that under certain appropriate conditions imposed on and , the sequence generated by (1.22) strongly converges to the unique solution , where which extend and improve the result of Kumam [14].

Definition 1.2 (see [21]).

Let be a sequence of nonexpansive mappings of into itself, and let be a sequence of nonnegative numbers in [0,1]. For each , define a mapping of into itself as follows:

(1.23)

Such a mapping is nonexpansive from to and it is called the -mapping generated by and .

On the other hand, Colao et al. [28] introduced and considered an iterative scheme for finding a common element of the set of solutions of the equilibrium problem (1.13) and the set of common fixed points of infinitely many nonexpansive mappings on . Starting with an arbitrary initial , define a sequence recursively by

(1.24)

where is a sequence in . It is proved [28] that under certain appropriate conditions imposed on and , the sequence generated by (1.24) strongly converges to , where is an equilibrium point for and is the unique solution of the variational inequality (1.20), that is, .

In this paper, motivated by Wangkeeree [27], Plubtieng and Punpaeng [26], Marino and Xu [25], and Colao, et al. [28], we introduce a new iterative scheme in a Hilbert space which is mixed bythe iterative schemes of (1.18), (1.19), (1.22), and (1.24) as follows.

Let be a contraction of into itself, a strongly positive bounded linear operator on with coefficient and a -inverse-strongly monotone mapping of into define sequences , , and recursively by

(1.25)

where is the sequence generated by (1.23), and and satisfying appropriate conditions. We prove that the sequences , , and generated by the above iterative scheme (1.25) converge strongly to a common element of the set of solutions of the equilibrium problem (1.13), the set of common fixed points of infinitely family nonexpansive mappings, and the set of solutions of variational inequality (1.1) for a -inverse-strongly monotone mapping in Hilbert spaces. The results obtained in this paper improve and extend the recent ones announced by Wangkeeree [27], Plubtieng and Punpaeng [26], Marino and Xu [25], Colao, et al. [28], and many others.

2. Preliminaries

We now recall some well-known concepts and results.

Let be a real Hilbert space, whose inner product and norm are denoted by and , respectively. We denote weak convergence and strong convergence by notations and , respectively.

A space is said to satisfy Opial's condition [29] if for each sequence in which converges weakly to point , we have

(2.1)

Lemma 2.1 (see [25]).

Let be a nonempty closed convex subset of let be a contraction of into itself with , and let be a strongly positive linear bounded operator on with coefficient . Then , for ,

(2.2)

That is, is strongly monotone with coefficient .

Lemma 2.2 (see [25]).

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

For solving the equilibrium problem for a bifunction , let us assume that satisfies the following conditions:

(A 1) for all

(A2) is monotone, that is, for all

(A3)for each

(A4)for each is convex and lower semicontinuous.

The following lemma appears implicitly in [30].

Lemma 2.3 (see [30]).

Let be a nonempty closed convex subset of and let be a bifunction of into satisfying (A1)–(A4). Let and . Then, there exists such that

(2.3)

The following lemma was also given in [17].

Lemma 2.4 (see [17]).

Assume that satisfies (A1)–(A4). For and , define a mapping as follows:

(2.4)

for all . Then, the following holds:

(1) is single-valued;

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

(2.5)

(3)

(4) is closed and convex.

For each , let the mapping be defined by (1.23). Then we can have the following crucial conclusions concerning . You can find them in [31]. Now we only need the following similar version in Hilbert spaces.

Lemma 2.5 (see [31]).

Let be a nonempty closed convex subset of a real Hilbert space . Let be nonexpansive mappings of into itself such that is nonempty, and let be real numbers such that for every . Then, for every and , the limit exists.

Using Lemma 2.5, one can define a mapping of into itself as follows:

(2.6)

for every . Such a is called the -mapping generated by and . Throughout this paper, we will assume that for every . Then, we have the following results.

Lemma 2.6 (see [31]).

Let be a nonempty closed convex subset of a real Hilbert space . Let be nonexpansive mappings of into itself such that is nonempty, and let be real numbers such that for every . Then, .

Lemma 2.7 (see [32]).

If is a bounded sequence in , then .

Lemma 2.8 (see [33]).

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

Lemma 2.9 (see [34]).

Assume that is a sequence of nonnegative real numbers such that

(2.7)

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

(1)

(2) or

Then

Lemma 2.10.

Let be a real Hilbert space. Then for all

(1);

(2).

3. Main Results

In this section, we prove the strong convergence theorem for infinitely many nonexpansive mappings in a real Hilbert space.

Theorem 3.1.

Let be a nonempty closed convex subset of a real Hilbert space , let be a bifunction from to satisfying (A1)–(A4), let be an infinitely many nonexpansive of into itself, and let be an -inverse-strongly monotone mapping of into such that . Let be a contraction of into itself with and let be a strongly positive linear bounded operator on with coefficient and . Let , , and be sequences generated by (1.25), where is the sequence generated by (1.23), , are three sequences in and is a real sequence in satisfying the following conditions:

(i)

(ii) and

(iii)

(iv) and

(v) for some and .

Then, and converge strongly to a point which is the unique solution of the variational inequality

(3.1)

Equivalently, one has

Proof.

Note that from the condition (i), we may assume, without loss of generality, that for all . From Lemma 2.2, we know that if , then . We will assume that . First, we show that is nonexpansive. Indeed, from the -inverse-strongly monotone mapping definition on and condition (v), we have

(3.2)

which implies that the mapping is nonexpansive. On the other hand, since is a strongly positive bounded linear operator on H, we have

(3.3)

Observe that

(3.4)

and this show that is positive. It follows that

(3.5)

Let , where . Note that is a contraction of into itself with . Then, we have

(3.6)

Since , it follows that is a contraction of into itself. Therefore by the Banach Contraction Mapping Principle, which implies that there exists a unique element such that .

We will divide the proof into five steps.

Step 1.

We claim that is bounded. Indeed, pick any . From the definition of , we note that . If follows that

(3.7)

Since is nonexpansive and from (1.6), we have

(3.8)

Put . Since , we have . Substituting and in (1.5), we can write

(3.9)

Using the fact that is -inverse-strongly monotone mapping, and is a solution of the variational inequality problem , we also have

(3.10)

It follows from (3.9) and (3.10) that

(3.11)

Substituting by and in (1.4), we obtain

(3.12)

It follows that

(3.13)

Substituting (3.13) into (3.11), we have

(3.14)

Setting , we can calculate

(3.15)

By induction,

(3.16)

Hence, is bounded, so are , , , , , and .

Step 2.

We claim that .

Observing that and we get

(3.17)

(3.18)

Putting in (3.17) and in (3.18), we have

(3.19)

So, from (A2) we have

(3.20)

and hence

(3.21)

Without loss of generality, let us assume that there exists a real number such that for all Then, we have

(3.22)

and hence

(3.23)

where . Note that

(3.24)

Setting

(3.25)

we have . It follows that

(3.26)

It follows from (3.24) and (3.26) that

(3.27)

Since and are nonexpansive, we have

(3.28)

where is a constant such that for all

Combining (3.27) and (3.28), we have

(3.29)

which implies that (noting that (i), (ii), (iii), (iv), (v), and

(3.30)

Hence, by Lemma 2.8, we obtain

(3.31)

It follows that

(3.32)

Applying (3.32) and (ii), (iv), and (v) to (3.23) and (3.24), we obtain that

(3.33)

Since , we have

(3.34)

that is

(3.35)

By (i), (iii), and (3.32) it follows that

(3.36)

Step 3.

We claim that the following statements hold:

(i)

(ii)

For any and (3.14), we have

(3.37)

Observe that

(3.38)

where

(3.39)

It follows from condition (i) that

(3.40)

Substituting (3.37) into (3.38), and using (v), we have

(3.41)

It follows that

(3.42)

Since and from (3.32), we obtain

(3.43)

Note that

(3.44)

Since , we have

(3.45)

As is -Lipschitz continuous, we obtain

(3.46)

then, we get

(3.47)

from

(3.48)

Applying (3.43), (3.45), and (3.47), we have

(3.49)

For any , note that is firmly nonexpansive (Lemma 2.4), then we have

(3.50)

and hence

(3.51)

which together with (3.38) gives

(3.52)

So

(3.53)

Using , as , (3.32), and (3.49), we obtain

(3.54)

Since , we obtain

(3.55)

Observe that

(3.56)

Applying (3.36), (3.49), and (3.54) to the last inequality, we obtain

(3.57)

Let be the mapping defined by (2.6). Since is bounded, applying Lemma 2.7 and (3.57), we have

(3.58)

Step 4.

We claim that where is the unique solution of the variational inequality

Since is a unique solution of the variational inequality (3.1), to show this inequality, we choose a subsequence of such that

(3.59)

Since is bounded, there exists a subsequence of which converges weakly to . Without loss of generality, we can assume that From we obtain . Next, We show that , where . First, we show that . Since , we have

(3.60)

If follows from (A2) that

(3.61)

and hence

(3.62)

Since and it follows by (A4) that for all For with and let Since and we have and hence So, from (A1) and (A4) we have

(3.63)

and hence . From (A3), we have for all and hence

Next, we show that By Lemma 2.6, we have . Assume Since and it follows by the Opial's condition that

(3.64)

which derives a contradiction. Thus, we have . By the same argument as that in the proof of [35, Theorem  2.1, Pages 10–11], we can show that Hence . Since , it follows that

(3.65)

It follows from the last inequality, (3.36), and (3.54) that

(3.66)

Step 5.

Finally, weshow that and convergestrongly to . Indeed, from (1.25) , we have

(3.67)

Since , and are bounded, we can take a constant such that

(3.68)

for all . It then follows that

(3.69)

where

(3.70)

Using (i), (3.65), and (3.66), we get . Applying Lemma 2.9 to (3.69), we conclude that in norm. Finally, noticing

(3.71)

we also conclude that in norm. This completes the proof.

Corollary 3.2 ([28, Theorem 3.1]).

Let be nonempty closed convex subset of a real Hilbert space , let be a bifunction from to satisfying (A1)–(A4), and let be an infinitely many nonexpansive of into itself such that . Let be a contraction of into itself with and let be a strongly positive linear bounded operator on with coefficient and . Let and are the sequences generated by

(3.72)

where is the sequence generated by (1.23), is a sequences in and is a real sequence in satisfying the following conditions:

(i)

(ii) and

Then, and converge strongly to a point which is the unique solution of the variational inequality

(3.73)

Equivalently, one has

Proof.

Put , and in Theorem 3.1., then . The conclusion of Corollary 3.2 can obtain the desired result easily.

Corollary 3.3.

Let be nonempty closed convex subset of a real Hilbert space , let be a bifunction from to satisfying (A1)–(A4) and let be an -inverse-strongly monotone mapping of into such that . Let be a contraction of into itself with and let be a strongly positive linear bounded operator on with coefficient and . Let , , and be sequences generated by

(3.74)

where , and are three sequences in and is a real sequence in satisfying the following conditions:

(i) and

(ii) and

(iii)

(iv) and

(v) for some and . Then, and converge strongly to a point which is the unique solution of the variational inequality

(3.75)

Equivalently, one has

Proof.

Put for all and for all . Then for all . The conclusion follows from Theorem 3.1.

Corollary 3.4.

Let be nonempty closed convex subset of a real Hilbert space , let be an infinitely many nonexpansive of into itself, and let be a -inverse-strongly monotone mapping of into such that . Let be a contraction of into itself with and let be a strongly positive linear bounded operator on with coefficient and . Let , , and be sequences generated by

(3.76)

where is the sequences generated by (1.23), and , , are three sequences in satisfying the following conditions:

(i) and

(ii) and

(iii)

(iv) for some and .

Then, converges strongly to a point which is the unique solution of the variational inequality

(3.77)

Equivalently, one has

Proof.

Put for all and for all in Theorem 3.1. Then, we have . So, by Theorem 3.1, we can conclude the desired conclusion easily.

If and in Theorem 3.1, then we can obtain the following result immediately.

Corollary 3.5.

Let be nonempty closed convex subset of a real Hilbert space , let be a bifunction from to satisfying (A1)–(A4), let be an infinitely many nonexpansive of into itself, and let be an -inverse-strongly monotone mapping of into such that . Let be a contraction of into itself with . Let , , and be sequences generated by

(3.78)

where is the sequences generated by (1.23), , , are three sequences in and is a real sequence in satisfying the following conditions:

(i)

(ii) and

(iii) and ;

(iv)

(v) and

(vi) for some and .

Then, and converge strongly to a point which is the unique solution of the variational inequality

(3.79)

Equivalently, one has

Corollary 3.6.

Let be nonempty closed convex subset of a real Hilbert space , let be a bifunction from to satisfying (A1)–(A4) and let be an infinite family of nonexpansive of into itself such that . Let be a contraction of into itself with . Let and be sequences generated by

(3.80)

where , , and are three sequences in , and is a real sequence in satisfying the following conditions:

(i)

(ii) and

(iii)

(iv) and

Then, and converge strongly to a point which is the unique solution of the variational inequality

(3.81)

Equivalently, one has

Proof.

Put and in Corollary 3.5. then . The conclusion of Corollary 3.6 can obtain the desired result easily.

4. Application for Optimization Problem

In this section, we shall utilize the results presented in the paper to study the following optimization problem:

(4.1)

where is a convex and lower semicontinuous functional defined on a closed subset of a Hilbert space . We denote by the set of solution of (4.1). Let be a bifunction from to defined by . We consider the following equilibrium problem, that is, to find such that

(4.2)

It is obvious that , where denotes the set of solution of equilibrium problem (4.2). In addition, it is easy to see that satisfies the conditions (A 1)–(A 4) in Section 1. Therefore, from the Corollary 3.6, we know the following iterative sequence defined by

(4.3)

where , , and are three sequences in and is a real sequence in satisfying the following conditions:

(i)

(ii) and

(iii)

(iv) and

Then, converges strongly to a point of optimization problem (4.1).

In special case, we pick for all and , , for all , then , and from (4.3) we obtain a special iterative scheme

(4.4)

Then, converges strongly to a solution of optimization problem (4.1). In fact, the is the minimum norm point on the .

Therefore, we consider a special from of optimization problem (4.1) which is as follows: (i.e., is taking )

(4.5)

In fact, the solution of optimization problem (4.4) is named the minimum norm point on the closed convex set . From iterative algorithm (4.4) we obtain the following iterative algorithm (4.5), and is defined by

(4.6)

for any initial guess . Then, converges strongly to a minimum norm point on the closed convex set .