A New Approximation Scheme Combining the Viscosity Method with Extragradient Method for Mixed Equilibrium Problems

Abstract

We introduce a new approximation scheme combining the viscosity method with extragradient method for finding a common element of the set of solutions of a mixed equilibrium problem and the set of fixed points of a finite family of nonexpansive mappings and the set of the variational inequality for a monotone, Lipschitz continuous mapping. We obtain a strong convergence theorem for the sequences generated by these processes in Hilbert spaces. Based on this result, we also get some new and interesting results. The results in this paper generalize, extend, and improve some well-known results in the literature.

1. Introduction

Let be a real Hilbert space with inner product and induced norm and let be a nonempty closed convex subset of . Let be a function and let be a bifunction from to , where is the set of real numbers. Ceng and Yao [1] and Bigi et al. [2] considered the following mixed equilibrium problem:

(1.1)

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

If , then the mixed equilibrium problem (1.1) becomes the following equilibrium problem:

(1.2)

The set of solutions of (1.2) is denoted by .

If for all , the mixed equilibrium problem (1.1) becomes the following minimization problem:

(1.3)

The set of solutions of (1.3) is denoted by .

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, [1–4].

Recall that a mapping of a closed convex subset into itself is nonexpansive [5] if there holds that

(1.4)

We denote the set of fixed points of by . Ceng and Yao [1] introduced an iterative scheme for finding a common element of the set of solutions of problem (1.1) and the set of common fixed points of a finite family of nonexpansive mappings in a Hilbert space and obtained a strong convergence theorem.

Some methods have been proposed to solve the problem (1.2); see, for instance, [3, 4, 6–12] and the references therein. Recently, Combettes and Hirstoaga [6] introduced an iterative scheme of finding the best approximation to the initial data when is nonempty and proved a strong convergence theorem. Takahashi and Takahashi [7] introduced an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of problem (1.2) and the set of fixed points of a nonexpansive mapping in a Hilbert space and proved a strong convergence theorem. Su et al. [8] introduced the following iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of problem (1.2) and the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality problem for an -inverse strongly monotone mapping in a Hilbert space. Starting with an arbitrary , define sequences and by

(1.5)

They proved that under certain appropriate conditions imposed on , , and , the sequences and generated by (1.5) converge strongly to , where . Tada and Takahashi [9] introduced two iterative schemes for finding a common element of the set of solutions of problem (1.2) and the set of fixed points of a nonexpansive mapping in a Hilbert space and obtained both strong convergence theorem and weak convergence theorem.

On the other hand, for solving the variational inequality problem in the finite-dimensional Euclidean , Korpelevich [13] introduced the following so-called extragradient method:

(1.6)

for every where . She showed that if is nonempty, then the sequences and , generated by (1.6), converge to the same point . The idea of the extragradient iterative process introduced by Korpelevich was successfully generalized and extended not only in Euclidean but also in Hilbert and Banach spaces; see, for example, the recent papers of He et al. [14], Gárciga Otero and Iuzem [15], and Solodov and Svaiter [16], Solodov [17]. Moreover, Zeng and Yao [18] and Nadezhkina and Takahashi [19] introduced iterative processes based on the extragradient method for finding the common element of the set of fixed points of nonexpansive mappings and the set of solutions of variational inequality problem for a monotone, Lipschitz continuous mapping. Yao and Yao [20] introduced an iterative process based on the extragradient method for finding the common element of the set of fixed points of nonexpansive mappings and the set of solutions of variational inequality problem for an -inverse strongly monotone mapping. Plubtieng and Punpaeng [11] introduced an iterative process based on the extragradient method for finding the common element of the set of fixed points of nonexpansive mappings, the set of solutions of an equilibrium problem, and the set of solutions of variational inequality problem for -inverse strongly monotone mappings. Chang et al. [12] introduced some iterative processes based on the extragradient method for finding the common element of the set of fixed points of a infinite family of nonexpansive mappings, the set of solutions of an equilibrium problem, and the set of solutions of variational inequality problem for an -inverse strongly monotone mapping. Peng et al. [21] introduced a new approximation scheme combining the viscosity method with parallel method for finding a common element of the set of solutions of a generalized equilibrium problem and the set of fixed points of a finite family of strict pseudocontractions and obtain a strong convergence theorem for the sequences generated by these processes in Hilbert spaces.

In the present paper, we introduce a new approximation scheme combining the viscosity method with extragradient method for finding a common element of the set of solutions of a mixed equilibrium problem, the set of fixed points of a finite family of nonexpansive mappings, and the set of solutions of the variational inequality for a monotone, Lipschitz continuous mapping. We obtain a strong convergence theorem for the sequences generated by these processes. Based on this result, we also get some new and interesting results. The results in this paper generalize and improve some well-known results in the literature.

2. Preliminaries

Let be a real Hilbert space with inner product and norm . Let be a nonempty closed convex subset of . Let symbols and denote strong and weak convergence, respectively. In a real Hilbert space , it is well known that

(2.1)

for all and .

For any , there exists a unique nearest point in , denoted by , such that for all . The mapping is called the metric projection of onto . We know that is a nonexpansive mapping from onto . It is also known that and

(2.2)

for all and .

It is easy to see that (2.2) is equivalent to

(2.3)

for all and .

A mapping of into is called monotone if

(2.4)

for all . A mapping of into is called -inverse strongly monotone if there exists a positive real number such that

(2.5)

for all . A mapping is called -Lipschitz continuous if there exists a positive real number such that

(2.6)

for all . It is easy to see that if is -inverse strongly monotone mappings, then is monotone and Lipschitz continuous. The converse is not true in general. The class of -inverse strongly monotone mappings does not contain some important classes of mappings even in a finite-dimensional case. For example, if the matrix in the corresponding linear complementarity problem is positively semidefinite, but not positively definite, then the mapping is monotone and Lipschitz continuous, but not -inverse strongly monotone.

Let be a monotone mapping of into . In the context of the variational inequality problem the characterization of projection (2.2) implies the following:

(2.7)

It is also known that satisfies Opial's condition [22], that is, for any sequence with , the inequality

(2.8)

holds for every with .

A set-valued mapping is called monotone if for all , and imply . A monotone mapping is maximal if its graph 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 A be a monotone, -Lipschitz continuous mapping of into and let be normal cone to at , that is, . Define

(2.9)

Then is maximal monotone and if and only if (see [23]).

For solving the mixed equilibrium problem, let us give the following assumptions for the bifunction , and the set :

(A1) for all ;

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

(A3) for each , is weakly upper semicontinuous;

(A4) for each , is convex;

(A5) for each , is lower semicontinuous;

(B1) for each and , there exist a bounded subset and such that for any ,

(2.10)

(B2) is a bounded set;

(B3) for each and , there exist a bounded subset and such that for any ,

(2.11)

(B4) for each and , there exist a bounded subset and such that for any ,

(2.12)

We will use the following results in the sequel.

Lemma 2.1 (see [21, 24]).

Let be a nonempty closed convex subset of . Let be a bifunction from to satisfying (A1)–(A5) and let be a proper lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. For and define a mapping as follows:

(2.13)

for all . Then the following conclusions hold:

(1)for each , ;

(2) is single-valued;

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

(2.14)

(4)

(5) is closed and convex.

Lemma 2.2 (see [25, 26]).

Assume that is a sequence of nonnegative real numbers such that

(2.15)

where is a sequence in and is a sequence such that

(i)

(ii) or

Then,

Lemma 2.3.

In a real Hilbert space , there holds the following inequality:

(2.16)

for all

Lemma 2.4 (see [21]).

Let and be bounded sequences in a Banach space, and let be a sequence of real numbers such that for all Suppose that for all and . Then, .

Let be a finite family of nonexpansive mappings of into itself and let be real numbers such that for every . We define a mapping of into itself as follows:

(2.17)

Such a mapping is called the -mapping generated by and . It is easy to see that nonexpansivity of each ensures the nonexpansivity of The concept of -mappings was introduced in [27, 28]. It is now one of the main tools in studying convergence of iterative methods for approaching a common fixed point of nonlinear mappings; more recent progresses can be found in [10, 29, 30] and the references cited therein.

Lemma 2.5 (see [29]).

Let be a nonempty closed convex set of a strictly convex Banach space. Let be nonexpansive mappings of into itself such that and let be real numbers such that for every and . Let be the -mapping generated by and . Then .

Lemma 2.6 (see [10]).

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

(2.18)

3. Strong Convergence Theorems

In this section, we show a strong convergence of an iterative algorithm based on both viscosity approximation method and extragradient method 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 finite family of nonexpansive mappings, and the set of solutions of the variational inequality for a monotone, Lipschitz continuous mapping in a 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)–(A5) and let be a proper lower semicontinuous and convex function. Let be a monotone and -Lipschitz continuous mapping of into . Let be a finite family of nonexpansive mappings of into such that . Let be sequences in with . Let be the -mapping generated by and . Assume that either (B1) or (B2) holds. Let be a contraction of into itself and let , , and be sequences generated by

(3.1)

for every where , , , , , and are sequences of numbers satisfying the conditions:

(C1) and ;

(C2);

(C3);

(C4) and ;

(C5) for all .

Then, , , and converge strongly to .

Proof.

We show that is a contraction of into itself. In fact, there exists such that for all . So, we have

(3.2)

for all Since is complete, there exists a unique element such that .

Put for every Let and let be a sequence of mappings defined as in Lemma 2.1. Then . From , we have

(3.3)

From (2.3), the monotonicity of , and , we have

(3.4)

Further, Since and is -Lipschitz continuous, we have

(3.5)

So, it follows from (C3) that the following inequality holds for , where is a positive integer:

(3.6)

Put It is obvious that Suppose By Lemma 2.5, we know that is nonexpansive and . From (3.3), (3.6) and , we have and

(3.7)

for every . Therefore, is bounded. From (3.3) and (3.6), we also obtain that and are bounded.

From and the monotonicity and the Lipschitz continuity of , we have

(3.8)

Hence, we obtain that is bounded. It follows from the Lipschitz continuity of that , , and are bounded. Since and are nonexpansive, we know that and are also bounded. From the definition of , we get

(3.9)

where is an approximate constant such that

(3.10)

On the other hand, from and , we have

(3.11)

(3.12)

Putting in (3.11) and in (3.12), we have

(3.13)

So, from the monotonicity of , we get

(3.14)

and hence

(3.15)

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

(3.16)

and hence

(3.17)

where

It follows from (3.9) and (3.7) that

(3.18)

Define a sequence such that

(3.19)

Then, we have

(3.20)

Next we estimate . It follows from the definition of that

(3.21)

where is an approximate constant such that

(3.22)

Since for all and , we have

(3.23)

It follows that

(3.24)

Substituting (3.24) into (3.21) yields that

(3.25)

Hence, we have

(3.26)

From (3.20), (3.26), and (3.18), we have

(3.27)

It follows from (C1)–(C5) that

(3.28)

Hence by Lemma 2.4, we have . Consequently

(3.29)

Since , we have

(3.30)

and thus

(3.31)

It follows from (C1) and (C2) that .

Since for , it follows from (3.3) and (3.6) that

(3.32)

from which it follows that

(3.33)

It follows from (C1)–(C3) and that .

By the same argument as in (3.6), we also have

(3.34)

Combining the above inequality and (3.32), we have

(3.35)

and thus

(3.36)

which implies that .

From we also have . As is -Lipschitz continuous, we have .

For , we have, from Lemma 2.1,

(3.37)

Hence,

(3.38)

By(3.3), (3.6), (3.32), and (3.38), we have

(3.39)

Hence,

(3.40)

It follows from (C1), (C2), and that .

Since

(3.41)

It follows that

(3.42)

Next we show that

(3.43)

where . To show this inequality, we can choose a subsequence of such that

(3.44)

Since is bounded, there exists a subsequence of which converges weakly to . Without loss of generality, we can assume that From , we obtain that . From , we also obtain that . From , we also obtain that . Since and is closed and convex, we obtain .

In order to show that , we first show By we know that

(3.45)

It follows from (A2) that

(3.46)

Hence,

(3.47)

It follows from (A4), (A5), and the weakly lower semicontinuity of , and that

(3.48)

For with and , let Since and , we obtain and hence . So by (A4) and the convexity of , we have

(3.49)

Dividing by , we get

(3.50)

Letting , it follows from (A3) and the weakly lower semicontinuity of that

(3.51)

for all and hence .

Now we show that Put

(3.52)

where is the normal cone to at . We have already mentioned that in this case the mapping is maximal monotone, and if and only if . Let . Then and hence . So, we have for all . On the other hand, from and we have

(3.53)

and hence

(3.54)

Therefore, we have

(3.55)

Hence we obtain as . Since is maximal monotone, we have and hence .

We next show that To see this, we observe that we may assume (by passing to a further subsequence if necessary) for Let be the -mapping generated by and . By Lemma 2.5, we know that is nonexpansive and . it follows from Lemma 2.6 that

(3.56)

Assume Since and , it follows from the Opial condition, (3.42), and (3.56) that

(3.57)

which is a contradiction. Hence, we have . This implies Therefore, we have

(3.58)

Finally, we show that , where .

From Lemma 2.3, we have

(3.59)

and thus

(3.60)

It follows from Lemma 2.2, (3.58), and (3.60) that . From and , we have and . The proof is now complete.

4. Applications

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

Theorem 4.1.

Let be a nonempty closed convex subset of a real Hilbert space . Let be a bifunction from to satisfying (A1)–(A5) and let be a proper lower semicontinuous and convex function. Let be a finite family of nonexpansive mappings of into such that . Let be sequences in with . Let be the -mapping generated by and . Assume that either (B1) or (B2) holds. Let be a contraction of into itself and let , , and be sequences generated by

(4.1)

for every where , , , , , and are sequences of numbers satisfying the following conditions:

(C1) and ;

(C2);

(C4) and ;

(C5) for all .

Then, , , and converge strongly to .

Proof.

Putting , by Theorem 3.1 we obtain the desired result.

Theorem 4.2.

Let be a nonempty closed convex subset of a real Hilbert space . Let be a bifunction from to satisfying (A1)–(A5). Let be a monotone and -Lipschitz continuous mapping of into . Let be a finite family of nonexpansive mappings of into such that . Let be sequences in with . Let be the -mapping generated by and . Assume that either (B4) or (B2) holds. Let be a contraction of into itself and let , , and be sequences generated by

(4.2)

for every where , , , , , , and are sequences of numbers satisfying the following conditions:

(C1) and ;

(C2);

(C3);

(C4) and ;

(C5) for all .

Then, , , and converge strongly to .

Proof.

Putting , by Theorem 3.1 we obtain the desired result.

Theorem 4.3.

Let be a nonempty closed convex subset of a real Hilbert space . Let be a proper lower semicontinuous and convex function. Let be a monotone and -Lipschitz continuous mapping of into . Let be a finite family of nonexpansive mappings of into such that . Let be sequences in with . Let be the -mapping generated by and . Assume that either (B3) or (B2) holds. Let be a contraction of into itself and let , , and be sequences generated by

(4.3)

for every . where , , , , , , and are sequences of numbers satisfying the following conditions:

(C1) and ;

(C2);

(C3);

(C4) and ;

(C5) for all .

Then, , , and converge strongly to .

Proof.

Let for all , by Theorem 3.1 we obtain the desired result.

Theorem 4.4.

Let be a nonempty closed convex subset of a real Hilbert space . Let be a bifunction from to satisfying (A1)–(A5) and let be a proper lower semicontinuous and convex function. Let be a monotone and -Lipschitz continuous mapping of into . Let be a nonexpansive mapping of into such that . Assume that either (B1) or (B2) holds. Let be a contraction of into itself and let , , and be sequences generated by

(4.4)

for every where , , and are sequences of numbers satisfying the following conditions:

(C1) and ;

(C2);

(C3);

(C4) and .

Then, , , and converge strongly to .

Proof.

Let , by Theorem 3.1 we obtain the desired result.

Theorem 4.5.

Let be a nonempty closed convex subset of a real Hilbert space . Let be a monotone and -Lipschitz continuous mapping of into . Let be a finite family of nonexpansive mappings of into such that . Let be sequences in with . Let be the -mapping generated by and . Let and be sequences generated by

(4.5)

for every where , , , , , and are sequences of numbers satisfying the following conditions:

(C1) and ;

(C2);

(C3);

(C5) for all .

Then, and converge strongly to .

Proof.

Let and let for all . Then . By Theorem 3.1 we obtain the desired result.

Remark 4.6.

• (1) Since the -inverse-strongly-monotonicity of has been weakened by the monotonicity and Lipschitz continuity of . Theorems 3.1, 4.2, and 4.4 generalize and improve Theorem 3.1 in [11], Theorem 3.1 in [12], and Theorem 3.1 in [8] and the main results in [31]. Theorem 4.5 improves Theorem 3.1 in [20].

• (2) It is easy to see that Theorems 3.1, 4.2, and 4.4 also generalize and improve Theorems 3.1, and 4.2 in [9].

• (3) It is clear that Theorem 4.5 generalizes, extends, and improves Theorem 3.1 in [18] and Theorem 3.1 in [19].

• (4) Theorem 3.1 improves and extends Theorem 3.1 in [1].