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Abstract

We introduce an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of generalized equilibrium problems, the set of common fixed points of infinitely many nonexpansive mappings, and the set of solutions of the variational inequality for α-inverse-strongly monotone mappings in Hilbert spaces. We give some strong-convergence theorems under mild assumptions on parameters. The results presented in this paper improve and generalize the main result of Yao et al. (2007).

1. Introduction

Let be a nonempty closed convex subset of a real Hilbert space and let be a bifunction, where is the set of real numbers. Let be a nonlinear mapping. The generalized equilibrium problem (GEP) for and is to find such that (1.1) The set of solutions for the problem (1.1) is denoted by , that is, (1.2)

If in (1.1), then GEP(1.1) reduces to the classical equilibrium problem (EP) and is denoted by , that is, (1.3)

If in (1.1), then GEP(1.1) reduces to the classical variational inequality and is denoted by , that is, (1.4)

It is well known that GEP(1.1) contains as special cases, for instance, optimization problems, Nash equilibrium problems, complementarity problems, fixed point problems, and variational inequalities (see, e.g., [1–6] and the reference therein).

A mapping is called -inverse-strongly monotone [7], if there exists a positive real number such that (1.5) for all . It is obvious that any -inverse-strongly monotone mapping is monotone and Lipschitz continuous. A mapping is called nonexpansive if (1.6) for all . We denote by the set of fixed points of , that is, . If is bounded, closed and convex and is a nonexpansive mappings of into itself, then is nonempty (see [8]).

In 1997, Flåm and Antipin [9] introduced an iterative scheme of finding the best approximation to initial data when is nonempty and proved a strong convergence theorem. In 2003, Iusem and Sosa [10] presented some iterative algorithms for solving equilibrium problems in finite-dimensional spaces. They have also established the convergence of the algorithms. Recently, Huang et al. [11] studied the approximate method for solving the equilibrium problem and proved the strong convergence theorem for approximating the solutions of the equilibrium problem.

On the other hand, for finding an element of , Takahashi and Toyoda [12] introduced the following iterative scheme: (1.7) where is metric projection of onto is a sequence in and is a sequence in . Further, Iiduka and Takahashi [13] introduced the following iterative scheme: (1.8) where , and proved the strong convergence theorems for iterative scheme (1.8) under some conditions on parameters. In 2007, S. Takahashi and W. Takahashi [14] introduced an iterative scheme by the viscosity approximation method for finding a common element of the set of an equilibrium problem and the set of fixed points of a nonexpansive mapping in Hilbert spaces. They also proved a strong convergence theorem which is connected with Combettes and Hirstoaga's result [3] and Wittmann's result [15]. Tada and W. Takahashi [16] introduced the Mann type iterative algorithm for finding a common element of the set of solutions of the and the set of common fixed points of nonexpansive mapping and obtained the weak convergence of the Mann type iterative algorithm. Yao et al. [17] introduced an iteration process for finding a common element of the set of solutions of the and the set of common fixed points of infinitely many nonexpansive mappings in Hilbert spaces. They proved a strong-convergence theorem under mild assumptions on parameters. Very recently, Moudafi [18] proposed an iterative algorithm for finding a common element of , where is an -inverse-strongly monotone mapping, and obtained a weak convergence theorem. There are some related works, we refer to [19–22] and the references therein.

Inspired and motivated by the works mentioned above, in this paper, we introduce an iterative process for finding a common element of the set of common fixed points of infinitely many nonexpansive mappings, the set of solutions of GEP(1.1), and the solution set of the variational inequality problem for an -inverse-strongly monotone mapping in real Hilbert spaces. We give some strong-convergence theorems under mild assumptions on parameters. The results presented in this paper improve and generalize the main result of Yao et al. [17].

2. Preliminaries

Let be a real Hilbert space with inner product and norm , and let be a closed convex subset of . Then, for any , there exists a unique nearest point in , denoted by , such that (2.1) is called the metric projection of onto . It is well known that is a nonexpansive mapping and satisfies (2.2) for all . Furthermore, is characterized by the following properties: (2.3) for all and . It is easy to see that (2.4) where is a parameter in .

A set-valued mapping is called monotone if for all and imply . A monotone mapping is maximal if the graph of is not properly contained in the graph of any other monotone mappings. It is known that a monotone mapping is maximal if and only if for for all implies . Let be a monotone, -Lipschitz continuous mappings and let be the normal cone to at , that is, . Define (2.5) Then is the maximal monotone and if and only if ; see [23].

Let be a sequence of nonexpansive mappings of into itself and let be a sequence of nonnegative numbers in . For any , define a mapping of into itself as follows: (2.6) Such a mapping is called the -mapping generated by and see [24]. It is obvious that is nonexpansive and if then .

Lemma 2.1 (see [24]). Let be a nonempty closed convex subset of a real Hilbert space . Let be a sequence of nonexpansive mappings of into itself such that and let be a sequence in for some . Then, for every and , the limit exists.

Remark 2.2 (see [17]). It can be known from Lemma 2.1 that if is a nonempty bounded subset of , then for , there exists such that for all (2.7)

Using Lemma 2.1, one can define a mapping of into itself as follows: (2.8) for every . Such a mapping is called the -mapping generated by and Since is nonexpansive, is also nonexpansive. If is a bounded sequence in , then we put . Hence, it is clear from Remark 2.2 that for an arbitrary there exists such that for all (2.9) This implies that (2.10) Since and are nonexpansive, we deduce that, for each (2.11) for some constant .

Lemma 2.3 (see [24]). Let be a nonempty closed convex subset of a real Hilbert space . Let be a sequence of nonexpansive mappings of into itself such that and let be a sequence in for some . Then, .

For solving the generalized equilibrium problem, we assume that the bifunction satisfies the following conditions:

(a1) 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 [1].

Lemma 2.4 (see [1]). Let be a nonempty closed convex subset of and let be a bifunction from into satisfying (a1)–(a4). Let and . Then, there exists such that (2.12)

The following lemma was also given in [3].

Lemma 2.5 (see [3]). Assume that satisfies (a1)–(a4). For , define a mapping as follows: (2.13) for all . Then, the following hold: (b1) is single-valued; (b2) is firmly nonexpansive, that is, for any (b3) ; (b4) is closed and convex.

Remark 2.6. Replacing with in (2.12), then there exists such that (2.14)

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

Lemma 2.7 (see [25]). Let and be bounded sequences in Banach space and let be a sequence in . Suppose (2.15) for all integers . If (2.16) then .

Lemma 2.8 (see [26]). Assume is a sequence of nonnegative real numbers such that (2.17) where is a sequence in (0,1) and is a sequence in such that (1) ; (2) or . Then .

3. Main Results

In this section, we deal with an iterative scheme by the approximation method for finding a common element of the set of common fixed points of infinitely many nonexpansive mappings, the set of solutions of GEP(1.1), and the solution set of the variational inequality problem for an -inverse-strongly monotone mapping in real Hilbert spaces.

Theorem 3.1. Let be a nonempty closed convex subset of a real Hilbert space . Let be a bifunction from into satisfying (a1)–(a4), an inverse-strongly monotone mapping with constant an inverse-strongly monotone mapping with constant a contraction mapping with constant . Let be a -mapping generated by and and , where sequence is nonexpansive and is a sequence in for some . For , suppose that and are generated by (3.1) for all , where , and are three sequences in is a sequence in for some and for some satisfying (i) ; (ii) and ; (iii) ; (iv) and ; (v) and . Then , and converge strongly to the point , where .

Proof. For any and , we have (3.2) which implies that is nonexpansive. Remark 2.6 implies that the sequences and are well defined. In view of the iterative sequence (3.1), we have (3.3) It follows from Lemma 2.5 that for all . Let . For each , we have . By Lemma 2.5, (3.4) and so (3.2) implies that (3.5) For , we have from (2.4). Since is a nonexpansive mapping and is an inverse-strongly monotone mapping with constant , by (3.1), we have (3.6) Thus, (3.5) and (3.6) imply that (3.7) and so (3.8) This implies that is bounded. Therefore, and are also bounded.

From and , we have (3.9)(3.10) Putting in (3.9) and in (3.10), we get (3.11) Adding the above two inequalities, the monotonicity of implies that (3.12) and so (3.13) It follows from (3.2) that (3.14) and hence (3.15) From (3.1), (3.16) Putting (3.17) we have (3.18) Obviously, we get (3.19) From (2.11) and (3.16), we have (3.20) for some constant . Combining (3.15), (3.19), and (3.20), we deduce (3.21) It is easy to check that (3.22) and so (3.23) Thus, by Lemma 2.7, we obtain . It then follows that (3.24) By (3.15) and (3.16), we have (3.25) Since , we get (3.26) On the other hand, for (3.27) It follows that (3.28) It is easy to see that and hence .

From (3.5) and (3.6), we obtain (3.29) Since , without loss of generality, we may assume that there exists a real number such that for all . Therefore, we have (3.30) Since and is bounded, (3.30) implies that as . From (2.2), we have (3.31) and so (3.32) It follows that (3.33) which implies that (3.34) Since , and the sequences and are bounded, it follows from (3.34) that . On the other hand, from (3.5), we have (3.35) The same as in (3.30), we have as . Likewise, using (3.5), we find (3.36) The same as in (3.34), we have . Since (3.37) we get as . From (2.10) and (3.38) we get .

Next, we show , where . To show this inequality, we choose a subsequence of such that (3.39) Since is bounded, there exists a subsequence of which converges weakly to . Without loss of generality, we can assume that . From , we obtain . We now show that . Indeed, we observe that and (3.40) From (a2), we deduce that (3.41) and hence (3.42) Form , we get . Put for all and . Consequently, we get . From (3.42), it follows that (3.43) From the Lipschitz continuous of and , we obtain . Since is monotone, we know that . Further, . It follows from (a4) that (3.44) Owing to (a1) and (a4), we get that (3.45) and hence (3.46) Letting , we have (3.47) This implies that .

Furthermore, we prove that . Assume , since , we have . From Opial's theorem [27], we get (3.48) This is a contradiction. Hence, .

Now, we will show that . Let (3.49) Then is a maximal monotone [23]. Let , since and , we have . From , we have (3.50) This is, (3.51) Therefore, we obtain (3.52) Noting that and is Lipschitz continuous, we obtain (3.53) Since is maximal monotone, we have and so . Thus, . The property of the metric projection implies that (3.54) From (3.1) we obtain (3.55) which implies that (3.56) Setting (3.57) we have (3.58) Applying Lemma 2.8 to (3.56), we conclude that converges strongly to . Consequently, and converge strongly to . This completes the proof.

As direct consequences of Theorem 3.1, we have the following two corollaries.

Corollary 3.2. Let be a nonempty closed convex subset of a real Hilbert space . Let be a contraction mapping with Lipschitz constant and let be an inverse-strongly monotone mapping with constant . Suppose and generated by (3.59) for all , where are three sequences in is a sequence in for some satisfying conditions (i)–(iv). Then converges strongly to the point , where .

Proof. Let and for all and in Theorem 3.1. Then for Letting (the identity mapping) for all , then for It is easy to see that all conditions of Theorem 3.1 hold. Therefore, we know that the sequence generated by (3.59) converges strongly to . This completes the proof.

Remark 3.3. From Corollary 3.2, we can get an iterative scheme for finding the solution of the variational inequality involving the -inverse-strongly monotone mapping .

Corollary 3.4 (see [17, Theorem  3.5]). Let be a nonempty closed convex subset of a real Hilbert space . Let be a bifunction from into satisfying (a1)–(a4), a contraction mapping with constant . Let be an -mapping generated by and and , where sequence is nonexpansive and is a sequence in for some . Suppose and are generated by (3.60) for all , where are three sequences in , and is a sequence in satisfying conditions (i)–(iii) and (v). Then, the sequences and converge strongly to the point , where .

Proof. Let for and and for all in Theorem 3.1. Since , we get that . It follows from Theorem 3.1 that the sequences and converge strongly to the point . This completes the proof.

Remark 3.5. The main result of Yao et al. [17, Corollary  3.2] improved and extended the corresponding theorems in Combettes and Hirstoaga [3] and S. Takahashi and W. Takahashi [14]. Our Theorem 3.1 improves and extends Theorem  3.5 of Yao et al. [17] in the following aspects:(1)the equilibrium problem is extended to the generalized equilibrium problem;(2)our iterative process (3.1) is different from Yao et al. iterative process (3.60) because there are a project operator and an -inverse-strongly monotone mapping;(3)our iterative process (3.1) is more general than Yao et al. iterative process (3.60) because it can be applied to solving the problem of finding a common element of the set of solutions of generalized equilibrium problems, the set of common fixed points of infinitely many nonexpansive mappings, and the set of solutions of the variational inequality for -inverse-strongly monotone mapping.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (50674078, 50874096, 10671135, 70831005) and the Open Fund of State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation (Southwest Petroleum University).

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