A New Iterative Method for the Set of Solutions of Equilibrium Problems and of Operator Equations with Inverse-Strongly Monotone Mappings

Abstract

The purpose of the paper is to present a new iteration method for finding a common element for the set of solutions of equilibrium problems and of operator equations with a finite family of -inverse-strongly monotone mappings in Hilbert spaces.

1. Introduction

Let be a real Hilbert space with the inner product and the norm , respectively. Let be a nonempty closed convex subset of , and let be a bifunction from into . The equilibrium problem for is to find such that The set of solutions of (1) is denoted by .

Equilibrium problem (1) includes the numerous problems in physics, optimization, economics, transportation, and engineering, as special cases.

Assume that the bifunction satisfies the following standard properties.

Assumption A. Let be a bifunction satisfying the conditions (A1)–(A4):(A1),  ;(A2),  ;(A3)for each , is lower semicontinuous and convex;(A4), .

Let , , be a finite family of -strictly pseudocontractive mappings from into with the set of fixed points ; that is, Assume that

The problem of finding an element is studied intensively in [1–27].

Recall that a mapping in is said to be a -strictly pseudocontractive mapping in the terminology of Browder and Petryshyn [28] if there exists a constant such that for all , the domain of , where is the identity operator in . Clearly, if , then is nonexpansive; that is,

We know that the class of -strictly pseudocontractive mappings strictly includes the class of nonexpansive mappings.

In the case that , (4) is reduced to the equilibrium problem (1) and shown in [5, 23] to cover monotone inclusion problems, saddle point problems, variational inequality problems, minimization problems, Nash equilibria in noncooperative games, vector equilibrium problems, and certain fixed point problems (see also [29]). For finding approximative solutions of (1) there exist several methods: the regularization approach in [7, 9, 15, 24, 30, 31], the gap-function approach in [8, 15, 16, 18, 19], and the iterative procedure approach in [1–4, 6, 8, 11–14, 19–22, 32, 33].

In the case that and , (4) is a problem of finding a fixed point for a -strictly pseudocontractive mapping in and is given by Marino and Xu [17].

Theorem 1 (see [17]). Let be a nonempty closed convex subset of a real Hilbert space . Let be a -strictly pseudocontractive mapping for some , and assume that Let be the sequence generated by the following algorithm: Assume that the control sequence is chosen so that for all . Then converges strongly to , the projection of onto .

For the case that and , (4) is a problem of finding a common fixed point for a finite family of -strictly pseudocontractive mappings in and is studied in [27].

Let and , , and three sequences in satisfying for all , and let be a sequence in . Then the sequence generated by is called the implicit iteration process with mean errors for a finite family of strictly pseudocontractive mappings .

The scheme (9) can be expressed in the compact form as where .

Theorem 2 (see [27]). Let be a nonempty closed convex subset of a real Hilbert space . Let be a finite family of strictly pseudocontractive mappings of into itself such that Let and let be a bounded sequence in ; let , , and be three sequences in satisfying the following conditions:(i), ;(ii)there exist constants such that ,  ;(iii).
Then the implicit iterative sequence defined by (9) converges weakly to a common fixed point of the mappings . Moreover, if there exists such that is demicompact, then converges strongly.

If is an arbitrary bifunction satisfying Assumption A and , then (4) is a problem of finding a common element of the fixed point set for a -strictly pseudocontractive mapping in and of the solution set of equilibrium problem for (see [26]).

Theorem 3 (see [26]). Let be a nonempty closed convex subset of a real Hilbert space . Let be a bifunction from to satisfying Assumption A, and let be a nonexpansive mapping of into such that Let be a contraction of into itself and let and be sequences generated by and for all   N, where and satisfy Then, and converge strongly to , where

Set . Obviously, are -inverse-strongly monotone; that is,

From now on, let be a finite family of -inverse-strongly monotone mappings in with and , . On the other hand, if there exists such that , then is a contraction; that is, with . And hence, has only one solution and, consequently, the stated problem does not have sense. So, without loss of generality, assume that , .

Set where is the solution set of in .

Assume that .

Our problem is to find an element

Since the mapping is -inverse-strongly monotone for each nonexpansive mapping , the problem of finding an element , which is not only a solution of a variational inequality involving an inverse-strongly monotone mapping but also a fixed point of a nonexpansive mapping, is a particular case of (18).

For instance, the case that , where is some inverse-strongly monotone mapping and , is studied in [25].

Theorem 4 (see [25]). Let be a nonempty closed convex subset of a real Hilbert space . Let . Let be a -inverse-strongly monotone mapping of into , and let be a nonexpansive mapping of into itself such that where denotes the solution set of the following variational inequality: find such that Let be a sequence defined by for every , where for some and for some . Then, converges weakly to , where

The following theorem is an improvement of Theorem 4 for the case of nonself-mapping.

Theorem 5 (see [34]). Let be a nonempty closed convex subset of a real Hilbert space . Let be a -inverse-strongly monotone mapping of into , and let be a nonexpansive nonself-mapping of into such that Suppose that and is given by for every , where is a sequence in and is a sequence in . If and are chosen so that for some a, b with , then converges strongly to .

We know that -inverse-strongly monotone mapping is -Lipschitz continuous and monotone. Therefore, for the case that , where is not inverse-strongly monotone, but Lipschitz continuous and monotone, Nadezhkina and Takahashi [35] prove the following theorem.

Theorem 6 (see [35]). Let be a nonempty closed convex subset of a real Hilbert space . Let be a monotone and -Lipschitz continuous mapping of into , and let be a nonexpansive mapping of into itself such that Let , , and be sequences generated by for every , where for some and for some . Then the sequences , , and converge strongly to .

Some similar results are also considered in [36, 37].

Buong [38] introduced two new implicit iteration methods for solving problem (18).

We construct a regularization solution of the following single equilibrium problem: find such that where and is the positive sequence of regularization parameters that converges to , as .

The first one is the following theorem.

Theorem 7 (see [38]). For each , problem (28) has a unique solution such that(i), , ,  ;(ii) where is a positive constant.

Next, we introduce the second result. Let and be some sequences of positive numbers, and let and be two arbitrary elements in . Then, the sequence of iterations is defined by the following equilibrium problem: find such that

Theorem 8 (see [38]). Assume that the parameters , and are chosen such that(i), (ii), ,(iii), (iv).
Then, the sequence defined by (31) converges strongly to the element , as .

In this paper, we consider the new another iteration method: for an arbitrary element in , the sequence of iterations is defined by finding such that where is the metric projection of onto and and are sequences of positive numbers.

The strong convergence of the sequence defined by (32) is proved under some suitable conditions on and in the next section.

2. Main Results

We formulate the following lemmas for the proof of our main theorems.

Lemma 9 (see [9]). Let be a nonempty closed convex subset of a real Hilbert space and let be a bifunction of into satisfying Assumption A. Let and . Then, there exists such that

Lemma 10 (see [9]). Let be a nonempty closed convex subset of a real Hilbert space . Assume that satisfies Assumption A. For and , define a mapping as follows: Then, the following statements hold: (i) is single valued;(ii) is firmly nonexpansive; that is, for any , (iii);(iv) is closed and convex.

Lemma 11 (see [36]). Let , and be the sequences of positive numbers satisfying the following conditions: (i), (ii), , .
Then, .

Lemma 12 (see [38]). Let be any inverse-strongly monotone mapping from into with the solution set , and let be a closed convex subset of such that Then, the solution set of the following variational inequality is coincided with .

From Lemma 9, we can consider the firmly nonexpansive mapping defined by From Lemma 10, we know that is nonexpansive. Consequently, is -inverse-strongly monotone. Let Then, and problem (18) are equivalent to finding

Now, we construct a regularization solution for (40) by solving the following variational inequality problem: find such that where the positive regularization parameter , as .

Now we are in a position to introduce and prove the main results.

Theorem 13. Let be a nonempty closed convex subset of a real Hilbert space . Let be a bifunction from to satisfying Assumption A and let be a finite family of -inverse-strongly monotone mappings in with and , , such that where denotes the set of solutions for (1) and Then, for each , problem (41) has a unique solution such that (i), ,(ii), , (iii) where is some positive constant.

Proof. From Lemma 12, we know that is the set of solutions for the following variational inequality problem: find such that If we define the new bifunction by then problem (41) is the same as (28) with a new , and the proof for the theorem is a complete repetition of the proof for Theorem  2.1 in [38].
Set

Theorem 14. Let be a nonempty closed convex subset of a real Hilbert space . Let be a bifunction from to satisfying Assumption A and let be a finite family of -inverse-strongly monotone mappings in with and , , such that where denotes the set of solutions for (1) and Suppose that satisfy the following conditions: Then, the sequence defined by (32) converges strongly to ; that is,

Proof. Let be the solution of (41). Then, Set . Obviously, From the nonexpansivity of , the monotone and Lipschitz continuous properties of , , (41), (52), and , we have Thus, Therefore,
We note that, for , , , the inequality holds. Thus, applying inequality (57) for , we obtain
Set Then, it is not difficult to check that and satisfy the conditions in Lemma 11 for sufficiently large . Hence, . Since , we have This completes the proof.

Remark 15. The sequences , and with satisfy all the necessary conditions in Theorem 14.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Authors’ Contribution

The main idea of this paper was proposed by Jong Kyu Kim. Jong Kyu Kim and Nguyen Buong prepared the paper initially and performed all the steps of proof in this research. All authors read and approved the final paper.

Acknowledgment

This paper was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012R1A1A2042138).