Strong Convergence Algorithm for Split Equilibrium Problems and Hierarchical Fixed Point Problems

Abstract

The purpose of this paper is to investigate the problem of finding the approximate element of the common set of solutions of a split equilibrium problem and a hierarchical fixed point problem in a real Hilbert space. We establish the strong convergence of the proposed method under some mild conditions. Several special cases are also discussed. Our main result extends and improves some well-known results in the literature.

1. Introduction

Let be a real Hilbert space, whose inner product and norm are denoted by and . Let be a nonempty closed convex subset of . We introduce the following definitions which are useful in the following analysis.

Definition 1. The mapping is said to be (a)monotone, if (b)strongly monotone, if there exists such that (c)-inverse strongly monotone, if there exists such that (d)nonexpansive, if (e)-Lipschitz continuous, if there exists a constant such that (f)contraction on , if there exists a constant such that It is easy to observe that every -inverse strongly monotone is monotone and Lipschitz continuous. It is well known that every nonexpansive operator satisfies, for all , the inequality and therefore, we get, for all , See, for example, [1, Theorem 1], and [2, Theorem 3].

The fixed point problem for the mapping is to find such that We denote by the set of solutions of (9). It is well known that is closed and convex and is well defined (see [3]).

The equilibrium problem denoted by EP is to find such that The solution set of (10) is denoted by . Numerous problems in physics, optimization, and economics reduce to finding a solution of (10); see [4–7]. In 1997, Combettes and Hirstoaga [8] introduced an iterative scheme of finding the best approximation to the initial data when is nonempty. In 2007, Plubtieng and Punpaeng [6] introduced an iterative method for finding the common element of the set .

Recently, Censor et al. [9] introduced a new variational inequality problem which we call the split variational inequality problem (SVIP). Let and be two real Hilbert spaces. Given operators and , a bounded linear operator , and nonempty, closed, and convex subsets and , the SVIP is formulated as follows: find a point such that and such that In [10], Moudafi introduced an iterative method which can be regarded as an extension of the method given by Censor et al. [9] for the following split monotone variational inclusions: and such that where is a set-valued mapping for . Later Byrne et al. [11] generalized and extended the work of Censor et al. [9] and Moudafi [10].

Very recently, Kazmi and Rizvi [12] studied the following pair of equilibrium problems called split equilibrium problem: let and be nonlinear bifunctions and let be a bounded linear operator; then, the split equilibrium problem (SEP) is to find such that and such that The solution set of SEP (15)-(16) is denoted by .

Let be a nonexpansive mapping. The following problem is called a hierarchical fixed point problem: find such that It is known that the hierarchical fixed point problem (17) links with some monotone variational inequalities and convex programming problems; see [13, 14]. Various methods [15–20] have been proposed to solve the hierarchical fixed point problem. In 2010, Yao et al. [14] introduced the following strong convergence iterative algorithm to solve the problem (17): where is a contraction mapping and and are two sequences in . Under some certain restrictions on parameters, Yao et al. proved that the sequence generated by (18) converges strongly to , which is the unique solution of the following variational inequality: In 2011, Ceng et al. [21] investigated the following iterative method: where is a Lipschitzian mapping and is a Lipschitzian and strongly monotone mapping. They proved that under some approximate assumptions on the operators and parameters, the sequence generated by (20) converges strongly to the unique solution of the variational inequality In the present paper, inspired by the above cited works and by the recent works going in this direction, we give an iterative method for finding the approximate element of the common set of solutions of (15)-(16) and (17) in real Hilbert space. Strong convergence of the iterative algorithm is obtained in the framework of Hilbert space. We would like to mention that our proposed method is quite general and flexible and includes many known results for solving split equilibrium problems and hierarchical fixed point problems; see, for example, [13, 14, 17–19, 21–23] and relevant references cited therein.

2. Preliminaries

In this section, we recall some basic definitions and properties, which will be frequently used in our later analysis. Some useful results proved already in the literature are also summarized. The first lemma provides some basic properties of projection onto .

Lemma 2. Let denote the projection of onto . Then, one has the following inequalities:

Assumption 3 (see [24]). Let be a bifunction satisfying the following assumptions: (i), for all ;(ii) is monotone; that is, , for all ;(iii)for each , ;(iv)for each , is convex and lower semicontinuous;(v)for fixed and , there exists a bounded subset of and such that

Lemma 4 (see [8]). Assume that satisfies Assumption 3. For and for all , define a mapping as follows: Then the following hold: (i) is nonempty and single-valued;(ii) is firmly nonexpansive; that is, (iii);(iv) is closed and convex.

Assume that satisfies Assumption 3. For and for all , define a mapping as follows: Then satisfies conditions (i)–(iv) of Lemma 4. Consider , where is the solution set of the following equilibrium problem:

Lemma 5 (see [25]). Assume that satisfies Assumption 3, and let be defined as in Lemma 4. Let and . Then

Lemma 6 (see [26]). Let be a nonempty closed convex subset of a real Hilbert space . If   is a nonexpansive mapping with , then the mapping is demiclosed at 0; that is, if is a sequence in weakly converging to and if converges strongly to 0, then .

Lemma 7 (see [21]). Let be -Lipschitzian mapping and let be a -Lipschitzian and -strongly monotone mapping; then for , is -strongly monotone; that is,

Lemma 8 (see [27]). Suppose that and . Let be an -Lipschitzian and -strongly monotone operator. In association with nonexpansive mapping , define the mapping by Then is a contraction provided that ; that is, where .

Lemma 9 (see [28]). Assume that is a sequence of nonnegative real numbers such that where is a sequence in and is a sequence such that(1);(2) or .Then .

Lemma 10 (see [29]). Let be a closed convex subset of . Let be a bounded sequence in . Assume that (i)the weak -limit set where ;(ii)for each , exists. Then is weakly convergent to a point in .

3. The Proposed Method and Some Properties

In this section, we suggest and analyze our method and we prove a strong convergence theorem for finding the common solutions of the split equilibrium problem (15)-(16) and the hierarchical fixed point problem (17).

Let and be two real Hilbert spaces and let and be nonempty closed convex subsets of Hilbert spaces and , respectively. Let be a bounded linear operator. Assume that and are the bifunctions satisfying Assumption 3 and is upper semicontinuous in first argument. Let be a nonexpansive mapping such that . Let be an -Lipschitzian mapping and -strongly monotone and let be -Lipschitzian mapping. Now we introduce the proposed method as follows.

Algorithm 11. For a given arbitrarily, let the iterative sequences , , and be generated by where and , is the spectral radius of the operator , and is the adjoint of . Suppose that the parameters satisfy , , where . And and are sequences in satisfying the following conditions: (a) and ;(b);(c) and ;(d) and .

Remark 12. Our method can be viewed as extension and improvement for some well-known results as follows (i)The proposed method is an extension and improvement of the method of Wang and Xu [23] for finding the approximate element of the common set of solutions of a split equilibrium problem and a hierarchical fixed point problem in a real Hilbert space.(ii)If the Lipschitzian mapping , , , we obtain an extension and improvement of the method of Yao et al. [14] for finding the approximate element of the common set of solutions of a split equilibrium problem and a hierarchical fixed point problem in a real Hilbert space.(iii)The contractive mapping with a coefficient in other papers (see [14, 19, 22, 27]) is extended to the cases of the Lipschitzian mapping with a coefficient constant . This shows that Algorithm 11 is quite general and unifying.

Lemma 13. Let . Then ,  , and are bounded.

Proof. Let ; we have and . Then From the definition of , it follows that It follows from (8) that Applying (36) and (35) to (34) and from the definition of , we get Denote . Next, we prove that the sequence is bounded; without loss of generality we can assume that for all . From (33), we have where the third inequality follows from Lemma 8.
By induction on , we obtain , for and . Hence is bounded and, consequently, we deduce that , , , , , and are bounded.

Lemma 14. Let and the sequence generated by the Algorithm 11. Then one has (a);(b)the weak -limit set , ().

Proof. Since and   it follows from Lemma 5 that where and . Without loss of generality, let us assume that there exists a real number such that , for all positive integers . Then we get From (33) and the above inequality, we get Next, we estimate where the second inequality follows from Lemma 8. From (41) and (42), we have where It follows from conditions (a)–(d) of Algorithm 11 and Lemma 9 that Next, we show that . Since by using (34) and (37), we obtain where the last inequality follows from (37), which implies that Then from the above inequality, we get Since , , , and , we obtain Since is firmly nonexpansive, we have where the last inequality follows from (34) and (37). Hence, we get From (46) and the above inequality, we have which implies that Hence Since , , , and , we obtain Now, let ; since , we have Since ,  ,  ,  , and are bounded and , we obtain Since is bounded, without loss of generality, we can assume that . It follows from Lemma 6 that . Therefore .

Theorem 15. The sequence generated by Algorithm 11 converges strongly to , which is the unique solution of the variational inequality

Proof. Since is bounded and from Lemma 14, we have . Next, we show that . Since , we have It follows from monotonicity of that Since , , and , it easy to observe that . It follows by Assumption 3(iv) that , for all .
For any and , let ; we have . Then, from Assumptions 3((i) and (iv)), we have Therefore . From Assumption 3(iii), we have , which implies that .
Next, we show that . Since is bounded and , there exists a subsequence of   such that and since is a bounded linear operator, . Now set . It follows from (49) that and . Therefore from the definition of , we have Since is upper semicontinuous in first argument, taking lim sup to above inequality as and using Assumption 3(iv), we obtain which implies that and hence .
Thus we have Observe that the constants satisfy and Therefore from Lemma 7, the operator is strongly monotone, and we get the uniqueness of the solution of the variational inequality (58) and denote it by .
Next, we claim that . Since is bounded, there exists a subsequence of such that Next, we show that . Consider which implies that Let and + .
Since It follows that Thus all the conditions of Lemma 9 are satisfied. Hence we deduce that . This completes the proof.

Remark 16. In hierarchical fixed point problem (17), if , then we can get the variational inequality (58). In (58), if , then we get the variational inequality , for all , which is just the variational inequality studied by Suzuki [27] extending the common set of solutions of a system of variational inequalities, a split equilibrium problem, and a hierarchical fixed point problem.

4. Conclusions

In this paper, we suggest and analyze an iterative method for finding the approximate element of the common set of solutions of (15)-(16) and (17) in real Hilbert space, which can be viewed as a refinement and improvement of some existing methods for solving a split equilibrium problem and a hierarchical fixed point problem. Some existing methods (e.g., [13, 14, 17–19, 21–23]) can be viewed as special cases of Algorithm 11. Therefore, the new algorithm is expected to be widely applicable.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The author would like to thank Professor Omar Halli, Rector, Ibn Zohr University, for providing excellent research facilities.