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

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.


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.
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]). 2 The Scientific World Journal The equilibrium problem denoted by EP is to find ∈ such that ( , ) ≥ 0, ∀ ∈ . (10) The solution set of (10) is denoted by EP( ). Numerous problems in physics, optimization, and economics reduce to finding a solution of (10); see [4][5][6][7]. In 1997, Combettes and Hirstoaga [8] introduced an iterative scheme of finding the best approximation to the initial data when EP( ) is nonempty. In 2007, Plubtieng and Punpaeng [6] introduced an iterative method for finding the common element of the set ( ) ∩ EP( ).
The solution set of SEP (15)-(16) is denoted by Λ = { ∈ EP( 1 ): ∈ EP( 2 )}. 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][16][17][18][19][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 (0, 1). 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][18][19][21][22][23] and relevant references cited therein.

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 7 (see [21]). Let : → be -Lipschitzian mapping and let : → be a -Lipschitzian and -strongly monotone mapping; then for 0 ≤ < , − is −strongly monotone; that is, Lemma 8 (see [27]). Suppose that ∈ (0, 1) and > 0. Let : → be an -Lipschitzian and -strongly monotone operator. In association with nonexpansive mapping : → , define the mapping : → by Then is a contraction provided that < (2 / 2 ); that is, Lemma 9 (see [28]). Assume that { } is a sequence of nonnegative real numbers such that where { } is a sequence in (0, 1) and is a sequence such that Lemma 10 (see [29]). Let be a closed convex subset of . Let { } be a bounded sequence in . Assume that Then { } is weakly convergent to a point in .

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 1 and 2 be two real Hilbert spaces and let ⊆ 1 and ⊆ 2 be nonempty closed convex subsets of Hilbert spaces 1 and 2 , respectively. Let : 1 → 2 be a bounded linear operator. Assume that 1 : × → R 4 The Scientific World Journal and 2 : × → R are the bifunctions satisfying Assumption 3 and 2 is upper semicontinuous in first argument. Let , : → be a nonexpansive mapping such that Λ ∩ ( ) ̸ = 0. 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 0 ∈ arbitrarily, let the iterative sequences { }, { }, and { } be generated by where { } ⊂ (0, 2 ) and ∈ (0, 1/ ), is the spectral radius of the operator * , and * is the adjoint of . Suppose that the parameters satisfy 0 < < (2 / 2 ), 0 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 = , = , = = 1, 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.
This shows that Algorithm 11 is quite general and unifying. Proof. Let * ∈ Λ ∩ ( ); we have * = 1 ( * ) and * = 2 ( * ). 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 The Scientific World Journal 5 Denote = ( ) + ( − )( ( )). Next, we prove that the sequence { } is bounded; without loss of generality we can assume that ≤ for all ≥ 1. From (33), we have where the third inequality follows from Lemma 8.
Since 2 is upper semicontinuous in first argument, taking lim sup to above inequality as → ∞ and using Assumption 3(iv), we obtain which implies that ∈ EP( 2 ) and hence ∈ Λ. Thus we have ∈ Λ ∩ ( ) .
(65) 10 The Scientific World Journal Observe that the constants satisfy 0 ≤ < ] and ≥ ⇐⇒ 2 ≥ 2 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 ∈ Λ ∩ ( ).
Thus all the conditions of Lemma 9 are satisfied. Hence we deduce that → . This completes the proof.