An iterative algorithm for the system of split mixed equilibrium problem

Abstract In this article, a new problem that is called system of split mixed equilibrium problems is introduced. This problem is more general than many other equilibrium problems such as problems of system of equilibrium, system of split equilibrium, split mixed equilibrium, and system of split variational inequality. A new iterative algorithm is proposed, and it is shown that it satisfies the weak convergence conditions for nonexpansive mappings in real Hilbert spaces. Also, an application to system of split variational inequality problems and a numeric example are given to show the efficiency of the results. Finally, we compare its rate of convergence other algorithms and show that the proposed method converges faster.


Introduction
Let H be a real Hilbert space and C be a nonempty, closed, and convex subset of H. For = … i N 1, 2, , , let × → F C C : i be a family of bifunctions such that ( ) = F x x , 0 i for = … i N 1, 2, , . We define the following problems: 1. The equilibrium problem is to find ∈ * x C such that ( ) ≥ * F x x , 0 1 for all ∈ x C. 2. The system of equilibrium problems is to find ∈ * Although the theory of equilibrium problems was first introduced by Fan [1] in 1972, the most significant contributions to this problem were made by Blum and Oettli [2] and Noor and Oettli [3] in 1994. The equilibrium problem has a great impact on the development of several branches of pure and applied sciences, and it provides a natural and unified framework for solving several problems arising in physics, engineering, economics, game theory, image reconstruction, transportation, network, and elasticity. It can also be reformulated in the form of different mathematical problems such as an optimization problem, a convex feasibility problem (see [4]), a variational inequality problem (see [5]), a minimization problem (see [6]), a minimax inequality problem, a fixed point problem, a complementarity problem, a saddle point problem, or a Nash equilibrium problem in noncooperative games (see [2]). Therefore, it is natural to extend such a problem to more general problems in several ways. The system of equilibrium problems and mixed equilibrium problems was introduced and studied by some authors in, for instance, [15][16][17][18][19].
Recently, Moudafi [7] introduced a split equilibrium problem which is a generalization of several optimization problems such as split feasibility problem, split inclusion problem, split variational inequality problem, and split common fixed point problem, see, e.g., [8][9][10][11][12][13][14]. By combining the ideas of split equilibrium problem with the system of equilibrium problems, in 2016, the system of split equilibrium problems and mixed equilibrium problems was introduced by Ugwunnadi and Ali [20] and Onjai-uea and Phuengrattana [21], respectively, see also [22][23][24]. These problems are defined as follows: 1. The split equilibrium problem is to find ∈ * x C such that ( ) ≥ * F x x , 0 1 , for all ∈ x C, and such that = ∈ * * y Ax Q solves ( ) ≥ * G y y , 0 1 for all ∈ y Q; 2. The system of split equilibrium problems is to find ∈ * In recent years, many authors have made several efforts to develop implementable iterative methods for solving all these problems. In 2016, Suantai et al. [10] considered the split equilibrium problem and proposed the following iterative algorithm to find a common solution of fixed point problem for a nonspreading multivalued mapping and the split equilibrium problem: They proved a weak convergence theorem for the iterative sequence. In the same year, Ugwunnadi and Ali [20] established the following algorithm to solve the system of split equilibrium problems and showed that the sequence generated by their algorithm converges strongly to the common solution of considered problem and fixed point problem for a finite family of continuous pseudocontractive mappings.  , and One year later, Onjai-uea and Phuengrattana [21] proposed another iterative algorithm to find a solution for the split mixed equilibrium problem for λ-hybrid multivalued mappings. They proved that the sequence generated by the following iterative algorithm converges weakly to a common solution of fixed point problem and split mixed equilibrium problem.  Motivated and inspired by these problems and iterative methods, we introduce a new problem called system of split mixed equilibrium problems, which generalizes all these problems stated above and propose a new iterative algorithm to find a common solution of fixed point problem and system of split mixed equilibrium problems. We prove that sequence generated by our algorithm converges weakly to the solution. Also, we give some corollaries and numeric results to show that our results generalize and extend many results in the literature.

Preliminaries
Throughout this article, we use and to represent the set of natural and real numbers, respectively, "→" for strong convergence of a sequence and " ⇀ " for the weak convergence. Let C and Q be nonempty closed convex subsets of real We need the following assumptions to solve a mixed equilibrium problem for a bifunction × → F C C : and a mapping φ: , is convex and lower semicontinuous, (A5) for each ∈ x C, ∈ ( ] λ 0, 1 , and > r 0, there exist a bounded subset ⊆ D C and ∈ a C such that for any ∈ z C D \ , (A6) C is a bounded set.
Assume that either (A5) or (A6) holds. Then: and it is closed and convex.

Main results
First, we introduce the system of split mixed equilibrium problems in the following form: The solution set of system of split mixed equilibrium problems (3.1) and (3.2) is denoted by : MEP , and MEP , , In Definition 1, if 1. = N 1, then the system of split mixed equilibrium problems is reduced to the split mixed equilibrium problem studied in, e.g., [21]. 2. = = φ ϕ 0, then the system of split mixed equilibrium problems is reduced to the system of split equilibrium problems studied in, e.g., [20].

4.
i , then the system of split mixed equilibrium problems is reduced to the system of mixed equilibrium problems.
, then the system of split mixed equilibrium problems is reduced to the system of equilibrium problems studied in, e.g., [16,17].
, be nonempty closed convex subsets of real Hilbert spaces H 1 and H 2 , respectively, → A H H : 1 2 be a bounded linear operator and → S C C : a nonexpansive mapping, where proper lower semicontinuous and convex functions such Assume that the following conditions hold: Then the sequence { } x n generated by (3.3) converges weakly to ∈ p Γ.
Proof. We divide our proof into six steps.
Step 1. In the first step, we show that ( − ) *

A I T Ax A I T Ay A I T A x y A I T A x y I T A x y AA I T A x y L I T A x y I T A x y L I T A x y L A x y I T A x y L x y A I T Ax A I T Ay
is a nonexpansive mapping.
Step 2. In the second step, we show that sequences and (3.5) Using (3.4) and (3.5), we obtain Step 3. In this step, we show that∥ − ∥ → u x 0 n n . For this, we need to show∥ − . Also, we know that the mapping J i is nonexpansive mapping and ∈ q Γ is a fixed point of J i . Thus, we have 1 .  Also, since T r F n N is firmly nonexpansive, we get  Similarly, we have   This implies that Hence, we obtain (3.14) Since,  we get, Step 5. In this step, we show that  for all ∈ y C 1 . It follows from weakly convergence of u ni to p, Condition (iv), (3.9), (3.14) and the proper lower semicontinuity of φ 1 that 0, 1 and ∈ y C 1 . It is clear that ∈ y C λ 1 . So, last inequality holds for = y y λ , that is, By taking limit as → λ 0, we get ( ) + ( ) − ( ) ≥ ∀ ∈ F p y φ y φ p y C , 0 , , it follows from (3.9), (3.10), (3.12), and (3.14) that ∈ ( On the other hand, since A is a bounded linear operator, we get  , and S be chosen as in Theorem 1. Let the bifunctions × → F C C :  .3) converges weakly to a solution of system of split variational inequality problems which is to find a point ∈ * x C such that

Numerical examples
Now, we give a numerical example to support our proof.
,0, 1 0 ,0, 0 , : x C for the following system of mixed equilibrium problems: . Since we choose the mapping φ as 0, the point * x has to be a solution for the inequality ( − ) ≥ * * ix x x 0 for all ∈ [− ] x i, 0 . This problem has a unique solution = * x 0. It is obvious that the point = = * * y Ax 0 is a solution for the following system of mixed equilibrium problems: , that is, = *   In Table 1, we give some steps of Algorithm (4.1) for some initial values and special N. From the table, it is clear that sequence { } x n generated by Algorithm (4.1) converges weakly to common solution = * x 0.
In Figure 1, we give the graphics of the fitted curves, which are generated according to the values given in Table 1. For the fitting process third-degree polynomials were used.

Conclusion
In this article, we generalized several equilibrium problems by introducing the system of split mixed equilibrium problems. We established an iterative algorithm and proved that the iterative sequence generated by the algorithm converges weakly to the common solution of considered problems. Since our problem is fairly general, our results are very significant. Also, we substantiated our results by constructing a numerical model. In this model, we constructed an iterative sequence by choosing special mappings and sequences, which satisfies the conditions of our theorem and calculated its steps in Mathematica software. As can be seen from the table, iterative sequence converges strongly and hence weakly to the solution. Also, we compare the rate of convergence of our method with the method of Ugwunnadi and Ali [20] and show that our method converges faster than their method.