On solving split mixed equilibrium problems and fixed point problems of hybrid-type multivalued mappings in Hilbert spaces.

In this paper, we introduce and study iterative algorithms for solving split mixed equilibrium problems and fixed point problems of λ-hybrid multivalued mappings in real Hilbert spaces and prove that the proposed iterative algorithm converges weakly to a common solution of the considered problems. We also provide an example to illustrate the convergence behavior of the proposed iteration process.

Introduction

Let H be a real Hilbert space with inner product and induced norm . Let C be a nonempty closed convex subset of H, be a function, and be a bifunction. The mixed equilibrium problem is to find such that The solution set of mixed equilibrium problem is denoted by . In particular, if , this problem reduces to the equilibrium problem, which is to find such that . The solution set of equilibrium problem is denoted by EP(F).

The mixed equilibrium problem is very general in the sense that it includes, as special cases, optimization problems, variational inequality problems, minimization problems, fixed point problems, Nash equilibrium problems in noncooperative games, and others; see, e.g., [1-4].

In 1994, Censor and Elfving [5] firstly introduced the following split feasibility problem in finite-dimensional Hilbert spaces: Let , be two Hilbert spaces and C, Q be nonempty closed convex subsets of and , respectively, and let be a bounded linear operator. The split feasibility problem is formulated as finding a point with the property The split feasibility problem can extensively be applied in fields such as intensity-modulated radiation therapy, signal processing and image reconstruction, then the split feasibility problem has received so much attention by so many scholars; see [6-9].

In 2013, Kazmi and Rizvi [10] introduced and studied the following split equilibrium problem: let and . Let and be nonlinear bifunctions and let be a bounded linear operator. The split equilibrium problem is to find such that The solution set of the split equilibrium problem is denoted by The authors gave an iterative algorithm to find the common element of sets of solution of the split equilibrium problem and hierarchical fixed point problem; for more details refer to [11, 12].

In 2016, Suantai et al. [13] proposed the iterative algorithm to solve the problems for finding a common elements the set of solution of the split equilibrium problem and the fixed point of a nonspreading multivalued mapping in Hilbert space, given sequence by where , and such that L is the spectral radius of and is the adjoint of A, , , is a -nonspreading multivalued mapping, and are two bifunctions. The authors showed that under certain conditions, the sequence converges weakly to an element of .

Several iterative algorithms have been developed for solving split feasibility problems and related split equilibrium problems; see, e.g., [14-16].

Motivated and inspired by the above results and related literature, we propose an iterative algorithm for finding a common element of the set of solutions of split mixed equilibrium problems and the set of fixed points of λ-hybrid multivalued mappings in real Hilbert spaces. Then we prove some weak convergence theorems which extend and improve the corresponding results of Kazmi and Rizvi [10] and Suantai et al. [13] and many others. We finally provide numerical examples for supporting our main result.

Preliminaries

Let C be a nonempty closed convex subset of a real Hilbert space H. We denote the strong convergence and the weak convergence of the sequence to a point by and , respectively. It is also well known [17] that a Hilbert space H satisfies Opial’s condition, that is, for any sequence with , the inequality holds for every with .

The following two lemmas are useful for our main results.

Lemma 2.1

In a real Hilbert space H, the following inequalities hold:

;

;

;

If is a sequence in H which converges weakly to , then

Lemma 2.2

[18]

Let H be a Hilbert space and be a sequence in H. Let be such that and exist. If and are subsequences of which converge weakly to u and v, respectively, then .

A single-valued mapping is called δ-inverse strongly monotone [19] if there exists a positive real number δ such that For each , we see that is a nonexpansive single-valued mapping, that is,

We denote by and the collection of all nonempty closed bounded subsets and nonempty compact subsets of C, respectively. The Hausdorff metric on is defined by where is the distance from a point x to a subset B. Let be a multivalued mapping. An element is called a fixed point of S if . The set of all fixed points of S is denoted by , that is, . Recall that a multivalued mapping is called We note that 0-hybrid is nonexpansive, 1-hybrid is nonspreading, and if S is λ-hybrid with , then S is quasi-nonexpansive. It is well known [20] that if S is λ-hybrid, then is closed. In addition, if S satisfies the condition: for all , then is also convex.

nonexpansive if

quasi-nonexpansive if and

nonspreading [13] if

λ-hybrid [20] if there exists such that

The following result is a demiclosedness principle for λ-hybrid multivalued mapping in a real Hilbert space.

Lemma 2.3

[20]

Let C be a nonempty closed convex subset of a real Hilbert space H and be a λ-hybrid multivalued mapping. If is a sequence in C such that and with , then .

For solving the mixed equilibrium problem, we assume that the bifunction satisfies the following assumption:

Assumption 2.4

Let C be a nonempty closed and convex subset of a Hilbert space . Let be the bifunction, is convex and lower semicontinuous satisfies the following conditions:

for all ;

is monotone, i.e., ;

for each , ;

for each , is convex and lower semicontinuous;

for each and fixed , there exist a bounded subset and such that, for any ,

C is a bounded set.

Lemma 2.5

[21]

Let C be a nonempty closed and convex subset of a Hilbert space . Let be a bifunction satisfies Assumption  2.4 and let be a proper lower semicontinuous and convex function such that . For and . Define a mapping as follows: for all . Assume that either (B1) or (B2) holds. Then the following conclusions hold: Further, assume that satisfying Assumption  2.4 and is a proper lower semicontinuous and convex function such that , where Q is a nonempty closed and convex subset of a Hilbert space . For each and , define a mapping as follows: Then we have the following:

for each , ;

is single-valued;

is firmly nonexpansive, i.e., for any ,

;

is closed and convex.

for each , ;

is single-valued;

is firmly nonexpansive;

;

is closed and convex.

Main results

In this section, we prove the weak convergence theorems for finding a common element of the set of solutions of split mixed equilibrium problems and the set of fixed points of λ-hybrid multivalued mappings in real Hilbert spaces and give a numerical example to support our main result.

We introduce the definition of split mixed equilibrium problems in real Hilbert spaces as follows.

Definition 3.1

Let C be a nonempty closed convex subset of a real Hilbert space and Q be a nonempty closed convex subset of a real Hilbert space . Let and be nonlinear bifunctions, let and be proper lower semicontinuous and convex functions such that and , and let be a bounded linear operator. The split mixed equilibrium problem is to find such that and such that

The solution set of the split mixed equilibrium problem (3.1) and (3.2) is denoted by

We now get our main result.

Theorem 3.2

Let C be a nonempty closed convex subset of a real Hilbert space and Q be a nonempty closed convex subset of a real Hilbert space . Let be a bounded linear operator and a λ-hybrid multivalued mapping. Let , be bifunctions satisfying Assumption  2.4, let and be a proper lower semicontinuous and convex functions such that and , respectively, and is upper semicontinuous in the first argument. Assume that and for all . Let be a sequence generated by and where , , , and such that L is the spectral radius of and is the adjoint of A. Assume that the following conditions hold: Then the sequence generated by (3.3) converges weakly to .

;

;

.

Proof

First, we show that is a -inverse strongly monotone mapping. Since is firmly nonexpansive and is 1-inverse strongly monotone, we see that for all . This implies that is a -inverse strongly monotone mapping. Since , it follows that is a nonexpansive mapping.

Now, we divide the proof into five steps as follows:

Step 1. Show that is bounded.

Let . Then we have and . By nonexpansiveness of , it implies that This implies that and so It follows that By (3.5) and (3.7), we have This implies that is decreasing and bounded below, thus exists for all .

Step 2. Show that .

From Lemma 2.1(3), (3.5), (3.7), and , we have This implies that From Condition (C1) and exists, we have

Step 3. Show that and .

For , we see that Thus, by (3.5) and (3.7), we have Therefore, we have Since , it follows by Condition (C1) and the existence of that Since is firmly nonexpansive and is nonexpansive, we have which implies that This implies by (3.5) and (3.7) that Therefore, we have where . This implies by Condition (C1), (3.12), and the existence of that From (3.5), (3.7), and the definition of , we obtain This implies that From Conditions (C1), (C2), and the existence of , we have By (3.14) and (3.15), we get

Step 4. Show that , where . Since is bounded and is reflexive, is nonempty. Let be an arbitrary element. Then there exists a subsequence converging weakly to p. From (3.14), it implies that as . By (3.16) and Lemma 2.3, we have .

Next, we show that . Since , we have which implies that From Assumption 2.4(A2), we have and hence This implies by , Condition (C3), (3.12), (3.14), Assumption 2.4(A2), and the proper lower semicontinuity of φ that Put for all and . Consequently, we get and hence . So, by Assumption 2.4(A1)-(A4), we have Hence, we have Letting , by Assumption 2.4(A3) and the proper lower semicontinuity of φ, we have This implies that .

Since A is a bounded linear operator, we have . Then it follows from (3.12) that By the definition of , we have Since is upper semicontinuous in the first argument, it implies by (3.17) that This shows that . Therefore, and hence .

Step 5. Show that converges weakly to an element of Θ. It is sufficient to show that is a singleton set. Let and , be two subsequences of such that and . From (3.14), we also have and . By (3.16) and Lemma 2.3, we see that . Applying Lemma 2.2, we obtain . This completes the proof. □

If in (3.1) and (3.2), then the split mixed equilibrium problem reduces to split equilibrium problem. So, the following result can be obtained from Theorem 3.2 immediately.

Theorem 3.3

Let C be a nonempty closed convex subset of a real Hilbert space and Q be a nonempty closed convex subset of a real Hilbert space . Let be a bounded linear operator and a λ-hybrid multivalued mapping. Let , be bifunctions satisfying Assumption  2.4, and is upper semicontinuous in the first argument. Assume that and for all . Let be a sequence generated by and where , , , and such that L is the spectral radius of and is the adjoint of A. Assume that the following conditions hold: Then the sequence generated by (3.18) converges weakly to .

;

;

.

Remark 3.4

Theorems 3.2 and 3.3 extend the corresponding one of Suantai et al. [13] and Kazmi and Rizvi [10] to λ-hybrid multivalued mapping and to a split mixed equilibrium problem. In fact, we present a new iterative algorithm for finding a common element of the set of solutions of split mixed equilibrium problems and the set of fixed points of λ-hybrid multivalued mappings in a real Hilbert space.

It is well known that the class of λ-hybrid multivalued mappings contains the classes of nonexpansive multivalued mappings, nonspreading multivalued mappings. Thus, Theorems 3.2 and 3.3 can be applied to these classes of mappings.

We give an example to illustrate Theorem 3.2 as follows.

Example 3.5

Let , , , and . Let defined by for each . Then for each . So, is the spectral radius of . Define a multivalued mapping by It easy to see that S is 1-hybrid multivalued mapping with and . For each , define the bifunction by and define for each . For each , define the bifunction by and define for each .

Choose , , , and . It is easy to check that , , , , satisfy all conditions in Theorem 3.2.

For each , we compute . Find z such that for all . Thus, by Lemma 2.5(2), it follows that . That is, for each . Furthermore, we get Next, we find such that for all , where . Note that Thus, by Lemma 2.5(2), it follows that Then the algorithm (3.3) becomes where We choose if and if . By using SciLab, we compute the iterates of (3.19) for the initial point . The numerical experiment’s results of our iteration for approximating the point 0 are given in Table 1.

Table 1

Numerical results of Example for the algorithm ()

n	x_n	∥x_n−x_n−1∥
1	−3.0000000e + 00	-
2	−6.8786127e − 02	2.9312139e + 00
3	0.0000000e + 00	6.8786127e − 02
4	0.0000000e + 00	0.0000000e + 00

Conclusions

The results presented in this paper extend and generalize the work of Suantai et al. [13] and Kazmi and Rizvi [10]. The main aim of this paper is to propose an iterative algorithm to find an element for solving a class of split mixed equilibrium problems and fixed point problems for λ-hybrid multivalued mappings under weaker conditions. Some sufficient conditions for the weak convergence of such proposed algorithm are given. Also, in order to show the significance of the considered problem, some important applications are discussed.