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A modified projection method for a common solution of a system of variational inequalities, a split equilibrium problem and a hierarchical fixed-point problem
Fixed Point Theory and Applications volume 2014, Article number: 22 (2014)
In this paper, we suggest and analyze a modified projection method for finding a common solution of a system of variational inequalities, a split equilibrium problem, and a hierarchical fixed-point problem in the setting of real Hilbert spaces. We prove the strong convergence of the sequence generated by the proposed method to a common solution of a system of variational inequalities, a split equilibrium problem, and a hierarchical fixed-point problem. Several special cases are also discussed. The results presented in this paper extend and improve some well-known results in the literature.
MSC: 49J30, 47H09, 47J20.
Let H be a real Hilbert space, whose inner product and norm are denoted by and . Let C be a nonempty closed convex subset of H. Recently, Ceng et al.  considered the general system of variational inequalities, which involves finding such that
where is a nonlinear mapping for each . The solution set of (1.1) is denoted by . As pointed out in  the system of variational inequalities is used as a tool to study the Nash equilibrium problem, see, for example, [3–6] and the references therein. We believe that the problem (1.1) could be used to study the Nash equilibrium problem for a two players game. The theory of variational inequalities is well established and it has a wide range of applications in science, engineering, management, and social sciences, see, for example, [4–7] and the references therein.
Ceng et al.  transformed problem (1.1) into a fixed-point problem (see Lemma 2.2) and introduced an iterative method for finding the common element of the set . Based on the one-step iterative method  and the multi-step iterative method , Latif et al.  proposed a multi-step hybrid viscosity method that generates a sequence via an explicit iterative algorithm to compute the approximate solutions of a system of variational inequalities defined over the intersection of the set of solutions of an equilibrium problem, the set of common fixed points of a finite family of nonexpansive mappings, and the solution set of a nonexpansive mapping. Under very mild conditions, they proved that the sequence converges strongly to a unique solution of system of variational inequalities defined over the set consisting of the set of solutions of an equilibrium problem, the set of common fixed points of nonexpansive mappings, and the set of fixed points of a mapping, and to a unique solution of the triple hierarchical variational inequality problem.
On the other hand, by combining the regularization method, Korpelevich’s extragradient method, the hybrid steepest-descent method, and the viscosity approximation method, Ceng et al.  introduced and analyzed implicit and explicit iterative schemes for computing a common element of the solution set of system of variational inequalities and a set of zeros of an accretive operator in Banach space. Under suitable assumptions, they proved the strong convergence of the sequences generated by the proposed schemes.
If , then the problem (1.1) reduces to finding such that
If and , then the problem (1.2) collapses to the classical variational inequality: finding , such that
We introduce the following definitions, which are useful in the following analysis.
Definition 1.1 The mapping is said to be
strongly monotone, if there exists an such that
α-inverse strongly monotone, if there exists an such that
k-Lipschitz continuous, if there exists a constant such that
a contraction on C, if there exists a constant such that
It is easy to observe that every α-inverse strongly monotone T is monotone and Lipschitz continuous. It is well known that every nonexpansive operator satisfies, for all , the inequality
and therefore, we get, for all ,
The fixed-point problem for the mapping T is to find such that
We denote by the set of solutions of (1.5). It is well known that is closed and convex, and is well defined.
The equilibrium problem, denoted by EP, is to find such that
The solution set of (1.6) is denoted by . Numerous problems in physics, optimization, and economics reduce to finding a solution of (1.6), see [25, 38]. In 1994, Censor and Elfving  introduced and studied the following split feasibility problem.
Let C and Q be nonempty closed convex subsets of the infinite-dimensional real Hilbert spaces and , respectively, and let , where denotes the collection of all bounded linear operators from to . The problem is to find such that
Very recently, Ceng et al.  introduced and analyzed an extragradient method with regularization for finding a common element of the solution set of the split feasibility problem and the set of fixed points of a nonexpansive mapping S in the setting of infinite-dimensional Hilbert spaces. By combining Mann’s iterative method and the extragradient method, Ceng et al.  proposed three different kinds of Mann type iterative methods for finding a common element of the solution set of the split feasibility problem and the set of fixed points of a nonexpansive mapping S in the setting of infinite-dimensional Hilbert spaces.
Recently, Censor et al.  introduced a new variational inequality problem which we call the split variational inequality problem (SVIP). Let and be two real Hilbert spaces. Given the operators and , a bounded linear operator , and the nonempty, closed, and convex subsets and , the SVIP is formulated as follows: find a point such that
and such that
and such that
In this paper, we consider the following pair of equilibrium problems, called split equilibrium problems: Let and be nonlinear bifunctions and be a bounded linear operator, then the split equilibrium problem (SEP) is to find such that
and such that
The solution set of SEP (1.9)-(1.10) 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 (1.11) links with some monotone variational inequalities and convex programming problems; see . Various methods have been proposed to solve the hierarchical fixed-point problem; see Mainge and Moudafi  and Cianciaruso et al. . In 2010, Yao et al.  introduced the following strong convergence iterative algorithm to solve the problem (1.11):
where is a contraction mapping and and are two sequences in . Under some certain restrictions on the parameters, Yao et al. proved that the sequence generated by (1.12) converges strongly to , which is the unique solution of the following variational inequality:
In 2011, Ceng et al.  investigated the following iterative method:
where U is a Lipschitzian mapping, and F is a Lipschitzian and strongly monotone mapping. They proved that under some approximate assumptions on the operators and parameters, the sequence generated by (1.14) converges strongly to the unique solution of the variational inequality
In this paper, motivated by the work of Censor et al. , Moudafi , Byrne et al. , Yao et al. , Ceng et al. , Bnouhachem [15–17] and by the recent work going in this direction, we give an iterative method for finding the approximate element of the common set of solutions of (1.1), (1.9)-(1.10), and (1.11) in real Hilbert space. We establish a strong convergence theorem based on this method. We would like to mention that our proposed method is quite general and flexible and includes many known results for solving a system of variational inequality problems, split equilibrium problems, and hierarchical fixed-point problems, see, e.g. [20, 26, 34, 40, 45] and relevant references cited therein.
In this section, we list some fundamental lemmas that are useful in the consequent analysis. The first lemma provides some basic properties of projection onto C.
Lemma 2.1 Let denote the projection of H onto C. Then we have the following inequalities:
Lemma 2.2 
For any , is a solution of (1.1) if and only if is a fixed point of the mapping defined by
where , and is for the -inverse strongly monotone mappings for each .
Assumption 2.1 
Let be a bifunction satisfying the following assumptions:
F is monotone, i.e., , ;
for each , ;
for each , is convex and lower semicontinuous;
for fixed and , there exists a bounded subset K of and such that
Lemma 2.3 
Assume that satisfies Assumption 2.1. For and , define a mapping as follows:
Then the following hold:
is nonempty and single-valued;
is firmly nonexpansive, i.e.,
is closed and convex.
Assume that satisfies Assumption 2.1, and for and , define a mapping as follows:
Then satisfies conditions (i)-(iv) of Lemma 2.3. , where is the solution set of the following equilibrium problem:
Lemma 2.4 
Assume that satisfies Assumption 2.1, and let be defined as in Lemma 2.3. Let and . Then
Lemma 2.5 
Let C be a nonempty closed convex subset of a real Hilbert space H. If is a nonexpansive mapping with , then the mapping is demiclosed at 0, i.e., if is a sequence in C weakly converges to x and if converges strongly to 0, then .
Lemma 2.6 
Let be a τ-Lipschitzian mapping and let be a k-Lipschitzian and η-strongly monotone mapping, then for , is -strongly monotone i.e.,
Lemma 2.7 
Suppose that and . Let be a k-Lipschitzian and η-strongly monotone operator. In association with a nonexpansive mapping , define the mapping by
Then is a contraction provided , that is,
Lemma 2.8 
Assume is a sequence of nonnegative real numbers such that
where is a sequence in and is a sequence such that
Lemma 2.9 
Let C be a closed convex subset of H. Let be a bounded sequence in H. Assume that
the weak w-limit set where ;
for each , exists.
Then is weakly convergent to a point in C.
3 The proposed method and some properties
In this section, we suggest and analyze our method for finding the common solutions of the system of the variational inequality problem (1.1), the split equilibrium problem (1.9)-(1.10), and the hierarchical fixed-point problem (1.11).
Let and be two real Hilbert spaces and 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 2.1 and is upper semicontinuous in the first argument. Let be a -inverse strongly monotone mapping for each and a nonexpansive mappings such that . Let be a k-Lipschitzian mapping and be η-strongly monotone, and let be a τ-Lipschitzian mapping.
Algorithm 3.1 For an arbitrary given , let the iterative sequences , , , and be generated by
where for each , and , L is the spectral radius of the operator , and is the adjoint of A. Suppose the parameters satisfy , , where . Also, and are sequences in satisfying the following conditions:
Remark 3.1 Our method can be viewed as an extension and improvement for some well-known results, for example the following.
The proposed method is an extension and improvement of the method of Wang and Xu  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.
If we have the Lipschitzian mapping , , , and , we obtain an extension and improvement of the method of Yao et al. 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 3.1 is quite general and unifying.
Lemma 3.1 Let . Then , , , and are bounded.
Proof Let ; we have and . Then
From the definition of L it follows that
It follows from (1.4) that
Applying (3.4) and (3.3) to (3.2) and from the definition of γ, we get
Let ; we have
We set . Since is a -inverse strongly monotone mapping, it follows that
Since is -inverse strongly monotone mappings, for each , we get
We denote . Next, we prove that the sequence is bounded, and without loss of generality we can assume that for all . From (3.1), we have
where the third inequality follows from Lemma 2.7.
By induction on n, we obtain , for and . Hence is bounded and, consequently, we deduce that , , , , , , , and are bounded. □
Lemma 3.2 Let and be the sequence generated by Algorithm 3.1. Then we have:
The weak w-limit set ().
Proof Since and . It follows from Lemma 2.4 that
where and . Without loss of generality, let us assume that there exists a real number μ such that , for all positive integers n. Then we get
Next, we estimate
It follows from (3.8) and (3.9) that
From (3.1) and the above inequality, we get
Next, we estimate
where the second inequality follows from Lemma 2.7. From (3.10) and (3.11), we have
It follows by conditions (a)-(d) of Algorithm 3.1 and Lemma 2.8 that
Next, we show that . Since by using (3.2), (3.5), and (3.7), we obtain
which implies that
Then from the above inequality, we get
Since , , , , , and , we obtain
Since is firmly nonexpansive, we have
Hence, we get
From (3.13), (3.7), and the above inequality, we have
which implies that
Since , , , and , we obtain
From (2.2), we get
where the last inequality follows from (3.15). On the other hand, from (3.1) and (2.2), we obtain