A modified projection method for a common solution of a system of variational inequalities, a split equilibrium problem and a hierarchical fixed-point problem
© Bnouhachem; licensee Springer. 2014
Received: 2 September 2013
Accepted: 30 December 2013
Published: 23 January 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.
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.
We introduce the following definitions, which are useful in the following analysis.
- (a)monotone, if
- (b)strongly monotone, if there exists an such that
- (c)α-inverse strongly monotone, if there exists an such that
- (d)nonexpansive, if
- (e)k-Lipschitz continuous, if there exists a constant such that
- (f)a contraction on C, if there exists a constant 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 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.
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.
The solution set of SEP (1.9)-(1.10) is denoted by .
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 2.2 
where , and is for the -inverse strongly monotone mappings for each .
Assumption 2.1 
F is monotone, i.e., , ;
for each , ;
for each , is convex and lower semicontinuous;
- (v)for fixed and , there exists a bounded subset K of and such that
Lemma 2.3 
is nonempty and single-valued;
- (ii)is firmly nonexpansive, i.e.,
is closed and convex.
Lemma 2.4 
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 
Lemma 2.7 
Lemma 2.8 
Lemma 2.9 
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.
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.
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. □
The weak w-limit set ().