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A New Hybrid Extragradient Iterative Method for Approximating the Common Solutions of a System of Variational Inequalities, a Mixed Equilibrium Problem, and a Hierarchical Fixed Point Problem
We suggest and analyze an iterative scheme for finding the approximate element of the common set of solutions of a system of variational inequalities, a mixed equilibrium problem, and a hierarchical fixed point problem in a real Hilbert space. Strong convergence of the proposed method is proved under some conditions. The results presented in this paper extend and improve some well-known results in the literature.
Let be a real Hilbert space, whose inner product and norm are denoted by and . Let be a nonempty closed convex subset of . We consider the system of variational inequalities of finding such that where is a nonlinear mapping for each . The solution set of (1) is denoted by .
If and , then the problem (2) collapses to the classical variational inequality finding , such that is called the classical variational inequality problem, which was introduced by Stampacchia  in 1964. For the recent applications, numerical techniques, and physical formulation, see [1–33]. We now have a variety of techniques to suggest and analyze various iterative algorithms for solving the system of variational inequalities (1); see [1, 2, 7, 8, 12, 14, 24, 28, 30].
We introduce the following definitions which are useful in the following analysis.
Definition 1. The mapping is said to be (a)monotone, if
(b)strongly monotone, if there exists an such that
(c)-inverse strongly monotone, if there exists an such that
(e)-Lipschitz continuous, if there exists a constant such that
(f)contraction on , if there exists a constant such that
It is easy to observe that every -inverse strongly monotone is monotone and Lipschitz continuous. A mapping is called -strict pseudocontraction, if there exists a constant such that
The fixed-point problem for the mapping is to find such that
We denote by the set of solutions of (11). It is well-known that the class of strict pseudocontractions strictly includes the class of nonexpansive mappings; then is closed and convex and is well defined (see ).
The mixed equilibrium problem, denoted by , is to find such that where is bifunction and is a nonlinear mapping. This problem was introduced and studied by Moudafi and Théra  and Moudafi . The set of solutions of (12) is denoted by If , then it is reduced to the equilibrium problem is to find such that The solution set of (14) is denoted by . Numerous problems in physics, optimization, and economics reduce to find a solution of (14); see [9, 13, 25, 26]. In 1997, Flåm and Antipin  introduced an iterative scheme of finding the best approximation to the initial data when is nonempty. Recently, Plubtieng and Punpaeng  introduced an iterative method for finding the common element of the set .
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 (15) links with some monotone variational inequalities and convex programming problems; see [11, 31]. Various methods have been proposed to solve the hierarchical fixed point problem; see Moudafi , Maingé and Moudafi in , Marino and Xu in , and Cianciaruso et al. . Very recently, Yao et al.  introduced the following strong convergence iterative algorithm to solve the problem (15): where is a contraction mapping and and are two sequences in . Under some certain restrictions on parameters, Yao et al. proved that the sequence generated by (16) converges strongly to , which is the unique solution of the following variational inequality: By changing the restrictions on parameters, the authors obtained another result on the iterative scheme (16); the sequence generated by (16) converges strongly to a point , which is the unique solution of the following variational inequality:
Let be a nonexpansive mapping and a countable family of nonexpansive mappings. Very recently, Gu et al.  introduced the following iterative algorithm: where , is a strictly decreasing sequence in , and is a sequence in . Under some certain conditions on parameters, Gu et al. proved that the sequence generated by (19) converges strongly to , which is unique solution of one of the variational inequalities (17) and (18).
In this paper, motivated by the work of Yao et al.  and Gu et al.  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), (12), and (15) for a strictly pseudocontraction mapping in real Hilbert space. We establish a strong convergence theorem based on this method. The presented method improves and generalizes many known results for solving system of variational inequality problems, mixed equilibrium problems, and hierarchical fixed point problems; see, for example [6, 11, 17, 31] 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 .
Lemma 2. Let denote the projection of onto . Then, one has the following inequalities:
Lemma 4 (see ). Let be a bifunction satisfying the following assumptions. (i). (ii) is monotone; that is, .(iii)For each .(iv)For each is convex and lower semicontinuous. Let and . Then, there exists such that
Lemma 5 (see ). Assume that satisfies assumptions of Lemma 4. For and , define a mapping as follows: Then, the following hold. (i) is single valued.(ii) is firmly nonexpansive; that is, (iii). (iv) is closed and convex.
Lemma 6 (see ). Let be a nonempty closed convex subset of a real Hilbert space . If is a -strict pseudocontraction, then (i)the mapping is demiclosed at ; that is, if is a sequence in weakly converging to and if converges strongly to , then ;(ii)the set of is closed and convex so that the projection is well defined.
Lemma 7 (see ). Assume that is a sequence of nonnegative real numbers such that where is a sequence in and is a sequence such that(1); (2) or .Then, .
Lemma 8 (see ). Let be a closed convex subset of . Let be a bounded sequence in . Assume that (i)the weak -limit set where ,(ii)for each exists. Then, is weakly convergent to a point in .
Lemma 9 (see ). Let be a Hilbert space, a closed and convex subset of , and a -strict pseudocontraction mapping. Define a mapping by , . Then, as is a nonexpansive mapping such that .
Lemma 10 (see ). Let be a Hilbert space, a closed and convex subset of , and a nonexpansive mapping such that . Then,
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 variational inequality problem (1), the mixed equilibrium problem (12), and the hierarchical fixed point problem (15).
Let be a nonempty closed convex subset of a real Hilbert space . Let be -inverse strongly monotone mappings for each , respectively. Let be a bifunction satisfying the assumptions of Lemma 4, a nonexpansive mapping, and a countable family of -strict pseudocontraction mappings such that , where . Let be a -contraction mapping.
Algorithm 11. For a given arbitrarily, let the iterative sequences , and be generated by where for each is a strictly decreasing sequence in , and is a sequence in satisfying the following conditions: (a) and ,(b), (c) and ,(d) and .
Lemma 12. Let . Then, , and are bounded.
Proof. First, we show that the mapping is nonexpansive. For any , Similarly, we can show that the mapping is nonexpansive for each . It follows from Lemma 5 that . Let ; we have , and it follows that Let ; we have where Setting . Since is -inverse strongly monotone mapping, it follows that Since is -inverse strongly monotone mappings for each , we get Next, we prove that the sequence is bounded; without loss of generality we can assume that for all . From (29), we have By induction on , we obtain , for and . Hence, is bounded and, consequently, we deduce that , and are bounded.
Lemma 13. Let and be the sequence generated by Algorithm 11. Then one has(a). (b)The weak -limit set .
Proof. Since and , we have
Take in (37) and in (38), we get
Adding (39) and using the monotonicity of , we have
which implies that
Without loss of generality, let us assume that there exists a real number such that , for all positive integers . Then, we get
Next, we estimate
It follows from (43) and (44) that
From (29) and the previous inequality, we get
Next, we estimate
From (46) and (47), we have
It follows by conditions of Algorithm 11 and Lemma 7 that
Next, we show that . Since and , by using (31) and (35), we obtain
Then, from the previous inequality, we get
Since , and , we obtain , and .
Since is firmly nonexpansive, we have Hence, From (51), (35), and the previous inequality, we have Hence, Since , and , we obtain From (21), we get Hence, where the last inequality follows from (54). On the other hand, from (29) and (21), we obtain