Abstract and Applied Analysis

Volume 2013 (2013), Article ID 737590, 19 pages

http://dx.doi.org/10.1155/2013/737590

Research Article

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

Abdellah Bnouhachem1,2 and Abdelouahed Hamdi³

¹School of Management Science and Engineering, Nanjing University, Nanjing 210093, China

²Ibn Zohr University, ENSA, BP 1136, Agadir, Morocco

³Department of Mathematics, Statistics and Physics College of Arts and Sciences, Qatar University, P.O. Box 2713, Doha, Qatar

Received 31 August 2013; Accepted 4 October 2013

Academic Editor: Erdal Karapınar

Copyright © 2013 Abdellah Bnouhachem and Abdelouahed Hamdi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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.

1. Introduction

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 , then the problem (1) reduces finding such that

which has been introduced and studied by Verma [1, 2].

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 [3] 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

(d)nonexpansive, if

(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 [33]).

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 [21] and Moudafi [22]. 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 [10] introduced an iterative scheme of finding the best approximation to the initial data when is nonempty. Recently, Plubtieng and Punpaeng [25] 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 [23], Maingé and Moudafi in [17], Marino and Xu in [19], and Cianciaruso et al. [6]. Very recently, Yao et al. [31] 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. [11] 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. [31] and Gu et al. [11] 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.

2. Preliminaries

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 3 (see [7]). For any is a solution of (1) if and only if is a fixed point of the mapping de fined by

where , and is -inverse strongly monotone mappings for each .

Lemma 4 (see [5]). 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 [10]). 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 [32]). 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 [29]). 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 [4]). 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 [33]). 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 [11]). 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

and then

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

where

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

Hence,

where the last inequality follows from (59). From (51) and the previous inequality, we have

Hence,

Since , and , , , we obtain

Since

we get

It follows from (57) and (66) that

Now, let ; since for each , and , we have . And

It follows that

From Lemma 10 and the previous inequality, we get

Since , and , we obtain

Since and is strictly decreasing, we have

Hence, we obtain

Since is bounded, without loss of generality, we can assume that . It follows from Lemma 6 that . Therefore, .

Theorem 14. The sequence generated by Algorithm 11 converges strongly to , which is the unique solution of the variational inequality

Proof. Since is bounded and from Lemma 13, we have . Next, we show that . Since , we have

It follows from monotonicity of that

Since and , it easy to observe that . For any and , let ; we have . Then from (76), we obtain

Since is Lipschitz continuous and , we obtain . From the monotonicity of and , it follows from (77) that

Hence, from assumptions of Lemma 4 and (78), we have

which implies that . Letting , we have

which implies that . Next, we show that . Since and there exists a subsequence of such that , it easy to observe that . For any , using (23), we have

This implies that is nonexpansive. On the other hand,

Since (see (66)), we have . It follows from Lemma 6 that , which implies from Lemma 3 that .

Thus, we have

Next, we claim that where .

Since is bounded, there exists a subsequence of such that

Next, we show that .

One has

which implies that

Let and .

Since

It follows that

Thus, all the conditions of Lemma 7 are satisfied. Hence, we deduce that .

Since is a contraction, there exists a unique such that . From (20), it follows that is the unique solution of the problem (74). This completes the proof.

Theorem 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. 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), (e)there exists a constant such that ,(f) and .

Then, sequence generated by Algorithm (89) converges strongly to , which is the unique solution of the variational inequality

Proof. From , without loss of generality, we can assume that for all . Hence . By similar argument as that lemmas 12 and 13, we can deduce that is bounded, (see (67)), and . Then, we have

It follows that, for all ,

From (91) and (92), we have

Set . From (47) and (48), we obtain

Let and . From conditions (a) and (d), we have

By Lemma 7, we obtain

From (89), we have

Hence, it follows that

and hence

Let . For any , we have

Since is nonexpansive mapping, is -contraction mapping and is -strict pseudocontraction mapping. Then, and are monotones, and is strongly monotone with coefficient . We can deduce that

From (20), we get

Then, from (100)–(102), we have

Then, we obtain

By condition (e) of Theorem 15, there exists a constant such that . Since , , and as , then every weak cluster point of is also a strong cluster point. Since is bounded, by Lemma 13 there exists a subsequence of converging to a point ; in similar argument as that Theorem 14 we can show that .

From (100)–(102), it follows that, for any ,

Since ,  ,   , and ; letting in (105), we obtain

that is,

In the following, we show that (90) has unique solution. Assume that is another solution. Then, we have

Adding (108), we get

Then, . Since (90) has unique solution, it follows that . Since every weak cluster point of is also a strong cluster point, we conclude that . This completes the proof.

4. Applications

In this section, we obtain the following results by using a special case of the proposed method. The first result can be viewed as extension and improvement of the method of Gu et al. [11] for finding the approximate element of the common set of solutions of a generalized equilibrium problem and a hierarchical fixed point problem in a real Hilbert space.

Corollary 16. Let be a nonempty closed convex subset of a real Hilbert space . Let be -inverse strongly monotone mappings, 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. For a given arbitrarily, let the iterative sequences , and be generated by

where , is a strictly decreasing sequence in , and is a sequence in satisfying the following conditions:

(a) and ,(b), (c) and ,(d), (e)there exists a constant such that ,(f) and .

Then, sequence generated by Algorithm (110) converges strongly to , which is the unique solution of the variational inequality

Proof. Putting and in Theorem 15, then conclusion of Corollary 16 is obtained.

The following result can be viewed as extension and improvement of the method of Yao et al. [31] for finding the approximate element of the common set of solutions of a generalized equilibrium problem and a hierarchical fixed point problem in a real Hilbert space.

Corollary 17. Let be a nonempty closed convex subset of a real Hilbert space . Let be -inverse strongly monotone mappings, respectively. Let be a bifunction satisfying the assumptions of Lemma 4, a nonexpansive mapping, and a countable family of -strict pseudocontraction mappings such that . Let be a -contraction mapping. For a given arbitrarily, let the iterative sequences , and be generated by

where , is a strictly decreasing sequence in and is a sequence in satisfying the following conditions:

(a) and ,(b), (c) and ,(d), (e)there exists a constant such that ,(f) and .

Then, sequence generated by Algorithm (112) converges strongly to , which is the unique solution of the variational inequality

Proof. Putting ,  , and in Theorem 15, then conclusion of Corollary 17 is obtained.

5. Conclusions

In this paper, we suggest and analyze 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, which can be viewed as a refinement and improvement of some existing methods for solving a system of variational inequality problem, a mixed equilibrium problem, and a hierarchical fixed point problem. It is easy to verify that Algorithm 11 includes some existing methods (e.g., [6, 11, 17, 31]) as special cases. Therefore, the new algorithm is expected to be widely applicable.

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