Journal of Applied MathematicsVolume 2012 (2012), Article ID 580158, 18 pagesdoi:10.1155/2012/580158

Research Article

Algorithms for a System of General Variational Inequalities in Banach Spaces

Jin-Hua Zhu,¹ Shih-Sen Chang,² and Min Liu¹

¹Department of Mathematics, Yibin University, Sichuan, Yibin 644007, China²College of Statistics and Mathematics, Yunnan University of Finance and Economics, Yunnan, Kunming 650221, China

Received 21 December 2011; Accepted 6 February 2012

Academic Editor: Zhenyu Huang

Copyright © 2012 Jin-Hua Zhu et al. 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.

The purpose of this paper is using Korpelevich's extragradient method to study the existence problem of solutions and approximation solvability problem for a class of systems of finite family of general nonlinear variational inequality in Banach spaces, which includes many kinds of variational inequality problems as special cases. Under suitable conditions, some existence theorems and approximation solvability theorems are proved. The results presented in the paper improve and extend some recent results.

1. Introduction

Throughout this paper, we denote by and the sets of positive integers and real numbers, respectively. We also assume that is a real Banach space, is the dual space of is a nonempty closed convex subset of , and is the pairing between and .

In this paper, we are concerned a finite family of a general system of nonlinear variational inequalities in Banach spaces, which involves finding such that where is a finite family of nonlinear mappings and are positive real numbers.

As special cases of the problem (1.1), we have the following.

(I) If is a real Hilbert space and , then (1.1) reduces to which was considered by Ceng et al. [1]. In particular, if , then the problem (1.2) reduces to finding such that which is defined by Verma [2]. Furthermore, if , then (1.3) reduces to the following variational inequality (VI) of finding such that

This problem is a fundamental problem in variational analysis and, in particular, in optimization theory. Many algorithms for solving this problem are projection algorithms that employ projections onto the feasible set of the VI or onto some related set, in order to iteratively reach a solution. In particular, Korpelevich’s extragradient method which was introduced by Korpelevich [3] in 1976 generates a sequence via the recursion where is the metric projection from onto , is a monotone operator, and is a constant. Korpelevich [3] proved that the sequence converges strongly to a solution of . Note that the setting of the space is Euclid space .

The literature on the VI is vast, and Korpelevich’s extragradient method has received great attention by many authors, who improved it in various ways. See, for example, [4–16] and references therein.

(II) If is still a real Banach space and , then the problem (1.1) reduces to finding such that which was considered by Aoyama et al. [17]. Note that this problem is connected with the fixed point problem for nonlinear mapping, the problem of finding a zero point of a nonlinear operator, and so on. It is clear that problem (1.6) extends problem (1.4) from Hilbert spaces to Banach spaces.

In order to find a solution for problem (1.6), Aoyama et al. [17] introduced the following iterative scheme for an accretive operator in a Banach space : where is a sunny nonexpansive retraction from to . Then they proved a weak convergence theorem in a Banach space. For related works, please see [18] and the references therein.

It is an interesting problem of constructing some algorithms with strong convergence for solving problem (1.1) which contains problem (1.6) as a special case.

Our aim in this paper is to construct two algorithms for solving problem (1.1). For this purpose, we first prove that the system of variational inequalities (1.1) is equivalent to a fixed point problem of some nonexpansive mapping. Finally, we prove the strong convergence of the proposed methods which solve problem (1.1).

2. Preliminaries

In the sequel, we denote the strong convergence and weak convergence of the sequence by and , respectively.

For , the generalized duality mapping is defined by for all . In particular, is called the normalized duality mapping. It is known that for all . If is a Hilbert space, then , the identity mapping. Let . A Banach space is said to be uniformly convex if, for any , there exists such that, for any ,

It is known that a uniformly convex Banach space is reflexive and strictly convex. A Banach space is said to be smooth if the limit exists for all . It is also said to be uniformly smooth if the previous limit is attained uniformly for . The norm of is said to be Fréchet differentiable if, for each , the previous limit is attained uniformly for all . The modulus of smoothness of is defined by where is function. It is known that is uniformly smooth if and only if . Let be a fixed real number with . Then a Banach space is said to be -uniformly smooth if there exists a constant such that for all . Note the following.(1) is a uniformly smooth Banach space if and only if is single valued and uniformly continuous on any bounded subset of .(2)All Hilbert spaces, (or ) spaces () and the Sobolev spaces are 2-uniformly smooth, while (or ) and spaces are -uniformly smooth.(3)Typical examples of both uniformly convex and uniformly smooth Banach spaces are , where . More precisely, is min-uniformly smooth for every .

In our paper, we focus on a 2-uniformly smooth Banach space with the smooth constant .

Let be a real Banach space, a nonempty closed convex subset of , a mapping, and the set of fixed points of .

Recall that a mapping is called nonexpansive if A bounded linear operator is called strongly positive if there exists a constant with the property A mapping is said to be accretive if there exists such that for all , where is the duality mapping.

A mapping of into is said to be -strongly accretive if, for , for all .

A mapping of into is said to be -inverse-strongly accretive if, for , for all .

Remark 2.1. Evidently, the definition of the inverse strongly accretive mapping is based on that of the inverse strongly monotone mapping, which was studied by so many authors; see, for instance, [6, 19, 20].Let be a subset of , and let be a mapping of into . Then is said to be sunny if whenever for and . A mapping of into itself is called a retraction if . If a mapping of into itself is a retraction, then for every , where is the range of . A subset of is called a sunny nonexpansive retract of if there exists a sunny nonexpansive retraction from onto . Then following lemma concerns the sunny nonexpansive retraction.

Lemma 2.2 (see [21]). Let be a closed convex subset of a smooth Banach space , let be a nonempty subset of , and let be a retraction from onto . Then is sunny and nonexpansive if and only if for all and .

Remark 2.3. (1) It is well known that if is a Hilbert space, then a sunny nonexpansive retraction is coincident with the metric projection from onto .(2) Let be a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space , and let be a nonexpansive mapping of into itself with the set . Then the set is a sunny nonexpansive retract of .

In what follows, we need the following lemmas for proof of our main results.

Lemma 2.4 (see [22]). Assume that is a sequence of nonnegative real numbers such that where is a sequence in and is a sequence such that (a),(b) or .Then .

Lemma 2.5 (see [23]). Let be a Banach space, be two bounded sequences in and be a sequence in satisfying Suppose that , for all and then .

Lemma 2.6 (see [24]). Let be a real 2-uniformly smooth Banach space with the best smooth constant . Then the following inequality holds:

Lemma 2.7 (see [25]). Let be a nonempty bounded closed convex subset of a uniformly convex Banach space , and let be a nonexpansive mapping of into itself. If is a sequence of such that and , then is a fixed point of .

Lemma 2.8 (see [26]). Let be a nonempty closed convex subset of a real Banach space . Assume that the mapping is accretive and weakly continuous along segments (i.e., as ). Then the variational inequality is equivalent to the dual variational inequality

Lemma 2.9. Let be a nonempty closed convex subset of a real 2-uniformly smooth Banach space . Let be a sunny nonexpansive retraction from onto . Let be a finite family of -inverse-strongly accretive. For given , where , then is a solution of the problem (1.1) if and only if is a fixed point of the mapping defined by where are real numbers.

Proof. We can rewrite (1.1) as By Lemma 2.2, we can check (2.19) is equivalent to This completes the proof.

Throughout this paper, the set of fixed points of the mapping is denoted by .

Lemma 2.10. Let be a nonempty closed convex subset of a real 2-uniformly smooth Banach space . Let be a sunny nonexpansive retraction from onto . Let be a finite family of -inverse-strongly accretive. Let be defined as Lemma 2.9. If , then is nonexpansive.

Proof. First, we show that for all , the mapping is nonexpansive. Indeed, for all , from the condition and Lemma 2.6, we have which implies for all , the mapping is nonexpansive, so is the mapping .

3. Main Results

In this section, we introduce our algorithms and show the strong convergence theorems.

Algorithm 3.1. Let be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space . Let be a sunny nonexpansive retraction from to . Let be a finite family of -inverse-strongly accretive. Let be a strongly positive bounded linear operator with coefficient and be a strongly positive bounded linear operator with coefficient . For any , define a net as follows: where, for any is a real number.

Remark 3.2. We notice that the net defined by (3.1) is well defined. In fact, we can define a self-mapping as follows: From Lemma 2.10, we know that if, for any , the mapping is nonexpansive and . Then, for any , we have This shows that the mapping is contraction. By Banach contractive mapping principle, we immediately deduce that the net (3.1) is well defined.

Theorem 3.3. Let be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space . Let be a sunny nonexpansive retraction from to . Let be a finite family of -inverse-strongly accretive. Let be a strongly positive bounded linear operator with coefficient , and let be a strongly positive bounded linear operator with coefficient . Assume that and . Then the net generated by the implicit method (3.1) converges in norm, as to the unique solution of VI

Proof. We divide the proof of Theorem 3.3 into four steps.(I) Next we prove that the net is bounded.Take that , we have It follows that Therefore, is bounded. Hence, , and are also bounded. We observe that From Lemma 2.10, we know that is nonexpansive. Thus, we have Therefore, (II) is relatively norm-compact as .Let be any subsequence such that as . Then, there exists a positive integer such that , for all . Let . It follows from (3.9) that We can rewrite (3.1) as For any , by Lemma 2.2, we have With this fact, we derive that It turns out that In particular, Since is bounded, without loss of generality, can be assumed. Noticing (3.10), we can use Lemma 2.7 to get . Therefore, we can substitute for in (**) to get Consequently, the weak convergence of to actually implies that strongly. This has proved the relative norm compactness of the net as .(III) Now, we prove that solves the variational inequality (3.4). From (3.1), we have For any , we obtain Now we prove that . In fact, we can write . At the same time, we note that , so Since is accretive (this is due to the nonexpansivity of ), we can deduce immediately that Therefore, Since is strongly positive, we have It follows that Combining (3.20) and (3.22), we get Now replacing in (3.23) with and letting , noticing that , we obtain which is equivalent to its dual variational inequality (see Lemma 2.8) that is, is a solution of (3.4).(IV) Now we show that the solution set of (3.4) is singleton.As a matter of fact, we assume that is also a solution of (3.4) Then, we have From (3.25), we have So, Therefore, . In summary, we have shown that each cluster point of (as ) equals . Therefore, as . This completes the proof.

Next, we introduce our explicit method which is the discretization of the implicit method (3.1).

Algorithm 3.4. Let be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space . Let be a sunny nonexpansive retraction from to . Let be a finite family of -inverse-strongly accretive. Let be a strongly positive bounded linear operator with coefficient , and let be a strongly positive bounded linear operator with coefficient . For arbitrarily given , let the sequence be generated iteratively by where and are two sequences in and, for any is a real number.

Theorem 3.5. Let be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space , and let be a sunny nonexpansive retraction from to . Let be a finite family of -inverse-strongly accretive.Let be a strongly positive bounded linear operator with coefficient , and let be a strongly positive bounded linear operator with coefficient . Assume that . For given , let be generated iteratively by (3.29). Suppose the sequences and satisfy the following conditions: (1) and ,(2).Then converges strongly to which solves the variational inequality (3.4).

Proof. Set for all . Then for all . Pick up .From Lemma 2.10, we have Hence, it follows that By induction, we deduce that Therefore, is bounded. Hence, , and are also bounded. We observe that Set for all . Then . It follows that This implies that Hence, by Lemma 2.5, we obtain . Consequently, At the same time, we note that It follows that From Lemma 2.10, we know that is nonexpansive. Thus, we have Thus, . We note that Next, we show that where is the unique solution of VI(3.4).To see this, we take a subsequence of such that We may also assume that . Note that in virtue of Lemma 2.7 and (3.40). It follows from the variational inequality (3.4) that Since , according to Lemma 2.2, we have From (3.44), we have It follows that Finally, we prove . From and (3.46), we have We can apply Lemma 2.4 to the relation (3.47) and conclude that . This completes the proof.

Acknowledgments

The authors would like to express their thanks to the referees and the editor for their helpful suggestion and comments. This work was supported by the Scientific Reserch Fund of Sichuan Provincial Education Department (09ZB102,11ZB146) and Yunnan University of Finance and Economics.

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