Journal of Function Spaces

Volume 2014 (2014), Article ID 606109, 8 pages

http://dx.doi.org/10.1155/2014/606109

Research Article

Projection Methods for a System of Nonlinear Mixed Variational Inequalities in Banach Spaces

Zhong-Bao Wang,¹ Guo-Ji Tang,² and Hong-Ling Zhang¹

¹Department of Mathematics, Southwest Jiaotong University, Chengdu 610031, China

²School of Science, Guangxi University for Nationalities, Nanning, Guangxi 530006, China

Received 29 October 2013; Accepted 8 December 2013; Published 15 May 2014

Academic Editor: M. Van Nguyen

Copyright © 2014 Zhong-Bao Wang 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 existence and uniqueness of solution for a system of nonlinear mixed variational inequality in Banach spaces is given firstly. A Mann iterative sequences with errors for this system of nonlinear mixed variational inequalities in Banach spaces is studied, by using the generalized f-projection operator . Our results extend the main results in (Verma (2005); Verma (2001)) from Hilbert spaces to Banach spaces.

1. Introduction

Throughout this paper, unless otherwise stated, we let be a real Banach space with the norm , we let be the topological dual space of , we let be the duality between , and we let and be a nonempty closed convex set.

Let be single-valued mappings. Let be the normalized duality mapping, and let be a proper lower semicontinuous mapping. We will consider the following system of nonlinear mixed variational inequalities problems (SNMVIP): find satisfying

where and .

If is a real Hilbert space, , , and , then SNMVIP (1) reduces to the following problem:

The problem (2) was introduced and studied by Verma [1, 2]. We note that for the problem (2) reduces to the nonlinear variational inequalities problem: determine an element such that

In brief, the system of nonlinear mixed variational inequalities (1) is more general and includes many systems of variational inequalities and variational inequalities as special cases.

It is well known that the projection method and its variant forms have represented an important tool for solving variational inequalities and the system of variational inequalities; see [1–21] and the references therein. Wu and Huang [10] introduced and studied a new class of generalized -projection operators in Banach spaces, which extends the definition of the generalized projection operators introduced and studied by Alber [3, 4] and Li [7]. Some properties of the generalized -projection operator are given in [10]. By using the generalized projection operators and the generalized -projection operators, some authors studied the existence theorems of solution and some iterative algorithms of approximating solutions for variational inequalities; see [3, 4, 6–8, 10–12].

In addition, Verma [1, 2] applies the metric projection operator technique and the fixed theorem to suggest some iterative algorithms for the system of variational inequalities with the monotone mappings in Hilbert spaces. By using the sunny nonexpansive retraction, Yao et al. [21] prove that the suggested two-step projection method converges strongly to the solution of a system of variational inequality, which extend the main results in Verma [1] from Hilbert spaces to Banach spaces. Recently, Wang et al. show that the Lipschitz continuity of the generalized -projection operator and the normalized duality mapping and study a class of system of generalized mixed variational inequalities in Banach spaces.

Motivated and inspired by the research work going on this field, in this paper, we firstly show the existence and uniqueness of solution for a system of nonlinear mixed variational inequality in Banach spaces. By using the generalized -projection operator , we study Mann iterative sequences with errors for this system of nonlinear mixed variational inequalities in Banach spaces. Our results extend and improve the main results in Verma [1, 2] from Hilbert spaces to Banach spaces.

2. Preliminaries

Recall that is said to be strictly convex if for all , with and . It is also said to be uniformly convex if for any two sequences , in with and . The function

is called the modulus of convexity of . It is well known that is -uniformly convex if and only if there exists a constant such that for all . Recall that is said to be smooth if exists for all with . It is also said to be uniformly smooth if the limit is attained uniformly for . The function

is called the modulus of smoothness of . It is known that is said to be -uniformly smooth if and only if there exists a constant such that . Moreover, is uniformly convex if and only if is uniformly smooth. In this case, is reflexive by the Milman theorem.

Next we recall the concept of the normalized duality mapping. The normalized duality mapping is defined by

Many properties of the normalized duality mapping can be found in [18].

We list some properties of as follows:

if is a smooth Banach space, then is single-valued and continuous from the strong topology of to the weak topology of ; is the identity operator in Hilbert spaces;if is a reflexive, smooth, and strictly convex Banach space, and is the normalized duality mapping on , then , and .

For any fixed , let be a function defined as follows:

where , and is proper, convex, and lower semicontinuous. It is easy to see that for all and .

Definition 1. We say that has the property if weakly and implies .

Remark 2. It is well known that any locally uniformly convex Banach space has the property and has a Fréchet differentiable norm if and only if is reflexive and strictly convex and has the property (see, e.g., [19]).

Definition 3 (see [10]). We say that is a generalized -projection operator if

Remark 4. (i) If for all , then the generalized -projection operator reduces to the generalized projection operator defined by Alber [4] and Li [7]; that is,

where for all and .

(ii) If for all and is a Hilbert space, then the generalized -projection operator is equivalent to the following metric projection operator:

Definition 5. Let be a reflexive, smooth, and strictly convex Banach space with the dual space , and denote the family of all nonempty subsets of . is said to be -strongly monotone if there exists a constant satisfying

Remark 6. If is a Hilbert space, then -strongly monotonicity reduce to strongly monotonicity in [15].

Theorem 7 7 (see [6, 10]). If is a reflexive Banach space with dual space and is a nonempty closed convex subset of , then the following conclusions hold:

(i)for any given , is a nonempty, closed, and convex subset of ;(ii)if B is smooth, then for any given , if and only if

(iii)if is strictly convex, then the operator is single-valued.

Lemma 8 (see [13]). is -uniformly smooth if and only if there exists a constant such that, for all ,

Lemma 9 9 (see [20]). Let , , and be three sequences of nonnegative numbers satisfying the following conditions: there exists such that

where

Then as .

Proposition 10 10 (see [14]). Let be a reflexive, smooth, and strictly convex Banach space with the dual space . Then for any ,

where .

Proposition 11 11 (see [14]). Let be a real uniformly convex and uniformly smooth Banach space. Then

where and .

Proposition 12. Let be a reflexive, smooth, and strictly convex Banach space with the dual space , let have the property (h), and let be a nonempty closed convex subset of . For any , let and . Then

where .

Proof. According to Proposition 10,

where . Since and , Theorem 7 yields

It follows from (20) that

and so

By (19), we have

where . This completes the proof.

Theorem 13. Let be a reflexive, smooth, and -uniformly convex Banach space with the dual space and for some . Then there exist a constant such that

Proof. From Proposition 12, it follows that

where . Since , (25) implies that

and so

This completes the proof.

Theorem 14. Let be a real -uniformly smooth and uniformly convex Banach space with the dual space and for some . Then there exist a constant such that

Proof. From Proposition 11, it follows that

where and . Since ,

This completes the proof.

3. Main Results

From Theorem 7, we know that the following theorem holds.

Theorem 15. Let be a -uniformly smooth and -uniformly convex Banach space. is a solution of the (1) if and only if satisfying

In the sequel, we shall show the existence and uniqueness of solution for the problems (1), (2), and (3), respectively. Next, we construct some new iterative algorithms for the problems (1), (2), and (3). We also give the convergence analysis of the iterative sequences generated by the algorithms.

Theorem 16. Let be a -uniformly smooth and -uniformly convex Banach space with for some and for some . Let be single-valued mappings. Let and be the normalized duality mapping, and let be a proper lower semicontinuous mapping. Suppose that the following conditions are satisfied:

(i)for each , is -strongly monotone with constants and -Lipschitz continuous;(ii)for each , is -Lipschitz continuous;

and that there exist constants satisfying

Then SNMVIP (1) has a unique solution .

Proof. First, we prove the existence of the solution. Define a mapping as follows:

For any , by Theorem 13, we know that there exists such that

Theorem 14 implies that there exists such that

From condition (i) and Lemma 8, it follows that

By condition (ii) and Theorem 13, we know that

By (35), condition (i), and Lemma 8, we have

Thus

Condition (ii) and (39) imply that

From (34)–(40), we have

It follows from (32) that and . Thus, (41) implies that is a contractive mapping and so there exists a point such that . Let

From the definition of , we have

By Theorem 15, we know that is a solution of SNMVIP (1).

Next, we show the uniqueness of the solution. Let be another solution of SNMVIP (1). It follows from Theorem 15 that

As the proof of (41), we have

Since and , it follows that and so . This completes the proof.

It is clear that if is a Hilbert space, then we can take in Theorem 16. According to Theorem 16, it is easy to get the following corollaries.

Corollary 17. Let be a real Hilbert space, let be a nonempty closed convex set, and let be strongly monotone with constants and -Lipschitz continuous. There exist satisfying

Then the problem (2) has a unique solution .

Corollary 18. Let be a real Hilbert space, let be a nonempty closed convex set, and let be strongly monotone with constants and -Lipschitz continuous. There exists satisfying

Then the problem (3) has a unique solution .

Algorithm 19. For any given , define the Mann iterative sequences and with errors as follows:

where is a sequence in and , , and are three sequences in satisfying the following conditions:

If is a real Hilbert space, , , and , then Algorithm 19 reduces to the following iterative algorithm.

Algorithm 20 (see [2, Algorithm 2.1]). For any given , define the iterative sequences and as follows:

where is a sequence in satisfying the following conditions:

If is a real Hilbert space, , , , and , then Algorithm 19 reduces to the following iterative algorithm.

Algorithm 21 21 (see [1, Algorithm 2.2]). For any given , define the iterative sequences as follows:

where is a sequence in satisfying the following conditions:

Theorem 22. Let , , , , , and be the same as in Theorem 16, and let be the iterative sequence generated by Algorithm 19. If the condition (32) holds, then converges strongly to the unique solution of SNMVIP (1).

Proof. By Theorem 16, we know that SNMVIP (1) has a unique solution . It follows from Theorem 15 that

By Lipschitz property of , Theorem 13, (48), and (54), we have

Since is -strongly monotone with constants and -Lipschitz continuous, by Theorem 14 and Lemma 8, we obtain

Combining (55) and (56), we obtain

From Lipschitz property of , (48), and (54), it follows that

Since is -strongly monotone with constants and -Lipschitz continuous, Theorem 14 and Lemma 8 imply that

It follows from (57)–(59) that

Taking , , , and , Lemma 9 implies that and so as . Since , by (58) and (59), we know that as . This completes the proof.

Corollary 23 (see [2, Theorem 2.1]). Let , , and be the same as in Corollary 17, and let and be the iterative sequence generated by Algorithm 20. If there exist constants satisfying

then converges strongly to the unique solution of the problem (2).

Corollary 24 (see [1, Theorem 3.3]). Let , , and be the same as in Corollary 18, and let be the iterative sequence generated by Algorithm 21. If there exists satisfying

then converges strongly to the unique solution of the problem (3).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The second author was partially supported by the Guangxi Natural Science Foundation (2013GXNSFBA019015), the Scientific Research Foundation of Guangxi University for Nationalities (2012QD015), and the Open fund of Guangxi key laboratory of hybrid computation and IC design analysis (2013HCIC06). The third author was partially supported by the Key Program of the Fundamental Research Funds for the Central Universities (A0920502051202-111).

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