Journal of Applied Mathematics

Volume 2012, Article ID 413468, 15 pages

http://dx.doi.org/10.1155/2012/413468

## Existence and Algorithm for Solving the System of Mixed Variational Inequalities in Banach Spaces

Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangmod, Bangkok 10140, Thailand

Received 22 December 2011; Accepted 29 January 2012

Academic Editor: Hong-Kun Xu

Copyright © 2012 Siwaporn Saewan and Poom Kumam. 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 to study the existence and convergence analysis of the solutions of the system of mixed variational inequalities in Banach spaces by using the generalized *f* projection operator. The results presented in this paper improve and extend important recent results of Zhang et al. (2011) and Wu and Huang (2007) and some recent results.

#### 1. Introduction

Let be a real Banach space with norm , let be a nonempty closed and convex subset of , and let denote the dual of . Let denote the duality pairing of and . If is a Hilbert space, denotes an inner product on . It is well known that the metric projection operator plays an important role in nonlinear functional analysis, optimization theory, fixed point theory, nonlinear programming, game theory, variational inequality, and complementarity problems, and so forth (see, e.g., [1, 2] and the references therein). In 1993, Alber [3] introduced and studied the generalized projections and from Hilbert spaces to uniformly convex and uniformly smooth Banach spaces. Moreover, Alber [1] presented some applications of the generalized projections to approximately solving variational inequalities and von Neumann intersection problem in Banach spaces. In 2005, Li [2] extended the generalized projection operator from uniformly convex and uniformly smooth Banach spaces to reflexive Banach spaces and studied some properties of the generalized projection operator with applications to solving the variational inequality in Banach spaces. Later, Wu and Huang [4] introduced a new generalized -projection operator in Banach spaces which extended the definition of the generalized projection operators introduced by Abler [3] and proved some properties of the generalized -projection operator. As an application, they studied the existence of solution for a class of variational inequalities in Banach spaces. In 2007, Wu and Huang [5] proved some properties of the generalized -projection operator and proposed iterative method of approximating solutions for a class of generalized variational inequalities in Banach spaces. In 2009, Fan et al. [6] presented some basic results for the generalized -projection operator and discussed the existence of solutions and approximation of the solutions for generalized variational inequalities in noncompact subsets of Banach spaces. In 2011, Zhang et al. [7] introduced and considered the system of mixed variational inequalities in Banach spaces. Using the generalized -projection operator technique, they introduced some iterative methods for solving the system of mixed variational inequalities and proved the convergence of the proposed iterative methods under suitable conditions in Banach spaces. Recently, many authors studied methods for solving the system of generalized (mixed) variational inequalities and the system of nonlinear variational inequalities problems (see, e.g., [8–17] and references therein).

We first introduce and consider the *system of mixed variational inequalities* (SMVI) which is to find such that
where for are mappings and is the normalized duality mapping from to .

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

If for , for all , (1.1) is equivalent to find , and such that
The problem (1.2) is called the *system of variational inequalities* we denote by (SVI).

If , for all and , then (1.1) is reduced to find such that which is studied by Zhang et al. [7].

If , for all and , (1.1) is reduced to find such that This iterative method is studied by Wu and Huang [5].

If , for all , (1.4) is reduced to find such that
which is studied by Alber [1, 18], Li [2], and Fan [19]. If is a Hilbert space, (1.5) holds which is known as the *classical variational inequality* introduced and studied by Stampacchia [20].

If is a Hilbert space, then (1.1) is reduced to find such that If for , for all , (1.6) reduces to the following (SVI): The purpose of this paper is to study the existence and convergence analysis of solutions of the system of mixed variational inequalities in Banach spaces by using the generalized -projection operator. The results presented in this paper improve and extend important recent results in the literature.

#### 2. Preliminaries

A Banach space is said to be *strictly convex* if for all with and . Let be the unit sphere of . Then, a Banach space is said to be *smooth* if the limit exists for each . It is also said to be *uniformly smooth* if the limit exists uniformly in . Let be a Banach space. The *modulus of smoothness* of is the function defined by . The *modulus of convexity* of is the function defined by . The *normalized duality mapping * is defined by . If is a Hilbert space, then , where is the identity mapping.

If is a reflexive smooth and strictly convex Banach space and is the normalized duality mapping on , then , and , where and are the identity mappings on and . If is a uniformly smooth and uniformly convex Banach space, then is uniformly norm-to-norm continuous on bounded subsets of and is also uniformly norm-to-norm continuous on bounded subsets of .

Let and be Banach spaces, , the operator is said to be *compact* if it is continuous and maps the bounded subsets of onto the relatively compact subsets of ; the operator is said to be *weak to norm continuous* if it is continuous from the weak topology of to the strong topology of .

We also need the following lemmas for the proof of our main results.

Lemma 2.1 (Xu [21]). *Let and be two fixed real numbers. Let be a -uniformly convex Banach space if and only if there exists a continuous strictly increasing and convex function , , such that
**
for all and , where .*

For case , we have

Lemma 2.2 (Chang [22]). *Let be a uniformly convex and uniformly smooth Banach space. The following holds:
*

Next we recall the concept of the generalized -projection operator. Let be a functional defined as follows: where is positive number and is proper, convex, and lower semicontinuous. From definitions of and , it is easy to see the following properties:(1);(2) is convex and continuous with respect to when is fixed;(3) is convex and lower semicontinuous with respect to when is fixed.

*Definition 2.3. *Let be a real Banach space with its dual . Let be a nonempty closed convex subset of . It is said that is the *generalized **-projection operator* if

In this paper, we fixed , we have

For the generalized -projection operator, Wu and Hung [5] proved the following basic properties.

Lemma 2.4 (Wu and Hung [4]). *Let be a reflexive Banach space with its dual and is a nonempty closed convex subset of . The following statements hold:*(1)* is nonempty closed convex subset of for all ;*(2)*if is smooth, then for all , if and only if
*(3)*if is smooth, then for any , , where is the subdifferential of the proper convex and lower semicontinuous functional .*

Lemma 2.5 (Wu and Hung [4]). *If for all , then for any ,
*

Lemma 2.6 (Fan et al. [6]). *Let be a reflexive strictly convex Banach space with its dual and is a nonempty closed convex subset of . If is proper, convex, and lower semicontinuous, then*(1)* is single valued and norm to weak continuous;*(2)*if has the property (h), that is, for any sequence and , imply that , then is continuous.*

Defined the functional by

#### 3. Generalized Projection Algorithms

Proposition 3.1. *Let be a nonempty closed and convex subset of a reflexive strictly convex and smooth Banach space . If for is proper, convex, and lower semicontinuous, then is a solution of (SMVI) equivalent to finding such that
*

*Proof. *From Lemma 2.4 (2) and is a reflexive strictly convex and smooth Banach space, we known that is single valued and for is well defined and single valued. So, we can conclude that Proposition 3.1 holds.

For solving the system of mixed variational inequality (1.1), we defined some projection algorithms as follow.

*Algorithm 3.2. **For an initial point **, define the sequences *,* and ** as follows:*
where .

If , for all , then Algorithm 3.2 reduces to the following iterative method for solving the system of variational inequalities (1.2).

*Algorithm 3.3. **For an initial point **, define the sequences *,* and ** as follows:*
where .

For solving the problem (1.6), we defined the algorithm as follows:

If is a Hilbert space, then Algorithm 3.2 reduces to the following.

*Algorithm 3.4. **For an initial point **, define the sequences *,* and ** as follows:*
where .

If , for all , then Algorithm 3.4 reduces to the following iterative method for solving the problem (1.7) as follows.

*Algorithm 3.5. **For an initial point **, define the sequences *,
* and ** as follows:*
where .

#### 4. Existence and Convergence Analysis

Theorem 4.1. *Let be a nonempty closed and convex subset of a uniformly convex and uniformly smooth Banach space with dual space . If the mapping and which is convex lower semicontinuous mappings for satisfying the following conditions:*(i)*, for all for ;*(ii)* are compact for ;*(iii)* and , for all and ;** then the system of mixed variational inequality (1.1) has a solution and sequences , and defined by Algorithm 3.2 have convergent subsequences , and such that
*

*Proof. *Since is a uniformly convex and uniform smooth Banach space, we known that is bijection from to and uniformly continuous on any bounded subsets of . Hence, for is well-defined and single-value implies that , and are well defined. Let , for any and , we have
By (4.2) and Lemma 2.5, we have
From Lemma 2.2, and for all , , so for , we obtain
Again by Lemma 2.2, for all , and for , we have
In similar way, for all , , and , we also have
It follows from (4.5) and (4.6) that
From (4.5) and (4.6), we compute
This implies that the sequences , and are bounded. For a positive number such that , by Lemma 2.1, for , there exists a continuous, strictly increasing, and convex function with such that for , we have
Applying (4.3), (4.4), and (4.7), we have
Summing (4.10), for , we have
taking , we get
This shows that series (4.12) is converge, we obtain that
From for all , thus and (4.13), we have
By property of functional , we have
Since is bounded sequence and is compact on , then sequence has a convergence subsequence such that
By the continuity of the , we have
Again since are bounded and are compact on , then sequences and have convergence subsequences such that
By the continuity of and , we have
Let
By using the triangle inequality, we have
From (4.15) and (4.17), we have
By definition of , we get
It follows by (4.20) and (4.23), we obtain
In the same way, we also have
By the continuity properties of , and for . We conclude that
This completes of proof.

Theorem 4.2. *Let be a nonempty compact and convex subset of a uniformly convex and uniformly smooth Banach space with dual space . If the mapping and which is convex lower semicontinuous mappings for satisfy the following conditions:*(i)* for all for ;*(ii)* and , for all for ;** then the system of mixed variational inequality (1.1) has a solution and sequences , and defined by Algorithm 3.2 have a convergent subsequences , and such that
*

*Proof. *In the same way to the proof in Theorem 4.1, we have
Hence, there exist subsequences and such that
From the compactness of , we have that
where are points in . Also, for a sequence , where is a points in . By the continuity properties of , and , we obtain that
From definition of , we get
By (4.25) and (4.31), we have
This completes of proof.

Corollary 4.3. *Let be a nonempty closed and convex subset of a uniformly convex and uniformly smooth Banach space with dual space . If the mapping for satisfy the following conditions:*(i)*, for all for ;*(ii)* are compact for ;** then the system of mixed variational inequality (1.2) has a solution and sequences , and defined by Algorithm 3.3 have convergent subsequences , and such that , and .*

If is a Hilbert space, then , so one obtains the following corollary.

Corollary 4.4. *Let be a nonempty closed and convex subset of a Hilbert space . If the mapping and which is convex lower semicontinuous mappings for satisfy the following conditions:*(i)* for ;*(ii)* and for all for ;** then the system of mixed variational inequality (1.6) has a solution and sequences , and defined by Algorithm 3.4 have a convergent subsequences , and such that , and .*

Corollary 4.5. *Let be a nonempty closed and convex subset of a Hilbert space . If the mapping for satisfy the conditions: for ; then the system of mixed variational inequality (1.7) has a solution and sequences , and defined by Algorithm 3.5 have a convergent subsequences , and such that , and .*

*Remark 4.6. *Theorems 4.1 and 4.2 and Corollary 4.3 extend and improve the results of Zhang et al. [7] and Wu and Huang [5].

#### Acknowledgments

This research was supported by grant from under the Program Strategic Scholarships for Frontier Research Network for the Joint Ph.D. Program Thai Doctoral degree from the Office of the Higher Education Commission, Thailand. Furthermore, we would like to thank the King Mongkuts Diamond Scholarship for Ph.D. Program at King Mongkuts University of Technology Thonburi (KMUTT) and the National Research University Project of Thailand’s Office of the Higher Education Commission (under CSEC project no.54000267) for their financial support during the preparation of this paper. P. Kumam was supported by the Commission on Higher Education and the Thailand Research Fund (Grant no. MRG5380044).

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