# Existence Results for System of Variational Inequality Problems with Semimonotone Operators

- Somyot Plubtieng
^{1}Email author and - Kamonrat Sombut
^{1}

**2010**:251510

https://doi.org/10.1155/2010/251510

© Somyot Plubtieng and Kamonrat Sombut. 2010

**Received: **14 July 2010

**Accepted: **29 October 2010

**Published: **31 October 2010

## Abstract

We introduce the system of variational inequality problems for semimonotone operators in reflexive Banach space. Using the Kakutani-Fan-Glicksberg fixed point theorem, we obtain some existence results for system of variational inequality problems for semimonotone with finite-dimensional continuous operators in real reflexive Banach spaces. The results presented in this paper extend and improve the corresponding results for variational inequality problems studied in recent years.

## Keywords

## 1. Introduction

*The classical variational inequality problem*, denoted by , is to find such that

The variational inequality problem (VIP) has been recognized as suitable mathematical models for dealing with many problems arising in different fields, such as optimization theory, game theory, economic equilibrium, mechanics. In the last four decades, since the time of the celebrated Hartman Stampacchia theorem (see [1, 2]), solution existence of variational inequality and other related problems has become a basic research topic which continues to attract attention of researchers in applied mathematics (see, e.g., [3–14] and the references therein). Related to the variational inequalities, we have the problem of finding the fixed points of the nonexpansive mappings, which is the current interest in functional analysis. It is natural to consider a unified approach to these different problems; see, for example, [10, 11].

There are many kinds of mappings in the literature of recent years; see, for example, [12, 13, 15–18]. In 1999, Chen [19] introduced the concept of semimonotonicity for a single valued mapping, which occurred in the study of nonlinear partial differential equations of divergence type. Recently, Fang and Huang [20] introduced two classes of variational-like inequalities with generalized monotone mappings in Banach spaces. Using the KKM technique, they obtained the existence of solutions for variational-like inequalities with relaxed monotone mappings in reflexive Banach spaces. Moreover, they present the solvability of variational-like inequalities with relaxed semimonotone mappings in arbitrary Banach spaces by means of the Kakutani-Fan-Glicksberg fixed point theorem.

On the other hand, some interesting and important problems related to variational inequalities and complementarity problems were considered in recent papers. In 2004, Cho et al. [21], introduced, and studied a system of nonlinear variational inequalities. They proved the existence and uniqueness of solution for this problem and constructed an iterative algorithm for approximating the solution of system of nonlinear variational inequalities. In 2000, [22], systems of variational inequalities were introduced and an existence theorem was obtained by Ky Fan lemma. In 2002, Kassay et al. [23] introduced and studied Minty and Stampacchia variational inequality systems by the Kakutani-Fan-Glicksberg fixed point theorem. Very recently, Fang and Huang [20] introduced and studied systems of strong implicit vector variational inequalities by the same fixed point theorem. Zhao and Xia [24] introduced and established some existence results for systems of vector variational-like inequalities in Banach spaces by also using the Kakutani-Fan-Glicksberg fixed point theorem.

*variational inequality problem*(VIP) is to find such that

*the system of variational inequality problem*(SVIP). Find such that

In particular, if setting
by
for all
, then the system of variational inequality problem reduces to *the variational inequality problem* (VIP).

In this paper, we introduce the system of generalized variational inequality in real reflexive Banach space. By using the Kakutani-Fan-Glicksberg fixed point theorem, we obtain some existence results for system of generalized variational inequality for semimonotone and finite dimensional continuous in real reflexive Banach spaces. The results presented in this paper extend and improve the corresponding results of Chen [19] and many others.

## 2. Preliminaries

*strictly convex*if, for any ,

*modulus of convexity of*as follows:

exists for all
. It is also said to be *uniformly smooth* if the limit (2.4) is attained uniformly for
. We recall that
is uniformly convex if and only if
is uniformly smooth. It is well known that
is smooth if and only if
is strictly convex.

Definition 2.1 (KKM mapping).

where denotes the convex hull of .

Lemma 2.2 (Fan-KKM Theorem).

Let be a nonempty convex subset of a Hausdorff topological vector space , and let be a KKM mapping with closed values. If there exists a point such that is a compact subset of , then .

Definition 2.3.

Let and be two topological vector spaces and a nonempty convex subset of . A set-valued mapping is said to be properly -quasiconvex if, for any and , we have either or .

Definition 2.4.

Let and be two topological vector spaces, and be a set-valued mapping.

(i) is said to be upper semicontinuous at if, for any open set containing , there exists an open set containing such that, for all , ; is said to be upper semicontinuous on if it is upper semicontinuous at all .

(ii) is said to be lower semicontinuous at if, for any open set with , there exists an open set containing such that, for all , ; is said to be lower semicontinuous on if it is lower semicontinuous at all .

(iii) is said to be continuous on if it is at the same time upper semicontinuous and lower semicontinuous on .

(iv) is said to be closed if the graph, , of , that is, , is a closed set in .

Lemma 2.5 (see [25]).

Let and be two Hausdorff topological vector spaces, and let be a set-valued mapping. Then the following properties hold.

(i)If is closed and is compact, then is upper semicontinuous, where and denotes the closure of the set .

(ii)If is upper semicontinuous and, for any , is closed, then is closed.

(iii) is lower semicontinuous at if and only if for any and any net , there exists a net such that and .

Lemma 2.6 (see Kakutani-Fan-Glicksberg [26]).

Let be a nonempty compact subset of locally convex Hausdorff vector topology space . If is upper semicontinuous and, for any , is nonempty convex and closed, then there exists an such that .

Lemma 2.7 (see [27]).

*monotone*if it satisfies

Definition 2.8 (see [19]).

A mapping
is said to be *semimonotone* if it satisfies the following:

(a)for each , is monotone; that is, , for all ;

(b)For each fixed , is completely continuous; that is, if in weak topology of , then has a subsequence in norm topology of .

An operator
is said to be *hemicontinuous at*
, if, for any
,
with
, we have
for all
at
.

Lemma 2.9.

Proof.

This completes the proof.

## 3. The Existence of the System of Generalized Variational Inequality

In this section, we prove two existence theorems for system of variational inequality problems for semimonotone with finite dimensional continuous operators in real reflexive Banach spaces. First, we prove an existence theorem for system of variational inequality problems for continuous mappings as follows.

Theorem 3.1.

Let be a reflexive Banach spac, let be a compact convex subset of , and let be two continuous mappings. Then the problem (1.2) has a solution and the set of solutions of (1.2) is closed.

Proof.

Step 1 (Show that and are nonempty compact convex subsets of ).

For any , we note that and . Thus, and are nonempty subsets of . Moreover, it follows from the definitions of and that both of them are compact convex subsets of .

Step 2 (Show that and are KKM mappings).

So, there is at least one number such that . Therefore, . Similarly, we obtain that . Hence, we have and . This implies that and are KKM mappings.

Step 3 (Show that and are closed for all ).

Since and are continuous, we have for all . Thus, we see that . This implies that is closed for all . Similarly, we note that is closed for all .

Since and are closed subsets of and is compact, it follows that and are compact subsets of . It follows from Lemma 2.2 that . Moreover, we note that and are closed and convex.

Step 5 (Show that the problem (1.2) has a solution).

Hence, are the solutions of problem (1.2).

Step 6 (Show that the set of solutions of problem (1.2) is closed).

This mean that belongs to the set of solution of problem (1.2). Hence, the set of solution of problem (1.2) is closed set. This completes the proof.

Theorem 3.2.

Let be a nonempty bounded closed convex subset of a real reflexive Banach space with dual space . Suppose that is a lower semicontinuous convex function with . Let and be two mappings satisfying the following conditions:

(i)for each , and are monotone;

(ii)for each , and are completely continuous;

(iii)for any given , and are finite dimensional continuous; that is, for any finite dimensional subspace , and are continuous.

Proof.

Next, we will show that this mapping has at least one fixed point in . To prove this, we need the following conditions.

(1)For all , as has a solution.

Since is bounded, it is obvious that is bounded.

(5)The mapping is upper semicontinuous.

By the completely continuity of , and being semicontinuous, we note that is upper semicontinuous.

By (3.10), we know that is nonempty bounded. We observe that , where is the weak* closure of in . Since is reflexive, it follows that is weak* compact. For any , we note that . So has the finite intersection property. Therefore, it follows that . Let , then we have .

This completes the proof.

Setting in Theorem 3.2, we have the following result.

Corollary 3.3.

Let be a nonempty bounded closed convex subset of a real reflexive Banach space with dual space . Let be two mappings satisfying the following conditions:

(i)for each , and are monotone;

(ii)for each , and are completely continuous;

(iii)for any given , and are finite dimensional continuous; that is, for any finite dimensional subspace , and are continuous.

Corollary 3.4 (see [19]).

Proof.

Next, we consider the system of generalized variational inequality in which is unbounded. We have the following result.

Theorem 3.5.

Let be a real reflexive Banach space with dual space , and let be a nonempty unbounded closed convex subset with . Suppose that is a lower semicontinuous convex function with . Let , be two mappings satisfying the following conditions:

(i)for each , and are monotone;

(ii)for each , and are completely continuous;

(iii)for any given , and are finite dimensional continuous;

Proof.

This completes the proof.

Setting in Theorem 3.5, we have the following result.

Corollary 3.6.

Let be a real reflexive Banach space with dual space , and let be a nonempty unbounded closed convex subset with . Let be two mappings satisfying the following conditions:

(i)for each , and are monotone;

(ii)for each , and are completely continuous;

(iii)for any given , and are finite dimensional continuous; that is, for any finite dimensional subspace , and are continuous;

Similarly as in the proof of Corollary 3.4, we have the following result.

Corollary 3.7 (see [19]).

## Declarations

### Acknowledgments

The first author is thankful to the Thailand Research Fund for financial support under Grant BRG5280016. Moreover, the second author would like to thank the Office of the Higher Education Commission, Thailand for supporting by grant fund under Grant CHE-Ph.D-THA- SUP/86/2550, Thailand.

## Authors’ Affiliations

## References

- Kinderlehrer D, Stampacchia G:
*An Introduction to Variational Inequalities and their Applications, Pure and Applied Mathematics*.*Volume 88*. Academic Press, New York, NY, USA; 1980:xiv+313.MATHGoogle Scholar - Hartman P, Stampacchia G: On some non-linear elliptic differential-functional equations.
*Acta Mathematica*1966, 115: 153–188.MathSciNetView ArticleMATHGoogle Scholar - Chang S-s, Joseph Lee HW, Chan CK: A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization.
*Nonlinear Analysis: Theory, Methods & Applications*2009, 70(9):3307–3319. 10.1016/j.na.2008.04.035MathSciNetView ArticleMATHGoogle Scholar - Chang S-s, Lee BS, Chen Y-Q: Variational inequalities for monotone operators in nonreflexive Banach spaces.
*Applied Mathematics Letters*1995, 8(6):29–34. 10.1016/0893-9659(95)00081-ZMathSciNetView ArticleMATHGoogle Scholar - Fang YP, Huang NJ: Variational-like inequalities with generalized monotone mappings in Banach spaces.
*Journal of Optimization Theory and Applications*2003, 118(2):327–338. 10.1023/A:1025499305742MathSciNetView ArticleMATHGoogle Scholar - Huang N-J, Fang Y-P: Fixed point theorems and a new system of multivalued generalized order complementarity problems.
*Positivity*2003, 7(3):257–265. 10.1023/A:1026222030596MathSciNetView ArticleMATHGoogle Scholar - Isac G:
*Complementarity Problems, Lecture Notes in Mathematics*.*Volume 1528*. Springer, Berlin, Germany; 1992:vi+297.MATHGoogle Scholar - Isac G, Sehgal VM, Singh SP: An alternate version of a variational inequality.
*Indian Journal of Mathematics*1999, 41(1):25–31.MathSciNetMATHGoogle Scholar - Junlouchai P, Plubtieng S: Existence of solutions for generalized variational inequality problems in Banach spaces.
*Nonlinear Analysis, Theory Methods & Applications*2011, 74(3):999–1004. 10.1016/j.na.2010.09.058MathSciNetView ArticleMATHGoogle Scholar - Nadezhkina N, Takahashi W: Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings.
*Journal of Optimization Theory and Applications*2006, 128(1):191–201. 10.1007/s10957-005-7564-zMathSciNetView ArticleMATHGoogle Scholar - Takahashi W, Toyoda M: Weak convergence theorems for nonexpansive mappings and monotone mappings.
*Journal of Optimization Theory and Applications*2003, 118(2):417–428. 10.1023/A:1025407607560MathSciNetView ArticleMATHGoogle Scholar - Verma RU: On a new system of nonlinear variational inequalities and associated iterative algorithms.
*Mathematical Sciences Research Hot-Line*1999, 3(8):65–68.MathSciNetMATHGoogle Scholar - Verma RU: Iterative algorithms and a new system of nonlinear quasivariational inequalities.
*Advances in Nonlinear Variational Inequalities*2001, 4(1):117–124.MathSciNetMATHGoogle Scholar - Yao J-C, Chadli O: Pseudomonotone complementarity problems and variational inequalities. In
*Handbook of Generalized Convexity and Generalized Monotonicity*.*Volume 76*. Edited by: Crouzeix JP, Haddjissas N, Schaible S. Springer, New York, NY, USA; 2005:501–558. 10.1007/0-387-23393-8_12View ArticleGoogle Scholar - Ansari QH, Yao J-C: A fixed point theorem and its applications to a system of variational inequalities.
*Bulletin of the Australian Mathematical Society*1999, 59(3):433–442. 10.1017/S0004972700033116MathSciNetView ArticleMATHGoogle Scholar - Ansari QH, Yao J-C: Systems of generalized variational inequalities and their applications.
*Applicable Analysis*2000, 76(3–4):203–217. 10.1080/00036810008840877MathSciNetView ArticleMATHGoogle Scholar - Ansari QH, Schaible S, Yao J-C: The system of generalized vector equilibrium problems with applications.
*Journal of Global Optimization*2002, 22(1–4):3–16.MathSciNetView ArticleMATHGoogle Scholar - Verma RU: Generalized system for relaxed cocoercive variational inequalities and projection methods.
*Journal of Optimization Theory and Applications*2004, 121(1):203–210.MathSciNetView ArticleMATHGoogle Scholar - Chen Y-Q: On the semi-monotone operator theory and applications.
*Journal of Mathematical Analysis and Applications*1999, 231(1):177–192. 10.1006/jmaa.1998.6245MathSciNetView ArticleMATHGoogle Scholar - Fang Y-P, Huang N-J: Existence results for systems of strong implicit vector variational inequalities.
*Acta Mathematica Hungarica*2004, 103(4):265–277.MathSciNetView ArticleMATHGoogle Scholar - Cho YJ, Fang YP, Huang NJ, Hwang HJ: Algorithms for systems of nonlinear variational inequalities.
*Journal of the Korean Mathematical Society*2004, 41(3):489–499.MathSciNetView ArticleMATHGoogle Scholar - Kassay G, Kolumbán J: System of multi-valued variational inequalities.
*Publicationes Mathematicae Debrecen*2000, 56(1–2):185–195.MathSciNetMATHGoogle Scholar - Kassay G, Kolumbán J, Páles Z: Factorization of Minty and Stampacchia variational inequality systems.
*European Journal of Operational Research*2002, 143(2):377–389. 10.1016/S0377-2217(02)00290-4MathSciNetView ArticleMATHGoogle Scholar - Zhao Y, Xia Z: Existence results for systems of vector variational-like inequalities.
*Nonlinear Analysis: Real World Applications*2007, 8(5):1370–1378. 10.1016/j.nonrwa.2005.10.007MathSciNetView ArticleMATHGoogle Scholar - Aubin J-P, Ekeland I:
*Applied Nonlinear Analysis*. John Wiley & Sons, New York, NY, USA; 1984:xi+518.MATHGoogle Scholar - Holmes RB:
*Geometric Functional Analysis and Its Application*. Springer, New York, NY, USA; 1975:x+246.View ArticleGoogle Scholar - Yamada Y: On evolution equations generated by subdifferential operators.
*Journal of the Faculty of Science of Tokyo*1975, 4: 491–502.MATHGoogle Scholar

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