Existence Results for System of Variational Inequality Problems with Semimonotone Operators

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.

1. Introduction

Let be a Banach space, let be the dual space of , and let denote the duality pairing of and . If is a Hilbert space and is a nonempty, closed, and convex subset of , then let, be a nonempty, closed, and convex subset of a Hilbert space and let be a mapping. The classical variational inequality problem, denoted by , is to find such that

(1.1)

for all .

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].

Let be a nonempty, closed, and convex subset of and let be single valued. For the system of generalized variational inequality problem (SGVIP), find such that

(1.2)

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.

Let be a semimonotone mapping and a closed convex set. The variational inequality problem (VIP) is to find such that

(1.3)

Let be a nonempty closed convex subset of a real reflexive Banach space with dual space , and let be two semimonotone mappings. We consider the following as the system of variational inequality problem (SVIP). Find such that

(1.4)

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

In this section, let be a real Banach space, and let be the unit sphere of . A Banach space is said to be strictly convex if, for any ,

(2.1)

It is also said to be uniformly convex if, for each , there exists such that, for any ,

(2.2)

It is known that a uniformly convex Banach space is reflexive and strictly convex, and we define a function called the modulus of convexity of as follows:

(2.3)

Then is uniformly convex if and only if for all . A Banach space is said to be smooth if the limit

(2.4)

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).

Let be a nonempty subset of a linear space . A set-valued mapping is said to be a KKM mapping if, for any finite subset of , we have

(2.5)

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]).

For each , is a Fréchet differentiable convex function on , and the Fréchet derivative of is equal to Yosida approximation of . More precisely,

(2.6)

holds for and .

Let be a nonempty closed convex subset of a real reflexive Banach space with dual space . A mapping is said to be monotone if it satisfies

(2.7)

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.

Let be a hemicontinuous monotone operator, let be a convex subset, and let be a convex function and a given point. Then

(2.8)

if and only if

(2.9)

Proof.

Let such that

(2.10)

By the monotonicity of , we have

(2.11)

On the other hand, suppose that

(2.12)

For any given and any , taking since is convex and Replacing by into the above inequality, one has

(2.13)

It follows from the convexity of on that

(2.14)

Letting and using the hemicontinuity of , we have

(2.15)

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.

Fix , for each , the sets and are defined as follows:

(3.1)

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).

For any finite set , we claim that and . Let . Then , where and . We observe that

(3.2)

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 ).

Let be a sequence in such that . Then

(3.3)

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 .

Step 4 (Show that ).

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).

Define the set-valued mapping by

(3.4)

From Step 4, we note that is nonempty closed convex subset of for all . Since , and is compact, and are compact. It follows from Lemma 2.5(i) that is upper semicontinuous. Hence, by the Kakutani-Fan-Glicksberg theorem, there exists a point ; that is, and for all . By definition of and , we get

(3.5)

Hence, are the solutions of problem (1.2).

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

Let be a net in the set of solutions of problem (1.2) such that . By definition of the set of solutions of problem (1.2) we obtain that

(3.6)

Since are continuous and is compact, it follows that and

(3.7)

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.

Then, there exists such that

(3.8)

Proof.

Let be the subdifferential of and the Yosida approximation of . Let be a finite dimensional subspace of with . For any given , we consider the following systems of variational inequalities . Find such that

(3.9)

Since is nonempty bounded closed convex and and are continuous on for each fixed , it follows from Theorem 3.1 that has a solution . By Lemma 2.9 and the Yosida approximation of , we have

(3.10)

Now, we defined a mapping by the following:

(3.11)

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.

(2) is convex for all .

Let and . Thus, we have

(3.12)

Adding (3.12), we get

(3.13)

It implies that is convex.

(3) is closed for all .

In fact, leting such that , we have

(3.14)

Since , and are continuous and is lower semicontinuous, we get

(3.15)

Thus implies that is closed.

(4) is bounded for .

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.

Hence, by the Kakutani-Fan-Glicksberg fixed point theorem, has a fixed point; that is, there exists such that . Thus, we have

(3.16)

Let is finite dimensional and . For each , we let be the set of all solutions of the following problem. Find such that

(3.17)

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 .

Next, we claim that

(3.18)

Indeed, for each , we choose such that . Since , there exists a sequence such that converge weakly to . This implies that

(3.19)

for all . Since is lower semicontinuous, it follows that is weakly lower semicontinuous. Therefore, by using the completely continuity of and , respectively, and letting , we have

(3.20)

By Lemma 2.9, we have

(3.21)

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.

Then, there exists such that

(3.22)

Corollary 3.4 (see [19]).

Let be a real reflexive Banach space and a bounded closed convex subset. Suppose that is a lower semicontinuous convex function with is semimonotone, and is finite dimensional continuous for each . Then there exists such that

(3.23)

Proof.

Define a mapping by for all . We observe that is monotone and is completely continuous for all . Moreover, is finite dimensional continuous. Therefore, by Theorem 3.2, there exists such that

(3.24)

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;

(iv) and .

Then, there exists such that

(3.25)

Proof.

Let be the closed ball in at center zero with radius such that . By Theorem 3.2, there exists such that

(3.26)

Leting in (3.26), we get

(3.27)

which implies that

(3.28)

By condition , we know that is bounded. So, we may assume that converge weakly to as . From (3.26), it follows by Lemma 2.9 that

(3.29)

Since are complete continuous and is weakly lower semicontinuous, it follows by letting that

(3.30)

Using Lemma 2.9 again, we obtain

(3.31)

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;

(iv) and

Then, there exists such that

(3.32)

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

Corollary 3.7 (see [19]).

Let be a real reflexive Banach space and an unbounded closed convex subset with . Suppose that is a lower semicontinuous convex function with , is semimonotone, and is finite dimensional continuous for each . Assume that the following condition holds:

(3.33)

Then there exists , such that

(3.34)