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International Journal of Analysis
Volume 2013 (2013), Article ID 762380, 7 pages
http://dx.doi.org/10.1155/2013/762380
Research Article

Vector Variational-Like Inequalities with Generalized Semimonotone Mappings

Department of Mathematics, BITS Pilani, Dubai Campus, 345055 Dubai, UAE

Received 14 August 2012; Accepted 20 October 2012

Academic Editor: Ying Hu

Copyright © 2013 Suhel Ahmad Khan. 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

We introduce the concepts of generalized relaxed monotonicity and generalized relaxed semimonotonicity. We consider a class of generalized vector variationa-llike inequality problem involving generalized relaxed semimonotone mapping. By using Kakutani-Fan-Glicksberg’s fixed-point theorem, we prove the solvability for this class of vector variational-like inequality with relaxed monotonicity assumptions. The results presented in this paper generalize some known results for vector variational inequality in recent years.

1. Introduction

Vector variational inequalities were initially introduced and considered by Giannessi [1] in a finite-dimensional Euclidean space in 1980. Due to its wide application the theory of the vector variational inequality is generalized in different directions and many existence results and algorithms for vector variational inequality problems have been established under various conditions; see for examples [27] and references therein.

The concept monotonicity and the compactness operators are very useful in nonlinear functional analysis and its applications. In 1968, Browder [8] first combined the compactness and accretion of operators and posed the concept of a semiaccretive operator. Motivated by this idea, Chen [9] studied the concept of a semimonotone operator, which combines the compactness and monotonicity of an operator and posed it to the study of variational inequalities. Recently in 2003, Fang and Huang [10] introduced relaxed --semimonotone mapping, a generalized concept of semimonotonicity, and they established several existence results for the variational-like inequality problem.

In this paper, we pose two new concepts of generalized relaxed monotonicity and generalized relaxed semimonotonicity as well as two classes of generalized vector variational-like inequalities with generalized relaxed monotone mappings and generalized relaxed semimonotone mappings. We investigate the solvability of vector variational-like inequalities with generalized relaxed semimonotone mappings by means of the Kakutani-Fan-Glicksberg fixed-point theorem. The results presented in this paper generalize the results of Chen [9], Fang and Huang [10], Usman and Khan [11], and Zheng [7].

2. Preliminaries

Throughout the paper unless otherwise specified, let and be two real Banach spaces, be a nonempty closed and convex subset of . is said to be a closed convex and pointed cone with its apex at the origin, if the following conditions hold: (i), for all , (ii),(iii).

The partial order in , induced by the pointed cone , is defined by declaring if and only if for all in . An ordered Banach space is a pair with the partial order induced by . The weak order in an ordered Banach space with is defined as if and only if for all in , where denotes the interior of . Let denote the space of all continuous linear mappings from into . A set-valued mapping be such that for each is a proper, closed, convex cone with and let .

First, we give the concept of generalized relaxed monotone mapping. In order to do so, the definition of a vector monotone mapping is needed, which was posed by Chen [9].

Definition 1 (see [9]). Let be a mapping, be a nonempty, closed and convex subset in . Let be such that for each is a proper, closed, convex cone with . is said to be -monotone on if and only if it satisfies the following condition: where .

We now give the concept of generalized relaxed monotone mapping.

Definition 2.  Let be such that for each is a proper, closed, convex cone with and a mapping . A mapping is said to be generalized relaxed monotone, if where is a mapping such that .

Remark 3. (i) If , where with for , then inclusion (2) reduces to Also if , for all , then . As a result the above inclusion reduces to then is said to be relaxed --monotone, introduced and studied by Fang and Huang [10].
(ii) In the above inclusion (5), if we take , for all , then it reduces to then is said to be -relaxed monotone.
(iii) In the above inclusion (6), if we take , where is constant, then it reduces to
Then is called -monotone, see for example [1214].

We recall the following concepts and results which are needed in the sequel.

Definition 4. A mapping is said to be (i)-convex, if , for all ;(ii)-concave, if is -convex.

Definition 5. A mapping is said to be affine in the first argument if for any and , with and any ,

Lemma 6 (see [3]). Let be an ordered Banach space with a proper, closed, convex and solid cone then for all , one has (i) and ,(ii) and .

Lemma 7 (see [15]). Let be a subset of a topological vector space and let be a KKM mapping. If for each , is closed and for at least one , is compact, then

Definition 8. A mapping is said to be -hemicontinuous, if for any , the mapping is continuous at .

Now, we have the following Minty’s type Lemma.

Lemma 9.  Let be a nonempty, closed, and convex subset of . Let be such that for each is a proper, closed, convex cone with . Let be -convex in the first argument with the condition , for all . Suppose the following conditions hold: (i) is an affine in the first argument with the condition , for all ; (ii)the set-valued mapping defined as for all is weakly upper semicontinuous; (iii) is -hemicontinuous and generalized relaxed monotone mapping.
Then the following two problems are equivalent: , for  all ,, for  all .

Proof. Following the lines of proof given by Chen [9], one can easily prove.

Theorem 10. Let be real reflexive Banach space, and be a Banach space. Let be a nonempty, bounded, closed, and convex subset of . Let be such that for each is a proper, closed, convex cone with . Let be -convex and upper semicontinuous in the first and second arguments, respectively, with the condition for all . Let conditions (i)–(iii) of Lemma 9 be satisfied and suppose the following conditions hold: (i) is lower semicontinuous in the second argument; (ii) is weakly lower semicontinuous and -convex in the first and second arguments, respectively. Then there exist , such that

Proof. Define two set-valued mappings as follows: Then and are nonempty since . We claim that is a KKM mapping. If this is not true, then there exist a finite set and with such that . Now by the definition of , we have Now we have which leads to a contradiction since is a proper cone. Thus our claim is verified. So is a KKM mapping.
Now we prove for every . Indeed let . Then, we have Since is generalized relaxed monotone, we have From Lemma 6, we have that is, , which follows , for each and so is also a KKM mapping. Now we claim that for each is closed in the weak topology of .
Indeed suppose , the weak closure of . Since is reflexive, there is a sequence in such that converges weakly to . Then which implies that Since and are lower and upper semicontinuous and and are weakly lower semicontinuous, therefore Thus we get and so . This shows that is weakly closed for each . Our claim is then verified. Since is reflexive and is nonempty, bounded, closed, and convex, is a weakly compact subset of and so is also weakly compact. According to Lemma 7, This implies that there exists such that Therefore by Minty’s type Lemma 9, we conclude that there exist such that This completes the proof.

3. Main Results

Throughout this section, let be real reflexive Banach space, a Banach space. Let be a nonempty, bounded, closed, and convex subset of . Let be such that for each is a proper, closed, convex cone with and let .

Some nonlinear mapping consisting with two variables, may be monotone with respect to the first variable and compact with respect to the second one. However, we cannot always expect for them to be monotone or compact with respect to the two variables simultaneously. Keeping this complexity in mind we are interested in the so called semimonotone mapping. We now give the concept of a generalized relaxed semimonotone mapping.

Definition 11. Let be a nonempty, closed, and convex subset of . Let be a mapping and let be a bifunction such that and . A mapping is said to be generalized relaxed semimonotone mapping, if the following conditions hold: (i)for every is a generalized relaxed monotone mapping, that is, (ii)for every is completely continuous, that is, when (by the norm of operators), where denotes the weak convergence.

Remark 12. When , and , for  all and , then this is exactly the concept which was introduced by Chen [9].

Example 13. Let . Let be defined by So Let be defined by Also , are define by , for  all and , for  all , respectively.
Let us suppose is defined by First, we show that is a generalized relaxed monotone mapping. Indeed, for each , it follows that
The norm of is defined as , for  all , and .
Now, for fixed , if , and , it is easy to prove that Hence, for every is completely continuous.
Therefore, the mapping defined as above is a generalized relaxed semimonotone mapping.

Now we will pose the main problem of our study. In this paper, we investigate the following generalized vector variational-like inequality problem (for short, GVVLIP) is to find a vector satisfying where is a nonlinear mapping and , are the two vector-valued bimappings.

The GVVLIP (31) includes many variational inequality problems as special cases.

Some special cases of GVVLIP (31) are as follows. (I)If and , then the GVVLIP (31) reduces to the following variational inequality problem of finding such that which was introduced and studied by Zheng [7] (II)If , for  all . Also if and is the dual space of , then the GVVLIP (31) reduces to the following variational inequality problem of finding such that which was introduced and investigated by Chen [9]. He obtained some existence results and discussed their applications in partial differential equations of divergence form.

We recall the following fixed-point theorem, by Zeidle [16], which will play an important role in establishing our existing results for GVVLIP (31).

Theorem 14 (see [16]). The set-valued mapping has a fixed point, if the following conditions are satisfied: (1) is compact, convex, and nonempty set in locally convex space; (2) is convex, closed, and nonempty for every ; (3) is upper semicontinuous on .

Now, we have the following existence results for GVVLIP (31) involving a generalized relaxed semimonotone mapping in reflexive Banach spaces.

Theorem 15. Let be real reflexive Banach space, and let be a Banach space. Let be a nonempty, bounded, closed, and convex subset of . Let be such that for each is a proper, closed, convex cone with . Let be a generalized relaxed semimonotone mapping. Suppose the following conditions hold: (i) is -convex with , for  all and upper semicontinuous in the second argument; (ii) is an affine mapping with , for  all and lower semicontinuous in the second argument; (iii)The set-valued mapping defined as , for  all , is weakly upper semicontinuous and concave; (iv) is -concave and weakly lower semicontinuous in the second and first argument, respectively; (v)For each fixed , is continuous on each finite dimensional subspace of . Then there exists a such that

Proof. Let be a finite dimensional subspace of and . For each , we consider the following generalized vector variational-like inequality problem. Find such that Since is bounded, closed, and convex, is continuous on and generalized relaxed monotone for each fixed . From Theorem 10, we know our problem has solution .
Define a set-valued mapping as follows: It follows from Minty’s type Lemma 9, that for each fixed Now we will use the fixed-point theorem to verify the existence of the solution of the problem in a finite dimensional. Since is of finite dimensional, hence is compact. First, we claim that is convex. Indeed, let and , such that Since is affine and are -convex, then from preceding two inclusions we have Since is concave, we have , that is, is convex and our claim is then verified. Now, we claim that is closed. Let such that , then Since is upper semicontinuous, also and is upper semicontinuous; therefore This implies ; hence is closed. Next, we claim that is upper semicontinuous. Let , and , then we have From the complete continuity of and the lower semicontinuity of , we have which implies that that is, ; thus our claim is then verified. Hence is upper semicontinuous. By Fan-Glicksberg fixed-point theorem, there exists a that is, there exists a such that Now we generalize this result to whole space.
Let and let .
From the above we know that . Let denote the weak closure of . For any , we know that . Therefore, . Since is weakly compact, from the finite intersection property, we have . Let , we see that Indeed, for each , let , such that , . From , there exists , that is, such that , from the complete continuity of , upper semicontinuity of , and lower semicontinuity of , we have From Lemma 9, we have This completes the proof.

If the boundedness of is dropped off, then we have the following theorem under certain coercivity condition.

Theorem 16. Let be real reflexive Banach space and be a Banach space. Let be a nonempty, closed, and convex subset of . Let be such that for each is a proper, closed, convex cone with . Let be a generalized relaxed semimonotone mapping. Let conditions (i)–(v) of Theorem 15 be satisfied and then there exist such that

Proof. For each , let denote the closed ball in the Banach space with center and radius . By Theorem 15, for each , there exists such that Since , we have From condition (vi), we know that is bounded. So, we may suppose that converges to as . It follows from Lemma 9, that Since is completely continuous, , are lower semicontinuous, and are upper semicontinuous. Therefore, it follows that Again from Lemma 9, we get This completes the proof.

Remark 17. Theorems 15 and 16 improve and generalize Theorems 3.1 and 3.2 of Fang and Huang [10] and Theorem 2.1 to Theorem 2.6 of Chen [9].

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