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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 439394, 7 pages
http://dx.doi.org/10.1155/2013/439394
Research Article

Vector and Ordered Variational Inequalities and Applications to Order-Optimization Problems on Banach Lattices

1Department of Mathematics, Lishui University, Lishui, Zhejiang 323000, China
2Department of Mathematics, Shawnee State University, Portsmouth Ohio 45662, USA
3Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321000, China

Received 28 May 2013; Accepted 15 July 2013

Academic Editor: Shih-sen Chang

Copyright © 2013 Linsen Xie 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

We investigate the connections between vector variational inequalities and ordered variational inequalities in finite dimensional real vector spaces. We also use some fixed point theorems to prove the solvability of ordered variational inequality problems and their application to some order-optimization problems on the Banach lattices.

1. Introduction

Let be a real Banach space with its norm dual . Let be a nonempty convex subset of and a single valued mapping. The variational inequality problem associated with and , simply denoted as VI, is to find an such that The variational inequality problem VI has been extensively studied by many authors. This theory has been widely applied to optimization theory, game theory, economic equilibrium, mechanics, and so forth. It has been recognized as an important branch in nonlinear analysis (see, e.g., [15]).

The theory of variational inequality was first extended by Giannessi [6] in 1980 to vector variational inequality problems in finite dimensional vector spaces. Since then, it has been deeply studied by many authors such as Giannessi [610], Agarwal et al. [11], Mastroeni and Pellegrini [12], Li and Huang [13], Luc [14], and Facchinei and Pang [15] and also has been applied to many fields such as vector optimization problems and vector equilibrium problems. In 2005, Huang and Fang [10] generalized the vector variational inequality problems from finite dimensional vector spaces to the Banach spaces. We recall the extension as follows.

For any positive integer , let denote the -dimensional Euclidean space. Let be a convex and pointed cone of and let be the partial order on induced by . Let be a matrix-valued function. Let be a closed convex subset of . The vector variational inequality problem associated with , , and is to find an such that As a special case, one may consider finding an such that In some economics circumstances, the preferences of a certain type of outcomes may be described by a special partial order, in particular a lattice order, on the space of outcomes. In this case, any preference inequalities and optimal problems must be defined under the given partial order that describes the preferences. Based on this motivation, Li and Wen [16] recently extended the variational inequality problem (1) to the following case: let be a Banach space and let be a Banach lattice, where is considered as the income domain and is considered as the production outcome space. Let be a nonempty closed convex subset of and let be a mapping from to . Then the ordered variational inequality problem associated with , , and , denoted by VOI, is to find an such that More general, similar to problem (2) is to find an such that From the Choquet-Kendall theorem, for a given convex and pointed cone in the -dimensional Euclidean space , the partial order on induced by may not be a lattice order on . Furthermore, we can provide counterexamples to show that even in the case that on induced by is a lattice order on , may not be a Banach lattice. Hence problems (5) and (4) are generalizations of (2) and (3), respectively, only if is considered as a Banach lattice. In this paper, we investigate the connections between problems (2), (3) and problems (4), (5).

In [16], Li and Wen proved some solvability theorems about the ordered variational inequality problems (4) and (5) by applying the KKM mappings and the Fan-KKM theorem. In this paper, we use some fixed point theorem to prove the solvability of the ordered variational inequality problems (4) and (5) and the existence of solutions to some order-optimization problems in the Banach lattices.

2. Preliminaries

In this section, we recall some concepts and properties of the Banach lattices. These properties will be frequently used throughout this paper. For more details, the readers are referred to [17]. We also recall the concept of vector variational inequality problems in the -dimensional Euclidean space and the variational order inequality problems in the Banach lattices.

A Banach space equipped with a lattice order is called a Banach lattice, which is written as , if the following properties hold: (b1) implies , for all ,(b2) implies , for all and ,(b3) implies , for every ,where, as usual, the origin of is denoted by 0, , , and , for all , and denotes the norm of the Banach space . Let be a Banach lattice. The positive cone of (), denoted by , is the subset . Analogously, we define the negative cone of , denoted by , as the subset . Then the properties of the lattice order in immediately imply From Section 4.1 in [17], we see that every normed Riesz space is a special example of locally convex-solid Riesz spaces. Meanwhile, the Banach lattices are normed Riesz spaces; therefore, as a special case of Theorem 3.46 [17], we have the following closeness properties for the positive cones of the Banach lattices, which are very useful in the contents of this paper.

Theorem 1 (see [16, 17]). Both of the positive and negative cones of any Banach lattice are norm closed, so they are weakly closed.

Lemma 2 (see [16]). Let be an arbitrary Banach lattice. Then both of the positive and negative cones X+ and X- are order-closed. That is, for any net in (or in ), which order-converges to x, (or in ).

For any pair of elements and in a Banach lattice , we say that whenever and . Define the strictly positive and the strictly negative cones of the Banach lattice , respectively, by

These two cones possess the following properties:

In general, both of and are neither closed nor open in . This can be demonstrated by the following example.

Example 3. Let be the Hilbert lattice with the coordinate lattice order on ; that is, for any pair of points , if and only if and . The positive cone of is the set and , which is closed in . The strictly positive cone of is the set and , which is neither closed nor open in . Similar to , the strictly negative cone of is the set and , which is neither closed nor open in .
Any two elements and in a Banach lattice are said to be noncomparable under the order , if neither nor holds, which is denoted by . The nonzero-comparable set of a Banach lattice , denoted by , is the subset .

From Theorem 3.46 [17] and Lemma 2.3 [16] recalled in this section, the union of the positive and negative cones of a Banach lattice is norm closed. It immediately implies the following lemma.

Lemma 4. The nonzero-comparable set of a Banach lattice is open in .

Proof. It is clear that . Then the lemma follows from that both are closed.

Definition 5. Let be a Banach space and let be a Banach lattice. Let be a nonempty convex subset of and a mapping, where denotes the Banach space of continuous linear operators from to . The ordered variational inequality problem associated with , , and , denoted by VOI, is to find an such that where, as usual, 0 denotes the origin of . If is linear, then the problem VOI is called a linear ordered variational inequality problem; otherwise, it is called a nonlinear ordered variational inequality problem.

Many authors study the solvability of variational inequalities by applying the Fan-KKM theorem and fixed point theorems (see [18]). In [16], Li and Wen used the Fan-KKM theorem to prove the existence of solutions for some ordered variational inequalities. In this paper, we use fixed point theorems to show the solvability of some ordered variational inequalities. We first recall the concept of upper semicontinuous mappings from some topological spaces to topological spaces.

Definition 6 (see [11]). Let be Hausdorff topological spaces. Let be a set valued mapping. For a point , if, for any neighborhood of the set in , there exists a neighborhood of the point in , such that then is said to be upper semicontinuous at point . If is upper semicontinuous at every point in , then is said to be upper semicontinuous on .

The following theorem provides a criterion to check if a mapping is upper semi-continuous, which is useful in the following contents (see Agarwal et al. [11], for details).

Theorem 7 (see [11]). Let be Hausdorff topological spaces with compact. Let be a set valued mapping. If has a closed graph, then is upper semi-continuous on .

Bohnenblust-Karlin Fixed Point Theorem (1950). Let be a Banach space and a nonempty compact convex subset of . Let be an upper semi-continuous mapping with closed convex values. Then has a fixed point.

3. Connections between Vector Variational Inequalities and Ordered Variational Inequalities

In finite dimensional real vector spaces, the vector variational inequalities defined by (2) and (3) and the ordered variational inequalities defined in (9) are all generalizations of variational inequalities defined in (1). In this section, we investigate the connections between these generalizations. We first recall a useful type of partial order on a vector space that is induced by a cone in this space.

Let be a nonempty subset of a topological linear space . is called a (pointed) cone in if it satisfies the following two conditions:(1) and , for any nonnegative number ;(2).

Let be a closed convex cone in a topological linear space . The partial order on induced by is defined as follows: Hence, the topological linear space equipped with the partial order is a partially ordered topological linear space, which is denoted by and is said to be induced by the cone .

It is important to notice that the partial order on the partially ordered topological linear space is far away from being a lattice order; that is, the partially ordered topological linear space may not be a Riesz space (vector lattice). This can be demonstrated by the following simple example.

Example 8. In , take to be the closed convex cone as follows: We claim that the partial order on is not a lattice order. In fact, we take two points and . One can see that the subset has no upper bound in . It immediately implies that does not exist. Hence is not a lattice order on and is not a vector lattice.
Here we consider some special cones in real vector spaces such that their induced partial order is a lattice order. For more details, the reader is referred to Kendall [19].
Let be a nonempty convex set in a real vector space . The subset of is called the minimal affine extension of . Suppose that the minimal affine extension of does not contain the zero vector. Let . Then is a pointed convex cone in .
Kinderlehrer and Stampacchia [4] calls a nonempty convex set in a real vector space a simplex when the intersection of two positively homothetic images of , is empty or is a set () of the same nature.
Choquet-Kendall Theorem (see [19]). Let   be a nonempty convex set spanning a real vector space   and suppose that its minimal affine extension does not contain the zero vector. Let   be the pointed positive convex cone   having   as a base (of ). Then , partly ordered by   , will be a vector lattice if and only if both ()  is a simplex, ()each B-segment   is contained in a maximal B-segment.Now by applying the Choquet-Kendall theorem, we prove the following theorem in .

Theorem 9. Let be a nonempty convex set spanning , for some positive integer n, and suppose that its minimal affine extension does not contain the zero vector. Let be the pointed positive convex cone, which has B as its base. Then is a Hilbert lattice if and only if, in addition to conditions (1°) and (2°) in the Choquet-Kendall theorem, the following condition also holds:()

Proof. Note that the partial order on induced by the cone always satisfies the first two conditions of Hilbert lattices (b1) and (b2) recalled in Section 2. From the Choquet-Kendall theorem, we only need to show that Now we just prove that previous statement.
” Suppose that the inequality holds for all . Then, for every with , we have and . It implies It implies , which is condition (b3) for the Hilbert lattices.
” Assume that is a Hilbert lattice satisfying that implies , for every . Then, for any , we have Since , the previous order inequalities imply Then from condition (b3) of the Hilbert lattices recalled in Section 2, we have . That is, Calculating the inner products in , it yields . It completes the proof.

The following example shows that condition (3°) in Theorem 9 is necessary for to be a Hilbert lattice.

Example 10. Let be the closed segment in with ending points at and . One can show that is a simplex in under Choquet’s definition (9). Then the closed pointed convex cone generated by is the closed convex cone in bounded by the two rays with origin as the ending point and passing through points and , respectively. It can be seen that is a convex subset spanning and satisfies the two conditions in the Choquet-Kendall theorem. Hence , partly ordered by , is a vector lattice (Riesz space). Take two vectors , in with origin as the initial point and with ending points at and , respectively. It is clear that . So does not satisfy condition (3°) in Theorem 9, and hence is not a Hilbert lattice. To more precisely show that is not a Hilbert lattice, we can show that does not satisfy condition (b3) of the Banach lattices recalled in Section 2. To this end, take to be the vector in with origin as the initial point and with ending points at (in fact, ). We have . It yields that . On the other hand, we see that and . Hence, the vectors and do not satisfy condition (b3) for the Hilbert lattices.
To summarize, we state the connections between vector variational inequalities and ordered variational inequalities as a proposition.

Proposition 11. If is a Hilbert lattice, that is, the convex cone in is generated by a simplex which spans and satisfies the three conditions in Theorem 9, then the ordered variational inequalities (5) and (4) in the Hilbert lattice coincide with vector variational inequalities (2) and (3), respectively.

4. Existence of Solutions of Ordered Variational Inequalities on the Banach Lattices

The convexity and concavity of functions play important roles in nonlinear analysis. In this section, we extend these concepts to order-convexity and order-concavity of mappings in Banach lattices, which will be applied in the proofs of the existence of solutions of ordered variational inequalities in the Banach lattices.

Definition 12. Let be a Banach space and let be a Banach lattice. Let be a nonempty convex subset of . A mapping is said to be (1) order-convex on if, for every pair of points , the following order inequality holds: (2) order-concave on if, for every pair of points , the following order inequality holds: The following result is the main theorem of this paper.

Theorem 13. Let be a Banach space and let be a Banach lattice. Let be a nonempty compact convex subset of . Let be a continuous (with respect to the norms) mapping. If has a lower order closed range, that is, exists and satisfies then the problem VOI has a solution.

Proof. Define a set valued mapping by The lower order closeness of the ranges of the mapping in this theorem implies that , for any . Next, we show that is closed for any . To this end, take any with in , as . Since is compact, then . For every fixed , from , this yields that is; . It is equivalent to , for all . Since , then is a linear and continuous mapping from to . From in , this implies , as . From Theorem 1 recalled in Section 2, is norm closed. It implies that . That is, , for any given . Noticing that , we obtain . Hence and therefore is closed.
To show that is convex, take any and for any from the linearity of and applying the linearity and transitive property of the lattice order in the Banach lattice , we have Since is convex, so . The previous order inequalities imply . Hence, is convex.
Next, we prove that the mapping is a upper semi-continuous set valued mapping. Since is compact, so is a set valued mapping from a compact topological space to . From Theorem 7 recalled in Section 2, it is sufficient to prove that the following set is closed: For this purpose, take any sequence with and in , as . We have Since is compact, then . For every fixed , from , this yields ; that is, . Thus, , for all . From the continuity condition of , since in , as , we have From in , as , and , for all , the previous limit implies Since is norm closed and , for all , this implies . That is, , for any given . The compactness of and in , as , and imply that . Then from , we get . We obtain that , and hence is closed.
To summarize, we obtain that is an upper semi-continuous set valued mapping with nonempty compact convex values. From the Bohnenblust-Karlin fixed point theorem, the set of fixed points of is nonempty. Taking a fixed point of , we have . That is, This implies Noticing that is linear, from the properties of the lattice order , it is equivalent to Hence is a solution to the ordered variational inequality problem VOI. This theorem is proved.

5. Applications to Order-Optimization Problems

As some applications of Theorem 13, we solve some order-optimization problems in the Banach lattices in this section.

Definition 14. Let be a Banach space and let be a Banach lattice. Let be a nonempty subset of . Let be a mapping. is said to take an associated order-minimum value at a point if it satisfies is said to take an associated order-maximum value at a point if it satisfies

From the proof of Theorem 13 and from the linearity of , for all , it is clear that (32) is equivalent to (9). Hence, a point is a solution to the problem VOI if and only if, at the point , takes an associated order-minimum value. So the following corollary is an immediate consequence of Theorem 13.

Corollary 15. Let be a Banach space and let be a Banach lattice. Let be a nonempty compact convex subset of . Let be a continuous (with respect to the norms) mapping. If has a lower order closed range, that is, exists and satisfies then there is a point , at which F takes an associated order-minimum value.

The existence of associated order-maximum value problem is also considered as a consequence of Theorem 13. We provide a solution of this problem as follows as a corollary of Theorem 13.

Corollary 16. Let be a Banach space and let be a Banach lattice. Let be a nonempty compact convex subset of . Let be a continuous (with respect to the norms) mapping. If has an upper order closed range, that is, exists and satisfies then there is a point , at which G takes an associated order-maximum value.

Proof. For the given mapping , define by , for every ,
Then is also a continuous (with respect to the norms) mapping. For every , we have From Corollary 15, there is a point , at which takes an associated order-minimum value; that is, Then we have It is equivalent to Hence, takes an associated order-maximum value at the point .

Acknowledgment

This research was partially supported by National Natural Science Foundation of China (11171137) and the National Natural Science Foundation of China (Grant 11101379).

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