A New Exceptional Family of Elements and Solvability of General Order Complementarity Problems

Huang, Na; Ma, Changfeng

doi:https://doi.org/10.1155/2014/248462

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Research Article | Open Access

Volume 2014 | Article ID 248462 | https://doi.org/10.1155/2014/248462

A New Exceptional Family of Elements and Solvability of General Order Complementarity Problems

Na Huang¹and Changfeng Ma¹

Academic Editor: Abdellah Bnouhachem

Received28 Mar 2014

Accepted13 May 2014

Published22 May 2014

Abstract

By using the concept of exceptional family, we propose a sufficient condition of a solution to general order complementarity problems (denoted by GOCP) in Banach space, which is weaker than that in Németh, 2010 (Theorem 3.1). Then we study some sufficient conditions for the nonexistence of exceptional family for GOCP in Hilbert space. Moreover, we prove that without exceptional family is a sufficient and necessary condition for the solvability of pseudomonotone general order complementarity problems.

1. Introduction

There are several types of order complementarity problems in real world applications. Among them, the linear order complementarity problem was systematically studied (see [1]). The problem was extended to the general linear order complementarity problem and some interesting results have been presented (see [2–4]). In [4], Sznajder extended the linear order complementarity problem to the nonlinear order complementarity problem. The notion of the general order complementarity problem considered in this paper is taken from [3, 5, 6].

There are many problems in engineering, management science, and other fields which can be reformulated as general order complementarity problems. But we are interested in the solvability of the problem. The concept of exceptional family is a powerful tool to study existence theorems of the solution to nonlinear complementarity problems and variational inequality problems (see [7–15]). Smith first introduced in [16] the notion of exceptional sequence of elements for continuous functions in order to investigate the solution existence of nonlinear complementarity problems. In 1997, Zhao first extended the concept of exceptional family for variational inequalities (see [17]). Several years later, Isac and Zhao extended the concept of exceptional family to variational inequalities in to general Hilbert space (see [18]). Using the more general notion of exceptional family of elements introduced by Isac et al. (see [19]) and Kalashnikov (see [20]), some existence theorems for complementarity problems are presented (see [19, 21]). In 2008, Zhang proposed an existence theorem for semidefinite complementarity problem (denoted by SDCP). He introduced generalizations of Isac-Carbone’s condition and proved that Isac-Carbone’s condition is the sufficient conditions for the solvability of SDCP (see [22]). In 2012, Hu et al. proposed an existence theorem for copositive complementarity problem (denoted by CCP) and extended the property of coercivity, -order coercivity, monotone, and (strictly) weakly proper to CCP (see [23]). In 2010, Németh first introduced the notion of exceptional family for general order complementarity problems in Banach space and used the notion to study the solvability of general order complementarity problems (see [6]).

Motivated and inspired by the works mentioned above, in this paper, by using the concept of exceptional family in [6], we propose a sufficient condition of a solution to general order complementarity problems (denoted by GOCP) in Banach space, which is weaker than that in [6, Theorem 3.1]. Then we study some sufficient conditions for the nonexistence of exceptional family for GOCP in Hilbert space. Moreover, we prove that the nonexistence of exceptional family is a sufficient and necessary condition for the solvability of pseudomonotone general order complementarity problems.

The remainder of this paper is organized as follows. The preliminary results which will be used in this paper are stated in Section 2. In Section 3, we recall the definition of general order complementarity problems (see [3, 5, 6]) and introduced the concept of exceptional family for the general order complementarity problems (see [6]), then we prove an essential result. In Section 4, we discuss the conditions for the nonexistence of exceptional family. Conclusions are drawn in Section 5.

2. Preliminaries

In this section, we recall some background materials and preliminary results used in the subsequent sections. Firstly, we give some concepts from [6].

Let be a Banach space whose norm is denoted by . Let be a closed set. is called a wedge, if for any and , and . A wedge is called a cone if .

Definition 1 (see [6]). A relation on is called an order if it meets(1)reflexivity; that is, for all ;(2)antisymmetry; that is, if and , then ;(3)transitivity; that is, if and , then .

We say a relation on is induced by a cone ; that is, if and only if . Hence by using the relation on . Then we denote an ordered Banach space by .

Property 1 (see [6]). A relation on is induced by a cone if and only if it is(1)translation invariant; that is, if , then for all ;(2)scale invariant; that is, if , then for any ;(3)continuous; that is, if for any two convergent sequences and in with for all , then , where and are the limits of and , respectively.

The ordered Banach space is called a vector lattice if for every , there exists with respect to the order induced by . In this case we say that the cone is latticial. By the above concepts, we give the following property from [6].

Property 2. Let be a Banach space ordered by the latticial cone . For each we denote ; then the following two equalities hold for all :(1);(2).

A continuous mapping is called completely continuous mapping if for every bounded set the set is relatively compact. The notation deg is the topological degree associated with , , and (see [24, 25]). Now we recall briefly the notation and some key properties of topological degree that will be used below.

Theorem 2 (see [8, Theorem 1.1]). Let be an open bounded subset, and let be an identity mapping; that is, , . Then , .

Theorem 3 (Poincaré-Bohl theorem, see [26]). Let be an open bounded subset, and let be a completely continuous mapping. If , then is a constant for , where .

Theorem 4 (Kronecker theorem, see [27]). Let be an open bounded subset, an identity mapping, and f : a completely continuous mapping. If , then equation has at least one solution in .

3. Exceptional Family for GOCP

First we recall the definition of general order complementarity problems (see [3, 5, 6]) and next we recall the concept of exceptional family for general order complementarity problems (GOCPs) (see [6]).

Definition 5. Let be a Banach space ordered by the latticial cone and a nonempty closed convex set. Consider mappings . The general order complementarity problem defined by the family of mappings and the set is

Definition 6. Let be a Banach space ordered by the latticial cone and a nonempty closed convex set. Consider mappings . A sequence is said to be an exceptional family for GOCP( if the following conditions are satisfied:(1) as ,(2)for every real number , there exists a real number such that , with , for .

The following lemma comes from the property proved in [6, Theorem 3.1]. Here we recall the property and the proof which are the same as the proof in [6, Theorem 3.1].

Lemma 7. Let . If and are completely continuous for all , so is .

Proof. Let for all and . We will prove by induction that are completely continuous mappings for all . If , then . From the condition we see that is a completely continuous mapping immediately. Suppose that is a completely continuous mapping where . By (2), the definition of , and Property 2, we have Hence, are completely continuous mappings, for all . In particular, is a completely continuous mapping.

In what follows, we will establish an important theorem for GOCP(.

Theorem 8. Let be a Banach space ordered by the latticial cone and an unbounded closed convex set. If and are completely continuous for all , then GOCP has either a solution or an exceptional family.

Proof. From Definition 5 we know that the solvability of the problem GOCP is equivalent to the problem of finding an such that . Let , . We get that is completely continuous mapping from Lemma 7. Consider a family of spheres and open balls : Since is unbounded, we have and for all . We consider the mapping , . If there exists an such that It follows from Theorem 3 that deg is constant for . This together with Theorem 2 implies that deg. Therefore, we know that problem is solvable from Theorem 4; that is, the problem GOCP is solvable.
On the other hand, for every , there exist a vector and a scalar such that ; that is, .
If , then , which again implies solvability of the problem GOCP. If , then , which contradict with the fact . Hence . If , then from the definition of we get that is, Dividing both parts by , we obtain where . Let , ; then where the second equality follows from Property 2. Thus, is an exceptional family for GOCP. The proof is complete.

Remark 9. Notice that, in [6], they used the condition of completely continuous field instead of and being completely continuous operators for all . Moreover, [6, Theorem 3.1] required that the condition holds, which does not need this condition in Theorem 8 in our paper. Hence, our condition in Theorem 8 is weaker than the condition of [6, Theorem 3.1].

4. Existence Conditions of a Solution to GOCP

In this section, we consider the general order complementarity problems in Hilbert space whose inner product and norm are denoted by and , respectively. We propose some sufficient conditions and prove that they guarantee existence of solutions to the general order complementarity problem. Firstly, we give the condition as follows.

Condition 1. Let be a Hilbert space ordered by the latticial cone and a nonempty set. satisfy the following condition: there exists such that for all with , there exists with such that

Theorem 10. Let be a Hilbert space ordered by the latticial cone , an unbounded closed convex set and , are completely continuous for all . If Condition 1 holds, then there exists no exceptional family for GOCP and hence, GOCP is solvable.

Proof. Suppose that GOCP( has an exceptional family . By Definition 6, we have
Take such that . Since satisfy Condition 1, there exists with such that . We have which is impossible. Hence, there exists no exceptional family for GOCP. Then the problem is solvable.

Condition 2. Let be a Hilbert space ordered by the latticial cone and a nonempty set. satisfy the following condition: there exists a nonempty bounded subset such that for every , there exists such that

Corollary 11. Let be a Hilbert space ordered by the latticial cone and an unbounded closed convex set and , are completely continuous for all . If Condition 2 holds, then there exists no exceptional family for GOCP and hence, GOCP is solvable.

Proof. Let be the set defined by Condition 2. Since is bounded, then there exists such that , where . For any such that , there exists such that . Hence Condition 1 is satisfied. This together with Theorem 10 completes the proof.

We extend the coercivity condition and -order coercivity condition (see [15, 23]) to GOCP as follows.

Definition 12. Let be a Hilbert space ordered by the latticial cone and an unbounded set. Consider mappings . is said to be -order coercive with respect to , if there exists and such that

Theorem 13. Let be a Hilbert space ordered by the latticial cone and an unbounded closed convex set and , are completely continuous for all . If there exists some such that is -order coercive with respect to , then there exists no exceptional family for GOCP and hence, GOCP is solvable.

Proof. Since is -order coercive with respect to for some , we get This together with and the definition of an infinite limit yields that for sufficiently large and . Hence Condition 1 is satisfied. This together with Theorem 10 completes the proof.

The following results extend monotone property and (strictly) weakly proper to GOCP.

Definition 14. Let be a Hilbert space ordered by the latticial cone and a nonempty set. Consider mappings . is said to be(a)pseudomonotone on if, for every , , , one has (b)quasiomonotone on if, for every , , , one has

Definition 15. Let be a Hilbert space ordered by the latticial cone and an unbounded set. Consider mappings . is said to be(a)weakly proper on , if for every sequence with , there exists a and some such that (b)strictly weakly proper on , if for every sequence with , there exists a and some such that

Theorem 16. Let be a Hilbert space ordered by the latticial cone and an unbounded closed convex set and , are completely continuous for all . If is pseudomonotone on , then the following conditions are equivalent: (1)GOCP has no exceptional family;(2)GOCP has at least a solution;(3) is weakly proper on .

Proof. follows from Theorem 13.
. Since has at least a solution, there exists such that . Then for every sequence with , we have which implies that is weakly proper on .
. Suppose that GOCP has an exceptional family. Then there exists , , and such that From Property 2, we obtain namely, Since is weakly proper on , then there exists a and some such that This together with the fact that is pseudomonotone on yields By (24) we get which is impossible. Hence GOCP has no exceptional family. From the above, we complete the proof.

Remark 17. The above theorem shows that if is pseudomonotone on , GOCP has no exceptional family GOCP which has at least a solution.

Theorem 18. Let be a Hilbert space ordered by the latticial cone and an unbounded closed convex set and , are completely continuous for all . If is quasimonotone on , then there exists no exceptional family for GOCP and hence, GOCP is solvable.

Proof. Suppose that GOCP has an exceptional family. Then from the proof of Theorem 16 we obtain (24). Since is strictly weakly proper on , then there exists a and some such that This together with the fact that is quasimonotone on yields By (24) we get which is impossible. Hence GOCP has no exceptional family. From the above, we complete the proof.

5. Conclusion

In this paper, by using the concept of exceptional family in [6], we propose an existence theorem of a solution to general order complementarity problems in Banach space. Then we study some sufficient conditions for the nonexistence of exceptional family in Hilbert space. Moreover, we prove that nonexistence of exceptional family is a sufficient and necessary condition for the solvability of pseudomonotone general order complementarity problems.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Authors’ Contribution

All authors contributed equally and significantly in writing this paper. All authors read and approved the final paper.

Acknowledgments

The project is supported by National Natural Science Foundation of China (Grant no. 11071041), Fujian Natural Science Foundation (Grant no. 2013J01006), and The University Special Fund Project of Fujian (Grant no. JK2013060).

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Copyright © 2014 Na Huang and Changfeng Ma. 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.

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