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

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.


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][3][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][8][9][10][11][12][13][14][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.

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The Scientific World Journal 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.

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 ≥ 0 and , ∈ , ∈ and + ∈ . A wedge is called a cone if ∩ (− ) = {0}.
We say a relation ⪯ on is induced by a cone ⊂ ; that is, ⪯ if and only if − ∈ . Hence = { ∈ : 0 ⪯ } by using the relation ⪯ on . Then we denote an ordered Banach space by ( , ‖ ⋅ ‖, ).
The ordered Banach space ( , ‖ ⋅ ‖, ) is called a vector lattice if for every , ∈ there exists ∧ := inf{ , } 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].

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

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]).

Remark 9.
Notice that, in [6], they used the condition of completely continuous field instead of and = − being completely continuous operators for all = 1, 2, . . . , . 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].
From Property 2(1), we obtain namely, Since 1 ( ) ∧ ⋅ ⋅ ⋅ ∧ ( ) is weakly proper on , then there exists a ∈ and some such that This together with the fact that 1 ( ) ∧ ⋅ ⋅ ⋅ ∧ ( ) is pseudomonotone on yields By (24) we get which is impossible. Hence GOCP({ } =1 , ) has no exceptional family. From the above, we complete the proof. Proof. Suppose that GOCP({ } =1 , ) has an exceptional family. Then from the proof of Theorem 16 we obtain (24). Since 1 ( ) ∧ ⋅ ⋅ ⋅ ∧ ( ) is strictly weakly proper on , then there exists a ∈ and some such that which is impossible. Hence GOCP({ } =1 , ) has no exceptional family. From the above, we complete the proof.

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.