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

Authors: Na Huang; Changfeng Ma
Journal: The Scientific World Journal
 (2014)

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## 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.


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## Body

## 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 Rn 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, p-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 Section2. 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].LetX be a Banach space whose norm is denoted by ∥·∥. Let K⊂H be a closed set. K is called a wedge, if for any λ≥0 and x,y∈K, λx∈K and x+y∈K. A wedge K is called a cone if K∩(-K)={0}.Definition 1 (see [6]).
A relation⪯ on X is called an order if it meets(1)
reflexivity; that is,x⪯x for all x∈X;(2)
antisymmetry; that is, ifx⪯y and y⪯x, then x=y;(3)
transitivity; that is, ifx⪯y and y⪯z, then x⪯z.We say a relation⪯ on X is induced by a cone K⊂X; that is, x⪯y if and only if y-x∈K. Hence K={x∈X:0⪯x} by using the relation ⪯ on X. Then we denote an ordered Banach space by (X,∥·∥,K).Property 1 (see [6]).
A relation⪯ on X is induced by a cone K⊂X if and only if it is(1)
translation invariant; that is, ifx⪯y, then x+z⪯y+z for all z∈X;(2)
scale invariant; that is, ifx⪯y, then λx⪯λy for any λ>0;(3)
continuous; that is, if for any two convergent sequences{xn}n>0 and {yn}n>0 in X with xn⪯yn for all n>0, then x*⪯y*, where x* and y* are the limits of {xn}n>0 and {yn}n>0, respectively.The ordered Banach space(X,∥·∥,K) is called a vector lattice if for every x, y∈X there exists x∧y:=inf⁡{x,y} with respect to the order induced by K. In this case we say that the cone K is latticial. By the above concepts, we give the following property from [6].Property 2.
Let(X,∥·∥) be a Banach space ordered by the latticial cone K⊂X. For each x∈X we denote x+=-0∧(-x); then the following two equalities hold for all x,y,z∈X:(1)
(x+z)∧(y+z)=x∧y+z;(2)
x∧y=y-(y-x)+.A continuous mappingF:Ω⊆X→X is called completely continuous mapping if for every bounded set Δ⊆Ω the set F(Δ) is relatively compact. The notation deg(F,Ω,y) is the topological degree associated with F, Ω, and y (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]).
LetΩ⊆X be an open bounded subset, and let I:X→X be an identity mapping; that is, I(x)=x, ∀x∈X. Then deg⁡(I,Ω,p)=1, ∀p∈Ω.Theorem 3 (Poincaré-Bohl theorem, see [26]).
LetΩ⊆X be an open bounded subset, and let H:[0,1]×Ω→X be a completely continuous mapping. If y∉ht(∂Ω), then deg⁡(ht,Ω,y) is a constant for 0≤t≤1, where ht(x)=x-H(t,x).Theorem 4 (Kronecker theorem, see [27]).
LetΩ⊆X be an open bounded subset, I:X→X an identity mapping, and f  : Ω-→X a completely continuous mapping. If deg⁡(I-f,Ω,y)≠0, then equation x-f(x)=y has at least one solution in Ω.


## 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(X,∥·∥) be a Banach space ordered by the latticial cone K⊂X and D⊂X a nonempty closed convex set. Consider m mappings f1,f2,…,fm:X→X. The general order complementarity problem defined by the family of mappings {fi}i=1m and the set D is
(1)GOCP({fi}i=1m,D):{findx*∈Dsuchthatf1(x*)∧⋯∧fm(x*)=0.Definition 6.
Let(X,∥·∥) be a Banach space ordered by the latticial cone K⊂X and D⊂X a nonempty closed convex set. Consider m mappings f1,f2,…,fm:X→X. A sequence {xr}r>0⊆D is said to be an exceptional family for GOCP({fi}i=1m,D) if the following conditions are satisfied:(1)
∥xr∥→+∞ as r→+∞,(2)
for every real numberr>0, there exists a real number μr>0 such that ur1∧⋯∧urm=0, with uri=μrxr+fi(xr), for i=1,2,…,m.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.
Letϕ(x)=x-f1(x)∧⋯∧fm(x). If fi and Si=I-fi are completely continuous for all i=1,2,…,m, so is ϕ(x).Proof.
Let(2)ϕk(x):=x-f1(x)∧⋯∧fk(x),
for all x∈H and k=1,2,…,m. We will prove by induction that ϕk are completely continuous mappings for all k=1,2,…,m. If k=1, then ϕ1(x)=x-f1(x)=S1(x). From the condition we see that ϕ1 is a completely continuous mapping immediately. Suppose that ϕk is a completely continuous mapping where k∈{1,2,…,m-1}. By (2), the definition of Sk+1(x), and Property 2(2), we have
(3)ϕk+1(x)=x-f1(x)∧⋯∧fk(x)∧fk+1(x)=x-(x-ϕk(x))∧fk+1(x)=x-(x-ϕk(x))∧(x-Sk+1(x))=x-[(x-Sk+1(x))-(x-Sk+1(x)-x+ϕk(x))+]=Sk+1(x)+(ϕk(x)-Sk+1(x))+.
Hence, ϕk are completely continuous mappings, for all k=1,2,…,m. In particular, ϕ(x)=ϕm(x) is a completely continuous mapping.In what follows, we will establish an important theorem for GOCP({fi}i=1m,D).Theorem 8.
Let(X,∥·∥) be a Banach space ordered by the latticial cone K⊂X and D⊂X an unbounded closed convex set. If fi and Si=I-fi are completely continuous for all i=1,2,…,m, then GOCP({fi}i=1m,D) has either a solution or an exceptional family.Proof.
From Definition5 we know that the solvability of the problem GOCP is equivalent to the problem of finding an x∈D such that ϕ(x)=x. Let H(t,x)=(1-t)ϕ(x), 0≤t≤1. We get that H(t,x) is completely continuous mapping from Lemma 7. Consider a family of spheres Br and open balls Ur:
(4)Br={x∈D:∥x∥=r},Ur={x∈D:∥x∥<r}.
Since D is unbounded, we have Br≠∅ and Ur≠∅ for all r>0. We consider the mapping ht(x)=x-H(t,x), t∈[0,1]. If there exists an r>0 such that
(5)0∉{ht(x):x∈Br,t∈[0,1]}.
It follows from Theorem 3 that deg(ht(x),Ur,0) is constant for t∈[0,1]. This together with Theorem 2 implies that deg(x-ϕ(x),Ur,0)=deg⁡(x,Ur,0)=1. Therefore, we know that problem ϕ(x)=x is solvable from Theorem 4; that is, the problem GOCP is solvable.
On the other hand, for everyr>0, there exist a vector xr∈Br and a scalar tr∈[0,1] such that htr(xr)=0; that is, xr-(1-tr)ϕ(xr)=0.
Iftr=0, then xr-ϕ(xr)=0, which again implies solvability of the problem GOCP. If tr=1, then xr=0, which contradict with the fact xr∈Br. Hence tr≠1. If 0<tr<1, then from the definition of ϕ(x) we get
(6)xr-(1-tr)(xr-f1(xr)∧⋯∧fm(xr))=0;
that is,
(7)trxr+(1-tr)f1(xr)∧⋯∧fm(xr)=0.
Dividing both parts by 1-tr, we obtain
(8)μrxr+f1(xr)∧⋯∧fm(xr)=0,
where μr=tr/(1-tr). Let uri=μrxr+fi(xr), i=1,2,…,m; then
(9)ur1∧⋯∧urm=(μrxr+f1(xr))∧⋯∧(μrxr+fm(xr))=μrxr+f1(xr)∧⋯∧fm(xr)=0,
where the second equality follows from Property 2(1). Thus, {xr}r>0 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 fi and Si=I-fi being completely continuous operators for all i=1,2,…,m. Moreover, [6, Theorem  3.1] required that the condition Sm(D)+K⊂D 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 spaceH 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(H,〈·,·〉) be a Hilbert space ordered by the latticial cone K⊂H and D⊂H a nonempty set. f1,f2,…,fm:H→H satisfy the following condition: there exists ρ>0 such that for all x∈D with ∥x∥>ρ, there exists y∈H with ∥y∥<∥x∥ such that
(10)〈x-y,f1(x)∧⋯∧fm(x)〉≥0.Theorem 10.
Let(H,〈·,·〉) be a Hilbert space ordered by the latticial cone K⊂H, D⊂H an unbounded closed convex set and fi, Si=I-fi are completely continuous for all i=1,2,…,m. If Condition 1 holds, then there exists no exceptional family for GOCP({fi}i=1m,D) and hence, GOCP({fi}i=1m,D) is solvable.Proof.
Suppose that GOCP({fi}i=1m,D) has an exceptional family {xr}r>0⊂D. By Definition 6, we have
(11)uri=μrxr+fi(xr),μr>0,i=1,2,…,m(∀r>0),ur1∧⋯∧urm=0(∀r>0),∥xr∥⟶+∞asr⟶+∞.
Taker>0 such that ∥xr∥>ρ. Since f1,f2,…,fm satisfy Condition 1, there exists yr∈H with ∥yr∥<∥xr∥ such that 〈xr-yr,f1(xr)∧⋯∧fm(xr)〉≥0. We have
(12)0≤〈xr-yr,f1(xr)∧⋯∧fm(xr)〉=〈xr-yr,-μrxr〉=-μr∥xr∥2+μr〈yr,xr〉≤-μr∥xr∥2+μr∥yr∥∥xr∥=-μr∥xr∥(∥xr∥-∥yr∥)<0,
which is impossible. Hence, there exists no exceptional family for GOCP({fi}i=1m,D). Then the problem is solvable.Condition 2.
Let(H,〈·,·〉) be a Hilbert space ordered by the latticial cone K⊂H and D⊂H a nonempty set. f1,f2,…,fm:H→H satisfy the following condition: there exists a nonempty bounded subset C⊂D such that for every x∈D∖C, there exists y∈C such that
(13)〈x-y,f1(x)∧⋯∧fm(x)〉≥0.Corollary 11.
Let(H,〈·,·〉) be a Hilbert space ordered by the latticial cone K⊂H and D⊂H an unbounded closed convex set and fi, Si=I-fi are completely continuous for all i=1,2,…,m. If Condition 2 holds, then there exists no exceptional family for GOCP({fi}i=1m,D) and hence, GOCP({fi}i=1m,D) is solvable.Proof.
LetC⊂D be the set defined by Condition 2. Since C is bounded, then there exists ρ>0 such that C⊂Uρ∩D, where Uρ={x∈H:∥x∥≤ρ}. For any x such that ∥x∥>ρ, there exists y∈C(∥y∥≤ρ<∥x∥) such that 〈x-y,f1(x)∧⋯∧fm(x)〉≥0. Hence Condition 1 is satisfied. This together with Theorem 10 completes the proof.We extend the coercivity condition andp-order coercivity condition (see [15, 23]) to GOCP as follows.Definition 12.
Let(H,〈·,·〉) be a Hilbert space ordered by the latticial cone K⊂H and D⊂H an unbounded set. Consider m mappings f1,f2,…,fm:H→H. f1(x)∧⋯∧fm(x) is said to be p-order coercive with respect to D, if there exists p∈(-∞,1] and y∈H such that
(14)lim⁡x∈D,∥x∥→+∞〈x-y,f1(x)∧⋯∧fm(x)〉∥x∥p=+∞.Theorem 13.
Let(H,〈·,·〉) be a Hilbert space ordered by the latticial cone K⊂H and D⊂H an unbounded closed convex set and fi, Si=I-fi are completely continuous for all i=1,2,…,m. If there exists some p∈(-∞,1] such that f1(x)∧⋯∧fm(x) is p-order coercive with respect to D, then there exists no exceptional family for GOCP({fi}i=1m,D) and hence, GOCP({fi}i=1m,D) is solvable.Proof.
Sincef1(x)∧⋯∧fm(x) is p-order coercive with respect to D for some p∈(-∞,1], we get
(15)lim⁡x∈D,∥x∥→+∞〈x-y,f1(x)∧⋯∧fm(x)〉∥x∥p=+∞.
This together with ∥x∥p≥0 and the definition of an infinite limit yields that
(16)〈x-y,f1(x)∧⋯∧fm(x)〉≥0,
for sufficiently large ρ and ∥x∥>ρ. 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(H,〈·,·〉) be a Hilbert space ordered by the latticial cone K⊂H and D⊂H a nonempty set. Consider m mappings f1,f2,…,fm:H→H. f1(x)∧⋯∧fm(x) is said to be(a)
pseudomonotone onD if, for every x, y∈D, x≠y, one has
(17)〈y-x,f1(x)∧⋯∧fm(x)〉≥0⟹〈y-x,f1(y)∧⋯∧fm(y)〉≥0;(b)
quasiomonotone onD if, for every x, y∈D, x≠y, one has
(18)〈y-x,f1(x)∧⋯∧fm(x)〉>0⟹〈y-x,f1(y)∧⋯∧fm(y)〉≥0.Definition 15.
Let(H,〈·,·〉) be a Hilbert space ordered by the latticial cone K⊂H and D⊂H an unbounded set. Consider m mappings f1,f2,…,fm:H→H. f1(x)∧⋯∧fm(x) is said to be(a)
weakly proper onD, if for every sequence {xr}⊂D with lim⁡r→+∞∥xr∥=+∞, there exists a y∈H and some r such that
(19)〈xr-y,f1(y)∧⋯∧fm(y)〉≥0,∥xr∥>∥y∥;(b)
strictly weakly proper onD, if for every sequence {xr}⊂D with lim⁡r→+∞∥xr∥=+∞, there exists a y∈H and some r such that
(20)〈xr-y,f1(y)∧⋯∧fm(y)〉>0,∥xr∥>∥y∥.Theorem 16.
Let(H,〈·,·〉) be a Hilbert space ordered by the latticial cone K⊂H and D⊂H an unbounded closed convex set and fi, Si=I-fi are completely continuous for all i=1,2,…,m. If f1(x)∧⋯∧fm(x) is pseudomonotone on D, then the following conditions are equivalent: (1)
GOCP({fi}i=1m,D) has no exceptional family;(2)
GOCP({fi}i=1m,D) has at least a solution;(3)
f1(x)∧⋯∧fm(x) is weakly proper on D.Proof.
(
1
)
⇒
(
2
) follows from Theorem 13.
(
2
)
⇒
(
3
). Since GOCP({fi}i=1m,D) has at least a solution, there exists x*∈D such that f1(x*)∧⋯∧fm(x*)=0. Then for every sequence {xr}⊂D with lim⁡r→+∞∥xr∥=+∞, we have
(21)〈xr-x*,f1(x*)∧⋯∧fm(x*)〉=0,
which implies that f1(x)∧⋯∧fm(x) is weakly proper on D.
(
3
)
⇒
(
1
). Suppose that GOCP({fi}i=1m,D) has an exceptional family. Then there exists {xr}r>0⊆D, ∥xr∥→+∞(r→+∞), and μr>0 such that
(22)uri=μrxr+fi(xr),i=1,2…,m,ur1∧⋯∧urm=0.
From Property 2(1), we obtain
(23)ur1∧⋯∧urm=(μrxr+f1(xr))∧⋯∧(μrxr+fm(xr))=μrxr+f1(xr)∧⋯∧fm(xr)=0;
namely,
(24)f1(xr)∧⋯∧fm(xr)=-μrxr.
Since f1(x)∧⋯∧fm(x) is weakly proper on D, then there exists a y∈H and some r such that
(25)〈xr-y,f1(y)∧⋯∧fm(y)〉≥0,∥xr∥>∥y∥.
This together with the fact that f1(x)∧⋯∧fm(x) is pseudomonotone on D yields
(26)〈xr-y,f1(xr)∧⋯∧fm(xr)〉≥0.
By (24) we get
(27)0≤〈xr-y,f1(xr)∧⋯∧fm(xr)〉=〈xr-y,-μrxr〉=-μr∥xr∥2+μr〈yr,xr〉≤-μr∥xr∥2+μr∥yr∥∥xr∥=-μr∥xr∥(∥xr∥-∥yr∥)<0,
which is impossible. Hence GOCP({fi}i=1m,D) has no exceptional family. From the above, we complete the proof.Remark 17.
The above theorem shows that iff1(x)∧⋯∧fm(x) is pseudomonotone on D, GOCP({fi}i=1m,D) has no exceptional family ⇔ GOCP({fi}i=1m,D) which has at least a solution.Theorem 18.
Let(H,〈·,·〉) be a Hilbert space ordered by the latticial cone K⊂H and D⊂H an unbounded closed convex set and fi, Si=I-fi are completely continuous for all i=1,2,…,m. If f1(x)∧⋯∧fm(x) is quasimonotone on D, then there exists no exceptional family for GOCP({fi}i=1m,D) and hence, GOCP({fi}i=1m,D) is solvable.Proof.
Suppose that GOCP({fi}i=1m,D) has an exceptional family. Then from the proof of Theorem 16 we obtain (24). Since f1(x)∧⋯∧fm(x) is strictly weakly proper on D, then there exists a y∈H and some r such that
(28)〈xr-y,f1(y)∧⋯∧fm(y)〉>0,∥xr∥>∥y∥.
This together with the fact that f1(x)∧⋯∧fm(x) is quasimonotone on D yields
(29)〈xr-y,f1(xr)∧⋯∧fm(xr)〉≥0.
By (24) we get
(30)0≤〈xr-y,f1(xr)∧⋯∧fm(xr)〉=〈xr-y,-μrxr〉=-μr∥xr∥2+μr〈yr,xr〉≤-μr∥xr∥2+μr∥yr∥∥xr∥=-μr∥xr∥(∥xr∥-∥yr∥)<0,
which is impossible. Hence GOCP({fi}i=1m,D) 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.



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