# The Existence and Stability of Solutions for Vector Quasiequilibrium Problems in Topological Order Spaces

Authors: Qi-Qing Song
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

In a topological sup-semilattice, we established a new existence result for vector quasiequilibrium problems. By the analysis of essential stabilities of maximal elements in a topological sup-semilattice, we prove that for solutions of each vector quasi-equilibrium problem, there exists a connected minimal essential set which can resist the perturbation of the vector quasi-equilibrium problem.


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## Body

## 1. Introduction

Vector equilibrium problems can unify many nonlinear problems such as vector optimization, vector variational inequality [1], and vector complementarity problems [2]. Recently, not only vector equilibrium problems [3–7] but also vector quasiequilibrium problems [8–14] and the system of vector quasi-equilibrium problems have attracted much attention [15–19].Topological vector spaces provide the usual mathematical framework in the study of many problems. To avoid the linear feature, sup-semilattices may be good choices. In fact, some results like the existence of KKM points in topological spaces were established in topological sup-semilattices [20], where a two-tuple (X,≤) is said to be a sup-semilattice, if X is a partially ordered set with the partial ordering ≤, in which every pair (x,x′) has a least upper bound x∨x′.The aim of this paper is to study the existence and essential stability of vector quasi-equilibrium problems in topological sup-semilattices. In order to achieve this, firstly, we give a stability result in relation to maximal elements in a topological sup-semilattice. Secondly, a new existence result for vector quasi-equilibrium problems is established, and we show that each vector quasi-equilibrium problem has a connected minimal essential set in its solution set.


## 2. Preliminaries

Let(X,≤) be a sup-semilattice. If x and x′ are two elements in (X,≤) and x≤x′, the set [x,x′]={y∈X:x≤y≤x′} is called an order interval. Let A, A′ be two nonempty finite subsets of X. Then the set ΔA=∪x∈A[x,supA] is well defined and has the properties: A⊆ΔA and ΔA⊆ΔA′ if A⊆A′.Definition 1 (see [20]).
A subsetE⊆X is Δ-convex, if for any nonempty finite subset A⊆E, we have ΔA⊆E. E being a Δ-convex set is equivalent to the following conditions:(a)
ifx, x′∈E, then its least upper bound x∨x′∈E.(b)
ifx, x′∈E and x≤x′, then the order interval [x,x′]⊂E.It is easy to check that the intersection of twoΔ-convex sets is Δ-convex as well.A topological spaceX is said to be a topological sup-semilattice if X is equipped with a sup-semilattice as its partial ordering denoted by ≤, for which f:X×X→X with (x,x′)→x∨x′ is a continuous function.LetY be a topological vector space and θ the zero element in Y. A subset C⊂Y is called a cone if, for any y∈C and real number t>0,ty∈C. A cone C is convex if C is a convex set. If C∩-C={θ}, it is called a pointed cone.Definition 2 (see [18]).
LetX be a topological sup-semilattice, Y a topological vector space with a cone C⊂Y, φ:X→Y a vector-valued function.(a)
φ:X→Y is CΔ-quasiconcave if, for any nonempty two points subset A={x1,x2}⊂X and y∈Y, φ(A)⊂y+C⇒φ(ΔA)⊂y+C.(b)
φ:X→Y is said to be CΔ-quasiconcave-like if, for any x1,x2∈X, φ(Δ{x1,x2})∈φ(x1)+C or φ(Δ{x1,x2})∈φ(x2)+C.Remark 3.
In general cases,CΔ-quasiconcave, CΔ-quasiconcave-like, and usual quasiconcave functions are independent of each other. See examples in [18]. Let φ:ℝ→ℝ, C=-ℝ+. Then the partial order on ℝ is “≤” (less than or equal to);, hence, the CΔ-quasiconcave, CΔ-quasiconcave-like, and usual quasiconcave property of φ coincide (the usual quasiconcave function φ means that for any x1, x2, y∈ℝ, φ(x1)≤y and φ(x2)≤y⇒φ(λx1+(1-λ)x2)≤y, forallλ∈[0,1]).Lemma 4 (see [18]).
LetX be a topological sup-semilattice, Y a Hausdorff locally convex topological vector space with a closed, convex, and pointed cone C⊂Y. If the vector-valued function φ:X→Y is CΔ-quasiconcave or CΔ-quasiconcave-like, then the set A={x:φ(x)∈intC} is Δ-convex.Now we introduce the vector quasiequilibrium problem (VQEP) that we will consider in this paper.LetX be a topological sup-semilattice and Y a topological vector space. C⊂Y is a closed, convex, and pointed cone with intC≠∅. φ:X×X→Y is a vector-valued function, and G is a multivalued mapping on X. The vector quasi-equilibrium problem ϕ with ϕ=(X,Y,C,φ,G) is to find x-∈X, such that
(1)x-∈G(x-):φ(x-,y)∉intC,∀y∈G(x-).LetG(x)=X, forallx∈X; then the VQEP is just a vector equilibrium problem (X,Y,C,φ) (VEP). That is to find x-∈X, such that
(2)φ(x-,y)∉intC,∀y∈X.Definition 5 (see [21, 22]).
A vector-valued functionφ:X→Y is said to be C-continuous on X if, for each x∈X and any open neighborhood V(θ) of θ in Y, there exists an open neighborhood O(x) of x in X such that
(3)∀x′∈O(x),φ(x′)∈φ(x)+V(θ)+C.Remark 6.
For a functionφ:X→ℝ, φ is ℝ+-continuous on X if and only if φ is lower semicontinuous on X.A maximal element version of the Browder fixed point theorem in a topological sup-semilattice can be found in [18]. We limit it in a metric space as the following lemma.Lemma 7 (see [18]).
Let(X,d) be a compact sup-semilattice with path connected interval, where d is the metric on X, S:X→2X a multivalued map on X with the conditions: (i)forallx∈X, S(x) is Δ-convex; (ii)forally∈X,S-1(y)={x∈X:y∈S(x)} is open in X; (iii)forallx∈X, x∉S(x). Then there exists an x-∈X, such that S(x-)=∅.Remark 8.
The existence of a metric space with a sup-semilattice can be guaranteed. For instance, letxj=(x1j,…,xij,…,xnj)∈ℝn, j=1,2, if x1≤x2 means that x2∈x1+ℝ+n, then x1∨x2=x-, where x-i=max{xi1,xi2}. Clearly, (ℝn,≤) with the usual Euclidean metric is a topological sup-semilattice.LetM denote the collection of S satisfying all the conditions of Lemma 7. For any S1,S2∈M, define the metric between S1 and S2 as
(4)ρ(S1,S2)=supy∈Xh(X∖S1-1(y),X∖S2-1(y)),
where h is the Hausdorff metric induced by d. For each S∈M and each y∈X, since y∉S(y), we have y∈X∖S-1(y), that is, X∖S-1(y)≠∅. Noting that X∖S-1(y) is closed, the metric ρ on M is well defined. Then (M,ρ) is a metric space.For eachS∈M, denote by F(S) the set of all maximal elements of S. Then F defines a multivalued mapping from M to X and F(S)=∩y∈X(X∖S-1(y)).Definition 9.
For eachS∈M, a set e(S) is called an essential set of F(S) if it satisfies the following conditions:(1)
e(S) is closed subset of F(S).(2)
For any open setU⊃e(S), there exists an open neighborhood O(S) of S∈M such that U∩F(S′)≠∅, for any S′∈O(S).
A setm(S) is called a minimal essential set of F(S) if it is a minimal element of all essential sets ordered by set inclusion in F(S). A connected component in F(S) is called an essential component, if it includes at least one minimal essential set of F(S).We recall some notions about multi-valued mappings. LetG:Y→2P be a multi-valued mapping, where Y, P are two topological vector spaces. Then (i)G is said to be upper semicontinuous at y∈Y, if for each open set U⊃G(y), there exists an open neighborhood O(y) of y such that U⊃G(y′) for any y′∈O(y). (ii)G is lower semi-continuous at y∈Y, if for each open set U∩G(y)≠ϕ, there exists an open neighborhood O(y) of y such that U∩G(y′)≠ϕ for any y′∈O(y).Remark 10.
For eachS∈M, a set e(S)⊂F(S) is essential if F is lower semi-continuous at S. If F is upper semi-continuous at S, then F(S) itself is an essential set. For any two closed sets A, B⊂F(S) with A⊂B, if A is essential, then B is also essential. For each S∈M and each y∈X, if S-1(y) is open, then X∖S-1(y) is closed; hence, F(S) is closed because we have F(S)=∩y∈XX∖S-1(y); consequently, F(S) is compact.Lemma 11 (see [23]).
Let(X,d) be a metric space, K1 and K2 two nonempty compact subsets of X, U1, and U2 two nonempty disjoint open subsets of X. If h(K1,K2)<d(U1,U2), then h(K1,(K1∖U2)∪(K2∖U1))≤h(K1,K2), where h is the Hausdorff metric defined on X.


## 3.  The Stability of Maximal Elements on Topological Semilattices

Theorem 12.
F
:
M
→
2
X is an upper semi-continuous mapping with compact values.Proof.
For eachS∈M, by Remark 10, F(S) is compact. Suppose that F is not upper semi-continuous. Then there is a S∈M, an open set U with U⊃F(S) and Sn, such that Sn→S and F(Sn)⊄U, n=1,2,…. That is, there exists a point xn∈F(Sn) such that xn∉U. Without loss of generality, we may assume that xn→x*. Since Sn→S, it holds that X∖Sn-1(y)→X∖S-1(y), forally∈X. Since xn∈F(Sn), we have xn∈X∖Sn-1(y), forally∈X. As n gets close to infinity, we can obtain that x*∈X∖S-1(y), forally∈X, that is, x*∈F(S)⊂U. This results in the fact that xn∈U while n is large enough, a contradiction with xn∉U. Therefore, F is definitely upper semi-continuous.Theorem 13.
For eachS∈M, there exists at least a minimal essential set of F(S). If m(S) is a minimal essential set of F(S), then m(S) is connected.Proof.
For the existence, by Remark10, each decreasing chain, consisting of essential subsets of F(S), has a minimal element, which is the intersection of the chain. By the Zorn’s lemma, the minimal element is just a minimal essential set. For the connectedness, by way of contradiction, suppose that m(S) is not connected. There exist two disjoint closed sets C1(S), C2(S) such that m(S)=C1(S)∪C2(S).
SinceCi(S) is not essential, there is an open set Wi with Wi⊃Ci(S) such that for any ε>0, there exists a Si∈M with ρ(S,Si)<ε and F(Si)∩Wi=∅, i=1,2. Clearly, Ci(S) is compact, then there is an open set Ui with Ci(S)⊂Ui⊂Wi, i=1,2, such that U1∩U2=∅. For U1∪U2⊃m(S), because m(S) is essential, there is a number δ<2d(U1,U2), such that F(T)∩(U1∪U2)≠∅ for each T satisfying ρ(T,S)<δ. Therefore, we can select a Si∈M such that ρ(S,Si)<δ/4 and F(Si)∩Ui=∅, i=1,2. Then ρ(S1,S2)<ρ(S1,S)+ρ(S,S2)<δ/2<d(U1,U2).
Define a multi-valued mappingS′:M→2X as
(5)S′={S1(x),x∈U1,S2(x),x∈U2,S1(x)∩S2(x),x∈X∖(U1∪U2).
We show thatS′∈M.(a)
For eachx∈X, since x∉S1(x) and x∉S2(x), we have x∉S′(x);(b)
for eachx∈X, because S1(x) and S2(x) are Δ-convex sets, it follows that S′(x) is Δ-convex;(c)
for eachy∈X, we have
(6)S′-1(y)=(S1-1(y)∩S2-1(y))∪(S2-1(y)∩U2)∪(S1-1(y)∩U1).Noting that S1-1(y),S2-1(y), U1, and U2 are open sets, it follows that S′-1(y) is open.
Through a direct calculation,X∖S′-1(y) can be written as
(7)((X∖S1-1(y))∩(X∖S2-1(y)))∪((X∖S1-1(y))∩(X∖U2))∪((X∖S2-1(y))∩(X∖U1)).
Take any x∈X∖S′-1(y). Note that if x∈U1, then x∈X∖U2; if x∈U2, then x∈X∖U1; if x∈X∖(U1∪U2), then x∈(X∖U1)∩(X∖U2). Consequently, we can obtain that if x∈(X∖S1-1(y))∩(X∖S2-1(y)), then
(8)x∈((X∖S1-1(y))∩(X∖U2))∪((X∖S2-1(y))∩(X∖U1)).
Therefore, we have
(9)X∖S′-1(y)=((X∖S1-1(y))∩(X∖U2))∪((X∖S2-1(y))∩(X∖U1)).
Since ρ(S1,S2)<d(U1,U2), by Lemma 11, we have
(10)h(X∖S′-1(y),X∖S1-1(y))≤h(X∖S2-1(y),X∖S1-1(y))≤ρ(S2,S)+ρ(S,S1)<δ2.
That is, ρ(S′,S1)<δ/2. This results in the fact that
(11)ρ(S′,S)<ρ(S′,S1)+ρ(S1,S)<δ.
Consequently, we have F(S′)∩(U1∪U2)≠∅. If there is a point x∈Ui∩F(S′), then x∈Ui and x∈X∖Si-1(y), forally∈X, that is, x∈Ui and x∈F(Si) which contradicts with F(Si)∩Ui=∅, i=1,2. Therefore, m(S) is connected.


## 4. The Existence and Stability of Solutions for VQEP

This section gives an existence result in relation to VQEP in topological sup-semilattices and induces the existence of minimal essentially stable sets for each VQEP in the set of its solutions.Theorem 14.
Letϕ=(X,Y,C,φ,G) be a VQEP, where X is a compact topological sup-semilattice with path connected intervals, Y is a Hausdorff locally convex topological vector space, and G:X→2X is a multi-valued mapping with nonempty and Δ-convex values. If the VQEP satisfies that(i)
forallx∈X, φ(x,x)∉intC;(ii)
forally∈X, x→φ(x,y) is C-continuous;(iii)
forallx∈X, y→φ(x,y) is CΔ-quasiconcave-like or CΔ-quasiconcave;(iv)
forally∈X, {x∈X:y∈G(x)} is open in X,(v)
{x∈X:x∈G(x)} is closed in X,
then the VQEP has a solution.Proof.
Denote{x∈X:x∈G(x)} by K. Let B:X→2X such that B(x)={y∈X:φ(x,y)∈intC}, forallx∈X. Define
(12)S(x)={B(x)∩G(x),x∈K,G(x),x∈X∖K.
Then for each x∈X, if x∈K, we have S(x)=B(x)∩G(x), by the condition (i), x∉B(x), hence, x∉S(x); if x∈X∖K, from the definition of K, we have x∉G(x)=S(x).
Sincey→φ(x,y) is CΔ-quasiconcave-like or CΔ-quasiconcave, by Lemma 4, we have that B(x) is Δ-convex. Then B(x)∩G(x) is a Δ-convex set, noting that G has Δ-convex values, we have that S(x) is also Δ-convex.
For eachy∈X, we can check that
(13)S-1(y)=(B-1(y)∩G-1(y))∪(G-1(y)∩(X∖K)).
Take a point x∈B-1(y)={x∈X:φ(x,y)∈intC}, since intC is open, there is an open set V(θ) such that V(θ)+φ(x,y)⊂intC, then, by the condition (ii), there exists an open neighborhood O(x) in X such that for all x′∈O(x),
(14)φ(x′,y)∈φ(x,y)+V(θ)+C⊂intC+C⊂intC.
That is, O(x)⊂B-1(y); hence, B-1(y) is open. Noting that X∖K and G-1(y) are open sets in X. We can obtain that S-1(y) is also open in X.
Thus, there is anx-∈X such that S(x-)=∅ by Lemma 7. If x-∈X∖K, then G(x-)=∅, a contradiction to the fact that G has nonempty values. Therefore, x-∈K and B(x-)∩G(x-)=∅, that is, x-∈G(x-), φ(x-,y)∉intC, forally∈X.By Theorem14 and its proof, we can also obtain the existence result for VQEP as the following.Corollary 15.
Letϕ=(X,Y,C,φ,G) be a VQEP, where X is a compact topological sup-semilattice with path connected intervals, Y is a Hausdorff topological vector space, and G:X→2X is a multi-valued mapping with nonempty and Δ-convex values. If the VQEP satisfies that(i)
forallx∈X, φ(x,x)∉intC;(ii)
forally∈X, {x∈X:φ(x,y)∈intC} is open in X;(iii)
forallx∈X, {y∈X:φ(x,y)∈intC} is Δ-convex;(iv)
forally∈X, {x∈X:y∈G(x)} is open in X,(v)
{x∈X:x∈G(x)} is closed in X,
then the VQEP has a solution.By Theorem14, for the special case of VQEP without the feasible mapping G, we can obtain the existence result concerning VEP as the following.Corollary 16.
Let(X,Y,C,φ) be a vector equilibrium problem, where X is a compact topological sup-semilattice with path connected intervals, Y is a Hausdorff locally convex topological vector space. If the VEP satisfies the following conditions:(i)
forallx∈X, y→φ(x,y) is CΔ-quasiconcave or CΔ-quasiconcave-like;(ii)
forally∈X, x→φ(x,y) is C-continuous;(iii)
forallx∈X, φ(x,x)∉intC,
then this VEP has a solution.Example 17.
LetX=[0,1]×[0,1]⊂ℝ2, C=-ℝ+. The (X,≤) is a sup-semilattice, in which x1≤x2 means that x2∈x1+ℝ+2, forallx1,x2∈X.
(a) For anyx=(x1,x2), y=(y1,y2)∈X, the function φ is defined as
(15)φ(x,y)=(1-y1)(1-y2)-(1-x1)(1-x2).
It can be easily checked that for eachx∈X, φ(x,·) is CΔ quasiconcave and CΔ quasiconcave-like but not a usual quasiconcave function.
Denote byD the set (1×[0,1])∪([0,1]×1). For each x=(x1,x2)∈X, the multi-valued mapping G satisfies that
(16)G(x)={(x1,1]×[0,1]∪[0,1]×(x2,1],x∈X∖D,(1,1),x∈D.
Note that G is not a usual convex but a Δ-convex multi-valued mapping. For each y=(y1,y2)∈X, if y∈X∖(1,1), then G-1(y)=[0,y1)×[0,y2); if y=(1,1), then G-1(y)=X. Thus, G-1(y) is open in X for each y∈X. Then φ and G satisfy all the conditions in Theorem 14. We can find that x-=(1,1) is the unique solution for the VQEP, (X,ℝ,C,φ,G).
(b) For anyx=(x1,x2), y=(y1,y2)∈X, let T(x)=Bε(x), where Bε(x) is the ε-neighborhood of x in X,
(17)φ(x,y)=(1+x1-y1)(1+x2-y2),
and G is the same as the setting in (a). Then the function φ and the mappings G and T meet all the conditions in Theorem 14. The set of solutions for the VEP (X,ℝ,C,φ) is the overall X which is also the set of solutions for (X,ℝ,C,φ,T). The solution of (X,ℝ,C,φ,G) is just one point x-=(1,1).
To study the stability of vector quasi-equilibrium problems, let(X,d) be a metric space and define the set M′ as
(18)M′={ϕ=(X,Y,C,φ,G):ϕsatisfiesalltheconditionsinTheorem14}
For eachϕ∈M′, by the proof of Theorem 14, we can find that a point x-∈X is a solution of ϕ if and only if x- is a maximal element of S defined in the proof. Let F′(ϕ) denote all the solutions of ϕ. Then F′ is a multi-valued mapping from M′ to X. For any two ϕ1, ϕ2, define the metric ρ′(ϕ1,ϕ2) between ϕ1 and ϕ2 as
(19)ρ′(ϕ1,ϕ2)=ρ(S1,S2),
where S1 and S2 are multi-valued mappings corresponding to ϕ1 and ϕ2 in the proof of Theorem 14. Then (M′,ρ′) is a metric space. Instead of M, S and F(S) by M′, ϕ and F′(ϕ) in Definition 9, we can also define essential sets e(ϕ), minimal essential sets m(ϕ) of F′(ϕ), and essential component in F′(ϕ). If an essential set e(ϕ) is singleton set {x*}, x* is called an essential solution of ϕ.From Theorems12 and 13, we have the following results.Theorem 18.
F
′
:
M
′
→
2
X is an upper semi-continuous mapping with compact values. For each VQEP ϕ∈M′, there exists at least a connected minimal essential set m(ϕ) of F′(ϕ).Remark 19.
For eachϕ∈M′, y∈X, let
(20)Aϕ(y)={x:y∉G(x),orφ(x,y)∉intCandx∈G(x)}.
For anyϕ1,ϕ2∈M′, from the definition of the metric between S1 and S2, then
(21)ρ′(ϕ1,ϕ2)=supy∈Xh(Aϕ1(y),Aϕ2(y)),
which gives an overall consideration of φ and G. If ϕ1 and ϕ2 are two VEP, then
(22)ρ′(ϕ1,ϕ2)=supy∈Xh({x:φ1(x,y)∉intC},x{x:φ2(x,y)∉intC}).
For the essential stability of solutions for VQEP, clearly, the class of perturbations induced by the metricρ′ is different from the perturbation of uniform topology in [3, 14] and also different from the perturbation of best response defined in [16]. For example, the existence of essential sets of solutions for VQEP in topological vector spaces is proved in [3], and the uniform metric for two VQEP ϕ1=(X,X,C,φ1,G1) and ϕ2=(X,X,C,φ2,G2) is defined as
(23)ρ′′(ϕ1,ϕ2)=sup(x,y)∈X×X∥φ1(x,y)-φ2(x,y)∥ρ′′(ϕ1,ϕ2)=+supx∈Xh(G1(x),G2(x)),
where X is a compact convex subset of a Banach space. Naturally, the feasible mapping G requires closed values, which is not a requirement in Theorem 14, however, where each inverse image being open is necessary.By Theorem18, each connected component including a connected minimal essential set of solutions is essential; that is, the existence of essential components can be induced.Corollary 20.
Letϕ∈M′. There is an essential component in F′(ϕ). If F′(ϕ)={x*} is a singleton, then x* is an essential solution of ϕ.



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