# Parametric Set-Valued Vector Quasi-Equilibrium Problems

Authors: Liya Fan; Aiqin Li
Journal: Advances in Decision Sciences
 (2010)

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## Abstract

Two kinds of parametric set-valued vector quasi-equilibrium problems are introduced. The existence of solutions to these problems is studied.  The upper and lower semicontinuities of their solution maps with respect to the parameters are investigated.


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## Body

## 1. Introduction and Preliminaries

Equilibrium problems are a class of general problems that contains many other problems, such as optimization problems, variational inequality problems, saddle point problems, and complementarity problems, as special cases. Up to now, the main efforts for equilibrium problems have been made for the solution existence; see for example [1–6] and the references therein. A few results have been obtained for properties of solution sets, see [7–12].Motivated and inspired by works in [1, 5, 8–12], in this paper, we will introduce two kinds of parametric set-valued vector quasi-equilibrium problems and study the solution existence of these problems. In addition, we will investigate the upper and lower semicontinuities of their solution maps with respect to the parameters.Throughout this paper, letX,Y be real Hausdorff topological vector spaces, Λ,M real topological vector spaces, and A a nonempty compact convex subset of X. We denote by coA,intA,∂A, and clA the convex hull, interior, boundary, and closed hull of A, respectively. Let K:A×M→2X,T:A×Λ→2Y,F:A×X×Y→2Y, and C:A→2Y be set-valued mappings such that A∩K(x,μ)≠∅ for all x∈A and μ∈M and C(x) be a closed convex pointed cone of Y with intC(x)≠∅ for each x∈A.The mappingF is said to be Y∖-intC quasiconvex of type 2 with respect to T (see [1]) if for any nonempty finite subset {y1,…,yn}⊆A and any x∈co{y1,…,yn}, there exist i∈{1,…,n} and z∈T(x) such that F(x,yi,z)⊆Y∖-intC(x). F is said to be Y∖-intC quasi convex-like of type 2 with respect to T (see [1]) if for any nonempty finite subset {y1,…,yn}⊆A and any x∈co{y1,…,yn}, there exist i∈{1,…,n} and z∈T(x) such that(1.1)F(x,yi,z)∩(Y∖-intC(x))≠∅.LetB be a nonempty subset of X. A set-valued mapping G:B→2Y is said to be upper semicontinuous (shortly, u.s.c) at x0∈B if for any open set V⊇G(x0), there exists an open neighborhood U of x0 such that G(x)⊆V for each x∈U∩B. G is said to be u.s.c on B if it is u.s.c at each point in B.The mappingG:B→2Y is said to be lower semicontinuous (shortly, l.s.c) at x0∈B if for each y∈G(x0) and any open neighborhood V of y there exists an open neighborhood U of x0 such that G(z)∩V≠∅ for each z∈U∩B, or, equivalently, if for any net {xα} with xα→x0 and any y∈G(x0), there exists a net {yα} with yα∈G(xα) for each α such that yα→y. G is said to be l.s.c on B if it is l.s.c at each point in B.The mappingG:B→2Y is said to be closed at x0∈B if for any net {(xα,yα)}:(xα,yα)→(x0,y0) and yα∈G(xα) for each α, one has y0∈G(x0). G is said to be a closed set-valued mapping if its graph, denoted by graphG, is a closed set in X×Y, where graphG={(x,y):x∈B,y∈G(x)}. G is said to have closed values if G(x) is a closed set for each x∈B.A set-valued mappingG:B→2B is said to be a KKM mapping if for each nonempty finite subset {x1,…,xn} of B, one has co{x1,…,xn}⊆⋃i=inG(xi).Lemma 1.1 (Fan-KKM Theorem).
LetB a nonempty convex subset of X and G:B→2B be a KKM mapping. If G(x) is a closed set for every x∈B and there exists x0∈B such that G(x0) is a compact set, then ⋂x∈BG(x)≠∅.Lemma 1.2 (see [13]).
If a set-valued mappingG:X→2Y is u.s.c and has closed values, then it is a closed set-valued mapping.Lemma 1.3 (see [14]).
Let the set-valued mappingG:X→2Y have a compact value at x. Then G is u.s.c at x∈X if and only if for any nets {xα}⊆X:xα→x and {yα}:yα∈G(xα) for each α there exist y∈G(x) and a subnet {yβ} of {yα} such that yβ→y.For any given parametersλ∈Λ and μ∈M, in this paper, we consider the following two parametric set-valued vector quasi-equilibrium problems.PSVQEP.
Findx∈A∩clK(x,μ) such that for each y∈K(x,μ) there exists z∈T(x,λ) satisfying
(1.2)F(x,y,z)⊆Y∖-intC(x).PSVQEP.
Findx∈A∩clK(x,μ) such that for each y∈K(x,μ) there exists z∈T(x,λ) satisfying
(1.3)F(x,y,z)∩(Y∖-intC(x))≠∅.We denote their solution sets byS1(λ,μ) and S2(λ,μ), respectively. Obviously, S1(λ,μ)⊆S2(λ,μ).


## 2. Solution Existence

In this section, we will study the existence of solutions for PSVQEP 1 and PSVQEP 2 without any monotonicity. Since parameters play no role in considering solution existence, for the sake of convenience, we state and prove existence results without parameters. We denote the above problems without parameters by SVQEP1 and SVQEP2, and their solution sets byS1 and S2, respectively.Theorem 2.1.
Let(i)
coK(x)⊆clK(x) for all x∈A,(ii)
{x∈A:y∈K(x)} be an open set,(iii)
F be Y∖-intC quasi convex of type 2 with respect to T,(iv)
{y∈A:∃z∈T(x)s.t.F(x,y,z)⊆Y∖-intC(x)} be a closed set for each x∈A. Then (SVQEP1) has at least a solution.Proof.
PutE:={x∈A:x∈clK(x)} and define three set-valued mappings P:A→2A,H:A→2A, and Q:A→2A by
(2.1)P(x)={y∈A:F(x,y,z)∩-intC(x)≠∅,∀z∈T(x)},∀x∈A,H(x)={K(x)∩P(x),x∈E,A∩K(x),x∈A∖E,Q(y)=A∖{x∈A:y∈H(x)},∀y∈A.
Firstly, we show thatQ is a KKM mapping.
Suppose to the contrary thatQ is not a KKM mapping. Then there exist a nonempty finite subset {y1,…,yn}⊆A and a point x̂=∑j=1nαjyj∈co{y1,…,yn}, where αj≥0,j=1,…,n and ∑j=1nαj=1, such that x̂∉⋃j=1nQ(yj), which implies that yj∈H(x̂),j=1,…,n.
Ifx̂∈E, then F(x̂,yj,z)∩-intC(x̂)≠∅ for all z∈T(x̂) and j=1,…,n, which contradicts (iii).
Ifx̂∉E, then yj∈K(x̂),j=1,…,n, which indicates that x̂=∑j=1nαjyj∈coK(x̂)⊆clK(x̂) and then x̂∈E. This is a contradiction.
Thus,Q is a KKM mapping.
Secondly, we show that⋂y∈AQ(y)≠∅.
For any giveny∈A, we can deduce that
(2.2)Q(y)=A∖{x∈A:y∈H(x)}=A∖({x∈E:y∈K(x)∩P(x)}∪{x∈A∖E:y∈A∩K(x)})=A∖({x∈A:y∈K(x)}∩((A∖E)∪{x∈A:y∈P(x)}))=(A∖({x∈A:y∈K(x)})∪(E)∪{x∈A:∃z∈T(x),F(x,y,z)⊆Y∖-intC(x)}).
By (ii) and (iv), we can conclude that Q(y) is a closed set. Since X is a Hausdorff topological vector space and A is a compact set, we have that Q(y) is compact for each y∈A. By Lemma 1.1, we get ⋂y∈AQ(y)≠∅.
Finally, we prove that the assertion of the theorem holds.
Taking arbitrarilyx¯∈⋂y∈AQ(y), we have x¯∉{x∈A:y∈H(x)} for all y∈A, which indicates that H(x¯)=∅. As A∩K(x)≠∅ for all x, we know that x¯∈E and then K(x¯)∩P(x¯)=∅. Consequently, for each y¯∈K(x¯), there exists z¯∈T(x¯) such that F(x¯,y¯,z¯)⊆Y∖-intC(x¯), which shows that x¯∈S1.By a similar proof as for Theorem2.1, we obtain the following result.Theorem 2.2.
Let hypotheses (i) and (ii) in Theorem2.1 hold and let
(iii)F be Y∖-intC quasi convex-like of type 2 with respect to T,
(iv){y∈A:∃z∈T(x)s.t.F(x,y,z)∩(Y∖-intC(x))≠∅} be a closed set for each x∈A. Then (SVQEP2) has at least a solution.


## 3. Upper Semicontinuity of Solution Sets

In this section, we will study the upper semicontinuity of the solution setsS1(λ,μ) and S2(λ,μ) with respect to parameters (λ,μ). For this end, we assume that S1(λ,μ) and S2(λ,μ) are nonempty for any (λ,μ)∈Λ×M. Let x0∈A,λ0,λ∈Λ and μ0,μ∈M.Theorem 3.1.
Let(i)
E(·)={x∈A:x∈clK(x,·)} and W(·) be closed set-valued mappings, where W(x):=Y∖-intC(x) for each x∈A;(ii)
for any nets{λα}:λα→λ0,{μα}:μα→μ0,{xα}:xα→x0,{zα}:zα∈T(xα,λα) for each α and any y0∈K(x0,μ0), there exist nets {yα}:yα∈K(xα,μα) for each α, {zβ}⊆{zα} and z0∈T(x0,λ0) such that yα→y0 and zβ→z0;(iii)
F be l.s.c on A×X×Y. Then S1(·,·) is both closed and u.s.c at (λ0,μ0).Proof.
We first show thatS1(·,·) is closed at (λ0,μ0).
Suppose to the contrary thatS1(·,·) is not closed at (λ0,μ0). Then there exist nets {(λα,μα)}:(λα,μα)→(λ0,μ0) and {xα}:xα→x0 and xα∈S1(λα,μα) for each α such that x0∉S1(λ0,μ0).
xα∈S1(λα,μα) implies that (μα,xα)∈graphE for each α. By the closedness of A∩clK(·,·), we get x0∈A∩clK(x0,μ0), which together with x0∉S1(λ0,μ0) indicates that there exists y0∈K(x0,μ0) such that
(3.1)F(x0,y0,z)∩-intC(x0)≠∅,∀z∈T(x0,λ0).
Fory0∈K(x0,μ0), by (ii), there exists yα∈K(xα,μα) for each α such that yα→y0. Due to xα∈S1(λα,μα), for each yα∈K(xα,μα), there exists zα∈T(xα,λα) such that F(xα,yα,zα)⊆Y∖-intC(xα). Again by (ii), there exist a subnet {zβ}⊆{zα} and a point z0∈T(x0,λ0) such that zβ→z0 and
(3.2)F(xα,yα,zβ)⊆W(xα).
Forz0∈T(x0,λ0), by (3.1), there exists f0∈F(x0,y0,z0) such that
(3.3)f0∈-intC(x0).
By the lower semicontinuity of F, there exists fβ∈F(xβ,yβ,zβ) for each β such that fβ→f0, which together with the closedness of W(·) and (3.2) implies that f0∈Y∖-intC(x0). This contradicts (3.3). Hence, S1(·,·) is closed at (λ0,μ0).
Next, we show thatS1(·,·) is u.s.c at (λ0,μ0).
By the closedness ofS1(·,·) at (λ0,μ0), S1(λ0,μ0) is closed and hence compact as is A.
Suppose to the contrary thatS1(·,·) is not u.s.c at (λ0,μ0). By Lemma 1.3, there exist nets {(λα,μα)}:(λα,μα)→(λ0,μ0) and {xα}:xα∈S1(λα,μα) for each α such that for any x0∈S1(λ0,μ0) and any subnet {xβ}⊆{xα} one has
(3.4)xβ↛x0.
xα∈S1(λα,μα) implies that xα∈E(μα) for each α and {xα}⊆A. By the compactness of A, there exists a convergent subnet {xβ} of {xα} such that xβ→x¯∈A. By the closedness of E(·), we have x¯∈E(μ0). By (3.4), we get x¯∉S1(λ0,μ0), that is,
(3.5)F(x¯,y¯,z)∩-intC(x¯)≠∅,∀z∈T(x¯,λ0).
By using a similar argument as in part one, we can complete the proof.Theorem 3.2.
Let hypotheses (i) and (ii) in Theorem3.1 hold and let
(iii)F be u.s.c on A×X×Y.
ThenS2(·,·) is both u.s.c and closed at (λ0,μ0).Proof.
We first prove thatS2(·,·) is closed at (λ0,μ0).
Suppose to the contrary thatS2(·,·) is not closed at (λ0,μ0). Then there exist nets {(λα,μα)}:(λα,μα)→(λ0,μ0) and {xα}:xα→x0 and xα∈S2(λα,μα) for each α such that x0∉S2(λ0,μ0). By using a similar reasoning as in part one of the proof of Theorem 3.1, we can conclude that there exists a net {(xβ,yβ,zβ)} such that (xβ,yβ,zβ)→(x0,y0,z0) and
(3.6)F(x0,y0,z0)⊆-intC(x0),(3.7)F(xβ,yβ,zβ)∩(Y∖-intC(xβ))≠∅,∀β,
where yβ∈K(xβ,μβ) with yβ→y0∈K(x0,μ0) and zβ∈T(xβ,λβ) with zβ→z0∈T(x0,λ0). By the upper semicontinuity of F and (3.6), we know that there exists β0 such that
(3.8)F(xβ,yβ,zβ)⊆-intC(x0),∀β≥β0,
which contradicts (3.7). Hence, S2(·,·) is closed at (λ0,μ0).
Next, we prove thatS2(·,·) is u.s.c at (λ0,μ0).
By the closedness ofS2(·,·) at (λ0,μ0), S2(λ0,μ0) is closed and hence compact as is A.
Suppose to the contrary thatS2(·,·) is not u.s.c at (λ0,μ0). By Lemma 1.3, there exist nets {(λα,μα)}:(λα,μα)→(λ0,μ0) and {xα}:xα∈S2(λα,μα) for each α such that (3.4) holds for any x0∈S2(λ0,μ0) and any subnet {xβ}⊆{xα}.
xα∈S2(λα,μα) implies that xα∈E(μα) for each α and {xα}⊆A. By the compactness of A and the closedness of E(·), it follows that there exists a convergent subnet {xβ} of {xα} such that xβ→x¯∈E(μ0). By (3.4), we get x¯∉S2(λ0,μ0), that is,
(3.9)F(x¯,y¯,z)⊆-intC(x¯),∀z∈T(x¯,λ0).
By using a similar argument as in part one, we can complete the proof.


## 4. Lower Semicontinuity of Solution Sets

In this section, we will consider the lower semicontinuity of the solution setsS1(·,·) and S2(·,·) with respect to parameters (λ,μ).Theorem 4.1.
Let(i)
E(·):={x∈A|x∈clK(x,·)} be l.s.c on M and C(·)u.s.c at x0;(ii)
for any nets{λα}:λα→λ0,{μα}:μα→μ0,{xα}:xα→x0,{yα}:yα∈K(xα,μα) for each α and any z0∈T(x0,λ0), there exist nets {zα}:zα∈T(xα,λα) for each α, {yβ}⊆{yα} and a point y0∈K(x0,μ0) such that zα→z0 and yβ→y0;(iii)
F be u.s.c and have compact values on A×X×Y;(iv)
F(x0,y0,z0)∩-∂C(x0)=∅ for all x0∈S1(λ0,μ0),y0∈K(x0,μ0), and z0∈T(x0,λ0). Then S1(·,·) is l.s.c at (λ0,μ0).Proof.
Suppose to the contrary thatS1(·,·) is not l.s.c at (λ0,μ0). Then there exist a net {(λα,μα)}:(λα,μα)→(λ0,μ0) and a point x0∈S1(λ0,μ0) such that for any net {x̃α}:x̃α∈S1(λα,μα) for each α one has
(4.1)x̃α↛x0.
x0∈S1(λ0,μ0) implies that x0∈E(μ0). By the lower semicontinuity of E, there exists a net {xα}:xα∈E(μα) for each α such that xα→x0, which combining with (4.1) shows that there exists a subnet {xβ} of {xα}  such that xβ∉S1(λβ,μβ) for all β. Consequently, for each β, there exists yβ∈K(xβ,μβ) satisfying
(4.2)F(xβ,yβ,zβ)∩-intC(xβ)≠∅,∀zβ∈T(xβ,λβ).
By (ii), there exist a subnet{y¯β}⊆{yβ} and a point y0∈K(x0,μ0) such that y¯β→y0, which together with x0∈S1(λ0,μ0) and (ii) indicates that there exist z0∈T(x0,λ0) and z¯β∈T(x¯β,λ¯β) such that z¯β→z0,F(x¯β,y¯β,z¯β)∩-intC(x¯β)≠∅ for all β and
(4.3)F(x0,y0,z0)⊆Y∖-intC(x0).
Take arbitrarilyfβ∈F(x¯β,y¯β,z¯β)∩-intC(x¯β) for each β. By Lemma 1.3, there exist f0∈F(x0,y0,z0) and a subset {fβ¯} of {fβ} such that fβ¯→f0.
Sincefβ¯∈-C(x¯β¯) for each β¯, by the upper semicontinuity of C(·) and Lemma 1.2, we know that f0∈-C(x0), which together with (iv) shows that f0∈-intC(x0). This contradicts (4.3). Hence, S1(·,·) is l.s.c at (λ0,μ0).Theorem 4.2.
Let hypotheses (i) and (ii) in Theorem4.1 hold and let
(iii)F(·,·,·) be l.s.c on A×X×Y;
(iv)F(x0,y0,z0)∩-∂C(x0)=∅ for all x0∈S2(λ0,μ0),y0∈K(x0,μ0), and z0∈T(x0,λ0). Then S2(·,·) is l.s.c at (λ0,μ0).Proof.
By arguments similar to those for Theorem4.1, we can conclude that there exists a net {(x¯β,y¯β,z¯β)} such that (x¯β,y¯β,z¯β)→(x0,y0,z0),F(x0,y0,z0)∩(Y∖-intC(x0))≠∅, and
(4.4)F(x¯β,y¯β,z¯β)⊆-intC(x¯β),∀β,
where x¯β∈E(μ¯β),y¯β∈K(x¯β,μ¯β),z¯β∈T(x¯β,λ¯β) for all β and x0∈S2(λ0,μ0),y0∈K(x0,μ0) and z0∈T(x0,λ0).
For any givenf0∈F(x0,y0,z0)∩(Y∖-intC(x0)), by the lower semicontinuity of F, there exists f¯β∈F(x¯β,y¯β,z¯β) for each β such that f¯β→f0. By (4.4), we have f¯β∈-C(x¯β) for each β. By the upper semicontinuity of C(·) and Lemma 1.2, it follows that f¯0∈-C(x¯0), which together with (iv) implies that f0∈-intC(x0). This is a contradiction. Hence, S2(·,·) is l.s.c at (λ0,μ0).



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