# Generalized Well-Posedness for Symmetric Vector Quasi-Equilibrium Problems

Authors: Wei-bing Zhang; Nan-jing Huang; Donal O’Regan
Journal: Journal of Applied Mathematics
 (2015)

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## Abstract

We introduce and study well-posedness in connection with the symmetric vector quasi-equilibrium problem, which unifies its Hadamard and Levitin-Polyak well-posedness. Using the nonlinear scalarization function, we give some sufficient conditions to guarantee the existence of well-posedness for the symmetric vector quasi-equilibrium problem.


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## Body

## 1. Introduction

Vector equilibrium problem provides a very general model for many problems like the vector variational inequality problem, the vector complementarity problem, the vector optimization problem, the multiobjective game problem, the vector network equilibrium problem, and the vector saddle point problem (see, e.g., [1–3] and the references therein).Well-posedness plays an important role in the theory and numerical methods for optimization. The first concept of well-posedness was introduced by Tykhonov [4] for a global minimization problem having a unique solution. Lucchetti and Patrone [5] introduced the notion of well-posedness for variational inequalities. Lignola and Morgan [6] studied the well-posedness of optimization problems with variational inequality constraints. Fang et al. [7] investigated well-posedness for equilibrium problems and optimization problems with equilibrium constraints. Hu et al. [8] studied well-posedness of systems of equilibrium problems.The notions of well-posedness can be mainly divided into three groups, namely, Hadamard type, Tykhonov type, and Levitin-Polyak type. Researchers have studied the relations between the Hadamard well-posedness and Tykhonov well-posedness for different problems (see [9, 10]). Most of the literature deals with directly specific notions of well-posedness. Huang et al. [11] investigated the Levitin-Polyak well-posedness of variational inequalities problems with functional constraints. S. J. Li and M. H. Li [12] studied the Levitin-Polyak well-posedness of vector equilibrium problems. Li et al. [13] investigated Levitin-Polyak well-posedness of generalized vector quasi-equilibrium problems. Peng et al. [14] studied Levitin-Polyak well-posedness of generalized vector equilibrium problems with both abstract set constraints and functional constraints. Salamon [15] considered the Hadamard well-posedness by using the vector topological pseudomonotonicity. Peng et al. [16] investigated the Hadamard well-posedness of vector equilibrium problems by considering the perturbations of both vector-valued functions and feasible sets. Li and Zhang [17] studied the Hadamard well-posedness for the vector-valued optimization problems. Long and Huang [18] considered the α-well posedness for the symmetric quasi-equilibrium problems. Recently, Han and Gong [19] studied the generalized Levitin-Polyak well-posedness of symmetric strong vector quasi-equilibrium problems. Deng and Xiang [20] introduced and studied the generalized well-posedness for the generalized vector equilibrium, which unifies its Hadamard and Levitin-Polyak well-posedness.Motivated and inspired by the papers mentioned above, in this paper, we introduce a well-posedness concept for the symmetric vector quasi-equilibrium problem, which unifies its Hadamard and Levitin-polyak well-posedness. By employing the scalarization function, we give some sufficient conditions to guarantee the existence of the well-posedness for the symmetric vector quasi-equilibrium problem in real locally convex Hausdorff topological vector spaces. The results presented in this paper generalize and extend Theorem 4.2 of [16] and Theorem 4.1 of [20].


## 2. Preliminaries

LetX and Y be nonempty subsets of real locally convex Hausdorff topological vector spaces F and W, respectively. Assume S:X×Y⇉X and G:X×Y⇉Y are two set-valued mappings. Let Z be a real topological vector space and C⊂Z be a closed convex pointed cone with intC≠∅, where intC denotes the topological interior of C. It is well known that the cone C can induce the following orders:
(1)z1≤z2iffz2-z1∈C,z1<z2iffz2-z1∈intC.Letφ:X×X×Y→Z and ψ:X×Y×Y→Z be two vector valued functions. Let C1 and C2 be two closed convex pointed cones of Z with intC1≠∅ and intC2≠∅. In this paper, we consider the following the symmetric vector quasi-equilibrium problem (in short SVQEP): find (x0,y0)∈X×Y such that x0∈S(x0,y0), y0∈G(x0,y0), and
(2)φx0,u,y0∉-intC1,∀u∈S(x0,y0),ψx0,v,y0∉-intC2,∀v∈Gx0,y0.Some special cases of SVQEP are as follows.(I) IfC1=C2=C, φ(x,u,y)=f(u,y)-f(x,y), and ψ(x,v,y)=g(x,v)-g(x,y) for all (x,y)∈X×Y, where f,g:X×Y→Z are two mappings, then (SVQEP) reduces to the problem of finding (x0,y0)∈X×Y such that x0∈S(x0,y0), y0∈G(x0,y0), and
(3)f(x,y0)-f(x0,y0)∉-intC,∀x∈S(x0,y0),g(x0,y)-g(x0,y0)∉-intC,∀x∈G(x0,y0).
This problem was studied in Fu [21] and Han and Gong [19].(II) IfZ=(-∞,+∞) and C1=C2=[0,+∞), then (SVQEP) reduces to the following problem: find (x0,y0)∈X×Y such that x0∈S(x0,y0), y0∈G(x0,y0), and
(4)φ(x0,u,y0)≥0,∀u∈S(x0,y0),ψ(x0,v,y0)≥,∀v∈G(x0,y0).(III) IfZ=(-∞,+∞), C1=C2=[0,+∞), φ(x,u,y)=f(u,y)-f(x,y), and ψ(x,v,y)=g(x,v)-g(x,y) for all (x,y)∈X×Y, where f,g:X×Y→Z are two mappings, then (SVQEP) reduces to the symmetric quasi-equilibrium problem: find (x0,y0)∈X×Y such that x0∈S(x0,y0), y0∈G(x0,y0), and
(5)f(x,y0)≥f(x0,y0),∀x∈S(x0,y0),g(x0,y)≥g(x0,y0),∀x∈G(x0,y0).
This problem was considered in Long and Huang [18].It is well known that SVQEP includes many important problems as special cases, such as equilibrium problems, Nash equilibrium problems, quasivariational inequalities, variational inequalities, and fixed point problems.Now we recall some useful definitions and lemmas.Let(X,d) be a metric space. Denote a family of all nonempty compact subsets of X by K(X). For any A,B∈K(X), let
(6)H(A,B)=max⁡sup⁡a∈Ainf⁡b∈Bd(a,b),sup⁡b∈Binf⁡a∈Ad(a,b)
denote the Hausdorff metric on K(X). It is well known that (K(X),H) is complete if and only if (X,d) is complete.Definition 1.
A sequence{(xn,yn)}⊂X×Y is called a Levitin-Polyak (in short LP) approximating solution sequence converging to (x0,y0) for SVQEP, if there exists ɛn>0 with ɛn→0 such that
(7)dxn,Sxn,yn≤ɛn,dyn,Gxn,yn≤ɛn,φ(x0,u,y0)+ɛne1∉-intC1,∀u∈S(x0,y0),ψ(x0,v,y0)+ɛne2∉-intC2,∀v∈G(x0,y0),
where e1∈intC1 and e2∈intC2 are given points.Definition 2 (see [3]).
LetE, Z be two real Hausdorff topological spaces, X⊂E a nonempty subset, and C⊂Z a closed convex pointed cone. A mapping F:X→Z is said to be(i)
C-upper semicontinuous (C-u.s.c.) (resp., C-lower semicontinuous (C-l.s.c.) at x0∈X if, for any neighborhood V of zero in Z, there exists a neighborhood U of zero in E such that
(8)F(x)∈F(x0)+V-C,∀x∈U∩X,(resp.,F(x)∈F(x0)+V+C,∀x∈U∩X).(ii)
C-u.s.c. (resp., C-l.s.c.) on X if it is C-u.s.c. (resp., C-l.s.c.) at every point x∈X.Definition 3 (see [22]).
LetX and Y be two topological spaces. A set-valued mapping F:X⇉Y is said to be(i)
upper semicontinuous (u.s.c.) atx0 if, for any open set V⊃F(x0), there is an open neighborhood Ox0 of x0 such that F(x′)⊂V for each x′∈Ox0,(ii)
lower semicontinuous (l.s.c.) atx0 if, for any open set V∩F(x0)≠∅, there is an open neighborhood Ox0 of x0 such that F(x′)∩V≠∅ for each x′∈Ox0,(iii)
continuous atx0 if it is both upper and lower semicontinuous at x0,(iv)
upper semicontinuous (lower semicontinuous or continuous) onX if it is upper semicontinuous (lower semicontinuous or continuous) at every x∈X,(v)
an usco mapping ifF is upper semicontinuous on X and F(x) is compact for each x∈X,(vi)
closed if and only if its graphGraph⁡(F):={(x,y)∈X×Y:y∈F(x)} is closed.Lemma 4 (see [23]).
LetX and Y be two topological spaces and F:X⇉Y a set-valued mapping. Suppose that Y is compact. Then F is closed if and only if it is upper semicontinuous.Lemma 5 (see [24]).
LetA and An(n=1,2,…) all be nonempty compact subsets of a metric space X with An→A in the Hausdorff metric topology. Then the following statements hold.(i)
⋃n=1+∞‍An⋃‍A is also nonempty compact subset of X.(ii)
Ifxn∈An,xn→x, then x∈A.(iii)
For anyx∈A, there exists xn∈An such that xn→x.Lemma 6 (see [25]).
LetX and Y be two topological spaces and F:X⇉Y be an usco mapping. Then for any net {xα}⊂X with xα→x and yα∈F(xα), there exists a subnet {yα′}⊂yα such that yα′→y∈F(x).Lemma 7 (see [26]).
For any fixed pointe∈intC, the nonlinear scalarization function ξe:Y→R is defined by
(9)ξe(y)=inf⁡{r∈R:y∈re-C},∀y∈Y.
The nonlinear scalarization function ξe has the following properties:(i)
ξe(y)<r⇔y∈re-intC,(ii)
ξe(y)≤r⇔y∈re-C,(iii)
ξe(y)≥r⇔y∉re-intC,(iv)
ξe(y)>r⇔y∉re-C,(v)
ξe(y)=r⇔y∈re-∂C, where ∂C denotes the topological boundary of  C.Lemma 8.
Ifφ:X×X×Y→Z is C-upper semicontinuous, then ξeφ is upper semicontinuous.Proof.
For any fixedr∈R=(-∞,+∞), let
(10)B={(x,u,y)∈X×X×Y:ξe(φ(x,u,y))≥r}.
In order to show that ξeφ is upper semicontinuous, we only need to show that B is closed. Letting (xn,un,yn)∈B with (xn,un,yn)→(x,u,y), then
(11)ξeφ(xn,un,yn)≥r
and Lemma 7 shows that
(12)φ(xn,un,yn)∉re-intC.
Suppose to the contrary that (x,u,y)∉B. We have
(13)ξe(φ(x,u,y))<r.
It follows from Lemma 7 that
(14)φ(x,u,y)-re∈-intC.
This implies that there exists a neighborhood V of zero element in Z such that
(15)φ(x,u,y)-re+V∈-intC.
Since φ is C-upper semicontinuous, for the above V, there exists a positive integral number N such that, when n>N, one has
(16)φ(xn,un,yn)∈φ(x,u,y)+V-C.
From (15) and (16), when n>N, we have
(17)φxn,un,yn∈φ(x,u,y)+V-C⊂re-intC-C⊂re-intC,
which is in contradiction with (12). Thus, (x,u,y)∈B and so ξeφ is upper semicontinuous. This completes the proof.Example 9.
LetX=Y=R=(-∞,+∞) and C=[0,+∞). Let
(18)φ(x,y,z)=2,ifx=y=z=1;x+y-z,otherwise.
Then it is easy to check that φ is C-upper semicontinuous and so Lemma 8 shows that ξeφ is upper semicontinuous for any e∈intC.


## 3. Bounded Rationality Model and Definition of Well-Posedness for SVQEP

Let(X,d1) and (Y,d2) be two metric spaces and Z be a Banach space with a norm ∥·∥. Let C1 and C2 be nonempty closed convex pointed cones of Z with apex at the origin such that intC1≠∅ and intC2≠∅.In this section, we first define the problem spaceΛ of SVQEP as follows: Λ={(φ,ψ,S,G):φ:X×X×Y→Z is C1-upper semicontinuous on X×X×Y,ψ:X×Y×Y→Z is C2-upper semicontinuous onX×Y×Y,φ(x,x,y)=0 and ψ(x,y,y)=0 for all (x,y)∈X×Y, sup⁡(x,u,y)∈X×X×Y‖φ(x,u,y)‖<+∞sup⁡(x,v,y)∈X×Y×Y‖ψ(x,v,y)‖<+∞, S:X×Y⇉X and G:X×Y⇉Y are continuous with compact values, and there exists (x,y)∈X×Y such that x∈S(x,y),y∈G(x,y) and φ(x,u,y)∉-intC1,ψ(x,v,y)∉-intC2 for  all u∈S(x,y) and v∈G(x,y)}.For anyλi=(φi,ψi,Si,Gi)∈Λ with i=1,2, define
(19)ρλ1,λ2=sup⁡(x,u,y)∈X×X×Yφ1x,u,y-φ2x,u,y+sup⁡(x,v,y)∈X×Y×Yψ1x,v,y-ψ2x,v,y+sup⁡(x,y)∈X×YH1S1x,y,S2x,y+sup⁡(x,y)∈X×YH2(G1(x,y),G2(x,y)),
where H1 and H2 denote the Hausdorff metric on K(X) and K(Y), respectively. Then it is easy to see that (Λ,ρ) is a metric space.Next we define the bounded rationality modelM={Λ,X,Y,f,Φ} for (SVQEP) as follows.(i)
(Λ,ρ), (X,d1) and (Y,d2) are metric spaces.(ii)
The feasible set of the symmetric vector quasi-equilibrium problemsλ∈Λ is defined by
(20)fλ:=x,y∈X×Y:x∈Sx,y,y∈Gx,y.(iii)
The solution set of problemλ∈Λ is defined by
(21)Eλ=x,y∈X×Y:x,y∈fλsuchthatφ(x,u,y)∉-intC1∀u∈Sx,y,ψ(x,v,y)∉-intC2∀v∈Gx,y.(iv)
For any(x,y)∈X×Y, define
(22)dx,y,fλ=d1x,Sx,y+d2y,Gx,y.(v)
The rationality function of the problemλ∈Λ is defined by
(23)Φ(λ,(x,y))=max⁡sup⁡u∈S(x,y)-ξe1φ(x,u,y),sup⁡v∈G(x,y)-ξe2ψ(x,v,y),
where e1∈intC1 and e2∈intC2.Lemma 10.
(i) For anyλ∈Λ and (x,y)∈f(λ), Φ(λ,(x,y))≥0.
(ii) For anyλ∈Λ, Eλ≠∅.
(iii) Forλ∈Λ and ϵ≥0 with (x,y)∈f(λ), Φ(λ,(x,y))≤ϵ if and only if
(24)φx,u,y+ϵe1∉-intC1,∀u∈Sx,y,ψ(x,v,y)+ϵe2∉-intC2,∀v∈G(x,y).
In particular,(x,y)∈E(λ) if and only if (x,y)∈f(λ) and Φ(λ,(x,y))=0.Proof.
(i) For anyλ∈Λ and (x,y)∈f(λ), we have x∈S(x,y) and y∈G(x,y). From Lemma 7(v), we know that ξe1φ(x,x,y)=0 and ξe2ψ(x,y,y)=0, since φ(x,x,y)=0 and ψ(x,y,y)=0. Thus, one has
(25)Φλ,x,y=max⁡sup⁡u∈S(x,y)-ξe1φ(x,u,y),sup⁡v∈G(x,y)-ξe2ψ(x,v,y)≥max⁡-ξe1φx,x,y,-ξe2ψx,y,y=0.
(ii) It is obvious thatE(λ)≠∅ for all λ∈Λ.
(iii) Assume thatλ∈Λ and ϵ≥0 such that Φ(λ,(x,y))≤ϵ. Then we have
(26)max⁡sup⁡u∈S(x,y)-ξe1φ(x,u,y),sup⁡v∈G(x,y)-ξe2ψ(x,v,y)≤ϵ
and so
(27)sup⁡u∈S(x,y)-ξe1φ(x,u,y)≤ϵ
with
(28)sup⁡v∈G(x,y)-ξe2ψ(x,v,y)≤ϵ.
It follows that
(29)ξe1φx,u,y≥-ϵ,∀u∈Sx,y,ξe2ψx,v,y≥-ϵ,∀v∈G(x,y).
By Lemma 7(iii), one has
(30)φx,u,y+ϵe1∉-intC1,∀u∈Sx,y,ψ(x,v,y)+ϵe2∉-intC2,∀v∈G(x,y).
Conversely, assume thatλ∈Λ and ϵ≥0 with (x,y)∈f(λ) such that
(31)φx,u,y+ϵe1∉-intC1,∀u∈Sx,y,ψ(x,v,y)+ϵe2∉-intC2,∀v∈G(x,y).
Then Lemma 7(iii) shows that
(32)ξe1φ(x,u,y)≥-ϵ,∀u∈S(x,y),ξe2φ(x,v,y)≥-ϵ,∀v∈G(x,y).
These inequalities imply that
(33)sup⁡u∈S(x,y)-ξe1φx,u,y≤ϵ,sup⁡v∈G(x,y)-ξe2ψ(x,v,y)≤ϵ.
Therefore, we get
(34)Φ(λ,(x,y))=max⁡sup⁡u∈S(x,y)-ξe1φx,u,y,sup⁡v∈G(x,y)-ξe2ψx,v,y≤ϵ.
This completes the proof.Remark 11.
Lemma10 is a generalization of Lemma 3.1 of Deng and Xiang [20].Example 12.
LetX=Y=(-∞,+∞), C1=C2=C=[0,+∞), and e1=e2=1. Assume that
(35)Sx,y=Gx,y=0,1,∀x,y∈X×Y,φ(x,u,y)=x-u,ψ(x,v,y)=y-v,∀x,y∈X×Y,u∈Sx,y,v∈G(x,y).
Then it is easy to see that
(36)fλ=x,y:x∈0,1,y∈0,1,Φ(λ,(x,y))=max⁡sup⁡u∈[0,1]-ξ1(x-u),sup⁡v∈[0,1]-ξ1(y-v).(i)
If(x,y)∈f(λ), then x∈[0,1] and y∈[0,1]. It is obvious that
(37)Φ(λ,(x,y))≥0.(ii)
For anyλ∈Λ, one has
(38)E(λ)={(x,y):x=1,y=1}≠∅.(iii)
It is easy to check that(x,y)∈E(λ) if and only if Φ(λ,(x,y))=0. Moreover, taking x=0,y=1/2, then (x,y)=(0,1/2)∉E(λ) and Φ(λ,(x,y))=1≠0.By Definition1 and Lemma 10, for all ɛn>0 with ɛn→0, the LP approximating solution set for the problem λ is defined as
(39)E(λ,ɛn)=x,y∈X×Y:dx,y,fλ≤ɛn,hh1111111hΦ(λ,(x,y))≤ɛn,
and the set of solution for the problem λ is defined as
(40)Eλ=E(λ,0)={(x,y)∈X×Y:(x,y)∈f(λ),Φ(λ,(x,y))=0}.Next we define the Levitin-Polyak well-posedness and the Hadamard well-posedness for (SVQEP) as follows.Definition 13.
(i) If, for any(xn,yn)∈E(λ,ɛn),ɛn>0 with ɛn→0, there exists a subsequence {(xnk,ynk)}⊂{(xn,yn)} such that (xnk,ynk)→(x,y)∈E(λ), then the problem λ∈Λ is said to be generalized LP well-posedness.
(ii) IfE(λ)={(x,y)} (a singleton), for any (xn,yn)∈E(λ,ɛn),ɛn>0 with ɛn→0, there exists a subsequence {(xnk,ynk)}⊂{(xn,yn)} such that (xnk,ynk)→(x,y)∈E(λ), then the problem λ∈Λ is said to be LP well-posedness.Definition 14.
(i) If, for anyλn∈Λ with λn→λ and any (xn,yn)∈E(λn), there exists a subsequence {(xnk,ynk)}⊂{(xn,xn)}, such that (xnk,ynk)→(x,y)∈E(λ), then the problem λ∈Λ is said to be generalized Hadamard well-posed.
(ii) IfE(λ)={(x,y)} (a singleton), for any λn∈Λ with λn→λ and any (xn,yn)∈E(λn), we have (xn,yn)→(x,y)∈E(λ), then the problem λ∈Λ is said to be Hadamard well-posed.By Definitions13 and 14, we introduce the definition of (generalized) well-posedness, which unifies its Levitin-Polyak well-posedness and Hadamard well-posedness.Definition 15.
(i) If for anyλn∈Λ with λn→λ, (xn,yn)∈E(λn,ɛn), ɛn>0 with ɛn→0, there exists a subsequence {(xnk,ynk)}⊂{(xn,yn)} such that (xnk,ynk)→(x,y)∈E(λ), then the problem λ∈Λ is said to be generalized well-posed.
(ii) IfE(λ)={(x,y)} (a singleton), for any λn∈Λ with λn→λ, (xn,yn)∈E(λn,ɛn), ɛn>0 with ɛn→0, we have (xn,yn)→(x,y)∈E(λ), then the problem λ∈Λ is said to be well-posed.Lemma 16.
If the problemλ∈Λ is (generalized) well-posed, then(i)
the problemλ∈Λ must be (generalized) LP well-posed;(ii)
the problemλ∈Λ must be (generalized) Hadamard well-posed.Proof.
We only prove the statement of generalized well-posedness. The proof of the well-posedness is similar to the generalized well-posedness.
(i) For any(xn,yn)∈E(λ,ɛn), letting λn=λ, we know that λn→λ and (xn,yn)∈E(λn,ɛn). Since the problem λ∈Λ is generalized well-posed, there exists a subsequence {(xnk,ynk)}⊂{(xn,yn)} such that (xnk,ynk)→(x,y)∈E(λ). Thus, it follows that λ is generalized LP well-posed.
(ii) For anyλn∈Λ with λn→λ and (xn,yn)∈E(λn), letting ɛn=0, it is easy to see that (xn,yn)∈E(λn,ɛn). Since the problem λ∈Λ is generalized well-posed, there exists a subsequence {(xnk,ynk)}⊂{(xn,yn)} such that (xnk,ynk)→(x,y)∈E(λ). Therefore, we know that λ is generalized Hadamard well-posed.
This completes the proof.


## 4. Some Sufficient Conditions for Well-Posedness of SVQEP

Assume(X,d1) and (Y,d2) are compact metric spaces. In this section, we give some sufficient conditions to guarantee the existence of the well-posedness for SVQEP.Lemma 17.
(
Λ
,
ρ
) is a complete metric space.Proof.
Let{λn=(φn,ψn,Sn,Gn)} be a Cauchy sequence in Λ. Then for any given ɛ>0, when integral numbers m and n are large enough, we have
(41)ρ(λn,λm)=sup⁡(x,u,y)∈X×X×Yφn(x,u,y)-φm(x,u,y)+sup⁡(x,v,y)∈X×Y×Yψnx,v,y-ψmx,v,y+sup⁡(x,y)∈X×YH1Snx,y,Smx,y+sup⁡(x,y)∈X×YH2(Gn(x,y),Gm(x,y))≤ɛ.
It follows that {φn(x,u,y)} and {ψn(x,u,y)} are both Cauchy sequences in Z for any fixed (x,u,y)∈X×X×Y and (x,v,y)∈X×Y×Y, respectively. Moreover, we know that {Sn(x,y)}, {Gn(x,y)} are both Cauchy sequences in K(X) and K(Y) for any fixed (x,y)∈X×Y, respectively.
(I) SinceZ is a complete space, there exists z∈Z such that limn→∞φn(x,u,y)=z. Let φ(x,u,y):=z. We show that φ is C1-upper semicontinuous and
(42)sup⁡(x,u,y)∈X×Y×Yφ(x,u,y)<+∞.
For any open neighborhood V of zero in Z, since φn is C1-upper semicontinuous, there exists an open neighborhood U of (x,u,y) such that
(43)φnx′,u′,y′∈φnx,u,y+V3-C1,∀x′,u′,y′∈U.
Since limn→∞φn(x,u,y)=φ(x,u,y), for n large enough, one has
(44)φx′,u′,y′∈φnx′,u′,y′+V3,φn(x,u,y)∈φ(x,u,y)+V3.
It follows from (43) and (44) that
(45)φ(x′,u′,y′)∈φ(x,u,y)+V-C1,∀(x′,u′,y′)∈U
and so φ is C1-upper semicontinuous.
On the other hand,λn∈Λ shows that
(46)sup⁡(x,u,y)∈X×X×Yφn(x,u,y)<+∞.
Since limn→∞φn(x,u,y)=φ(x,u,y), it is easy to see that
(47)sup⁡(x,u,y)∈X×X×Yφ(x,u,y)<+∞.
Moreover, from the fact thatlimn→∞φn(x,u,y)=φ(x,u,y) and φn(x,x,y)=0, we know that φ(x,x,y)=0.
(II) Similar to the proof of (I), we know that there exists aC2-upper semicontinuous mapping ψ such that limn→∞ψn(x,u,y)=ψ(x,u,y) with ψ(x,y,y)=0 and
(48)sup⁡(x,u,y)∈X×Y×Yψ(x,u,y)<+∞.
(III) SinceSn(x,y), Gn(x,y) are both Cauchy sequences, K(X) and K(Y) are complete, we know that there exist sets M∈K(X) and N∈K(Y) such that
(49)limn→∞Sn(x,y)=M,limn→∞Gn(x,y)=N.
Let S(x,y):=M and G(x,y):=N. Then the fact that Sn and Gn are both continuous with compact set-values shows that S and G are both continuous with compact set-values.
(IV) We prove that there existx∈S(x,y) and y∈G(x,y) such that
(50)φx,u,y∉-intC1,∀u∈Sx,y,ψx,v,y∉-intC2,∀v∈G(x,y).
In fact, since (φn,ψn,Sn,Gn)∈Λ, there exist sequences {xn} and {yn} such that xn∈Sn(xn,yn), yn∈Gn(xn,yn), and
(51)φn(xn,u,yn)∉-intC1,∀u∈Sn(xn,yn)
with
(52)ψ(xn,v,yn)∉-intC2,∀v∈G(xn,yn).
Since X and Y are compact, there exist subsequences {xnk}⊂{xn} and {ynk}⊂{yn} such that xnk→x∈X and ynk→y∈Y. From the continuities of S and G, we have
(53)H1(S(xnk,ynk),S(x,y))⟶0,H2(G(xnk,ynk),G(x,y))⟶0.
When n,m are large enough, one has
(54)sup⁡(x,y)∈X×YH1(Sn(x,y),Sm(x,y))≤ɛ
and so
(55)H1(Snk(xnk,ynk),S(xnk,ynk))≤ɛ
holds for nk large enough. By the definition of the Hausdorff metric, we have
(56)d1x,Sx,y≤d1(x,xnk)+d1(xnk,Snk(xnk,ynk))+H1(Snk(xnk,ynk),S(x,y))=d1x,xnk+H1Snkxnk,ynk,Sx,y≤d1x,xnk+H1Snkxnk,ynk,Sxnk,ynk+H1(S(xnk,ynk),S(x,y)).
This together with (55) shows that d1(x,S(x,y))=0 and so x∈S(x,y). Similarly, we can get y∈G(x,y).
Next we prove that(57)φ(x,u,y)∉-intC1,∀u∈S(x,y).
By contradiction, we assume that there exists u0∈S(x,y) such that
(58)φ(x,u0,y)∈-intC1.
Then there exists an open neighborhood V of zero in Z such that
(59)φ(x,u0,y)+V∈-intC1.
By Lemma 5, there is a sequence {unk} with unk∈Snk(xnk,ynk) such that unk→u0. Since φnk is C1-upper semicontinuous and φnk(x,u0,y)→φ(x,u0,y), we have
(60)φnk(xnk,unk,ynk)∈φnk(x,u0,y)+V2-C1,φnk(x,u0,y)∈φ(x,u0,y)+V2.
From (59) and (60), we get
(61)φnk(xnk,unk,ynk)∈-C1-intC1⊂-intC1,
which is in contradiction with the fact that φnk(xnk,u,ynk)∉-intC1 for all u∈Snk(xnk,ynk).
Similarly, we can show thaty∈G(x,y) satisfies
(62)ψ(x,v,y)∉-intC2,∀v∈G(x,y).
(V) Letλ=(φ,ψ,S,G). Then λ∈Λ and λn→λ. Therefore, (Λ,ρ) is a complete metric space. This completes the proof.Lemma 18.
f
:
Λ
⇉
X
×
Y is an usco mapping.Proof.
It is easy to see thatf(λ) is closed for any given λ∈Λ. In fact, for any
(63)(xn,yn)∈f(λ)={(x,y)∈X×Y:x∈S(x,y),y∈G(x,y)}
with xn→x and yn→y, we have xn∈S(xn,yn) and yn∈G(xn,yn) and so
(64)d1x,Sx,y≤d1(x,xn)+d1(xn,S(x,y))≤d1(x,xn)+H1(S(xn,yn),S(x,y)).
It follows from the continuity of S that d1(x,S(x,y))=0 and so x∈S(x,y). Similarly, we can get y∈G(x,y). Thus, (x,y)∈f(λ) and so f(λ) is closed. Since X×Y is compact, we know that f(λ) is compact for any given λ∈Λ. In order to show that f is an u.s.c. mapping, from Lemma 4, it is sufficient to show that Graph⁡(f)={(λ,(x,y)):(x,y)∈f(λ)} is closed.
Let(λn,(xn,yn))∈Graph⁡(f) with (λn,(xn,yn))→(λ,(x,y)). Then the completeness of (Λ,ρ) shows that λ∈Λ. Moreover, we have xn∈Sn(xn,yn) and yn∈Gn(xn,yn) and
(65)H1Snxn,yn,Sxn,yn⟶0,H2(Gn(xn,yn),G(xn,yn))⟶0,H1(S(xn,yn),S(x,y))⟶0,H2(G(xn,yn),G(x,y))⟶0.
By the definition of the Hausdorff metric, it follows that
(66)d1x,Sx,y≤d1(x,xn)+d1(xn,Sn(xn,yn))+H1Snxn,yn,Sxn,yn+H1(S(xn,yn),S(x,y))⟶0,d2y,Gx,y≤d2(y,yn)+d2(yn,Gn(xn,yn))+H2(Gn(xn,yn),G(xn,yn))+H2(G(xn,yn),G(x,y))⟶0.
From (66), we have (λ,(x,y))∈Graph⁡(f). This completes the proof.Lemma 19.
For any(λ,(x,y))∈Λ×X×Y, the rationality function Φ is lower semicontinuous at (λ,(x,y)).Proof.
For any givenr∈R=(-∞,+∞), let
(67)M=λ,x,y∈Λ×X×Y:Φλ,x,y≤r.
In order to show Φ(λ,(x,y)) is lower semicontinuous, we only need to show that M is closed. Let (λn,(xn,yn))∈M with (λn,(xn,yn))→(λ,(x,y)). We show that (λ,(x,y))∈M, that is, Φ(λ,(x,y))≤r, which is equivalent to
(68)max⁡sup⁡u∈S(x,y)-ξe1φ(x,u,y),sup⁡v∈G(x,y)-ξe2ψ(x,v,y)≤r,
where e1∈intC1 and e2∈intC2, that is,
(69)ξe1φx,u,y≥-r,∀u∈Sx,y,ξe2ψ(x,v,y)≥-r,∀v∈G(x,y),
which is equivalent to (by Lemma 7)
(70)φx,u,y∉-re1-intC1,∀u∈Sx,y,ψ(x,v,y)∉-re2-intC2,∀v∈G(x,y).
By way of contradiction, assume there exists u0∈S(x,y) or v0∈G(x,y) such that
(71)φ(x,u0,y)∈-re1-intC1
or
(72)ψ(x,v0,y)∈-re2-intC2.
Without loss of generality, assume that (71) holds. Then there exists a neighborhood V of zero in Z such that
(73)φ(x,u0,y)+V⊂-re1-intC1.
Foru0∈S(x,y), since S(x,y) and Sn(xn,yn) are compact with Sn(xn,yn)→S(x,y), by Lemma 5, there exists un∈Sn(xn,yn) with un→u0. Since φn is C1-upper semicontinuous and φn(x,u0,y)→φ(x,u0,y), we have
(74)φn(xn,un,yn)∈φn(x,u0,y)+V2-C1,φn(x,u0,y)∈φ(x,u0,y)+V2.
It follows from (73) and (74) that
(75)φn(xn,un,yn)∈-re1-intC1.
On the other hand, since(λn,(xn,yn))∈M, we know that Φ(λn,(xn,yn))≤r and so
(76)max⁡sup⁡u∈Sn(xn,yn)-ξe1φnxn,u,yn,hhhhhsup⁡v∈Gn(xn,yn)-ξe2ψnxn,v,yn≤r.
It follows that
(77)ξe1φn(xn,u,yn)≥-r,∀u∈Sn(xn,yn).
Now Lemma 10 implies that
(78)φn(xn,u,yn)∉-re1-intC1,∀u∈Sn(xn,yn),
which is in contradiction with (75). Thus, we know that Φ(λ,(x,y)) is lower semicontinuous. This completes the proof.Theorem 20.
For allλ∈Λ, the problem λ is generalized well-posed. Moreover, for all λ∈Λ, if E(λ)={(x,y)} (a singleton), then the problem λ is well-posed.Proof.
Letλn∈Λ with λn→λ and (xn,yn)∈E(λn,ɛn) with ɛn→0. Then
(79)dxn,yn,fλn≤ɛn,(80)Φ(λn,(xn,yn))≤ɛn.
By (79), there exists (x¯n,y¯n)∈f(λn) such that d((xn,yn),(x¯n,y¯n))→0. Since f is an usco mapping, by Lemma 6, there exists a subsequence {(x¯nk,y¯nk)}⊂{(x¯n,y¯n)} such that (x¯nk,y¯nk)→(x,y)∈f(λ). It follows that
(81)d((xnk,ynk),(x,y))≤d((xnk,ynk),(x¯nk,y¯nk))+d((x¯nk,y¯nk),(x,y))⟶0
and so
(82)(xnk,ynk)⟶(x,y)∈f(λ).
On the other hand, the lower semicontinuity of Φ shows that
(83)0≤Φλ,x,y≤liminf⁡Φλn,xnk,ynk≤liminf⁡ɛn=0,
which implies that Φ(λ,(x,y))=0. It follows from (82) and (83) that (x,y)∈E(λ) and so the problem λ is generalized well-posed.
Moreover, we show that, ifE(λ)={(x,y)}, then the problem λ is well-posed. If the sequence {(xn,yn)} does not converge to (x,y), then there exist an open neighborhood O of (x,y) and a subsequence {(xnk,ynk)} of {(xn,yn)} such that (xnk,ynk)∉O. By the proof of the first part, we know that (xnk,ynk)→(x,y). This is in contradiction with (xnk,ynk)∉O. This completes the proof.Example 21.
LetX=Y=[0,2]×[0,2], Z=(-∞,+∞)×(-∞,+∞), C=[0,+∞)×[0,+∞), and e=(1,1). Let
(84)Sx,y=Gx,y=0,1×0,1,∀x,y∈X×Y,φx,u,y=x-u,ψx,v,y=y-v,∀x,u∈X,∀y,v∈Y.
Then it is easy to see that λ=(φ,ψ,S,G)∈Λ and
(85)f(λ)={(x,y)∣x∈[0,1]×[0,1],y∈[0,1]×[0,1]}
with
(86)Eλ=(x,y)∈f(λ)∣x=(x1,x2),y=(y1,y2),hhhmax⁡{x1,x2}=1,max⁡{y1,y2}=1.
Furthermore, by Theorem 20, the problem λ is generalized well-posedness.Remark 22.
Whenφ=ψ and S=G, the well-posedness for the problem λ was studied by [20]. Theorem 20 presented in this paper can be considered as a generalization and extension of Theorem 4.1 in [20].From Theorem20, we have the following corollaries.Corollary 23.
For allλ∈Λ, the problem λ is (generalized) Hadamard well-posed.Corollary 24.
For allλ∈Λ, the problem λ is (generalized) LP well-posed.Remark 25.
Whenφ=ψ, S=G, the generalized Hadamard well-posedness for the problem λ was studied by [16]. Corollary 23 presented in this paper generalizes and extends Theorem 4.2 in [16].Remark 26.
We note that, when(87)φ(x,u,y)=f(u,y)-f(x,y),ψ(x,v,y)=g(x,v)-g(x,y),
the generalized LP well-posedness for the symmetric vector quasi-equilibrium problem was studied by [19].



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*Source: 108357-2015-02-24.xml*