0Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2011, Article ID 936428,15 pages doi:10.1155/2011/936428
Research Article
Boundedness and Nonemptiness of Solution Sets for Set-Valued Vector Equilibrium Problems with an Application
Ren-You Zhong,1 Nan-Jing Huang,1 and Yeol Je Cho2
1 Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China
2 Department of Mathematics Education and the RINS, Gyeongsang National University, Chinju 660-701, Republic of Korea
Correspondence should be addressed to YeolJe Cho, yjcho@gsnu.ac.kr
Received 25 October 2010; Accepted 19 January 2011
Academic Editor: K. Teo
Copyright © 2011 Ren-You Zhong et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper is devoted to the characterizations of the boundedness and nonemptiness of solution sets for set-valued vector equilibrium problems in reflexive Banach spaces, when both the mapping and the constraint set are perturbed by different parameters. By using the properties of recession cones, several equivalent characterizations are given for the set-valued vector equilibrium problems to have nonempty and bounded solution sets. As an application, the stability of solution set for the set-valued vector equilibrium problem in a reflexive Banach space is also given. The results presented in this paper generalize and extend some known results in Fan and Zhong (2008), He (2007), and Zhong and Huang (2010).
1. Introduction
Let X and Y be reflexive Banach spaces. Let K be a nonempty closed convex subset of X. Let F : K x K ^ 2Y be a set-valued mapping with nonempty values. Let P be a closed convex pointed cone in Y with int P / 0. The cone P induces a partial ordering in Y, which was defined by y1<Py2 if and only if y2 - y1 e P. We consider the following set-valued vector equilibrium problem, denoted by SVEP(F,K), which consists in finding x e K such that
F(x,y) n (-intP) = 0, Vy e K.
It is well known that (1.1) is closely related to the following dual set-valued vector equilibrium problem, denoted by DSVEP(F,K), which consists in finding x € K such that
F(y,x) c (-P), Vy € K. (1.2)
We denote the solution sets of SVEP(F,K) and DSVEP(F, K) by S and SD, respectively.
Let (Z1,d1) and (Z2,d2) be two metric spaces. Suppose that a nonempty closed convex set L c X is perturbed by a parameter u, which varies over (Z1,d1), that is, L : Z1 ^ 2X is a set-valued mapping with nonempty closed convex values. Assume that a set-valued mapping F : X x X ^ 2y is perturbed by a parameter v, which varies over (Z2,d2), that is, F : X x X x Z2 ^ 2y . We consider a parametric set-valued vector equilibrium problem, denoted by SVEP(F(■, -,v),L(u)), which consists in finding x € L(u) such that
F(x,y, v) n (-int P)= 0, Vy € L(u). (1.3)
Similarly, we consider the parameterized dual set-valued vector equilibrium problem, denoted by DSVEP(F(-, -,v),L(u)), which consists in finding x € L(u) such that
F(y,x,v) c (-P), Vy € L(u). (1.4)
We denote the solution sets of SVEP(F(■, ■, v), L(u)) and DSVEP(F(■, ■, v),L(u)) by S(u, v) and SD(u,v), respectively.
In 1980, Giannessi [1] extended classical variational inequalities to the case of vector-valued functions. Meanwhile, vector variational inequalities have been researched quite extensively (see, e.g., [2]). Inspired by the study of vector variational inequalities, more general equilibrium problems [3] have been extended to the case of vector-valued bifunctions, known as vector equilibrium problems. It is well known that the vector equilibrium problem provides a unified model of several problems, for example, vector optimization, vector variational inequality, vector complementarity problem, and vector saddle point problem (see [4-9]). In recent years, the vector equilibrium problem has been intensively studied by many authors (see, e.g., [1-3,10-26] and the references therein).
Among many desirable properties of the solution sets for vector equilibrium problems, stability analysis of solution set is of considerable interest (see, e.g, [27-33] and the references therein). Assuming that the barrier cone of K has nonempty interior, McLinden [34] presented a comprehensive study of the stability of the solution set of the variational inequality, when the mapping is a maximal monotone set-valued mapping. Adly [35], Adly et al. [36], and Addi et al. [37] discussed the stability of the solution set of a so-called semicoercive variational inequality. He [38] studied the stability of variational inequality problem with either the mapping or the constraint set perturbed in reflexive Banach spaces. Recently, Fan and Zhong [39] extended the corresponding results of He [38] to the case that the perturbation was imposed on the mapping and the constraint set simultaneously. Very recently, Zhong and Huang [40] studied the stability analysis for a class of Minty mixed variational inequalities in reflexive Banach spaces, when both the mapping and the constraint set are perturbed. They got a stability result for the Minty mixed variational inequality with ®-pseudomonotone mapping in a reflexive Banach space, when both the mapping and the constraint set are perturbed by different parameters, which generalized and extended some known results in [38, 39].
Inspired and motivated by the works mentioned above, in this paper, we further study the characterizations of the boundedness and nonemptiness of solution sets for set-valued vector equilibrium problems in reflexive Banach spaces, when both the mapping and the constraint set are perturbed. We present several equivalent characterizations for the vector equilibrium problem to have nonempty and bounded solution set by using the properties of recession cones. As an application, we show the stability of the solution set for the set-valued vector equilibrium problem in a reflexive Banach space, when both the mapping and the constraint set are perturbed by different parameters. The results presented in this paper extend some corresponding results of Fan and Zhong [39], He [38], Zhong and Huang [40] from the variational inequality to the vector equilibrium problem.
The rest of the paper is organized as follows. In Section 2, we recall some concepts in convex analysis and present some basic results. In Section 3, we present several equivalent characterizations for the set-valued vector equilibrium problems to have nonempty and bounded solution sets. In Section 4, we give an application to the stability of the solution sets for the set-valued vector equilibrium problem.
2. Preliminaries
In this section, we introduce some basic notations and preliminary results.
Let X be a reflexive Banach space and K be a nonempty closed convex subset of X. The symbols " ^ " and are used to denote strong and weak convergence, respectively. The barrier cone of K, denoted by barr(K), is defined by
barr(K) := ] x* e X* : sup(x*,x) < to (2.1)
The recession cone of K, denoted by Kto, is defined by
:= {d e X : 3tn —> 0,3xn e K,tnxn ^ d}.
It is known that for any given x0 e K,
Kto = {d e X : xo + Id e K,V! > 0}.
We give some basic properties of recession cones in the following result which will be used in the sequel. Let (Ki}iei be any family of nonempty sets in X. Then
PK*) cfl (Kt)c
■ /.
If, in addition, P|i€I Ki / 0 and each set Ki is closed and convex, then we obtain an equality in the previous inclusion, that is,
()kA = H(Ki)„. (2.5)
i€I / o i€I
Let O : K ^ R U {+o} be a proper convex and lower semicontinuous function. The recession function ®o of O is defined by
Oo (x) := lim °(x0 + tx) - °(x0), (2.6)
t ^ +o t
where xo is any point in Dom O. Then it follows that
Oo(x) := lim . (2.7)
t ^ +o t
The function Oo(-) turns out to be proper convex, lower semicontinuous and so weakly lower semicontinuous with the property that
O(u + v) < O(u)+Oo(v), Vu € Dom O,v € X. (2.8)
Definition 2.1. A set-valued mapping G : K ^ 2Y is said to be
(i) upper semicontinuous at x0 € K if, for any neighborhood N(G(x0)) of G(x0), there exists a neighborhood N(x0) of x0 such that
G(x) cN(G(xo)), Vx € N(xo); (2.9)
(ii) lower semicontinuous at x0 € K if, for any y0 € G(x0) and any neighborhood N(y0) of y0, there exists a neighborhood N(x0) of x0 such that
G(x^ N(yo) f 0, Vx €N(xo). (2.10)
We say G is continuous at x0 if it is both upper and lower semicontinuous at x0, and we say G is continuous on K if it is both upper and lower semicontinuous at every point of K.
It is evident that G is lower semicontinuous at x0 € K if and only if, for any sequence {xn} with xn ^ x0 and y0 € G(x0), there exists a sequence {yn} with yn € G(xn) such that yn ^ y0.
Definition 2.2. A set-valued mapping G : K ^ 2Y is said to be weakly lower semicontinuous at x0 € K if, for any y0 € G(x0) and for any sequence {xn} € K with xn ^ x0, there exists a sequence yn € G(xn) such that yn ^ y0.
We say G is weakly lower semicontinuous on K if it is weakly lower semicontinuous at every point of K. By Definition 2.2, we know that a weakly lower semicontinuous mapping is lower semicontinuous.
Definition 2.3. A set-valued mapping G : K ^ 2Y is said to be
(i) upper P-convex on K if for any x1 and x2 e K, t e [0,1],
tG(xi) + (1 - t)G(x2) c G(txi + (1 - t)x2) + P; (2.11)
(ii) lower P-convex on K if for any x1 and x2 e K, t e [0,1],
G(tx1 + (1 - t)x2) c tG(x1) + (1 - t)G(x2) - P. (2.12)
We say that G is P-convex if G is both upper P-convex and lower P-convex. Definition 2.4. Let {An} be a sequence of sets in X. We define
w-lim sup An := {x e X : 3{nk}, xnk e Ank such that xnk ^ x}. (2.13)
Lemma 2.5 (see [36]). Let K be a nonempty closed convex subset ofX with int(barr(K)) / 0. Then there exists no sequence {xn} c K such that \\xn\\ ^ to and xn/||xn|| ^ 0.
Lemma 2.6 (see [39]). Let K be a nonempty closed convex subset of X with int(barr(K)) / 0. Then there exists no sequence {dn} c with each \\dn\\ = 1 such that dn ^ 0.
Lemma 2.7 (see [39]). Let (Z,d) be a metric space and u0 e Z be a given point. Let L : Z ^ 2X be a set-valued mapping with nonempty values and let L be upper semicontinuous at u0. Then there exists a neighborhood U of u0 such that (L(u))TO c (L(u0for all u e U.
Lemma 2.8 (see [41]). Let K be a nonempty convex subset of a Hausdorff topological vector space E and G : K ^ 2E be a set-valued mapping from K into E satisfying the following properties:
(i) G is a KKM mapping, that is, for every finite subset A of K, co(A) c U xeA G(x);
(ii) G(x) is closed in E for every x e K;
(iii) G(xo) is compact in E for some x0 e K.
Then (~]xeK G(x) / 0.
3. Boundedness and Nonemptiness of Solution Sets
In this section, we present several equivalent characterizations for the set-valued vector equilibrium problem to have nonempty and bounded solution set. First of all, we give some assumptions which will be used for next theorems.
Let K be a nonempty convex and closed subset of X. Assume that F : K x K ^ 2Y is a set-valued mapping satisfying the following conditions:
(f0) for each x € K, F(x,x) = 0;
(fi) for each x,y € K, F(x,y) n (-int P) = 0 implies that F(y,x) c (-P);
(f2) for each x € K, F(x, ■) is P-convex on K;
(f3) for each x € K, F(x, ■) is weakly lower semicontinuous on K;
(f4) for each x,y € K, the set {£ € [x, y] : F(£,y) f|(-int P) = 0} is closed, here [x,y] stands for the closed line segment joining x and y.
Remark 3.1. If
F(x,y) = (Ax,y - x) + O(y) - O(x), Vx,y € K, (3.1)
where A:K ^ 2X* is a set-valued mapping, O : K ^ RU{+o} is a proper, convex, lower semicontinuous function and P = R+, then condition (f1) reduces to the following O-pseudomonotonicity assumption which was used in [40]. (See [40, Definition 2.2(iii)] of [40]): for all (x,x*), (y,y*) in the graph(A),
(x*,y - x) + O(y) - O(x) > 0 (y*,y - x) + O(y) - O(x) > 0. (3.2)
Remark 3.2. If, for each y € K, the mapping F(-,y) is lower semicontinuous in K, then condition (f4) is fulfilled. Indeed, for each x,y € K and for any sequence {¿n} c{ € [x,y] : F(¿,y) H(- int P) = 0} with In ^ h, we have ¿0 € [x,y] and F(&,y) n(- int P) = 0. By the lower semicontinuity of F(-,y), for any z € F(^0,y), there exists zn € F(¿n,y) such that Zn ^ z. Since F(ln,y) n(-int P) = 0, we have Zn € Y \ (-int P) and so z € Y \ (-int P) by the closedness of Y \ (-intP). This implies that F(^0,y)f|(-intP) = 0 and the set {¿, € [x,y] : F(ly) n(-int P) = 0} is closed.
The following example shows that conditions (f0)-(f4) can be satisfied.
Example 3.3. Let X = R, Y = R2, P = R+ and K = [1,2]. Let
F(x,y) = (y - x, [1,1 + x](y - x))T, Vx,y € K. (3.3)
It is obvious that (f0) holds. Since for each x,y € K, F(x, ■) and F(-,y) are lower semicontinuous on K, by Remark 3.2, we known that conditions (f3) and (f4) hold. For each x,y € K, if F(x, y) n (-R+) = 0, then we have y - x > 0. This implies that
F(y,x) = (x - y, [1,1 + y](x - y))Tc (-R+) (3.4)
and so (f1) holds. Moreover, for each x € K, y1,y2 € K and t1,t2 € [0,1] with t1 + t2 = 1, it is easy to verify that
F(x,t1y1 + t2y2) = t1F(x,y1) + t2F{x,y2)
which shows that F(x, ■) is R+-convex on K and so (/2) holds. Thus, F satisfies all conditions
(/0M/4).
Theorem 3.4. Let K be a nonempty closed convex subset of X and F : K x K ^ 2Y be a set-valued mapping satisfying assumptions (/0)-(/4). Then S = SD.
Proof. From the assumption (/1), it is easy to see that S c SD. We now prove that SD c S. Let x e SD. Then for all y e K, F(y,x) c (-P). Set xt = x + t(y - x), where t e [0,1]. Clearly, xt e K. From the upper P-convexity of F(x, ■), we have
(1 - t)F(xt,x) + tF(xt,y) c F(xt,xt) + P. (3.6)
Since F(xt,x) c (-P), we obtain
tF(xt,y) c -(1 - t)F(xt, x) + 0 + P c P + P c P. (3.7)
This implies that F(xt,y) c P and so F(xt,y) n (-int P) = 0. Letting t ^ 0+, by assumption (/4), we have F(x, y) n (- int P) = 0. Thus, x e S and SD c S. This completes the proof. □
Theorem 3.5. Let K be a nonempty closed convex subset of X and F : K x K ^ 2Y be a set-valued mapping satisfying assumptions (/0)-(/4). If the solution set S is nonempty, then
Sm = = R1 := p| {deK„ : F(y,y + id) c (-P), VX> 0}.
Proof. From the proof of Theorem 3.4, we know that
S = SD = {xeK : F(y,x) c (-P),Vy e K} = p) {x e K : F(y,x) c (-P)}.
Let Sy = {x eX : F(y,x) c (-P)}. Then S = SD = HyeK(K n Sy). By the assumptions (/2) and (/3), we know that the set Sy is nonempty closed and convex. It follows from (2.5) and Theorem 3.4 that
= Si = (PIK n Sy ) ^ K n S(y)) „
Ve K /„ yeK
= p K~n (S(y))M
= P{deK. : de (S(y)J) (3.10)
= 0 {deKoo : y + ide S(y), Vi > 0}
= 0 {deKo : F(y,y + id) c -P, Vi > 0}.
Then this completes the proof.
Remark 3.6. If
F(y,x) = {Ay,x - y) + ®(x) - ®(y), Vx,y e K,
(3.11)
where A : K ^ 2X' is a set-valued mapping, ® : K ^ RU{+o} is a proper, convex, lower semicontinuous function and P = R+, then it follows from (3.8) and (2.8) that
= K^n {d e X : {y*,y + Xd - y) + ®(y + id) - ®(y) < 0,Vy e K,y* e A(y), VX > 0} = Koo n {d e X : {y*,d) +®oo(d) < 0,Vy* e A(K)}.
Thus, we know that Theorem 3.5 is a generalization of [40, Theorem 3.1]. Moreover, by [40, Remark 3.1], Theorem 3.5 is also a generalization of [38, Lemma 3.1].
Theorem 3.7. Let K be a nonempty closed convex subset of X and F : K x K ^ 2Y be a set-valued mapping satisfying assumptions (f0 )-(f4). Suppose that int(barr(K) = 0. Then the following statements are equivalent:
(i) the solution set of SVEP(F,K) is nonempty and bounded;
(ii) the solution set of DSVEP(F, K) is nonempty and bounded;
(iv) there exists a bounded set C c K such that for every x e K \ C, there exists some y e C such that F(y,x)C(-P).
Proof. The implications (i)^(ii) and (ii)^(iii) follow immediately from Theorems 3.4 and 3.5 and the definition of recession cone.
Now we prove that (iii) implies (iv). If (iv) does not hold, then there exists a sequence {xn} c K such that for each n, \\xn\\ > n and F(y,xn) c (-P) for every y e K with ||y|| < n. Without loss of generality, we may assume that dn = xn/\\xn\\ weakly converges to d. Then d e Ko by the definition of the recession cone. Since int(barrK) = 0, by Lemma 2.5, we know that d = 0. Let y e K and X > 0 be any fixed points. For n sufficiently large, by the lower P-convexity of F(y, ■),
soo = n {d e Koo : F(y,y + id) c (-P), VX> 0}
(3.12)
(iii) R = DyeK{d e Koo : F(y,y + Xd) c (-P), VX > 0} = {0};
(3.14)
and F(y, ■) is weakly lower semicontinuous, we know that F (y,y + Xd) c -P and so d e R\. However, it contradicts the assumption that Ri = {0}. Thus (iv) holds.
Since (i) and (ii) are equivalent, it remains to prove that (iv) implies (ii). Let G : K ^ 2K be a set-valued mapping defined by
G(y) := {xeK : F(y,x) c (-P)}, Vy e K. (3.15)
We first prove that G(y) is a closed subset of K. Indeed, for any xn e G(y) with xn ^ x0, we have F(y,xn) c (-P). It follows from the weakly lower semicontinuity of F(y, ■) that F(y,x0) c (-P). This shows that x0 e G(y) and so G(y) is closed.
We next prove that G is a KKM mapping from K to K. Suppose to the contrary that there exist t1, t2,...,tne [0,1] with t1 + t2 + ••• + tn = 1, y1,y2,...,yn e K and y = t1y1 + t2y2 + ••• + tnyn e co{yi,y2,...,yn} such that y / U e{1/2,..,n}G(yi). Then
F (yj,y)c(-P), i = 1,2,...,n. (3.16)
By assumption (/1), we have
F(y,yi) n (-intP) =0, i = 1,2,...,n. (3.17)
It follows from the upper P-convexity of F (y, ■) that
fiF(y,yi) + t2F(y,y2) + ■■■ + tnF(y,yn) c F(y,y) + P c P, (3.18)
which is a contradiction with (3.17). Thus we know that G is a KKM mapping.
We may assume that C is a bounded closed convex set (otherwise, consider the closed convex hull of C instead of C). Let {y1, ...,ym} be finite number of points in K and let M := co(C U {y\,...,ym}). Then the reflexivity of the space X yields that M is weakly compact convex. Consider the set-valued mapping G' defined by G'(y) := G(y) n M for all y e M. Then each G'(y) is a weakly compact convex subset of M and G' is a KKM mapping. We claim that
0 / H G'(y) c C. (319)
Indeed, by Lemma 2.8, intersection in (3.19) is nonempty. Moreover, if there exists some x0 e nyeM G'(y) but xo / C, then by (iv), we have F(y, x0)C(-P) for some y e C. Thus, xo / G(y) and so x0 e G'(y), which is a contradiction to the choice of x0.
Let z e nyeM G'(y). Then zeC by (3.19) and so z e flmi(G(yi) n C). This shows that the collection {G(y) n C : y e K} has finite intersection property. For each y e K, it follows from the weak compactness of G(y) n C that fy eK(G(y) n C) is nonempty, which coincides with the solution set of DSVEP(F, K). □
Remark 3.8. Theorem 3.7 establishes the necessary and sufficient conditions for the vector equilibrium problem to have nonempty and bounded solution sets. If
F(y,x) = (Ay,x - y) +®(x) - ®(y), Vx,yeK, (3.20)
where A : K ^ 2X' is a set-valued mapping, ® : K ^ RU{+o} is a proper, convex, lower semicontinuous function and P = R+, then problem (1.2) reduces to the following Minty mixed variational inequality: finding x e K such that
(y*,y - x) + 0(y) - ®(x) > 0, Vy e K,y* e A(y), (3.21)
which was considered by Zhong and Huang [40]. Therefore, Theorem 3.7 is a generalization of [40, Theorem 3.2]. Moreover, by [40, Remark 3.2], Theorem 3.7 is also a generalization of Theorem 3.4 due to He [38].
Remark 3.9. By using a asymptotic analysis methods, many authors studied the necessary and sufficient conditions for the nonemptiness and boundedness of the solution sets to variational inequalities, optimization problems, and equilibrium problems, we refer the reader to references [42-49] for more details.
4. An Application
As an application, in this section, we will establish the stability of solution set for the set-valued vector equilibrium problem when the mapping and the constraint set are perturbed by different parameters.
Let (Z1,d1) and (Z2,d2) be two metric spaces. F : X x X x Z2 ^ 2Y is a set-valued mapping satisfying the following assumptions:
(f0) for each u e Z1, v e Z2, x e L(u), F(x, x, v) = 0;
(f 1) for each u e Z1, v e Z2, x,y e L(u), F(x, y, v) n (- int P) = 0 implies that F(y, x, v) c
(f2) for each u e Z1, v e Z2, x e L(u), F(x, -,v) is P-convex on L(u);
(f3) for each u e Z1,v e Z2, x,y e L(u) and z e F(x,y,v), for any sequences {xn}, {yn} and {vn} with xn ^ x, yn ^ y and vn ^ v, there exists a sequence {zn} with zn e F(xn, yn, vn) such that zn ^ z.
The following Theorem 4.1 plays an important role in proving our results.
Theorem 4.1. Let (Z1,d1) and (Z2, d2) be two metric spaces, u0 e Z1 and v0 e Z2 be given points. Let L : Z1 ^ 2X be a continuous set-valued mapping with nonempty closed convex values and int(barr(L(u0))) = 0. Suppose that F : X x X x Z2 ^ 2Y is a set-valued mapping satisfying the assumptions f -f3. If
R1(u0,v0)= 0 {d e L(u0)o : F(y,y + Xd,v0) c (-P),VX> 0} = {0}, (41)
yeL(u0)
then there exists a neighborhood U x V of (u0, v0) such that
R1(u,v)= p| {d e L(u)o : F(y,y + Xd,v) c (-P), VX> 0} = {0}, V(u,v) e U x V.
yeL(u)
Proof. Assume that the conclusion does not hold, then there exist a sequence {(un,vn)} in Z1 x Z2 with (un,vn) ^ (u0,v0) such that R1 (un,vn) = {0}.
Since R1(un,vn) is cone, we can select a sequence {dn} with dn e R1(un,vn) such that \\dn\\ = 1 for every n = 1,2,.... As X is reflexive, without loss of generality, we can assume that dn ^ d0, as n ^ +o. Since L is a continuous set-valued mapping, hence, L is upper semicontinuous and lower semicontinuous at u0. From the upper semicontinuity of L, by Lemma 2.7, we have (L(un))o c (L(u0))o as n large enough and hence dn e (L(u0))o as n large enough. Since (L(u0))o is a closed convex cone and hence weakly closed. This implies that d0 e (L(u0))o. Moreover, it follows from Lemma 2.6 that d0 / 0.
For any i > 0, ye L(u0) and y* e F(y,y + id0,v0), from the lower semicontinuity of L, there exists yn e L(un) such that yn ^ y. Since dn ^ d0, it follows that yn + idn ^ y + d0. Together with vn ^ v0, from assumption (/3), there exists y*n e F(yn,yn + idn,vn) such that y*n ^ y*. Since dn e R1(umvn), we have F(yn,yn + idn, vn) c (-P) and yn e -P. Letting n ^ o,weobtainthat y* e (-P). Since y e L(u0) and y* e F(y,y+id0,v0) are arbitrary, from the above discussion, we obtain d0 e R1(u0,v0) with d0 / 0. This contradicts our assumption that R1(u0,v0) = {0}. This completes the proof. □
Remark 4.2. If
F(y,x,v) = (A(y,v),x - y) +®(x) - ®(y), Vx,yeL(u), (4.3)
where A : X x Z2 ^ 2X' is a set-valued mapping, ® : X ^ RU{+o} is a proper, convex, lower semicontinuous function and P = R+, from Remark 3.6, we know that (4.1) and (4.2) in Theorem 4.1 reduce to (4.1) and (4.2) in [40, Theorem 4.1], respectively. Therefore, Theorem 4.1 is a generalization of [40, Theorem 4.1]. Moreover, by [40, Remark 4.1], Theorem 4.1 is also a generalization of [39, Theorem 3.1].
From Theorem 4.1, we derive the following stability result of the solution set for the vector equilibrium problem.
Theorem 4.3. Let (Z1,d1) and (Z2, d2) be two metric spaces, u0 e Z1 and v0 e Z2 be given points. Let L : Z1 ^ 2X be a continuous set-valued mapping with nonempty closed convex values and int(barr(L(u0))) / 0. Suppose that F : X x X x Z2 ^ 2Y is a set-valued mapping satisfying the assumptions (/'0)-(/'3). If S(u0,v0) is nonempty and bounded, then
(i) there exists a neighborhood U x V of (u0,v0) such that for every (u, v) eU x V, S(u, v) is nonempty and bounded;
(ii) W-limsup(u,v) ^ (u0rv0)S(u,v) c S(u0, v0).
Proof. If S(u0,v0) is nonempty and bounded, then by Theorem 3.7 we have R1 (u0,v0) = {0}. It follows from Theorem 4.1 that there exists a neighborhood U x V of (u0,v0), such that R1(u,v) = {0} for every (u,v) e U x V .By using Theorem 3.7 again, we have S(u,v) is nonempty and bounded for every (u,v) eU x V. This verifies the first assertion.
Next, we prove the second assertion o>-limsup(uv)^(uovo)S(u,v) c S(u0,v0). For any given sequence {(un,vn)} e U x V with (un,vn) ^ (u0,v0), we need to prove that o>-limsupn^oS(un,vn) c S(u0,v0). Let x e o>-limsupn^oS(un,vn). Then there exists a sequence {xnj} with each xnj e S(un, vnj) such that xnj weakly converges to x. We claim that there exists znj e L(u0) such that limy^o\\xnj - znj \\ = 0. Indeed, if the claim does hold, then
there exist that a subsequence {xn } of {xnj} and some s0 > 0, such that d(xn^,L(u0)) > s0, for all k = 1,2,.... This implies that xnjk / L(u0) + s0B(0,1) and so L(unjk)/L(u0) + s0B(0,1), which contradicts with the upper semicontinuity of L(-). Thus, we have the claim. Moreover, we obtain x e L(u0) as L(u0) is a closed convex subset of X and hence weakly closed.
Now we prove F(y,x,v0) c (-P) for all y e L(u0) and hence x e SD(u0,v0) = S(u0,v0). For any y e L(u0) and y* e F(y,x,v0), from the lower semicontinuity of L, there exist ynj e L(unj) such that limy= y. Moreover, from assumption (f), there exists a sequence of elements y*j e F(ynj,xnj,vnj) such that y*n. ^ y*. Since xnj e S(unj,vnj), we have F(ynj,xnj,vnj) c (-P) and so y* e -P. Letting j ^ to, we obtain that y* e (-P). Since y* e F(y,x,v0) is arbitrary, we have F(y,x,v0) c (-P). This yields that x e SD(u0,v0) = S(u0, v0). Thus, have the second assertion. This completes the proof. □
Remark 4.4. If
F(y,x,v) = (A(y,v),x - y) +®(x) - y), Vx,y e L(u), (4.4)
where A : X x Z2 ^ 2X* is a set-valued mapping, ® : X ^ RU{+to} is a proper, convex, lower semicontinuous function and P = R+, then problem (1.4) reduces to the following parametric Minty mixed variational inequality: finding x e L(u) such that
{y*,y - x) + ®(y) - ®(x) > 0, 'Vy e L(u),y* e A(y,v), (4.5)
which was considered by Zhong and Huang [40]. Therefore, Theorem 4.3 is a generalization of [40, Theorem 4.2]. Moreover, by [40, Remark 4.2], Theorem 4.3 ia also a generalizationof Theorems 4.1 and 4.4 due to He [38] and Theorem 3.5 due to Fan and Zhong [39].
The following examples show the necessity of the conditions of Theorem 4.3.
Example 4.5. Let X = Y = R, P = R+, Z1 = Z2 = [-1,1] and u0 = v0 = 0,
f {0}, v = 0,
L(u) = [0,1], F(x,y,v) = { (4.6)
[y2 - x2, v = 0.
Note that L(-) is continuous on Z1. However, F(■, ■, ■) is not lower semicontinuous at (1/2, 1/4,0) e X x X x Z2. Clearly, we have S(0,0) = {0} and S(0,v) = [0,1] for any v = 0. Thus,
lim supS(0,v) = [0,1]cS(0,0). (47)
Example 4.6. Let X = Y = R, P = R+, Z1 = Z2 = [-1,1] and u0 = v0 = 0,
([2,3], u = 0, 2 2
L(u) = < F(x,y,v) = y - x , forany x,y e L(u),v e Z2. (4.8)
[[1,3], u /0,
Note that F satisfies the assumptions (/0)-(/3), and L(u) is upper semicontinuous. However, L(u) is not lower semicontinuous at u = 0. Clearly, we have S(0,0) = {1} and S(u, 0) = {2} for any u / 0. Thus,
limsup S(u,0) = {2}cS(0,0). (49)
Example 4.7. Let X = Y = R, P = R+, Z1 = Z2 = [-1,1], u0 = v0 = 0,
f [2,3], u = 0, 2 2
L(u) = < F(x,y,v) = y - x , for any x,y e L(u),v e Z2. (4.10)
[[1,3], u / 0,
Note that F satisfies the assumptions (/0)-(/3) and L(u) is lower semicontinuous. However, L(u) is not upper semicontinuous at u = 0. Clearly, we have S(0,0) = {2} and S(u,0) = {1} for any u / 0. Thus,
limsup S(u, 0) = {1}cS(0,0). (411)
Acknowledgments
The authors are grateful to the editor and reviewers for their valuable comments and suggestions. This work was supported by the Key Program of NSFC (Grant no. 70831005), the National Natural Science Foundation of China (10671135) and the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-313-C00050).
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