Scholarly article on topic 'Existence Result of Generalized Vector Quasiequilibrium Problems in Locally G-Convex Spaces'

Existence Result of Generalized Vector Quasiequilibrium Problems in Locally G-Convex Spaces Academic research paper on "Mathematics"

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Academic research paper on topic "Existence Result of Generalized Vector Quasiequilibrium Problems in Locally G-Convex Spaces"

Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2011, Article ID 967515,13 pages doi:10.1155/2011/967515

Research Article

Existence Result of Generalized Vector Quasiequilibrium Problems in Locally G-Convex Spaces

Somyot Plubtieng and Kanokwan Sitthithakerngkiet

Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand Correspondence should be addressed to Somyot Plubtieng, somyotp@nu.ac.th Received 30 November 2010; Accepted 18 February 2011 Academic Editor: Yeol J. Cho

Copyright © 2011 S. Plubtieng and K. Sitthithakerngkiet. 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 deals with the generalized strong vector quasiequilibrium problems without convexity in locally G-convex spaces. Using the Kakutani-Fan-Glicksberg fixed point theorem for upper semicontinuous set-valued mapping with nonempty closed acyclic values, the existence theorems for them are established. Moreover, we also discuss the closedness of strong solution set for the generalized strong vector quasiequilibrium problems.

1. Introduction

Let X be real topological vector space, and let C be a nonempty closed convex subset of X. Let F : C x C ^ R be a bifunction, where R is the set of real numbers. The equilibrium problem for F is to find x e C such that

F{x,y) > 0 Vy e C.

Problem (1.1) was studied by Blum and Oettli [1]. The set of solution of (1.1) is denoted by EP(F). The equilibrium problem contains many important problems as special cases, including optimization, Nash equilibrium, complementarity, and fixed point problems (see [1-3] and the references therein). Recently, there has been an increasing interest in the study of vector equilibrium problems. Many results on the existence of solutions for vector variational inequalities and vector equilibrium problems have been established (see, e.g., [416]).

Let X and Y be real topological vector spaces and K a nonempty subset of X. Let C be a closed and convex cone in Y with int C(x) / 0, where int C(x) denotes the topological interior of C. For a bifunction F : K x K ^ Y, the vector equilibrium problem (for short, VEP) is to find x e K such that

F(x,y) / - int C, Vy e K, (1.2)

which is a unified model of several known problems, for instance, vector variational and variational-like inequality problems, vector complementarity problem, vector optimization problem, and vector saddle point problem; see, for example, [3, 8, 17, 18] and references therein. In 2003, Ansari and Yao [19] introduced vector quasiequilibrium problem (for short, VQEP) to find x e K such that

x e A(x) : F(x,y) / - int C Vy e A(x), (1.3)

where A : K ^ 2K is a multivalued map with nonempty values.

Recently, Ansari et al. [4] considered a more general problem which contains VEP and generalized vector variational inequality problems as special cases. Let X and Z be real locally convex Hausdorff space, K c X a nonempty subset and C c Z a closed convex pointed cone. Let F : K x K ^ 2Z be a given set-valued mapping. Ansari et al. [4] introduced the following problems, to find x e K such that

F(x,y)C- intC Vy e K, (1.4)

or to find x e K such that

F(x,y) c C Vy e K. (1.5)

It is called generalized vector equilibrium problem (for short, GVEP), and it has been studied by many authors; see, for example, [20-22] and references therein. For other possible ways to generalize VEP, we refer to [23-25]. If int C is nonempty and x satisfies (1.4), then we call x a weak efficient solution for VEP, and if x satisfies (1.5), then we call x a strong solution for VEP. Moreover, they also proved an existence theorem for a strong vector equilibrium problem (1.5) (see [4]).

On the other hand, it is well known that a strong solution of vector equilibrium problem is an ideal solution; it is better than other solutions such as efficient solution, weak efficient solution, proper efficient solution, and supper efficient solution (see [12]). Thus, it is important to study the existence of strong solution and properties of the strong solution set. In 2003, Ansari and Flores-Bazan [26] considered the following generalized vector quasiequilibrium problem (for short, GVQEP): to find x e K such that

x e A(x) : F(x,y)C- intC Vy e A(x). (1.6)

Very recently, the generalized strong vector quasiequilibrium problem (in short, GSVQEP) is introduced by Hou et al. [27] and Long et al. [16]. Let X, Y, and Z be real

locally convex Hausdorff topological vector spaces, K c X and D c Y nonempty compact convex subsets, and C c Z a nonempty closed convex cone. Let S : K ^ 2K, T : K ^ 2D and F : K x D x K ^ 2Z be three set-valued mappings. They considered the GSVQEP, finding x e K, y e T(X) such that x e S(x) and

F(x,y,x) c C, Vx e S(x). (1.7)

Moreover, they gave an existence theorem for a generalized strong vector quasiequilibrium problem without assuming that the dual of the ordering cone has a weak* compact base.

Throughout this paper, motivated and inspired by Hou et al. [27], Long et al. [16], and Yuan [28], we will introduce and study the generalized vector quasiequilibrium problem on locally G-convex Hausdorff topological vector spaces. Let X, Y, and Z be real locally G-convex Hausdorff topological vector spaces, K c X and D c Y nonempty compact subsets, and C c Z a nonempty closed convex cone. We also suppose that F : K x D x K ^ 2Z, S : K ^ 2K and T : K ^ 2D are set-valued mappings.

The generalized vector quasiequilibrium problem of type (I) (GSVQEP I) is to find x* e K and y* e T(x*) such that

x* e S(x*), F(x*,y*,z) c C Vz e S(x*). (1.8)

The generalized vector quasiequilibrium problem of type (II) (GSVQEP II) is to find x* e K and y* e T(x*) such that

x* e S(x*), F(x*,y*,z)cC Vz e S(x*). (1.9)

We denote the set of all solution to the (GSVQEP I) and (GSVQEP II) by Vs(F) and Vro(F), respectively. The main motivation of this paper is to prove the existence theorems of the generalized strong vector quasiequilibrium problems in locally G-convex spaces, by using Kakutani-Fan-Glicksberg fixed point theorem for upper semicontinuous set-valued mapping with nonempty closed acyclic values, and the closedness of Vs(F) and Vw (F). The results in this paper generalize, extend, and unify some well-known some existence theorems in the literature.

2. Preliminaries

Let An be the standard n-dimensional simplex in Rn+1 with vertices e0,e1,e2,...,en. For any nonempty subset J of {0,1,2,..., n}, we denote Aj by the convex hull of the vertices {ej : j e J}. The following definition was essentially given by Park and Kim [29].

Definition 2.1. A generalised convex space, or say, a G-convex space (X,D, r) consists of a topological space X, a nonempty subset D of X and a function r : F ^ X \{0} such that

(i) for each A,B e F(X), r(A) c r(B) if A c B,

(ii) for each A e F(X) with |A| = n + 1, there exists a continuous function : An ^ r(A) such that $a(Aj) c r(J) for each 0 = J c {0,1,2,...,n}, where A = {x0 ,xi,x2,...,xn} and A j denotes the face of An corresponding to the subindex of J in {0,1,2,...,n}.

A subset C of the G-convex space (X, D, r) is said to be G-convex if for each A e F(D), rA c C for all A c C. For the convenience of our discussion, we also denote r(A) by rA or rN if there is no confusion for A = (xo,xi,x2,...,xn) e F(X), where N is the set of all indices for the set A; that is, N = (0,1,2,...,n). A space X is said to have a G-convex structure if and only if X is a G-convex space.

In order to cover general economic models without linear convex structures, Park and Kim [29] introduced another abstract convexity notion called a G-convex space, which includes many abstract convexity notions such as H-convex spaces as special cases. For the details on G-convex spaces, see [30-34], where basic theory was extensively developed.

Definition 2.2. A G-convex X is said to be a locally G-convex space if X is a uniform topological space with uniformity U, which has an open base B := (Vi : i e I) of symmetric entourages such that for each V eB, the set V(x) := (y e X : (y,x) e V) is a G-convex set for each x e X.

We recall that a nonempty space is said to be acyclic if all of its reduced Cech homology groups over the rationals vanish.

Definition 2.3 (see [35]). Let E be a topological space. A subset D of E is called contractible at v e D, if there is a continuous mapping F : D x [0,1] ^ D such that F(u, 0) = u for all u e D and F(u, 1) = v for all u e D.

In particular, each contractible space is acyclic and thus any nonempty convex or star-shaped set is acyclic. Moreover, by the definition of contractible set, we see that each convex space is contractible.

Definition 2.4. Let X and Y be two topological vector spaces and K a nonempty subset of X, and let F : K ^ 2Y be a set-valued mapping.

(i) F is called upper C-continuous at x0 e K if, for any neighbourhood U of the origin in Y, there is a neighbourhood V of x0 such that, for all x e V,

F(x) c F(x0) + U + C. (2.1)

(ii) F is called lower C-continuous at x0 e K if, for any neighbourhood U of the origin in Y, there is a neighbourhood V of x0 such that for all x e V,

F(x0) c F(x) + U - C. (2.2)

Definition 2.5. Let X and Y be two topological vector spaces and K a nonempty convex subset of X. A set-valued mapping F : K ^ 2Y is said to be properly C-quasiconvex if, for any x,y e K and t e [0,1], we have

either F(x) c F(tx + (1 - t)y) + C or F(y) c F(tx + (1 - t)y) + C. (2.3)

Definition 2.6. Let X and Y be two topological vector spaces and T : X ^ 2Y a set-valued mapping.

(i) T is said to be upper semicontinuous at x e X if, for any open set V containing T(x), there exists an open set U containing x such that for all t e U, T(t) c V; T is said to be upper semicontinuous on X if it is upper semicontinuous at all x e X.

(ii) T is said to be lower semicontinuous at x e X if, for any open set V with T(x)nV = 0, there exists an open set U containing x such that for all t e U, T(t) n V / 0; T is said to be lower semicontinuous on X if it is lower semicontinuous at all x e X.

(iii) T is said to be continuous on X if it is at the same time upper semicontinuous and lower semicontinuous on X.

(iv) T is said to be closed if the graph, Graph(T), of T, that is, Graph(T) = {(x,y) : x e X and y e T(x)}, is a closed set in X x Y.

Lemma 2.7 (see [36]). Let X and Y be two Hausdorff topological vector spaces and T : X ^ 2Y a set-valued mapping. Then, the following properties hold:

(i) if T is closed and T(X) is compact, then T is upper semicontinuous, where T(X) = uxeXT(x) and E denotes the closure of the set E,

(ii) if T is upper semicontinuous and for any x e X, T (x) is closed, then T is closed,

(iii) T is lower semicontinuous at x e X if and only if for any y e T (x) and any net {xa}, xa ^ x, there exists a net {ya} such that ya e T(xa) and y

We now have the following fixed point theorem in locally G-convex spaces given by Yuan [28] which is a generalization of the Fan-Glickberg-type fixed point theorems for upper semicontinuous set-valued mapping with nonempty closed acyclic values given in several places (e.g., see Kirk and Shin [37], Park and Kim [29], and others in locally convex spaces).

Lemma 2.8 (see [28]). Let X be a compact locally G-convex space and F : X ^ 2X an upper semicontinuous set-valued mappings with nonempty closed acyclic values. Then, F has a fixed point; that is, there exists an x* e X such that x* e F(x*).

3. Main Results

In this section, we apply the Kakutani-Fan-Glicksberg fixed point theorem for upper semicontinuous set-valued mapping with nonempty closed acyclic values to establish two existence theorems of strong solutions and obtain the closedness of the strong solutions set for generalized strong vector quasiequilibrium problem.

Theorem 3.1. Let X, Y, and Z be real locally G-convex topological vector spaces, K c X and D c Y nonempty compact subsets, and C c Z a nonempty closed convex cone. Let S : K ^ 2K be a continuous set-valued mapping such that for any x e K, the set S(x) is a nonempty closed contractible subset of K. Let T : K ^ 2D be an upper semicontinuous set-valued mapping with nonempty closed acyclic values and F : K x D x K ^ 2Z a set-valued mapping satisfy the following conditions:

(i) for all (x, y) e K x D,F(x,y,S(x)) c C,

(ii) for all (y, z) e D x K, F(■, y, z) are properly C-quasiconvex,

(iii) F(v ■, ■) are upper C-continuous,

(iv) for all y e D,F(^,yr) are lower (-C)-continuous. Then, the solutions set VS(F) is nonempty and closed subset of K.

Proof. For any (x, y) e K x D, we define a set-valued mapping G : K x D ^ 2K by

G(x,y) = {u e S(x) : F(u,y,z) c C, Vz e S(x)}. (3.1)

Since for any (x,y) e K x D, S(x) is nonempty. So, by assumption (i), we have that G(x,y) is nonempty. Next, we divide the proof into five steps.

Step 1 (to show that G(x,y) is acyclic). Since every contractible set is acyclic, it is enough to show that G(x,y) is contractible. Let u e G(x,y), thus u e S(x) and F(u,y,z) c C for all z e S(x). Since S(x) is contractible, there exists a continuous mapping h : S(x) x [0,1] ^ S(x) such that h(s, 0) = s for all s e S(x) and h(s, 1) = u for all s e S(x). Now, we set H(s,t) = tu+(1-t)h(s,t) for all (s,t) e G(x,y)x[0,1]. Then, H is a continuous mapping, and we see that H(s, 0) = s for all s e G(x,y) and H(s, 1) = u for all s e G(x,y). Let (s,t) e G(x,y) x [0,1]. We claim that H(s,t) e G(x,y). In fact, if H(s,t) / G(x,y), then there exists z* e S(x) such that

F (H (s,t),y,z* )CC. (3.2)

Since (■, y, z*) is properly C-quasiconvex, we can assume that

F(u,y,z*) c F(tu + (1 - t)h(s, t),y, z*) + C. (3.3)

It follows that

F(u, y, z*) c F(H(s,t),y,z*) + CCC + C c C, (3.4)

which contradicts u e G(x,y). Therefore, H(s,t) e G(x,y), and hence G(x,y) is contractible.

Step 2 (to show that G(x,y) is a closed subset of K). Let {aa} be a sequence in G(x,y) such that aa ^ a*. Then, aa e S(x). Since S(x) is a closed subset of K, a* e S(x). Since S is a lower semicontinuous, it follows by Lemma 2.7(iii) that for any z* e S(x) and any net {xa} ^ x, there exists a net {za} such that za e S(xa) and za ^ z*. This implies that

F(aa,y,za) c C. (3.5)

Since F(■, y, ■) are lower (-C)-continuous, we note that for any neighbourhood U of the origin in Z, there exists a subnet {ap,zp} of {aa,za} such that

F (a*,y,z*) c F (ap,y,z()) + U + C. (3.6)

From (3.5) and (3.6), we have

F(a*,y,z*) c U + C.

We claim that F(a*,y,z*) c C. Assume that there exists p e F(a*,y,z*) and p / C. Then, we note that 0 / (C - p), and the set C - p is closed. Thus, Z \ (C - p) is open, and 0 e Z \ (C - p). Since Z is a locally G-convex space, there exists a neighbourhood U0 of the origin such that U0 c Z \ (C - p) and U0 = -U0. Thus, we note that 0 / U0 + (C - p), and hence p / U0 + C, which contradicts to (3.7). Hence, F(a*,y,z*) c C, and therefore, a* e G(x,y). Then, G(x,y) is a closed subset of K.

Step 3 (to show that G(x,y) is upper semicontinuous). Let {(xa,ya) : a e I} c K x D be given such that (xa,ya) ^ (x,y) e K x D, and let aa e G(xa,ya) such that aa ^ a. Since aa e S(xa) and S is upper semicontinuous, it follows by Lemma 2.7(ii) that a e S(x). We claim that a e G(x,y). Assume that a / G(x,y). Then, there exists z* e S(x) such that

F(a,y,z*)/C, (3.8)

which implies that there is a neighbourhood U0 of the origin in Z such that

F(a,y,z*) + U0/C. (3.9)

Since F is upper C-continuous, it follows that for any neighbourhood U of the origin in Z, there exists a neighbourhood U1 of (a,y,z*) such that

F(a, y,z) c F(a,y,z*) + U + C, V(a,y,z) e U1. (3.10)

Without loss of generality, we can assume that U0 = U. This implies that

F(a,y,z) c F(a,y,z*) + U0 + C/C + C c C, V(a,y,z) e U1. (3.11)

Thus, there is a0 e I such that

F(aa,ya,za)/C, Va > a0, (3.12)

it is a contradiction to aa e G(xa,ya). Hence, a e G(x,y), and therefore, G is a closed mapping. Since K is a compact set and G(x,y) is a closed subset of K, G(x,y) is compact. This implies that G(x,y) is compact. Then, by Lemma 2.7(i), we have G(x,y) is upper semicontinuous.

Step 4 (to show that the solutions set VS(F) is nonempty). Define the set-valued mapping Q : K x D ^ 2KxD by

Q(x,y) = (G(x,y),T(x)) V(x,y) e K x D. (3.13)

Then, Q is an upper semicontinuous mpping. Moreover, we note that Q(x, y) is a nonempty closed acyclic subset of K x D for all (x, y) e K x D. By Lemma 2.8, there exists a point

(x,y) e (K x D) such that (x,y) e Q(x,y). Thus, we have x e G(x,y), y e T(x). It follows that there exists x e K and y e T(x) such that x e S(x) and

F(x,y,z) c C Vz e S(x). (3.14)

Hence, the solutions set VS(F) f 0.

Step 5 (to show that the solutions set VS(F) is closed). Let {xa : a e I} be a net in VS(F) such that xa ^ x . By definition of the solutions set VS(F), we note that xa e S(xa), and there exist ya e T(xa) satisfying

F (xa, ya, z) c C Vz e S(xa). (3.15)

Since S is a continuous closed valued mapping, x* e S(x*). From the compactness of D, we can assume that ya ^ y*. Since T is an upper semicontinuous closed valued mapping, it follows by Lemma 2.7(ii) that T is closed. Thus, we have y* e T(x*). Since F(-, y*, ■) is a lower (-C)-continuous, we have

F(x*,y*,z) c C Vz e S(x*). (3.16)

This means that x* belongs to VS(F). Therefore, the solutions set VS(F) is closed. This completes the proof. □

Theorem 3.1 extends Theorem 3.1 of Long et al. [16] to locally G-convex which includes locally convex Hausdorff topological vector spaces.

Corollary 3.2. Let X, Y and Z be real locally convex Hausdorff topological vector spaces, K c X and D c Y two nonempty compact convex subsets, and C c Z a nonempty closed convex cone. Let S : K ^ 2K be a continuous set-valued mapping such that for any x e K, S(x) is a nonempty closed convex subset of K. Let T : K ^ 2D be an upper semicontinuous set-valued mapping such that for any x e K, T(x) is a nonempty closed convex subset of D. Let F : K x D x K ^ 2Z be a set-valued mapping satisfying the following conditions:

(i) for all (x, y) e K x D,F(x,y,S(x)) c C,

(ii) for all (y, z) e D x K, F(■, y, z) are properly C-quasiconvex,

(iii) F(■, ■, ■) are upper C-continuous,

(iv) for all y e D,F(,yr) are lower (-C)-continuous.

Then, the solutions set VS(F) is nonempty and closed subset of K.

Theorem 3.3. Let X, Y and Z be real locally G-convex topological vector spaces, K c X and D c Y nonempty compact subsets, and C c Z a nonempty closed convex cone. Let S : K ^ 2K be a continuous set-valued mapping such that for any x e K, the set S(x) is a nonempty closed contractible

subset of K. Let T : K ^ 2D be an upper semicontinuous set-valued mapping with nonempty closed acyclic values and F : K x D x K ^ 2Z a set-valued mapping satisfying the following conditions:

(i) for all (x, y) e K x D,F(x,y,S(x))CC,

(ii) for all (y, z) e D x K, F(■, y, z) are properly C-quasiconvex,

(iii) F(, ■, ■) are upper C-continuous,

(iv) for all y e D,F(,y,) are lower (-C)-continuous.

Then, the solutions set VW(F) is nonempty and closed subset of K.

Proof. For any (x, y) e K x D, define a set-valued mapping B : K x D ^ 2K by

B(x,y) = {u e S(x) : F(u,y,z)/C, Vz e S(x)}. (3.17)

Proceeding as in the proof of Theorem 3.1, we need to prove that B(x,y) is closed acyclic subset of K x D for all (x, y) e K x D. We divide the remainder of the proof into three steps.

Step 1 (to show that B(x,y) is a closed subset of K). Let {aa} be a sequence in B(x,y) such that aa ^ a*. Then, aa e S(x) and F(aa, y, z)CC for all z e S(x). Since S(x) is a closed subset of K, we have a* e S(x). By the lower semicontinuity of S and Lemma 2.7(iii), we note that for any z e S(x) and any net {xa} ^ x, there exists a net {za} such that za e S(xa) and za ^ z. Thus, we have

F(aa,y,za)£C, (3.18)

which implies that there exists a neighbourhood U0 of the origin in Z such that

F(aa,y,za) + U0CC. (3.19)

Since F(■,y,■) are lower (-C)-continuous, it follows that for any neighbourhood U of the origin in Z, there exists a subnet {ap, zp} of {aa, za} such that

F (a*,y, z) c F (ap,y,zp) + U + C. (3.20)

Without loss of generality, we can assume that U = U0. Then, by (3.18), (3.19), and (3.20), we have

F (a*,y,z) c F (aa,y,za) + U0 + C/C + C c C. (3.21)

This means that a* e B(x,y) and so B(x,y) is a closed subset of K.

Step 2 (to show that B(x,y) is upper semicontinuous). Let {(xa,ya) : a e I} cK x D be given such that (xa,ya) ^ (x,y) e K x D, and let aa e B(xa,ya) such that aa ^ a. Then, aa e S(xa) and F(aa,y,z)CC, for all z e S(xa). Since S is upper semicontinuous closed valued

mapping, it follows by Lemma 2.7(ii) that a e S(x). We claim that a e B(x,y). Indeed, if a / B(x, y), then there exists a z0 e S(x) such that

F (a,y,z0) c C (3.22)

Since F is upper C-continuous, we note that for any neighbourhood U of the origin in Z, there exists a neighbourhood U0 of (a,y,z0) such that

F(a*,y*,z*) c F(a,y,ZQ) + U + C, V(a*,y*,z*) e U0. (3.23)

From (3.22) and (3.23), we obtain

F(a*,y*,z*) c U + C, V(a*,y*,z*) e U0. (3.24)

As in the proof of Step 2 in Theorem 3.1, we can show that F(a*, y*, z*) c C for all (a*, y*, z*) e U0. Hence, there is a0 e I such that

F(aa,ya,za) c C, Va > a0, (3.25)

it is a contradiction to aa e B(xa,ya). Hence, a e B(x,y), and therefore, B is a closed mapping. Since K is a compact set and B(x,y) is a closed subset of K, B(x,y) is compact. This implies that B(x, y) is compact. Then, by Lemma 2.7(i), we have that B(x, y) is upper semicontinuous.

Step 3 (to show that the solutions set (F) is nonempty and closed). Define the set-valued mapping P : K x D ^ 2KxD by

P(x,y) = (B(x,y),T(x)) V(x,y) e K x D. (3.26)

Then, P is an upper semicontinuous mapping. Moreover, we note that P(x,y) is a nonempty closed acyclic subset of K x D for all (x, y) e K x D. Hence, by Lemma 2.8, there exists a point (x,y) e (K x D) such that (x,y) e P(x,y). Thus, we have x e B(x,y) and y e T(x). This implies that there exists x e K and y e T(x) such that x e S(x) and

F(x,y,z)/C Vz e S(x). (3.27)

Hence, (F) / 0. Similarly, by the proof of Step 5 in Theorem 3.1, we have (F) is closed. This completes the proof. i-.

4. Stability

In this section, we discuss the stability of the solutions for the generalized strong vector quasiequilibrium problem (GSVQEPII).

Throughout this section, let X, Y be Banach spaces, and let Z be a real locally G-convex Hausdorff topological vector space. Let K c X and D c Y be nonempty compact subsets, and

let C c Z be a nonempty closed convex cone. Let E := {(S, T) | S : K ^ 2K is a continuous set-valued mapping with nonempty closed contractible values, and T : K ^ 2D is an upper semicontinuous set-valued mapping with nonempty closed acyclic values}.

Let B\,B2 be compact sets in a normed space. Recall that the Hausdorff metric is defined by

H(Bi, B2) := maxi sup d(b, B2), sup d(b, Bi) L (4.1)

[beBi beB2 J

where d(b,B2) := infaeB2||b - ay.

For (S1, T1), (S2, T2) e E, we define

p((Si, Ti), (S2, T2)) := sup Hi(Si(x), S2(x)) + sup H2(Ti(x), T2(x)), (4 2)

xeK xeK

where Hi,H2 being the appropriate Hausdorff metrics. Obviously, (E,p) is a metric space. Now, we assume that F satisfies the assumptions of Theorem 3.3. Then, for each (S,T) e E, (GSVQEPII) has a solution x*. Let

y(S,T) = {x e K : x e S(x), 3y e T(x), F(x,y,z)/C Vz e S(x)}. (4.3)

Thus, y(S, T) = 0, which conclude that y defines a set-valued mapping from E into K. We also need the following lemma in the sequel.

Lemma 4.1 (see [8,38]). Let W be a metric space, and let A,An (n = i, 2,...) be compact sets in W. Suppose that for any open set O d A, there exists n0 such that An d O for all n > n0. Then, any sequence {xn} satisfying xn e An has a convergent subsequence with limit in A.

In the following theorem, we replaced the convex set by the contractible set and acyclic set in Theorem 4.i in [i6]. The following theorem can acquire the same result appearing on the Theorem 4.i by utilized Lemma 4.i. Now, we need only to present stability theorem for the solution set mapping y for (GSVQEP II).

Theorem 4.2. y : E ^ 2K is an upper semicontinuous mapping with compact values.

Proof. Since K is compact, we need only to show that y is a closed mapping. In fact, let ((Sn,Tn),xn) e Graph(y) be such that ((Sn,Tn),xn) ^ ((S,T),x*). Since xn e y(Sn,Tn), we have xn e Sn(xn), and there exists yn e Tn(xn) such that

F(xn,yn,z)/C Vz e Sn(xn). (4.4)

By the same argument as in the proof of Theorem 4.i in [i6], we can show that x* e S(x*) and y* e T(x*).

Since S is lower semicontinuous at x* and xn ^ x*, it follows by Lemma 2.7(iii) that for any z e S(x*), there exists zn e S(xn) such that zn ^ z. To finish the proof of the theorem, we need to show that F(x*,y*,z)/C for all z e S(x*). Since p((Sn,Tn), (S,T)) ^ 0, it follows

by the same argument as in the proof of Theorem 4.1 in [16] that there exists a subsequence [Xnk} of {Xn} such that Xnk e Snk\xnk), ynk e Tnk(xnk), Znk e Snk(xnk), and

HXnk'ynk,Znk)/C. (45)

From the upper C-continuous of F, we have

F(x*,y*,z)/C Vz e S(x*). (4.6)

Then, ((S, T),x*) e Graph(^), and so Graph(^) is closed. The theorem is proved. □

Acknowledgments

S. Plubtieng would like to thank the Thailand Research Fund for financial support under Grant no. BRG5280016. Moreover, K. Sitthithakerngkiet would like to thank the Office of the Higher Education Commission, Thailand, for supporting by grant fund under Grant no. CHE-Ph.D-SW-RG/41/2550, Thailand.

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