KY FAN S INEQUALITIES FOR VECTOR-VALUED MULTIFUNCTIONS IN TOPOLOGICAL ORDERED SPACES

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1 Fixed Point They, 15(2014), No. 1, nodeacj/sfptcj.html KY FAN S INEQUALITIES FOR VECTOR-VALUED MULTIFUNCTIONS IN TOPOLOGICAL ORDERED SPACES NGUYEN THE VINH AND PHAM THI HOAI Department of Mathematical Analysis University of Transpt and Communications, Hanoi, Vietnam thevinhbn@gmail.com School of Applied Mathematics and Infmatics Ha Noi University of Science and Technology, Hanoi, Vietnam phamhoai051087@gmail.com Abstract. The aim of this paper is using the cone semicontinuity and cone quasiconvexity f multivalued mappings to present four variants of Ky Fan s type inequality f vect-valued multifunctions in topological dered spaces. Key Wds and Phrases: fixed point theem, multivalued mapping, Ky Fan minimax inequality, topological semilattices, C -quasiconvex (quasiconcave), C-upper (lower) semicontinuous Mathematics Subject Classification: 47H04, 49J53, 47H Introduction In 1972, Ky Fan [5] proved the following minimax inequality f real-valued functions. Theem 1.1. Let X be a Hausdff topological vect space, and let K be a nonempty compact convex subset of X. Suppose that f : K K R satisfies the following: (1) f(x, x) 0, x K; (2) y K, f(., y) is quasiconcave; (3) x K, f(x,.) is lower semicontinuous. Then there exists y K such that f(x, y ) 0, x K. The above Ky Fan minimax inequality is well known. It plays a very imptant role in many fields, such as variational inequalities, game they, mathematical economics, optimization they, and fixed point they. Because of wide applications, this inequality has been generalized in a number ways (e.g., see Allen [1], Aubin and Ekeland [2], Chang [3], Ding and Tan [4], Tian [14], Yen [16], Yuan [17], and Zhou and Chen [19], Hvath [7], Gegiev and Tanaka [6]). In the framewk of topological semilattices, Hvath and Llinares Ciscar (1996, [8]) first established an der theetical version of the classical result of Knaster-Kuratowski-Mazurkiewicz, as well as 253

2 254 NGUYEN THE VINH AND PHAM THI HOAI fixed point theems f multivalued mappings. In 2001, by using Hvath and Llinares Ciscar s results, Luo [10] proved a similar result to Theem 1.1 in topological semilattices. In 2006, Luo [11] studied Ky Fan s minimax inequalities f vect multivalued mappings in topological semilattices. However, his results does not imply scalar Ky Fan minimax inequality in the setting of topological semilattice spaces. Recently, Song and Wang [13] gave an extension of Ky Fan minimax inequality but only f vect single-valued mappings in topological semilattices. Motivated and inspired by research wks mentioned above, in this paper, we will use the cone semicontinuity and cone quasiconvexity f multivalued mappings to show four kinds of vect valued Ky Fan s type inequality f multivalued mappings. Any of our Theems implies the scalar Fan minimax inequality in topological semilattices, while the main result in [11] does not imply it in its full generality, but only f continuous functions. The rest of the paper is ganized as follows. In Section 2, we introduce about topological semilattices and recall some concepts of cone semicontinuity and cone convexity. In Section 3, we prove the existence of solutions f multivalued Ky Fan inequalities (SKFI) as an application by means of Browder-Fan fixed point theem in the setting of topological semilattices. We also give some examples to illustrate our results. 2. Preliminaries Definition 2.1. ([8]) A partially dered set (X, ) is called a sup-semilattice if any two elements x, y of X have a least upper bound, denoted by sup{x, y}. The partially dered set (X, ) is a topological semilattice if X is a sup-semilattice equipped with a topology such that the mapping is continuous. X X X (x, y) sup{x, y} We have given the definition of a sup-semilattice, we could obviously also consider inf-semilattices. When no confusion can arise we will simply use the wd semilattice. It is also evident that each nonempty finite set A of X will have a least upper bound, denoted by sup A. In a partially dered set (X, ), two arbitrary elements x and x do not have to be comparable but, in the case where x x, the set [x, x ] = {y X : x y x } is called an der interval simply, an interval. Now assume that (X, ) is a semilattice and A is a nonempty finite subset; then the set (A) = [a, sup A] a A is well defined and it has the following properties:

3 KY FAN S INEQUALITIES FOR VECTOR VALUED MULTIFUNCTIONS 255 (1) A (A); (2) if A A, then (A) (A ). We say that a subset E X is -convex if f any nonempty finite subset A E we have (A) E. Example 2.2. We consider R 2 with usual der defined by (a, b), (c, d) R 2, (a, b) (c, d) a c; b d. Clearly, (R 2, ) is a topological semilattice. (1) The set X = {(x, 1) : 0 x 1} {(1, y) : 0 y 1} is -convex but not convex in the usual sense. (2) The set X = {(x, y) : 0 x 1; y = 1 x} is convex in the usual sense but not -convex. Lemma 2.3. ([18], Lemma 1.1) Let Y be a topological vect space and C a closed, convex and pointed cone of Y with int C, where int C denotes the interi of C. Then we have int C + C int C. We now recall some concepts of generalized convexity of multivalued mappings. Let X be a nonempty convex subset of a vect space E, C be a convex cone of a vect space Y, and F : X 2 Y be a set-valued mapping with nonempty values. The mapping F is called C-quasiconvex if f all x i X, i = 1, 2 and x conv{x 1, x 2 }, either F (x) F (x 1 ) C, F (x) F (x 2 ) C. The mapping F is called C-quasiconcave if f all x i X, i = 1, 2, and x conv{x 1, x 2 }, either F (x 1 ) F (x) + C, F (x 2 ) F (x) + C. Similarly, in the setting of topological semilattices, we introduce the following definition. Definition 2.4. Let X be a topological semilattice a -convex subset of a topological semilattice, Y be a topological vect space, C Y be a convex cone. Let F : X 2 Y be a multivalued mapping with nonempty values.

4 256 NGUYEN THE VINH AND PHAM THI HOAI (1) F is called C -quasiconvex mapping if, f any pair x 1, x 2 X and f any x ({x 1, x 2 }), we have either F (x) F (x 1 ) C F (x) F (x 2 ) C. (2) F is called C -quasiconcave mapping if, f any pair x 1, x 2 X and f any x ({x 1, x 2 }), we have either F (x 1 ) F (x) + C, F (x 2 ) F (x) + C. We use instead of when F is single-valued. Remark 2.5. If Y = R = (, + ) and C = [0, + ), and F is a real function, then the C -quasiconvexity of ϕ is equivalent to the -quasiconvexity of ϕ (see [10]). Example 2.6. Let X = [0, 1] [0, 1]. We set x 1 x 2 denoting that x 2 x 1 + R 2 +, x 1, x 2 X, where R 2 + = {(y 1, y 2 ) R 2 : y 1 0, y 2 0}. It is obvious that (X, ) is a topological semilattice, in which x 1 x 2 = (max(x 1 1, x 2 1), max(x 1 2, x 2 2)), x i = (x i 1, x i 2) X, i = 1, 2. (1) Let F : X 2 R and C = R + such that F (x) = [(1 x 1 )(1 x 2 ), + ), x = (x 1, x 2 ) X. It is clear that F is C -quasiconcave mapping but not C-quasiconcave. Indeed, f x 1 = (0, 1), x 2 = (1, 0), x = 1 2 x x2 = ( 1 2, 1 2 ), we see that [ ) 1 F (x 1 ) = F (x 2 ) = [0, + ), F (x) = 4, + while F (x 1 ) = F (x 2 ) = [0, + ) F (x) + C = (2) Let F : X 2 R and C = R + such that F (x) = {x x 2 2}, x = (x 1, x 2 ) X. [ ) 1 4, +. It is easy to see that F is C-quasiconvex but not C -quasiconvex. Now, we recall the semicontinuous properties of multivalued mappings (see Ref. [2]). Let F : X 2 Y be a multivalued mapping between topological spaces X and Y. The domain of F is defined to be the set domf = {x D : F (x) }. The mapping F is upper semicontinuous (shtly, usc) at x 0 domf if, f any open set V of Y with F (x 0 ) V, there exists a neighbhood U of x 0 such that F (x) V f all x U. The mapping F is lower semicontinuous (shtly, lsc) at x 0 domf if, f any open set V of Y with F (x 0 ) V, there exists a neighbhood U of x 0 such that F (x) V f all x U.

5 KY FAN S INEQUALITIES FOR VECTOR VALUED MULTIFUNCTIONS 257 The mapping F is continuous at x 0 domf if it is both usc and lsc at x 0. The mapping F is continuous (resp. usc, lsc) if domf = X and if F is continuous (resp. usc, lsc) at each point x X. If Y is a partially dered topological vect space, then the above definitions of semicontinuous can be weakened. Me precisely, we can introduce the following definitions taken from Ref. [9, 12]. Definition 2.7. Let X be a topological space, Y be a topological vect space with a cone C. Let F : X 2 Y. We say that (1) F is C-upper semicontinuous (shtly, C-usc) at x 0 domf if f any open set V of Y with F (x 0 ) V there exists a neighbhood U of x 0 such that F (x) V + C f each x domf U. (2) F is C-lower semicontinuous (shtly, C-lsc) at x 0 domf if f any open set V of Y with F (x 0 ) V there exists a neighbhood U of x 0 such that F (x) [V C] f each x domf U. (3) F is C-usc (resp. C-lsc) if domf = X and if F is C-usc (resp. C-lsc) at each point of domf. Remark 2.8. If Y = R and C = R + = {x R : x 0} (resp. C = R + ), F is single-valued and C-usc at x 0, then F is lower semicontinuous (resp. upper semicontinuous) at x 0 in the usual sense. Remark 2.9. The upper (resp. lower) semicontinuity of F implies the C-upper (resp. C-lower) semicontinuity of F. Example 3.1 in Section 3 will show that the converse statement is no longer true. Definition Let X, Y be two topological spaces; F : X 2 Y is said to have open lower sections if F 1 (y) = {x X : y F (x)} is open f any y Y. The following lemma is a special case of [8, Collary 1, pp. 298]. Lemma (Browder-Fan fixed point theem) Let K be a nonempty compact - convex subset of a topological semilattice with path-connected intervals M, F : K 2 K with nonempty -convex values, and F 1 (y) K be open, f any y K. Then F has a fixed point. 3. Ky Fan s inequalities f vect-valued multifunctions Let X be a topological semilattice, K X a nonempty -convex subset, Y a topological vect space, A : K 2 K, f : K K 2 Y, C a closed, pointed and convex cone in Y with int C. We consider the following multivalued Ky Fan inequalities (SKFI): (SKFI1) Find x K such that x A(x), f(x, y) int C, y A(x).

6 258 NGUYEN THE VINH AND PHAM THI HOAI (SKFI2) Find x K such that (SKFI3) Find x K such that (SKFI4) Find x K such that x A(x), f(x, y) int C =, y A(x). x A(x), f(x, y) ( C), y A(x). x A(x), f(x, y) C, y A(x). The existence of solutions f the problems (SKFI1), (SKFI2), (SKFI4) were studied by Luo in [11]. However, he used either upper semicontinuous lower semicontinuous multifunctions. So, in the scalar case, the single-valued function f is continuous with respect to the first variable, and therefe, his results are weaker than the iginal fm. In this paper, we use the cone semicontinuity and cone convexity of multivalued mappings to give some genuine generalizations of scalar Ky Fan minimax inequality in the setting of topological semilattices. Theem 3.1. Let K be a nonempty compact -convex subset of a topological semilattice with path-connected intervals M, Y a topological vect space, A : K 2 K with nonempty -convex values, f : K K 2 Y, C a closed, pointed and convex cone in Y with int C. Assume that (1) A has open lower sections and B := {x K : x A(x)} is closed; (2) f(x, x) int C, x K; (3) x K, f(x,.) is C -quasiconvex; (4) y K, f(., y) is C-upper semicontinuous. Then there exists x K such that x A(x ) and f(x, y) int C, y A(x ). Proof. Define P : K 2 K by P (x) = {y K : f(x, y) int C}, x K. Suppose that there exists x K such that P (x ) is not -convex; then there exist y 1, y 2 P (x ) such that ({y 1, y 2 }) P (x ), i.e., there exists z ({y 1, y 2 }) and z P (x ); hence f(x, z) int C. By (3), we have either By Lemma 2.1, we have either f(x, z) f(x, y 1 ) + C f(x, z) f(x, y 2 ) + C. f(x, z) f(x, y 1 ) + C int C + C int C f(x, z) f(x, y 2 ) + C int C + C int C

7 KY FAN S INEQUALITIES FOR VECTOR VALUED MULTIFUNCTIONS 259 which is a contradiction. Therefe, f any x X, P (x) is -convex. Next, we prove that P 1 (y) is open f each y K. We have P 1 (y) = {x K : f(x, y) int C} F each y K and each x P 1 (y), we have f(x, y) int C. By (4), there exists a neighbhood U(x) of x such that f(x, y) int C + C int C whenever x U(x), which implies that U(x) P 1 (y), i.e., P 1 (y) is open. By Lemma 2.2, B is a nonempty set. Define S : K 2 K by { A(x) P (x), if x B, S(x) = A(x), if x K \ B. Then f any x K, S(x) is -convex. And f any y K, S 1 (y) = (A 1 (y) P 1 (y)) ((K \ B) A 1 (y)) is open. Suppose that x K, S(x) is nonempty. By Lemma 2.2, we deduce that S has a fixed point, i.e., there exists x 0 K such that x 0 S(x 0 ). If x 0 B, then x 0 S(x 0 ) = A(x 0 ) P (x 0 ). Hence x 0 P (x 0 ), f(x 0, x 0 ) int C which contradicts our assumption (2). If x 0 K \ B, then x 0 S(x 0 ) = A(x 0 ) A(x 0 ), and hence x 0 B which contradicts x 0 K \ B. Therefe, there exists x K such that S(x ) =. Since A(x) is nonempty f all x K, hence x B, S(x ) = A(x ) P (x ) =, i.e., x A(x ) and f any y A(x ), y P (x ), we have x A(x ), f(x, y) int C =, y A(x ). Therefe, the assertion of Theem 3.1 is true. In Theem 3.1, when f is single-valued, we have the following collary. Collary 3.2. Let K be a nonempty compact -convex subset of a topological semilattice with path-connected intervals M, Y a topological vect space, A : K 2 K with nonempty -convex values, f : K K Y, C a closed, pointed and convex cone in Y with int C. Assume that (1) A has open lower sections and B := {x K : x A(x)} is closed; (2) f(x, x) int C, x K; (3) x K, f(x,.) is C -quasiconvex; (4) y K, f(., y) is C-upper semicontinuous. Then there exists x K such that x A(x ) and f(x, y) int C, y A(x ). Now we give an example to explain that Collary 3.2 is applicable. Example 3.3. Let X be given in Example 2.6 and Y = R with C = R +. F each x X, let A(x) = [(0, 1), (1, 1)] [(1, 0), (1, 1)], where [(0, 1), (1, 1)] denotes the line segment joining points (0, 1) and (1, 1). Then we have: (1) f each x X, A(x) is nonempty and -convex;

8 260 NGUYEN THE VINH AND PHAM THI HOAI (2) f y = (y 1, y 2 ) X, A 1 (y) = { X if y [(0, 1), (1, 1)] [(1, 0), (1, 1)] if y X \ {[(0, 1), (1, 1)] [(1, 0), (1, 1)]} Therefe, f each y X, A 1 (y) is open in X. (3) The set B = {x X : x A(x)} = [(0, 1), (1, 1)] [(1, 0), (1, 1)] is closed. F any x = (x 1, x 2 ), y = (y 1, y 2 ) X, we define f : X X Y by { (1 + x 1 y 1 )(1 + x 2 y 2 ), if (x, y) (0, 0) f(x, y) = 2, if (x, y) = (0, 0) Then all the assumptions of Collary 3.2 are satisfied. So Collary 3.2 is applicable. The set of solutions f the (SKFI1) is the overall B. Remark 3.4. F every fixed x, following the same argument as Example 2.1 in Ref. [13], we see that f(x,.) is not a usual quasiconcave function. Indeed, f x = 0, we have { (1 y 1 )(1 y 2 ), if (y 1, y 2 ) (0, 0) f(0, y) = 2, if (y 1, y 2 ) = (0, 0) Clearly, f y 1 = (1, 0), y 2 = (0, 1), y = 1 2 y y2 = ( 1 2, 1 2 ), we see that f(0, y1 ) = 0, f(0, y 2 ) = 0, while f(0, y) = 1 4. When Y = (, + ), C = [0, + ) and A(x) = K, x K, from Collary 3.2, we get scalar Ky Fan inequality f real-valued functions in topological semilattices (see, f instance, [10, 15]). Collary 3.5. Let K be a nonempty compact -convex subset of a topological semilattice with path-connected intervals M and let f : K K R be such that (1) f(x, x) 0, x K; (2) x K, f(x,.) is -quasiconcave; (3) y K, f(., y) is lower semicontinuous. Then there exists x K such that f(x, y) 0, y K. Theem 3.6. Let K be a nonempty compact -convex subset of a topological semilattice with path-connected intervals M, Y a topological vect space, A : K 2 K with nonempty -convex values, f : K K 2 Y, C a closed, pointed and convex cone in Y with int C. Assume that (1) A has open lower sections and B := {x K : x A(x)} is closed; (2) f(x, x) int C =, x K; (3) x K, f(x,.) is C -quasiconvex; (4) y K, f(., y) is C-lower semicontinuous.

9 KY FAN S INEQUALITIES FOR VECTOR VALUED MULTIFUNCTIONS 261 Then there exists x K such that x A(x ) and f(x, y) int C =, y A(x ). Proof. Define P : K K by P (x) = {y K : f(x, y) int C }, x K. Suppose that there exists x K such that P (x ) is not -convex; then there exist y 1, y 2 P (x ) such that ({y 1, y 2 }) P (x ), i.e., there exists z ({y 1, y 2 }) and z P (x ); hence f(x, z) int C =. By (3), we have either f(x, y 1 ) f(x, z) C f(x, y 2 ) f(x, z) C. Since f(x, y i ) int C, take u i f(x, y i ) int C, i = 1, 2. Then there exist v i f(x, z) and w i C such that either u 1 = v 1 w 1 u 2 = v 2 w 2. By Lemma 2.1, we have either v 1 = u 1 + w 1 int C v 2 = u 2 + w 2 int C which contradicts f(x, z) int C =. Therefe, f any x X, P (x) is -convex. Next, we prove that P 1 (y) is open f each y K. We have P 1 (y) = {x K : f(x, y) int C }. Take arbitrarily x P 1 (y), we have f(x, y) int C. By assumption (4), there exists an open neighbhood U(x) such that f(x, y) (int C + C) = f(x, y) int C, f all x U(x). Let {x α } be any net in D converging to x, hence there exists β such that x α U, α β and then f(x α, y) int C, which contradicts x α D. Therefe, x D and D is closed. Consequently, we infer that P 1 (y) is open f each y K. The rest of the proof is similar to that of Theem 3.1. Hence the proof is complete. Theem 3.7. Let K be a nonempty compact -convex subset of a topological semilattice with path-connected intervals M, Y a topological vect space, A : K 2 K with nonempty -convex values, f : K K 2 Y, C a closed, pointed and convex cone in Y with int C. Assume that (1) A has open lower sections and B := {x K : x A(x)} is closed; (2) f(x, x) ( C), x K; (3) x K, f(x,.) is C -quasiconvex; (4) y K, f(., y) is C-upper semicontinuous. Then there exists x K such that x A(x ) and f(x, y) ( C), y A(x ). Proof. Define P : K 2 K by P (x) = {y K : f(x, y) ( C) = }, x K. Suppose that there exists x K such that P (x ) is not -convex; then there exist y 1, y 2 P (x ) such that ({y 1, y 2 }) P (x ), i.e., there exists z ({y 1, y 2 }) and

10 262 NGUYEN THE VINH AND PHAM THI HOAI z P (x ); hence f(x, z) ( C). Take u f(x, z) ( C). By (3), we have either f(x, z) f(x, y 1 ) + C f(x, z) f(x, y 2 ) + C. Since u f(x, z), then there exist v i f(x, y i ) and w i C such that either u = v 1 + w 1 u = v 2 + w 2. Therefe either v 1 = u w 1 C v 2 = u w 2 C, which is a contradiction. Therefe, f any x X, P (x) is -convex. Next, we prove that P 1 (y) is open f each y K. We have Take arbitrarily x P 1 (y), we have P 1 (y) = {x K : f(x, y) ( C) = }. f(x, y) ( C) = f(x, y) Y \ ( C). By assumption (4), there exists an open neighbhood U(x) such that f(x, y) Y \ ( C) + C Y \ ( C), f all x U(x), it means f(x, y) ( C) = f all x U(x). We infer that P 1 (y) is open f each y K. The rest of the proof is similar to that of Theem 3.1. Hence our proof is finished. Theem 3.8. Let K be a nonempty compact -convex subset of a topological semilattice with path-connected intervals M, Y a topological vect space, A : K 2 K with nonempty -convex values, f : K K 2 Y, C a closed, pointed and convex cone in Y with int C. Assume that (1) A has open lower sections and B := {x K : x A(x)} is closed; (2) f(x, x) C, x K; (3) x K, f(x,.) is C -quasiconcave; (4) y K, f(., y) is C-lower semicontinuous. Then there exists x K such that x A(x ) and f(x, y) C, y A(x ). Proof. Define P : K 2 K by P (x) = {y K : f(x, y) C}, x K. Suppose that there exists x K such that P (x ) is not -convex; then there exist y 1, y 2 P (x ) such that ({y 1, y 2 }) P (x ), i.e., there exists z ({y 1, y 2 }) and z P (x ); hence f(x, z) C. By (3), we have either Consequently, we have either f(x, y 1 ) f(x, z) C f(x, y 2 ) f(x, z) C. f(x, y 1 ) f(x, z) C C C C

11 KY FAN S INEQUALITIES FOR VECTOR VALUED MULTIFUNCTIONS 263 f(x, y 2 ) f(x, z) C C C C, which is a contradiction. Therefe, f any x X, P (x) is -convex. Next, we prove that P 1 (y) is open f each y K. We have P 1 (y) = {x K : f(x, y) C}. Take arbitrarily x P 1 (y), we have f(x, y) C. It is equivalent to f(x, y) [Y \( C)]. By assumption (4), there exists an open neighbhood U(x) such that f(x, y) [Y \ ( C) + C], f all x U(x). Since Y \ ( C) + C Y \ ( C), it follows that f(x, y) [Y \ ( C)], f all x U(x). Therefe f(x, y) C f all x U(x). We infer that P 1 (y) is open f each y K. The rest of the proof is similar to that of Theem 3.1. Hence our proof is finished. Acknowledgment. This research is funded by Vietnam National Foundation f Science and Technology Development (NAFOSTED). References [1] G. Allen, Variational inequalities, complementarity problems and duality theems, J. Math. Anal. Appl., 58(1977), [2] J.P. Aubin, I. Ekeland, Applied Nonlinear Analysis, John Wiley, New Yk, [3] S.S. Chang, Y. Zhang, Generalized KKM theem and variational inequalities, J. Math. Anal. Appl., 159(1991), [4] X.P. Ding, K.K. Tan, A minimax inequality with applications to existence of equilibrium point and fixed point theems, Colloq. Math., 63(1992), [5] K. Fan, A minimax inequality and applications, in: Inequalities III. (O. Shisha, ed.) Proc. of the Third Symposium on Inequalities, Academic Press, New Yk, [6] P.G. Gegiev, T. Tanaka, Fan s inequality f set-valued maps, Nonlinear Anal., 47(2001), [7] C.D. Hvath, Contractibility and generalized convexity, J. Math. Anal. Appl., 156(1991), [8] C.D. Hvath, J.V. Llinares Ciscar, Maximal elements and fixed points f binary relations on topological dered spaces, J. Math. Econom., 25(1996), [9] D.T. Luc, They of Vect Optimization, in Lecture Notes in Economics and Mathematical Systems, Vol. 319, Springer-Verlag, Berlin, [10] Q. Luo, KKM and Nash equilibria type theems in topological dered spaces, J. Math. Anal. Appl., 264(2001), [11] Q. Luo, The applications of the Fan-Browder fixed point theem in topological dered spaces, Appl. Math. Lett., 19(2006), [12] P.H. Sach, New nonlinear scalarization functions and applications, Nonlinear Anal., 75(2012), [13] Q.Q. Song, L.S. Wang, The existence of solutions f the system of vect quasi-equilibrium problems in topological der spaces, Comput. Math. Appl., 62(2011), [14] G. Tian, Generalized KKM theems, minimax inequalities, and their applications, J. Optim. They Appl., 83(1994), [15] N.T. Vinh, Matching theems, fixed point theems and minimax inequalities in topological dered spaces, Acta Math. Vietnam., 30(2005),

12 264 NGUYEN THE VINH AND PHAM THI HOAI [16] C.L. Yen, A minimax inequality and its applications to variational inequalities, Pacific J. Math., 97(1981), [17] X.Z. Yuan, KKM principle, Ky Fan minimax inequalities and fixed point theems, Nonlinear Wld, 2(1995), [18] H. Yang, J. Yu, Essential component of the set of weakly Pareto-Nash equilibrium points, Appl. Math. Lett., 15(2002), [19] J. Zhou, G. Chen, Diagonal convexity conditions f problems in convex analysis and quasivariational inequalities, J. Math. Anal. Appl., 132(1988), Received: May 2, 2012; Accepted: March 28, 2013.