0 Journal of Inequalities and Applications

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RESEARCH

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Bilevel minimax theorems for non-continuous set-valued mappings

Yen-Cherng Lin*

Correspondence: ydin@maiLamu.edu.tw Department of OccupationalSafety and Health, College of Public Health, China MedicalUniversity, Taichung, 40421, Taiwan

Abstract

We study new types for minimax theorems with a couple of set-valued mappings, and we propose several versions for minimax theorems in topological vector spaces setting. These problems arise naturally from some minimax theorems in the vector settings. Both the types of scalar minimax theorems and the set minimax theorems are discussed. Furthermore, we propose three versions of minimax theorems for the last type. Some examples are also proposed to illustrate our theorems. MSC: 49J35; 58C06

Keywords: minimax theorems; cone-convexities

1 Introduction and preliminaries

Let X, Y be two nonempty sets in two Hausdorff topological vector spaces, respectively, Z be a Hausdorff topological vector space, C c Z a closed convex and pointed cone with apex at the origin and int C = 0, this means that C is a closed set with nonempty interior and satisfies XC c C, VX > 0; C + C c C; and C n (-C) = {0}. The scalar bilevel minimax theorems stated as follows: given two set-valued mappings F, G: X x Y ^ R, under suitable conditions the following relation holds:

(s - B) min U max^J F(x, y) < max ^J min LJG(x, y).

xeX yeY yeY xeX

Given two mappings F, G : X x Y ^ Z, the first version of bilevel minimax theorems stated that under suitable conditions the following relation holds:

(Bi) Maxy Min„ U G(x,y) c Mini co UMax^F(x,y) ] + C.

yeY xeX ^ xeX yeY '

The second version of bilevel minimax theorems stated that under suitable conditions the following relation holds:

(B2) Max U Min„ U G(x,y) c Mi^ Max^ F(x,y) + C.

y&Y xeX xeX y&Y

©2014 Lin; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

ringer

The third version of bilevel minimax theorems stated that under suitable conditions the following relation holds:

(B3) Min U Maxw U F(x,y) C Max U Min^ G(x,y) + Z \ (C \ (Oj).

xeX yeY yeY xeX

The case G = F of (5 - B) and (Bi)-(B3) has been discussed in [1-3] for set-valued mapping and in [4] for vector-valued mapping, respectively. Scalar minimax theorems and set minimax theorems for non-continuous set-valued mappings were first proposed by Lin et al. [1]. These results can be compared with the recent existing results [2, 3]. In this paper, we establish bilevel minimax results with a couple of non-continuous set-valued mappings (Theorem 2.1 in Section 2, Theorems 3.1-3.3 in Section 3). These results might not hold for each individual non-continuous set-valued mapping since it always lack some conditions so that the existing minimax theorems are not applicable, such as Theorems 4.1-4.3 [1], Theorem 2.1 [2] or Proposition 2.1 [3]. We present some fundamental concepts which will be used in the sequel.

Definition 1.1 [1, 2, 4] Let A be a nonempty subset of Z. A point z e A is called a

(a) minimal point of A if A n (z - C) = (zj; Min A denotes the set of all minimal points ofA;

(b) maximal point of A if A n (z + C) = (zj; Max A denotes the set of all maximal points ofA;

(c) weakly minimal point of A if A n (z - int C) = 0; MinwA denotes the set of all weakly minimal points of A;

(d) weakly maximal point of A if A n (z + int C) = 0; Maxw A denotes the set of all weakly maximal points of A.

Following [2], we denote both Max and Maxw by max (both Min and Minw by min) in R since both Max and Maxw (both Min and Minw) are same in R. We note that, for a nonempty compact set A, the both sets Max A and Min A are nonempty. Furthermore, Min A C MinwA, Max A C MaxwA, A C Min A + C, and A C Max A - C. In the sequel we shall use the following geometric result.

Lemma 1.1 [5] Let X, Y be nonempty convex subsets of two real Hausdorff topological spaces, respectively, A C X x Y be a subset such that

(a) for each y e Y, the set (x e X: (x, y) e Aj is closed in X; and

(b) for each x e X, the set (y e Y :(x,y) e Aj is convex or empty.

Suppose that there exist a subset B of A and a compact convex subset K ofX such thatB is closed inX x Y and

(c) for each y e Y, the set (x e K: (x, y) e Bj is nonempty and convex. Then there exists a point xO e K such that (xOj x Y C A.

Definition 1.2 Let U, V be Hausdorff topological spaces. A set-valued map F: U ^ V with nonempty values is said to be (a) lower semi-continuous at xO e U if for any net (x^j C U such that x^ ^ xO and any yO e F(xO), there exists a net y^ e F(x^) such that y^ ^ yO;

(b) upper semi-continuous at xO e U if for every xO e U and for every open set N containing F(xO), there exists a neighborhood M of xO such that F(M) C N;

(c) continuous at xO e U if F is upper semi-continuous as well as lower semi-continuous at xO.

We note that T is upper semi-continuous at xO and T(xO) is compact, then for any net (xv j C U, xv ^ xO, and for any net yv e T(xv)foreachv,thereexistyO e T(xO) and a subnet (yVa j such that yVa ^ yO. For more details, we refer the reader to [6, 7].

Definition 1.3 [2, 8] Let k e int C and v e Z. The Gerstewitz function f^ : Z ^ R is defined by

fkv(u) = min(t e R: u e v + tk - Cj. We present some fundamental properties of the scalarization function.

Proposition 1.1 [2,8] Let k e int C and v e Z. The Gerstewitz function fkv: Z ^ R has the followingproperties:

(a) fkv(u) < r ■ u e v + rk - C;

(b) fkv(u) <r ■ u e v + rk - int C; and

(c) fkv() is a continuous convex increasing and strictly increasing function.

We also need the following different kinds of cone-convexities for set-valued mappings.

Definition 1.4 [1] Let X be a nonempty convex subset of a topological vector space. A set-valued mapping F: X ^ Z is said to be

(a) above-C-convex (respectively, above-C-concave) on X if for all x1,x2 e X and all X e [O,1],

F(Xx1 + (1 - k)x2) C XF(xi) + (1 - X)F(x2) - C (respectively, XF(x1) + (1 - X)F(x2) C F(Xx1 + (1 - X)x2) - C);

(b) above-naturally C-quasi-convex on X if for all x1, x2 e X and all X e [0,1],

F(Xx1 + (1 - X)x2) C co{F(x1) U F(x2)} - C, where co A denotes the convex hull of a set A;

Let C* = (g e Z* : g(c) > O for all c e Cj, where Z* is the set of all nonzero continuous linear functional on Z.

Proposition 1.2 Let A be a nonempty compact subset of Z, for any f e C*, we have f Maxw A C max fA - R+ and max fA e f Maxw A - R+.

Proof max f A exists since f A is compact. There is u e A such that f u = max f A. By the Proposition 3.14 of [1], we have u e MaxwA. Thus, max fA e f (MaxwA) c f (MaxwA) -R+. Furthermore, for any t e f MaxwA, there exists u e MaxwA C A such that t = f u < max f A. Thus, f MaxwA C max fA - R+. □

By using a similar argument as in Proposition 1.2, we can deduce the following conclusion.

Proposition 1.3 Let A be a nonempty compact subset of Z, for any f e C*, we have £ Minw A c min £A + R+ and min £A c £ Minw A + R+.

The following proposition can be derived from Definition 1.1 and Proposition 1.1, so we omit the proof.

Proposition 1.4 Suppose that UxeXF(x) is compact. For any given k e int C, v e Z and a Gerstewitz function £kv : Z ^ R. Then, for any d e MinwUxeXF(x), we have £kvd e minUxeX £kvF(x) + R+. Similarly, for any d e MaxwUxeXF(x), we have £kvd e maxUxeX fkvF(x) - R+.

We note that, if X is nonempty compact set and F: X ^ Z is upper semi-continuous with nonempty compact values, then Proposition 1.4 is also valid.

Proposition 1.5 Ifx ^ G(x, y) is above-naturally C-quasi-convex on Xfor each y e Y, and a Gerstewitz function £kv: Z ^ R, then x ^ £kvG(x, y) is above-naturally R+-quasi-convex on Xfor each y e Y.

In the proof of Proposition 1.5, we need to use the monotonicity and positive homogeneous property of fkv, and a similar technique of Proposition 3.13 [1], we leave the readers to prove it.

2 Scalar bilevel minimax theorems

We first establish the following scalar bilevel minimax theorem.

Theorem 2.1 LetX, Y be two nonempty compact convex subsets of real Hausdorff topological vector spaces, respectively. The set-valued mappings F, G: X x Y ^ R with F(x, y) c G(x, y) such that the sets\^JyeYF (x, y), UxeX G(x, y) and G(x, y) are compact for all (x, y) e X x Y, and they satisfy the following conditions:

(i) x ^ F(x, y) is lower semi-continuous on X for each y e Y and y ^ F(x, y) is above-R+ -concave on Y for each x e X;

(ii) x ^ G(x,y) is above-naturally R+-quasi-convex for each y e Y, and (x,y) ^ G(x,y) is lower semi-continuous on X x Y; and

(iii) for each w e Y, there is an xw e X such that

max G(xw, w) < max ^J min ^J G(x,y).

yeY xeX

Then the relation (s - B) holds.

Proof For each y e Y, the compactness of UxeXG(x, y) implies the existence of minUxeXG(x,y). By the lower semi-continuity of G and Lemma 3.1 [9], the mapping y ^ UxeX G(x,y) is lower semi-continuous with nonempty compact values. By Lemma 3.2 [9], the mapping y ^ min UxeX G(x, y) is upper semi-continuous function on Y. Since Y is nonempty and compact, the set UyeY min UxeX G(x,y) is nonempty and compact. This

implies that the maximal points of UyeY min UxeX G(x, y) exist. Another similar argument to explain the left-hand side of (s - B) exists. Therefore, both sides of the relation (s - B) make sense.

For any given t e R with t > maxUyeY minUxeX G(x,y). Define two sets A, B C X x Y by

A = {(x, y) e X x Y: Vf e F(x, y),f < t},

B = {(x,y) e X x Y: Vg e G(x,y),g < t}. Since F(x, y) C G(x, y) for all (x, y) e X x Y,we have

0= B C A.

The nonempty property of B can be deduced from the choice of t and (iii).

Choose any y1,y2 e Y \ A(x) = (y e Y: 3f e F(x,y),f > tj. There existf1 e F(x,y1) with f1 > t andf2 e F(x,y2) withf2 > t. Then, for any X e [0,1], t e XF(x,y1) + (1 - X)F(x,y2) - R+. By the above-R+-concavity of F, we see that there isf e F(x, Xy1 + (1-X)y2) suchthatf > t. Thus, Xy1 + (1 - X)y2 e Y \ A(x), and hence Y \ A(x) is convex for each x e X. Similarly, by the above-naturally R+-convexity of G, the set (x e X: (x,y) e Bj is convex for each y e Y. Furthermore, by the lower semi-continuity of G, we know that the set B is closed.

Since all conditions of Lemma 1.1 hold, by Lemma 1.1, there exists a point xO e X such that (xOj x Y C A, that is, there exists a point xO e X such that

Vf e F(xo,y), f < t,

for all y e Y. Thus, we know that max\^)yeYF(x0,y) < t and the relation (s - B) is valid.

We see that Theorem 2.1 includes the case G = F as a special case. We state the following.

Corollary 2.1 Let X, Y be two nonempty compact convex subsets of real Hausdorff topological vector spaces, respectively. The set-valued mapping F: X x Y ^ R such that the sets \^jyeYF (x, y), U xeXF (x, y) andF (x, y) are compact for all (x, y) e X x Y, and they satisfy the following conditions:

(i) y ^ F(x, y) is above-R+ -concave on Y for each x e X;

(ii) x ^ F(x,y) is above-naturally R+-quasi-convex for each y e Y, and (x,y) ^ F(x,y) is lower semi-continuous on X x Y; and

(iii) for each w e Y, there is an xw e X such that

maxF(xw, w) < max ^J min ^J F(x,y).

yeY xeX

Then we have the relation (s - B) with G = F holds.

If, in additional, the mapping (x, y) ^ F(x, y) is upper semi-continuous with nonempty compact values on X x Y in Corollary 2.1, then we can easy see that the both sets \^)yeYF(x,y) and UxeXF(x,y) are compact. Hence we can deduce the following result due to Li etal. ([3, Proposition 2.1]).

Corollary 2.2 Let X, Y be two nonempty compact convex subsets of real Hausdorff topological vector spaces, respectively. The set-valued mapping F: X x Y ^ R is continuous with nonempty compact values and satisfies the following conditions:

(i) y ^ F(x, y) is above-R+ -concave on Y for each x e X;

(ii) x ^ F(x,y) is above-naturally R+-quasi-convex for each y e Y; and

(iii) for each y e Y, there is an xy e X such that

maxF(xy,y) < max ^J min ^J F(x,y).

yeY xeX

Then we have the relation (s - B) with G = F holds.

Throughout the rest of this paper, we assume that X, Y are two nonempty compact convex subsets of real Hausdorff topological vector spaces, respectively, and Z is a complete locally convex Hausdorff topological vector space.

3 The bilevel minimax theorems

In this section, we will present three versions of bilevel minimax theorems. As the following result illustrates, the relation (B1) is true.

Theorem 3.1 Suppose that the set-valued mappings F, G : X x Y ^ Z with F(x,y) c G(x, y) for all (x, y) e X x Y, and they satisfy the following conditions:

(i) the mapping (x, y) ^ F(x, y) is upper semi-continuous with nonempty compact values, the mapping x ^ F(x, y) is lower semi-continuousfor each y e Y, and y ^ F(x,y) is above-C-concave on Yfor each x e X;

(ii) x ^ G(x, y) is above-naturally C-quasi-convex for each y e Y, (x, y) ^ G(x, y) is continuous with nonempty compact values on X x Y;

(iii) for each w e Y andfor each f e C*, there is an xw e X such that

max f G(xw, w) < max ^J min ^J f G(x, y);

yeY xeX

(iv) for each w e Y,

Max ^J Minw ^J G(x, y) c Minw ^J G(x, w) + C.

yeY xeX xeX

Then the relation (B1) is valid.

Proof Let A(x) := Maxw \^jyeYF(x,y) for allx e X. From Lemma 2.4 [1] and Proposition 3.5 [1], the mapping x ^ A (x) is upper semi-continuous with nonempty compact values on X. Hence UxeX A(x) is compact, and so is co(UxeX A(x)). Then co(UxeX A(x)) + C is closed

convex set with nonempty interior. Suppose that v é co(UxéX A(x)) + C. By separation theorem, there is a k é R, e > 0 and a nonzero continuous linear functional f : Z ^ R such that

f (v) < k - e < k < f (u + c) (1)

for all u é co(UxéX A(x)) and céC. From this we can see that f éC* and f (v)<f (u)

for all u é co(UxéX A(x)). By Proposition 3.14 of [1], for any x éX, there is a y* é Y and f (x, yX) é F (x, yX) with/(x, y*) é A(x) such that

ff(x, y*) = max U f F (x, y).

Let us choose c = 0 and u = f (x,yx) in equation (1), we have f (v) < f / (x, y*)) = maxU f F (x, y)

for all x éX. Therefore, f (v) < min ^J max

Uf F (x, y).

x X y Y

From conditions (i)-(iii), applying Proposition 3.9 and Proposition 3.13 in [1], all conditions of Theorem 2.1 hold. Hence we have

f (v) < max ^J min

Uf G(x, y).

y Y x X

Since Y is compact, there is a y1 é Y such that f (v) <minU f G(x,/).

v é U G(x,/) + C,

and hence

v é Min„ U G(x,y') + C. (2)

By (iv) and (2), we have

v é Max U Minw U G(x,y).

y Y x X

Hence, for every v e Max UyeY Minw [JxeX G(x,y), we have v e coi y A(xM + C.

That is, the relation (Bi) is valid. □

Remark 3.1 We note that Theorem 3.1 includes the case G = F as a special case, and it almost can be compared with Theorem 3.1 [2]. Neither F nor G is able to apply the theorems in [2, 3] to deduce the minimax properties since F is not continuous and G does not satisfy the conditions (iv)-(v) of Theorem 3.1 [2], (Hi)-(H2) of Theorem 3.1 [3] or (H3)-(H4) of Theorem 3.2 [3].

We note that the relation (Bi) does not hold for any two mappings satisfy the condition F(x, y) c G(x, y) for all (x, y) e X x Y, even though both of F and G are continuous set-valued mappings. For example, let F (x, y) = {x} x [i - ^/i -y2,i + ^/i -y2] and G(x, y) = {x} x [-i, i + -y2] for all x, y e [-1, i] = X = Y. Hence we propose the following example to illustrate the validity of Theorem 3.i.

Example 3.1 Let X = [0, i], Y = [-i, 0], Z = R2 and C = C* = R+. Define H : X ^ X by

H<y)=jr y

[{0}, y =-i. Define F, G : X x Y ^ Z by F (x, y) = {x2} x H (y),

G(x,y)= [0,x2] x [y,0]

for all (x, y) e X x Y. We can see that F (x, y) c G(x, y) for all (x, y) e X x Y, and conditions (i)-(ii) of Theorem 3.i hold. We now claim that the condition (iii) of Theorem 3.i is valid. Indeed, for each £ = (fi, £2) e C*, since

max £ G(x, y) = max{ fis + Ç2t : 0 < s < x2, y < t < 0} = £ix2

for all (x, y) e X x Y, and min y £ G(x, y) = £2y

xe[0,i]

for all y e Y,we have

max y min y £ G(x, y) = 0.

ye[-i,0] xe[0,i]

For each w e Y,we can choose xw = 0 such that the condition (iii) is valid. The reason so that the condition (iv) is valid can be explained as follows: for each w e Y,we see that

Minw U G(x, w) = ({0} x [w,0]) U ([0,1] x (wj),

Max U Minw U G(x, y) = {(1,0)}.

y eY x eX

This implies that

Max y Minw y G(x, y) c Minw y G(x, w) + C,

y Y x X x X

and so the condition (iv) holds. Finally, from the observation of

Min(coUMaxwUF(x, y)) = {(0,-1)},

x X y Y

the relation (B1) is valid.

Corollary 3.1 Suppose that the set-valued mapping F: X x Y ^ Z such that the following conditions are satisfied:

(i) the mapping (x, y) ^ F(x, y) is continuous with nonempty compact values, and y ^ F(x, y) is above-C-concave on Y for each xeX;

(ii) x ^ F (x, y) is above-naturally C-quasi-convex for each y eY;

(iii) for each we Y and for each % eC*, there is an xweX such that

max %F(xw, w) < max U min U %F(x,y);

y Y x X

(iv) for each w Y,

Max U Min^ y F(x, y) c Min^ y F(x, w) + C.

y Y x X x X

Then the relation (B1) with G = F is valid.

In the following result, we apply the Gerstewitz function %kv: Z ^ R to introduce the second version of bilevel minimax theorems, where k e int C and v e Z.

Theorem 3.2 Suppose that the set-valued mappings F, G: X x Y ^ Z such that F(x, y) c G(x, y) for all (x, y) eX x Y, and the following conditions are satisfied:

(i) the mapping (x, y) ^ F(x, y) is upper semi-continuous with nonempty compact values, the mapping x ^ F (x, y) is lower semi-continuous for all ye Y;

(ii) x ^ G(x, y) is above-naturally C-quasi-convex on Xfor each y eY, (x, y) ^ G(x, y) is continuous with nonempty compact values on X x Y;

(iii) given any Gerstewitzfunction %kv with v e UxeXMaxw \^jyeYF(x,y) + C satisfies the following conditions:

(iiia) y ^ %kvF(x, y) is above-R+ -concave for all xe X; and

(iiib) for each we Y, there is an xweX such that

max%kvG(xw, w) < max ^J min ^J %ivG(x,y); and

y Y x X

(iv) for each w Y,

Max U Minw U G(x, y) c Minw U G(x, w) + C.

y Y x X x X

Then the relation (B2) is valid.

Proof Let A(x) be defined in the same way as in Theorem 3.1 for all x e X. Using the same process in the proof of Theorem 3.1, we know that the set UxeX A(x) is nonempty and compact. For any v e UxeX A(x) + C, there is a Gerstewitz function %kv : Z ^ R with some k e int C such that

&v(«) > 0 (3)

for all u e (Jx X A(x). Then, for each x e X, there is y*xe Y and f (x, y*) e F(x, y£) with f (x, yX) e Max„ UygyF(x, y) such that

Çkvf(x, yx)) = ma^y %kvF (x, y).

Choosing u = f (x,yx) in equation (3), we have max^) ÇivF (x, y) > 0

for all x X. Therefore,

min U max ^J ÇkvF(x, y) > 0.

x X y Y

By Proposition 1.5 and combining conditions (i)-(iii), we know that all conditions of Theorem 2.1 hold, and by relation (s - B) we have

max U min ^J ^kvG(x,y) > 0. y Y x X

Since Y is compact, there is a y' e Y such that

min U&vG(x,/) >0.

v G(x, y') + C,

and hence

v é Min„ U G(x, y') + C. (4)

If v é Max Uyéy Minw [jxéX G(x,y), then, by (iv), we have v é Minw y G(x,/) + C,

which contradicts (4). Therefore, we can deduce the relation (B2) is valid. □

The following example illustrates the validity of Theorem 3.2.

Example 3.2 Let X = Y = [0,1], C = R+, Z = R2 and F(x, y) = {x} x{1-s(y -l)2:5 é [0, x]}, G(x, y) = {x} x{1 - s(y -1)2:5 é [0,1]} for all (x, y) é X x Y. Then F (x, y) c G(x, y) for all (x, y) é X x Y,

U Max„ U F (x, y)= {(s, t): 0 < s < 1,1- s < t < 1},

xéX yéY

Min y Max^y F (x, y) = {(s, t): 0 < s < 1, s + t = 1}

xéX yéY

Max y Minw U G(x, y) = {(1,1)}.

yéY xéX

We can easily see that the set-valued mappings F and G satisfy all of the continuities in the conditions (i) and (ii) of Theorem 3.2. Let T = {g]_(x,y),g2(x,y)}, where g1(x,y) = -x and g2(x,y) = -y for all (x,y) é X x Y. Let k = (1,1) é int C and choose v = (2,-1) é Ux X Max„ Uy éY F (x, y) + C. By Corollary 2.4 [10], we have

£kv(u) = maxJgiU) -g;(v)/gi(k)} = max{u1 - 2, U2 + 1}

for all u = (u1, u2) é Z. Then £kv(u) > 0 for all u é |Jx ^ A(x), and

£kvF(x,y) = {2 - s(y - 1)2 : 0 < s < x}

£kvG(x,y)= {2 - s(y -1)2: 0 < s < 1} for all (x, y) éX x Y.

We claim that the mapping y ^ Ç^F (x, y) is above-R+-concave for each xeX. Indeed, for eachf e Ç^F(x,y1) andf2 e Ç^F(x,y2), there exist s1,s2 e [0,x] such that

fi = 2 -si(yi -1)2, f2 = 2-S2(y2-1)2. Then, for each X e [0,1],

Xfi + (1 - X)f2 = 2 - Xs1(y1 - 1)2 -(1 - X)s2(y2 -1)2 = 2- (s1X(y1-1)2+ s2(1- X)(y2-1)2) < 2 - s3(Xy1 + (1 - X)y2 -1)2.

The last inequality holds by the facts that the mapping y ^ (y -1)2isa real-valued convex function and we take s3 = min{s1,s2}. Hence Xf1 + (1 - X)f2 e Ç^F(x, Xy1 + (1 - X)y2) - C and the mapping y ^ ÇkvF (x, y) is above-R+ -concave for each xeX. The above-naturally C -quasi-convexity for the mapping x ^ G(x,y), for each y e Y, can be deduced by a simple calculation, so we leave the proof to the readers.

Furthermore, the condition (iiib) holds since for each w eY and any xw e X, we have ÇkvG(xw, w) = {2 - s(w -1)2: 0 < s < 1}, and hence maxÇkvG(xw, w) = 2. On the other hand, maxUyeY min UxeX ÇkvG(x,y) = maxUy eY min Use[0,1]{2 - s(y - 1)2} = 2. Thus, the condition (iiib) is valid.

Since Minw Ux eX G(x, w) = ({0}x [1-(w -1)2,1]) U ([0,1] x{1-(w -1)2}) for each we Y, we have

Max U Minw UG(x, y)

y Y x X

= {(U)}

C ({0}x [1-(w-1)2,1]) u ([0,1] x {1-(w -1)2}) + C = Minw U G(x, w) + C

for each we Y. This tells us that condition (iv) of Theorem 3.2 holds. Therefore, all conditions of Theorem 3.2 hold, and the relation (B2) is valid since

Max U Minw UG(x, y)

y Y x X

= {(U)}

C {(s, t) : 0 < s < 1,1- s < t < ^ + C

= Mi^ Maxw U F (x, y) + C.

x X y Y

The third version of the bilevel minimax theorems is as follows. We remove the condition (iv) in Theorem 3.2 to deduce the relation (B3).

Theorem 3.3 Given any Gerstewitzfunction with v e Min y Maxw y F(x,y).

xeX yeY

Under the framework of Theorem 3.2 except the condition (iv). Then the relation (B3) is valid.

Proof For each x e X, let A(x) be defined the same as in Theorem 3.1. For any v e

MinUxeX A(xl

(ljA(x) \{v})n (v - C) = 0.

Then there is a Gerstewitz function £kv: Z ^ R with some k e int C such that

Hkv(u) > 0

fkv(v) = 0

for all u e UxeX A(x) \ {v}. Since £kv is continuous, by the compactness of (JyeYF(x,y), for each xeX, there exist y1 e Y andf e F (x, y1) such that

Zkvfi) = max y £kvF (x, y).

By Proposition 3.14 [1], f1 e Maxw (JyeYF(x,y). Thus, for each xe X, we have maxy £kvF(x, y) > 0,

min y max y £kvF (x, y) > 0.

x X y Y

From the conditions (i)-(iii) and according to similar arguments in Theorem 3.2, we know that all conditions of Theorem 2.1 hold for the mappings £kvF and £kvG. Hence, by Theorem 2.1, we have

max y min y £kvG(x,y) > 0.

y Y x X

Since X and Y are compact, there are x0 e X, y0 eY and g0 e G(x0,y0) such that

fkv(g0>) = min y £kvG(x,y0) > 0.

Applying Proposition 3.14 in [1], we have g0 e Minw (Jx eX G(x, y0). If go = v, we have v e g0 + (C \ {0}). If g0 = v, we have Çkvg) > 0, and hence g0 e v - C .Therefore, ve g0 + (C \{0}).

Thus, in any case, we have v e g0 + Z\ (C \ {0}). This implies that the relation (B3) is valid.

We illustrate Theorem 3.3 by the following example.

Example 3.3 Let X, Y, F, G, C, Z, g1, g2, r be given the same as in Example 3.2. Then F(x, y) C G(x, y) for all (x, y) e X x Y and

Min U Maxw U F (x, y) = {(t,1-t):te [0,1]}.

x X y Y

Let k = (1,1) e int C and choose v = (1,0) e MinUxeXMaxwUy eYF(x,y). By Corollary 2.4 [10], we have

Çkv(u) = ma«!«(«) -gi(v)/gi(k)} = max{u -1, U2}

for all u = (u1, u2) e Z. Then

Çkv(u) > 0

for all u e \JxeX Maxw Uy eYF (x, y) \ {v}, and ÇkvF(x,y) = {1 - s(y -1)2 : 0 < s < x}

ÇkvG(x,y)= {1 - s(y -1)2: 0 < s < 1} for all (x, y) X x Y.

By a similar discussion in Example 3.2, we know that the mapping y ^ ÇkvF(x, y) is above-R+-concave for each x eX, the mapping x ^ G(x, y) is above-naturally C-quasi-convex for each yeY and the condition (iiib) is valid. Therefore, all conditions of Theorem 3.3 hold, and the relation (B3) is valid since

Min U Maxw U F (x, y)

x X y Y

= {(t,1-t):te [0,1]} C {(1,1)} + Z\ (C\{0})

= Max U Minw U G(x,y) + Z \ (C \ {0}).

y Y x X

Remark 3.2 We note that Theorems 3.2-3.3 include the case G = F as a special case.

Competing interests

The author declares that they have no competing interests. Acknowledgements

This work was supported by 'Department of OccupationalSafety and Health, College of Public Health, China Medical University, Taiwan' that are gratefully acknowledged. The author would like to thank the editor and the reviewers for their valuable comments and suggestions to improve this paper.

Received: 17 January 2014 Accepted: 23 April 2014 Published: 12 May 2014

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10.1186/1029-242X-2014-182

Cite this article as: Lin: Bilevel minimax theorems for non-continuous set-valued mappings. Journal of Inequalities and Applications 2014, 2014:182

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