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Bilevel minimax theorems for noncontinuous setvalued mappings
Journal of Inequalities and Applications volume 2014, Article number: 182 (2014)
Abstract
We study new types for minimax theorems with a couple of setvalued 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.
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\subset Z a closed convex and pointed cone with apex at the origin and intC\ne \mathrm{\varnothing}, this means that C is a closed set with nonempty interior and satisfies \lambda C\subseteq C, \mathrm{\forall}\lambda >0; C+C\subseteq C; and C\cap (C)=\{0\}. The scalar bilevel minimax theorems stated as follows: given two setvalued mappings F,G:X\times Y\rightrightarrows \mathbb{R}, under suitable conditions the following relation holds:
Given two mappings F,G:X\times Y\rightrightarrows Z, the first version of bilevel minimax theorems stated that under suitable conditions the following relation holds:
The second version of bilevel minimax theorems stated that under suitable conditions the following relation holds:
The third version of bilevel minimax theorems stated that under suitable conditions the following relation holds:
The case G=F of (sB) and ({B}_{1})({B}_{3}) has been discussed in [1–3] for setvalued mapping and in [4] for vectorvalued mapping, respectively. Scalar minimax theorems and set minimax theorems for noncontinuous setvalued 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 noncontinuous setvalued mappings (Theorem 2.1 in Section 2, Theorems 3.13.3 in Section 3). These results might not hold for each individual noncontinuous setvalued mapping since it always lack some conditions so that the existing minimax theorems are not applicable, such as Theorems 4.14.3 [1], Theorem 2.1 [2] or Proposition 2.1 [3].
We present some fundamental concepts which will be used in the sequel.
Let A be a nonempty subset of Z. A point z\in A is called a

(a)
minimal point of A if A\cap (zC)=\{z\}; MinA denotes the set of all minimal points of A;

(b)
maximal point of A if A\cap (z+C)=\{z\}; MaxA denotes the set of all maximal points of A;

(c)
weakly minimal point of A if A\cap (zintC)=\mathrm{\varnothing}; {Min}_{w}A denotes the set of all weakly minimal points of A;

(d)
weakly maximal point of A if A\cap (z+intC)=\mathrm{\varnothing}; {Max}_{w}A denotes the set of all weakly maximal points of A.
Following [2], we denote both Max and {Max}_{w} by max (both Min and {Min}_{w} by min) in ℝ since both Max and {Max}_{w} (both Min and {Min}_{w}) are same in ℝ. We note that, for a nonempty compact set A, the both sets MaxA and MinA are nonempty. Furthermore, MinA\subset {Min}_{w}A, MaxA\subset {Max}_{w}A, A\subset MinA+C, and A\subset MaxAC.
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\subset X\times Y be a subset such that

(a)
for each y\in Y, the set \{x\in X:(x,y)\in A\} is closed in X; and

(b)
for each x\in X, the set \{y\in Y:(x,y)\notin A\} is convex or empty.
Suppose that there exist a subset B of A and a compact convex subset K of X such that B is closed in X\times Y and

(c)
for each y\in Y, the set \{x\in K:(x,y)\in B\} is nonempty and convex.
Then there exists a point {x}_{0}\in K such that \{{x}_{0}\}\times Y\subset A.
Definition 1.2 Let U, V be Hausdorff topological spaces. A setvalued map F:U\rightrightarrows V with nonempty values is said to be

(a)
lower semicontinuous at {x}_{0}\in U if for any net \{{x}_{\mu}\}\subset U such that {x}_{\mu}\to {x}_{0} and any {y}_{0}\in F({x}_{0}), there exists a net {y}_{\mu}\in F({x}_{\mu}) such that {y}_{\mu}\to {y}_{0};

(b)
upper semicontinuous at {x}_{0}\in U if for every {x}_{0}\in U and for every open set N containing F({x}_{0}), there exists a neighborhood M of {x}_{0} such that F(M)\subset N;

(c)
continuous at {x}_{0}\in U if F is upper semicontinuous as well as lower semicontinuous at {x}_{0}.
We note that T is upper semicontinuous at {x}_{0} and T({x}_{0}) is compact, then for any net \{{x}_{\nu}\}\subset U, {x}_{\nu}\to {x}_{0}, and for any net {y}_{\nu}\in T({x}_{\nu}) for each ν, there exist {y}_{0}\in T({x}_{0}) and a subnet \{{y}_{{\nu}_{\alpha}}\} such that {y}_{{\nu}_{\alpha}}\to {y}_{0}. For more details, we refer the reader to [6, 7].
Let k\in intC and v\in Z. The Gerstewitz function {\xi}_{kv}:Z\to \mathbb{R} is defined by
We present some fundamental properties of the scalarization function.
Let k\in intC and v\in Z. The Gerstewitz function {\xi}_{kv}:Z\to \mathbb{R} has the following properties:

(a)
{\xi}_{kv}(u)\le r\iff u\in v+rkC;

(b)
{\xi}_{kv}(u)<r\iff u\in v+rkintC; and

(c)
{\xi}_{kv}(\cdot ) is a continuous convex increasing and strictly increasing function.
We also need the following different kinds of coneconvexities for setvalued mappings.
Definition 1.4 [1]
Let X be a nonempty convex subset of a topological vector space. A setvalued mapping F:X\rightrightarrows Z is said to be

(a)
aboveCconvex (respectively, aboveCconcave) on X if for all {x}_{1},{x}_{2}\in X and all \lambda \in [0,1],
\begin{array}{c}F(\lambda {x}_{1}+(1\lambda ){x}_{2})\subset \lambda F({x}_{1})+(1\lambda )F({x}_{2})C\hfill \\ (\text{respectively},\phantom{\rule{0.25em}{0ex}}\lambda F({x}_{1})+(1\lambda )F({x}_{2})\subset F(\lambda {x}_{1}+(1\lambda ){x}_{2})C);\hfill \end{array} 
(b)
abovenaturally Cquasiconvex on X if for all {x}_{1},{x}_{2}\in X and all \lambda \in [0,1],
F(\lambda {x}_{1}+(1\lambda ){x}_{2})\subset co\{F({x}_{1})\cup F({x}_{2})\}C,
where coA denotes the convex hull of a set A;
Let {C}^{\star}=\{g\in {Z}^{\star}:g(c)\ge 0\text{for all}c\in C\}, where {Z}^{\star} 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 \xi \in {C}^{\star}, we have \xi {Max}_{w}A\subset max\xi A{\mathbb{R}}_{+} and max\xi A\in \xi {Max}_{w}A{\mathbb{R}}_{+}.
Proof max\xi A exists since ξA is compact. There is u\in A such that \xi u=max\xi A. By the Proposition 3.14 of [1], we have u\in {Max}_{w}A. Thus, max\xi A\in \xi ({Max}_{w}A)\subset \xi ({Max}_{w}A){\mathbb{R}}_{+}. Furthermore, for any t\in \xi {Max}_{w}A, there exists u\in {Max}_{w}A\subset A such that t=\xi u\le max\xi A. Thus, \xi {Max}_{w}A\subset max\xi A{\mathbb{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 \xi \in {C}^{\star}, we have \xi {Min}_{w}A\subset min\xi A+{\mathbb{R}}_{+} and min\xi A\subset \xi {Min}_{w}A+{\mathbb{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 {\bigcup}_{x\in X}F(x) is compact. For any given k\in intC, v\in Z and a Gerstewitz function {\xi}_{kv}:Z\to \mathbb{R}. Then, for any d\in {Min}_{w}{\bigcup}_{x\in X}F(x), we have {\xi}_{kv}d\in min{\bigcup}_{x\in X}{\xi}_{kv}F(x)+{\mathbb{R}}_{+}. Similarly, for any d\in {Max}_{w}{\bigcup}_{x\in X}F(x), we have {\xi}_{kv}d\in max{\bigcup}_{x\in X}{\xi}_{kv}F(x){\mathbb{R}}_{+}.
We note that, if X is nonempty compact set and F:X\rightrightarrows Z is upper semicontinuous with nonempty compact values, then Proposition 1.4 is also valid.
Proposition 1.5 If x\mapsto G(x,y) is abovenaturally Cquasiconvex on X for each y\in Y, and a Gerstewitz function {\xi}_{kv}:Z\to \mathbb{R}, then x\mapsto {\xi}_{kv}G(x,y) is abovenaturally {\mathbb{R}}_{+}quasiconvex on X for each y\in Y.
In the proof of Proposition 1.5, we need to use the monotonicity and positive homogeneous property of {\xi}_{kv}, 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 Let X, Y be two nonempty compact convex subsets of real Hausdorff topological vector spaces, respectively. The setvalued mappings F,G:X\times Y\rightrightarrows \mathbb{R} with F(x,y)\subset G(x,y) such that the sets {\bigcup}_{y\in Y}F(x,y), {\bigcup}_{x\in X}G(x,y) and G(x,y) are compact for all (x,y)\in X\times Y, and they satisfy the following conditions:

(i)
x\mapsto F(x,y) is lower semicontinuous on X for each y\in Y and y\mapsto F(x,y) is above{\mathbb{R}}_{+}concave on Y for each x\in X;

(ii)
x\mapsto G(x,y) is abovenaturally {\mathbb{R}}_{+}quasiconvex for each y\in Y, and (x,y)\mapsto G(x,y) is lower semicontinuous on X\times Y; and

(iii)
for each w\in Y, there is an {x}_{w}\in X such that
maxG({x}_{w},w)\le max\bigcup _{y\in Y}min\bigcup _{x\in X}G(x,y).
Then the relation (sB) holds.
Proof For each y\in Y, the compactness of {\bigcup}_{x\in X}G(x,y) implies the existence of min{\bigcup}_{x\in X}G(x,y). By the lower semicontinuity of G and Lemma 3.1 [9], the mapping y\mapsto {\bigcup}_{x\in X}G(x,y) is lower semicontinuous with nonempty compact values. By Lemma 3.2 [9], the mapping y\mapsto min{\bigcup}_{x\in X}G(x,y) is upper semicontinuous function on Y. Since Y is nonempty and compact, the set {\bigcup}_{y\in Y}min{\bigcup}_{x\in X}G(x,y) is nonempty and compact. This implies that the maximal points of {\bigcup}_{y\in Y}min{\bigcup}_{x\in X}G(x,y) exist. Another similar argument to explain the lefthand side of (sB) exists. Therefore, both sides of the relation (sB) make sense.
For any given t\in \mathbb{R} with t>max{\bigcup}_{y\in Y}min{\bigcup}_{x\in X}G(x,y). Define two sets A,B\subset X\times Y by
and
Since F(x,y)\subset G(x,y) for all (x,y)\in X\times Y, we have
The nonempty property of B can be deduced from the choice of t and (iii).
Choose any {y}_{1},{y}_{2}\in Y\setminus A(x)=\{y\in Y:\mathrm{\exists}f\in F(x,y),f>t\}. There exist {f}_{1}\in F(x,{y}_{1}) with {f}_{1}>t and {f}_{2}\in F(x,{y}_{2}) with {f}_{2}>t. Then, for any \lambda \in [0,1], t\in \lambda F(x,{y}_{1})+(1\lambda )F(x,{y}_{2}){\mathbb{R}}_{+}. By the above{\mathbb{R}}_{+}concavity of F, we see that there is {f}_{\lambda}\in F(x,\lambda {y}_{1}+(1\lambda ){y}_{2}) such that {f}_{\lambda}>t. Thus, \lambda {y}_{1}+(1\lambda ){y}_{2}\in Y\setminus A(x), and hence Y\setminus A(x) is convex for each x\in X. Similarly, by the abovenaturally {\mathbb{R}}_{+}convexity of G, the set \{x\in X:(x,y)\in B\} is convex for each y\in Y. Furthermore, by the lower semicontinuity 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 {x}_{0}\in X such that \{{x}_{0}\}\times Y\subset A, that is, there exists a point {x}_{0}\in X such that
for all y\in Y. Thus, we know that max{\bigcup}_{y\in Y}F({x}_{0},y)\le t and the relation (sB) 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 setvalued mapping F:X\times Y\rightrightarrows \mathbb{R} such that the sets {\bigcup}_{y\in Y}F(x,y), {\bigcup}_{x\in X}F(x,y) and F(x,y) are compact for all (x,y)\in X\times Y, and they satisfy the following conditions:

(i)
y\mapsto F(x,y) is above{\mathbb{R}}_{+}concave on Y for each x\in X;

(ii)
x\mapsto F(x,y) is abovenaturally {\mathbb{R}}_{+}quasiconvex for each y\in Y, and (x,y)\mapsto F(x,y) is lower semicontinuous on X\times Y; and

(iii)
for each w\in Y, there is an {x}_{w}\in X such that
maxF({x}_{w},w)\le max\bigcup _{y\in Y}min\bigcup _{x\in X}F(x,y).
Then we have the relation (sB) with G=F holds.
If, in additional, the mapping (x,y)\mapsto F(x,y) is upper semicontinuous with nonempty compact values on X\times Y in Corollary 2.1, then we can easy see that the both sets {\bigcup}_{y\in Y}F(x,y) and {\bigcup}_{x\in X}F(x,y) are compact. Hence we can deduce the following result due to Li et al. ([[3], Proposition 2.1]).
Corollary 2.2 Let X, Y be two nonempty compact convex subsets of real Hausdorff topological vector spaces, respectively. The setvalued mapping F:X\times Y\rightrightarrows \mathbb{R} is continuous with nonempty compact values and satisfies the following conditions:

(i)
y\mapsto F(x,y) is above{\mathbb{R}}_{+}concave on Y for each x\in X;

(ii)
x\mapsto F(x,y) is abovenaturally {\mathbb{R}}_{+}quasiconvex for each y\in Y; and

(iii)
for each y\in Y, there is an {x}_{y}\in X such that
maxF({x}_{y},y)\le max\bigcup _{y\in Y}min\bigcup _{x\in X}F(x,y).
Then we have the relation (sB) 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 ({B}_{1}) is true.
Theorem 3.1 Suppose that the setvalued mappings F,G:X\times Y\rightrightarrows Z with F(x,y)\subset G(x,y) for all (x,y)\in X\times Y, and they satisfy the following conditions:

(i)
the mapping (x,y)\mapsto F(x,y) is upper semicontinuous with nonempty compact values, the mapping x\mapsto F(x,y) is lower semicontinuous for each y\in Y, and y\mapsto F(x,y) is aboveCconcave on Y for each x\in X;

(ii)
x\mapsto G(x,y) is abovenaturally Cquasiconvex for each y\in Y, (x,y)\mapsto G(x,y) is continuous with nonempty compact values on X\times Y;

(iii)
for each w\in Y and for each \xi \in {C}^{\star}, there is an {x}_{w}\in X such that
max\xi G({x}_{w},w)\le max\bigcup _{y\in Y}min\bigcup _{x\in X}\xi G(x,y);
and

(iv)
for each w\in Y,
Max\bigcup _{y\in Y}{Min}_{w}\bigcup _{x\in X}G(x,y)\subset {Min}_{w}\bigcup _{x\in X}G(x,w)+C.
Then the relation ({B}_{1}) is valid.
Proof Let \mathrm{\Lambda}(x):={Max}_{w}{\bigcup}_{y\in Y}F(x,y) for all x\in X. From Lemma 2.4 [1] and Proposition 3.5 [1], the mapping x\mapsto \mathrm{\Lambda}(x) is upper semicontinuous with nonempty compact values on X. Hence {\bigcup}_{x\in X}\mathrm{\Lambda}(x) is compact, and so is co({\bigcup}_{x\in X}\mathrm{\Lambda}(x)). Then co({\bigcup}_{x\in X}\mathrm{\Lambda}(x))+C is closed convex set with nonempty interior. Suppose that v\notin co({\bigcup}_{x\in X}\mathrm{\Lambda}(x))+C. By separation theorem, there is a k\in \mathbb{R}, \u03f5>0 and a nonzero continuous linear functional \xi :Z\mapsto \mathbb{R} such that
for all u\in co({\bigcup}_{x\in X}\mathrm{\Lambda}(x)) and c\in C. From this we can see that \xi \in {C}^{\star} and
for all u\in co({\bigcup}_{x\in X}\mathrm{\Lambda}(x)). By Proposition 3.14 of [1], for any x\in X, there is a {y}_{x}^{\star}\in Y and f(x,{y}_{x}^{\star})\in F(x,{y}_{x}^{\star}) with f(x,{y}_{x}^{\star})\in \mathrm{\Lambda}(x) such that
Let us choose c=0 and u=f(x,{y}_{x}^{\star}) in equation (1), we have
for all x\in X. Therefore,
From conditions (i)(iii), applying Proposition 3.9 and Proposition 3.13 in [1], all conditions of Theorem 2.1 hold. Hence we have
Since Y is compact, there is a {y}^{\prime}\in Y such that
Thus,
and hence
By (iv) and (2), we have
Hence, for every v\in Max{\bigcup}_{y\in Y}{Min}_{w}{\bigcup}_{x\in X}G(x,y), we have
That is, the relation ({B}_{1}) 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], ({H}_{1})({H}_{2}) of Theorem 3.1 [3] or ({H}_{3})({H}_{4}) of Theorem 3.2 [3].
We note that the relation ({B}_{1}) does not hold for any two mappings satisfy the condition F(x,y)\subset G(x,y) for all (x,y)\in X\times Y, even though both of F and G are continuous setvalued mappings. For example, let F(x,y)=\{x\}\times [1\sqrt{1{y}^{2}},1+\sqrt{1{y}^{2}}] and G(x,y)=\{x\}\times [1,1+\sqrt{1{y}^{2}}] for all x,y\in [1,1]=X=Y. Hence we propose the following example to illustrate the validity of Theorem 3.1.
Example 3.1 Let X=[0,1], Y=[1,0], Z={\mathbb{R}}^{2} and C={C}^{\star}={\mathbb{R}}_{+}^{2}. Define H:X\rightrightarrows X by
Define F,G:X\times Y\mapsto Z by
and
for all (x,y)\in X\times Y. We can see that F(x,y)\subset G(x,y) for all (x,y)\in X\times Y, and conditions (i)(ii) of Theorem 3.1 hold. We now claim that the condition (iii) of Theorem 3.1 is valid. Indeed, for each \xi =({\xi}_{1},{\xi}_{2})\in {C}^{\star}, since
for all (x,y)\in X\times Y, and
for all y\in Y, we have
For each w\in Y, we can choose {x}_{w}=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\in Y, we see that
and
This implies that
and so the condition (iv) holds. Finally, from the observation of
the relation ({B}_{1}) is valid.
Corollary 3.1 Suppose that the setvalued mapping F:X\times Y\rightrightarrows Z such that the following conditions are satisfied:

(i)
the mapping (x,y)\mapsto F(x,y) is continuous with nonempty compact values, and y\mapsto F(x,y) is aboveCconcave on Y for each x\in X;

(ii)
x\mapsto F(x,y) is abovenaturally Cquasiconvex for each y\in Y;

(iii)
for each w\in Y and for each \xi \in {C}^{\star}, there is an {x}_{w}\in X such that
max\xi F({x}_{w},w)\le max\bigcup _{y\in Y}min\bigcup _{x\in X}\xi F(x,y);
and

(iv)
for each w\in Y,
Max\bigcup _{y\in Y}{Min}_{w}\bigcup _{x\in X}F(x,y)\subset {Min}_{w}\bigcup _{x\in X}F(x,w)+C.
Then the relation ({B}_{1}) with G=F is valid.
In the following result, we apply the Gerstewitz function {\xi}_{kv}:Z\mapsto \mathbb{R} to introduce the second version of bilevel minimax theorems, where k\in intC and v\in Z.
Theorem 3.2 Suppose that the setvalued mappings F,G:X\times Y\rightrightarrows Z such that F(x,y)\subset G(x,y) for all (x,y)\in X\times Y, and the following conditions are satisfied:

(i)
the mapping (x,y)\mapsto F(x,y) is upper semicontinuous with nonempty compact values, the mapping x\mapsto F(x,y) is lower semicontinuous for all y\in Y;

(ii)
x\mapsto G(x,y) is abovenaturally Cquasiconvex on X for each y\in Y, (x,y)\mapsto G(x,y) is continuous with nonempty compact values on X\times Y;

(iii)
given any Gerstewitz function {\xi}_{kv} with v\notin {\bigcup}_{x\in X}{Max}_{w}{\bigcup}_{y\in Y}F(x,y)+C satisfies the following conditions:
(iii_{a}) y\mapsto {\xi}_{kv}F(x,y) is above{\mathbb{R}}_{+}concave for all x\in X; and
(iii_{b}) for each w\in Y, there is an {x}_{w}\in X such that

(iv)
for each w\in Y,
Max\bigcup _{y\in Y}{Min}_{w}\bigcup _{x\in X}G(x,y)\subset {Min}_{w}\bigcup _{x\in X}G(x,w)+C.
Then the relation ({B}_{2}) is valid.
Proof Let \mathrm{\Lambda}(x) be defined in the same way as in Theorem 3.1 for all x\in X. Using the same process in the proof of Theorem 3.1, we know that the set {\bigcup}_{x\in X}\mathrm{\Lambda}(x) is nonempty and compact. For any v\notin {\bigcup}_{x\in X}\mathrm{\Lambda}(x)+C, there is a Gerstewitz function {\xi}_{kv}:Z\mapsto \mathbb{R} with some k\in intC such that
for all u\in {\bigcup}_{x\in X}\mathrm{\Lambda}(x). Then, for each x\in X, there is {y}_{x}^{\star}\in Y and f(x,{y}_{x}^{\star})\in F(x,{y}_{x}^{\star}) with f(x,{y}_{x}^{\star})\in {Max}_{w}{\bigcup}_{y\in Y}F(x,y) such that
Choosing u=f(x,{y}_{x}^{\star}) in equation (3), we have
for all x\in X. Therefore,
By Proposition 1.5 and combining conditions (i)(iii), we know that all conditions of Theorem 2.1 hold, and by relation (sB) we have
Since Y is compact, there is a {y}^{\prime}\in Y such that
Thus,
and hence
If v\in Max{\bigcup}_{y\in Y}{Min}_{w}{\bigcup}_{x\in X}G(x,y), then, by (iv), we have
which contradicts (4). Therefore, we can deduce the relation ({B}_{2}) is valid. □
The following example illustrates the validity of Theorem 3.2.
Example 3.2 Let X=Y=[0,1], C={\mathbb{R}}_{+}^{2}, Z={\mathbb{R}}^{2} and F(x,y)=\{x\}\times \{1s{(y1)}^{2}:s\in [0,x]\}, G(x,y)=\{x\}\times \{1s{(y1)}^{2}:s\in [0,1]\} for all (x,y)\in X\times Y. Then F(x,y)\subset G(x,y) for all (x,y)\in X\times Y,
and
We can easily see that the setvalued mappings F and G satisfy all of the continuities in the conditions (i) and (ii) of Theorem 3.2. Let \mathrm{\Gamma}=\{{g}_{1}(x,y),{g}_{2}(x,y)\}, where {g}_{1}(x,y)=x and {g}_{2}(x,y)=y for all (x,y)\in X\times Y. Let k=(1,1)\in intC and choose v=(2,1)\notin {\bigcup}_{x\in X}{Max}_{w}{\bigcup}_{y\in Y}F(x,y)+C. By Corollary 2.4 [10], we have
for all u=({u}_{1},{u}_{2})\in Z. Then {\xi}_{kv}(u)>0 for all u\in {\bigcup}_{x\in X}\mathrm{\Lambda}(x), and
and
for all (x,y)\in X\times Y.
We claim that the mapping y\mapsto {\xi}_{kv}F(x,y) is above{\mathbb{R}}_{+}concave for each x\in X. Indeed, for each {f}_{1}\in {\xi}_{kv}F(x,{y}_{1}) and {f}_{2}\in {\xi}_{kv}F(x,{y}_{2}), there exist {s}_{1},{s}_{2}\in [0,x] such that
Then, for each \lambda \in [0,1],
The last inequality holds by the facts that the mapping y\mapsto {(y1)}^{2} is a realvalued convex function and we take {s}_{3}=min\{{s}_{1},{s}_{2}\}. Hence \lambda {f}_{1}+(1\lambda ){f}_{2}\in {\xi}_{kv}F(x,\lambda {y}_{1}+(1\lambda ){y}_{2})C and the mapping y\mapsto {\xi}_{kv}F(x,y) is above{\mathbb{R}}_{+}concave for each x\in X. The abovenaturally Cquasiconvexity for the mapping x\mapsto G(x,y), for each y\in Y, can be deduced by a simple calculation, so we leave the proof to the readers.
Furthermore, the condition (iii_{b}) holds since for each w\in Y and any {x}_{w}\in X, we have {\xi}_{kv}G({x}_{w},w)=\{2s{(w1)}^{2}:0\le s\le 1\}, and hence max{\xi}_{kv}G({x}_{w},w)=2. On the other hand, max{\bigcup}_{y\in Y}min{\bigcup}_{x\in X}{\xi}_{kv}G(x,y)=max{\bigcup}_{y\in Y}min{\bigcup}_{s\in [0,1]}\{2s{(y1)}^{2}\}=2. Thus, the condition (iii_{b}) is valid.
Since {Min}_{w}{\bigcup}_{x\in X}G(x,w)=(\{0\}\times [1{(w1)}^{2},1])\cup ([0,1]\times \{1{(w1)}^{2}\}) for each w\in Y, we have
for each w\in Y. This tells us that condition (iv) of Theorem 3.2 holds.
Therefore, all conditions of Theorem 3.2 hold, and the relation ({B}_{2}) is valid since
The third version of the bilevel minimax theorems is as follows. We remove the condition (iv) in Theorem 3.2 to deduce the relation ({B}_{3}).
Theorem 3.3 Given any Gerstewitz function {\xi}_{kv} with
Under the framework of Theorem 3.2 except the condition (iv). Then the relation ({B}_{3}) is valid.
Proof For each x\in X, let \mathrm{\Lambda}(x) be defined the same as in Theorem 3.1. For any v\in Min{\bigcup}_{x\in X}\mathrm{\Lambda}(x),
Then there is a Gerstewitz function {\xi}_{kv}:Z\mapsto \mathbb{R} with some k\in intC such that
and
for all u\in {\bigcup}_{x\in X}\mathrm{\Lambda}(x)\setminus \{v\}. Since {\xi}_{kv} is continuous, by the compactness of {\bigcup}_{y\in Y}F(x,y), for each x\in X, there exist {y}_{1}\in Y and {f}_{1}\in F(x,{y}_{1}) such that
By Proposition 3.14 [1], {f}_{1}\in {Max}_{w}{\bigcup}_{y\in Y}F(x,y). Thus, for each x\in X, we have
or
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 {\xi}_{kv}F and {\xi}_{kv}G. Hence, by Theorem 2.1, we have
Since X and Y are compact, there are {x}_{0}\in X, {y}_{0}\in Y and {g}_{0}\in G({x}_{0},{y}_{0}) such that
Applying Proposition 3.14 in [1], we have {g}_{0}\in {Min}_{w}{\bigcup}_{x\in X}G(x,{y}_{0}). If {g}_{0}=v, we have v\notin {g}_{0}+(C\setminus \{0\}). If {g}_{0}\ne v, we have {\xi}_{kv}({g}_{0})>0, and hence {g}_{0}\notin vC. Therefore, v\notin {g}_{0}+(C\setminus \{0\}). Thus, in any case, we have v\in {g}_{0}+Z\setminus (C\setminus \{0\}). This implies that the relation ({B}_{3}) is valid. □
We illustrate Theorem 3.3 by the following example.
Example 3.3 Let X, Y, F, G, C, Z, {g}_{1}, {g}_{2}, Γ be given the same as in Example 3.2. Then F(x,y)\subset G(x,y) for all (x,y)\in X\times Y and
Let k=(1,1)\in intC and choose v=(1,0)\in Min{\bigcup}_{x\in X}{Max}_{w}{\bigcup}_{y\in Y}F(x,y). By Corollary 2.4 [10], we have
for all u=({u}_{1},{u}_{2})\in Z. Then
for all u\in {\bigcup}_{x\in X}{Max}_{w}{\bigcup}_{y\in Y}F(x,y)\setminus \{v\}, and
and
for all (x,y)\in X\times Y.
By a similar discussion in Example 3.2, we know that the mapping y\mapsto {\xi}_{kv}F(x,y) is above{\mathbb{R}}_{+}concave for each x\in X, the mapping x\mapsto G(x,y) is abovenaturally Cquasiconvex for each y\in Y and the condition (iii_{b}) is valid.
Therefore, all conditions of Theorem 3.3 hold, and the relation ({B}_{3}) is valid since
Remark 3.2 We note that Theorems 3.23.3 include the case G=F as a special case.
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Acknowledgements
This work was supported by ‘Department of Occupational Safety 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.
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Lin, YC. Bilevel minimax theorems for noncontinuous setvalued mappings. J Inequal Appl 2014, 182 (2014). https://doi.org/10.1186/1029242X2014182
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DOI: https://doi.org/10.1186/1029242X2014182