Multilayer approach to minimax theorems
- Yen-Cherng Lin^{1}Email author and
- Hang-Chin Lai^{2}
https://doi.org/10.1186/s13663-016-0580-x
© Lin and Lai 2016
Received: 31 December 2015
Accepted: 25 August 2016
Published: 31 August 2016
Abstract
In this research, a multihierarchical methodology works by taking a whole intersection property to establish a fixed point theorem. Using such a whole intersection property to let multihierarchical procedures of set-valued mappings can work on a Hausdorff topological vector space. Some examples are proposed in order to illustrate our theory.
Keywords
minimax theorems fixed points whole intersectionMSC
90C47 58C061 Introduction and preliminaries
In 1961, on the basis of the KKM theorem, Fan established the celebrated minimax theorem [1, 2] by employing a whole intersection theorem. Since then, fruitful extensions for Fan’s minimax theorem were established [3–7]. It is important in mathematical economics and game theory; in the meantime, it has been very useful in many applications in convex and nonlinear analysis. One of important extensional directions is the multihierarchical approach. A large number of papers [3, 4, 8–12] have developed along this direction and applied the concept of multihierarchical approach to either the whole intersection property or to minimax theorems for real-valued functions. In the meanwhile, Ha [3] recently established a minimax theorem ([3], Theorem 3.1), which uses the following whole intersection theorem.
Theorem A
- (a)
\(F(x)\) and \(H(x)\) are open in Y for each \(x \in X\), and \(F^{-1}(y)\) and \(H^{-1}(y)\) are convex for each \(y \in Y\); and
- (b)
\(G^{-1}(y)\) is open in X for each \(y \in Y\), and \(Y \setminus G(x)\) is convex for each \(x \in X\).
Theorem A can be deduced by using the powerful method of barycentric subdivision. By employing the same direction of Ha, we can derive the following refinement result.
Theorem B
- (a)
\(A(x)\) and \(E(x)\) are open in Y for each \(x \in X\), and \(B^{-1}(y)\) and \(F^{-1}(y)\) are convex for each \(y \in Y\); and
- (b)
\(D^{-1}(y)\) is open in X for each \(y \in Y\), and \(Y \setminus C(x)\) is convex for each \(x \in X\).
As we mentioned above, it inspires and encourages us to apply the multihierarchical approach to minimax theorems for scalar set-valued mappings in Section 3 and for set-valued mappings in Section 4. In this paper, we will continue in this direction and create a generalized aspect.
2 Fixed point theorems and whole intersection theorems
The following theorem is a generalized form of Lemma 2.2 in [3].
Theorem 1
- (i)
\(P^{-1}(y)\) is nonempty for each \(y \in Y\), \(Y \setminus Q(x)\) is convex for each \(x \in X\), and \(R^{-1}(y)\) is closed for each \(y \in Y\); and
- (ii)
\(S(x)\) is closed for each \(x \in X\), and \(T^{-1}(y)\) is convex for each \(y \in Y\).
Proof
By the same directional process of Lemma 2.2 in [3], we can deduce the theorem. So, we leave the proof to the readers. □
As a consequence of Theorem 1, we have the following fixed point theorem.
Corollary 1
Under the framework of Theorem 1, in addition, if \(Y=X\) is an n-simplex and f is the identity mapping on X, then T has a fixed point in X.
The following corollary is a variant form of Theorem 1.
Corollary 2
- (i)
\(P(x)\) is nonempty for each \(x \in Z\), \(Z \setminus Q^{-1}(y)\) is convex for each \(y \in Y\), and \(R(x)\) is closed for each \(x \in Z\); and
- (ii)
\(S^{-1}(y)\) is closed for each \(y \in Y\), and \(T(x)\) is convex for each \(x \in Z\).
Now, we can prove Theorem B.
Proof of Theorem B
The following corollary is a variant form of Theorem B.
Corollary 3
- (i)
\(A^{-1}(y)\) is open in X for each \(y \in Y\), \(B(x)\) is convex in Y for each \(x \in X\), and \(X \setminus C^{-1}(y)\) is convex for each \(y \in Y\);
- (ii)
\(D(x)\) is open in Y for each \(x \in X\), \(E^{-1}(y)\) is open in X for each \(y \in Y\), and \(F(x)\) is convex in Y for each \(x \in X\); and
- (iii)
\(A(x) \subset B(x) \subset C(x) \subset D(x) \subset E(x) \subset F(x) \subset G(x)\) for each \(x \in X\).
The following two corollaries can be derived from Corollary 3.
Corollary 4
Under the framework of Corollary 3, in addition, if \(A=B\), \(C=D\), and \(E=F=G\), then Corollary 3 is reduced to Theorem A.
Corollary 5
3 Multihierarchical structures for scalar set-valued mappings
In this section, we present minimax theorems for scalar set-valued mappings under multihierarchical structures. We first recall some definitions. Let V and W be two nonempty sets equipped with some suitable topologies. Then we say that \(\Phi:V\rightrightarrows W\) is upper semicontinuous on V iff for each \(v_{0} \in V\) and for every open set W̃ containing \(\Phi(v_{0})\), there exists a neighborhood Ṽ of \(v_{0}\) such that \(\Phi(\tilde{V}) \subset\tilde{W}\). We say that Φ is lower semicontinuous on V iff for each \(v_{0} \in V\) and for every open set W̃ with \(\Phi(v_{0}) \cap\tilde{W}\neq \emptyset\), there exists a neighborhood Ṽ of \(v_{0}\) such that \(\Phi(v) \cap\tilde{W}\neq\emptyset\) for all \(v \in \tilde{V}\). If Φ is both lower semicontinuous and upper semicontinuous on V, then we say that Φ is continuous on V. If W is a Hausdorff topological vector space, C is a closed convex pointed cone with \(\operatorname{int} C \neq\emptyset\) in W, and V is a convex subset of a vector space, then we say that \(\Phi:V \rightrightarrows W\) is convex [13] (concave [13], respectively) on V iff for any \(v_{1}, v_{2} \in V\) and any \(\lambda\in[0,1]\), \(\Phi(\lambda v_{1} + (1- \lambda) v_{2}) \subset\lambda\Phi( v_{1}) +(1- \lambda)\Phi(v_{2}) -C\) (\(\lambda\Phi(v_{1}) +(1- \lambda)\Phi(v_{2}) \subset\Phi(\lambda v_{1} + (1- \lambda) v_{2}) - C\), respectively). We say that \(\Phi:V \rightrightarrows Z\) is quasi-convex (quasi-concave, respectively) [13] on V iff the set \(\{v \in V: \Phi(v) \subset z-C\}\) (\(\{v \in V: \Phi(v) \subset z+C\}\), respectively) is convex for all \(z \in Z\). Whenever \(W=\mathbb{R}\), \(C=\mathbb{R_{+}}\), and Φ becomes a single-valued function, then the above concepts of mappings coincide with the classical ones. We propose a proposition and several examples to illustrate some relations between those mappings.
Proposition 1
For a set-valued mapping, the convexity implies the quasi-convexity (Proposition 3.9 in [13]). However, the concavity does not imply the quasi-concavity.
Proof
Example 1
We would like to construct three set-valued mappings from \(X \times Y\) to \(\mathbb{R}\) that satisfy the following conditions: The first mapping is lower semicontinuous in the first variable but neither upper semicontinuous in the second variable nor quasi-convex in the first variable. The second mapping is quasi-convex in the first variable but not upper semicontinuous in the second variable. The third mapping is upper semicontinuous in the second variable but not lower semicontinuous in the first variable.
Sol.
Example 2
We would like to construct two set-valued mappings from \(X \times Y\) to \(\mathbb{R}\) that satisfy following conditions: One is a mapping that is upper semicontinuous in the first variable but not concave in the second variable. The other mapping is concave in the second variable but not upper semicontinuous in the first variable.
Sol.
Example 3
We would like to construct two set-valued mappings from \(X \times Y\) to \(\mathbb{R}\) that satisfy following conditions: One is a mapping that is quasi-convex in the first variable but is not continuous in second variable. The other mapping is continuous in the second variable, and its union in the first variable is compact but not quasi-convex in the first variable.
Sol.
Let \(X=Y=[0,1]\), and let \(U(x,y):= R(x,y)+22\) for all \((x,y) \in X \times Y\), which means that we lift R up to 22, and let \(V(x,y):=[28,28+4(x-x^{2})y^{2}]\), where R is the same as in Example 1. Hence, U is quasi-convex in the first variable, upper semicontinuous but not continuous in the second variable, and V is continuous in the second variable, and its union \(\bigcup_{x \in X}V(x,y)=[28,28+y^{2}]\) is compact but not quasi-convex in the first variable.
The following two lemmas will help us to derive the main results. These two lemmas describe some relationships of convexities between a scalar set-valued mapping and real-valued function.
Lemma 1
Suppose that X is a nonempty convex subset of a topological vector space and \(G : X \rightrightarrows\mathbb{R}\) is a set-valued mapping such that \(\max G(x)\) exists for each \(x \in X\). Then the mapping \(G : X \rightrightarrows\mathbb{R}\) is quasi-convex if and only if the mapping \(x \mapsto\max G(x)\) is a quasi-convex function.
Proof
Conversely, if the mapping \(x \mapsto\max G(x)\) is quasi-convex, we need to show that the set \(\{ x \in X: G(x) \subset c -\mathbb{R}_{+} \}\) is convex for each \(c \in\mathbb{R}\). Indeed, for any \(x_{1}, x_{2} \in\{ x \in X: G(x) \subset c -\mathbb{R}_{+} \}\) and \(\lambda\in [0,1]\), we have \(G(x_{1}) \subset c -\mathbb{R}_{+}\) and \(G(x_{1}) \subset c -\mathbb{R}_{+}\). Thus, \(\max G(x_{1}) \leq c\) and \(\max G(x_{1}) \leq c\). Since the mapping \(x \mapsto\max G(x)\) is quasi-convex, \(\max G(\lambda x_{1}+(1-\lambda)x_{2}) \leq c\), and \(G(\lambda x_{1}+(1-\lambda)x_{2}) \subset\max G(\lambda x_{1}+(1-\lambda)x_{2}) -\mathbb{R}_{+}\). Hence, \(\lambda x_{1}+(1-\lambda)x_{2} \in\{ x \in X: G(x) \subset c -\mathbb{R}_{+} \}\). Therefore, \(G : X \mapsto \mathbb{R}\) is quasi-convex. □
Lemma 2
Suppose that X is a nonempty convex subset of a topological vector space and \(G : X \rightrightarrows\mathbb{R}\) is a set-valued mapping such that \(\max G(x)\) exists for each \(x \in X\). Then the mapping \(G : X \rightrightarrows\mathbb{R}\) is convex (concave, resp.) if and only if the mapping \(x \mapsto\max G(x)\) is a convex (concave, resp.) function.
Proof
We can deduce the conclusion directly from the definition. □
With the help of Lemmas 1 and 2, we derive the following scalar hierarchical minimax theorem.
Theorem 2
- (i)
\(y \mapsto V(x,y)\) is continuous on Y for each \(x \in X\), \(x \mapsto U(x,y)\) is quasi-convex on X for each \(y \in Y\), \(y \mapsto T(x,y)\) is concave for each \(x \in X\), and \(x \mapsto S(x,y)\) is upper semicontinuous on X for each \(y \in Y\);
- (ii)
\(y \mapsto R(x,y)\) is upper semicontinuous on Y for each \(x \in X\), and \(x \mapsto Q(x,y)\) is quasi-convex on X for each \(y \in Y\);
- (iii)
\(x \mapsto P(x,y)\) is lower semicontinuous on X for each \(y \in Y\), and \(y \mapsto P(x,y)\) is upper semicontinuous on Y for each \(x \in X\); and
- (iv)for each \(w \in Y\), there is \(x_{w} \in X\) such that$$\max V(x_{w},w) \leq\max\bigcup_{y \in Y} \min\bigcup_{x \in X} V(x,y). $$
Proof
From the upper semicontinuity of the mappings R, S, V and Lemma 2.5 in [13], the sets \(A(x)\), \(D^{-1}(y)\), and \(E(x)\) are open for all \(x \in X\) and \(y \in Y\). By the convexity of the mappings \(Q, T\), and U and Lemmas 1 and 2, the sets \(B^{-1}(y)\), \(F^{-1}(y)\), and \(Y \setminus C(x)\) are convex for each \(x \in X\) and for each \(y \in Y\).
If we let all mappings be equal and single-valued in Theorem 2, then Theorem 2 can be compared to [4–7]. The following example shows that Theorem 2 is valid.
Example 4
In Example 4, we cannot apply Theorem 2 in [15] to confirm whether relation (3) is true or not. The reason is that \(P \neq Q \neq R\), \(S \neq T\), and \(U \neq V\). If \(P=Q=R=S=T=U=V\), then Theorem 2 can be reduced to Corollary 2 in [15].
4 Multihierarchical structures for set-valued mappings
Theorem 3
- (i)
\((x,y) \mapsto P(x,y)\) is upper semicontinuous on \(X \times Y\), \(y \mapsto R(x,y)\) is upper semicontinuous on Y for each \(x \in X\), \(x \mapsto Q(x,y)\) is naturally quasi-convex [13], and \(x \mapsto P(x,y)\) is lower semicontinuous on X for each \(y \in Y\);
- (ii)
\(x \mapsto S(x,y)\) is upper semicontinuous on X for each \(y \in Y\), and \(y \mapsto T(x,y)\) is concave on Y for each \(x \in X\);
- (iii)
\((x,y) \mapsto V(x,y)\) is upper semicontinuous on \(X \times Y\), \(x \mapsto U(x,y)\) is naturally quasi-convex on X for each \(y \in Y\), and \(y \mapsto V(x,y)\) is continuous on Y for each \(x \in X\);
- (iv)for any \(\varphi\in C^{*}\) and \(y \in Y\), there is \(x_{y} \in X\) such that$$\max\varphi V(x_{y},y) \leq\max\bigcup_{y \in Y} \min\bigcup_{x \in X}\varphi V(x,y); $$
- (v)for each \(y \in Y\),and$$\operatorname{Max} \bigcup_{y \in Y} \operatorname{Min}_{w}\bigcup_{x \in X}V(x,y) \subset \operatorname{Min}_{w}\bigcup_{x \in X}V(x,y)+C; $$
- (vi)
for all \((x,y) \in X \times Y\), \(P(x,y) \preceq Q(x,y)\), \(Q(x,y) \preceq R(x,y)\), \(R(x,y) \preceq S(x,y)\), \(S(x,y) \preceq T(x,y)\), \(T(x,y) \preceq U(x,y)\), and \(U(x,y) \preceq V(x,y)\).
Proof
Some techniques of the proof are similar to those of Theorem 3 in [15]. For sake of completeness, we will shorten its proof as follows.
The following example is very suitable to illustrate Theorem 3.
Example 5
5 Conclusions
We construct successfully minimax theorems with multihierarchical procedures of set-valued mappings. Just like the rainbow has seven colors, the multihierarchical methodology can work by seven set-valued mappings. The multihierarchical structure can include some particular cases that appeared in the literatures.
Declarations
Acknowledgements
The first author was partially supported by grant MOST104-2115-M-039-001 of the Ministry of Science and Technology of Taiwan (Republic of China), which is gratefully acknowledged.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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