Generalized hierarchical minimax theorems for set-valued mappings
- Haijun Wang^{1}Email author
https://doi.org/10.1186/s13660-016-1050-z
© Wang 2016
Received: 17 November 2015
Accepted: 18 March 2016
Published: 31 March 2016
Abstract
In this paper, we discuss generalized hierarchical minimax theorems with four set-valued mappings and we propose some scalar hierarchical minimax theorems and generalized hierarchical minimax theorems in topological spaces. Some examples are given to illustrate our results.
Keywords
minimax theorems set-valued mappings cone-convexities1 Introduction
It is well known that minimax theorems are important in the areas of game theory, and mathematical economical and optimization theory (see [1–5]). Within recent years, many generalizations of minimax theorems have been successfully obtained. On the one hand, the minimax theorem of two functions has been studied based on the two-person non-zero-sum games (see [6, 7]); on the other hand, with the development of vector optimization, there are many authors paying their attention to minimax problems of vector-valued mappings (see [8–10]).
Since Kuroiwa [11] investigated minimax problems of set-valued mappings in 1996, many authors have devoted their efforts to the study of the minimax problems for set-valued mappings. Li et al. [12] proved some minimax theorems for set-valued by using section theorem and separation theorem. Some other minimax theorems for set-valued mappings can be found in [13–16]. Zhang et al. [17] established some minimax theorems for two set-valued mappings, which improved the corresponding results in [12, 13]. Lin et al. [18, 19] investigated some bilevel minimax theorems and hierarchical minimax theorems for set-valued mappings by using nonlinear scalarization function.
Recently, Balaj [20] proposed some minimax theorems for four real-valued functions by using some new alternative principles. Inspired by [17–20] we shall study some generalized hierarchical minimax theorems for set-valued mappings. The imposed conditions involve four set-valued mappings. In the second section, we introduce some notions and preliminary results. In the third section, we prove the hierarchical minimax theorem for scalar set-valued mappings. In the fourth section, we show some hierarchical minimax theorems for set-valued mappings in Hausdorff topological vector spaces by using the results obtained in the previous section.
2 Preliminary
In this section, we recall some notations and some known facts.
Let X, Y be two nonempty sets in two local convex Hausdorff topological vector spaces, respectively, Z be a local convex Hausdorff topological vector space, \(S \subset Z\) be a closed convex pointed cone with \(\operatorname {int}S \neq\emptyset\), and let \(Z^{\ast}\) denote the topological dual space of Z. A set-valued mapping \(F: X\rightarrow 2^{Z} \) are associated with other two mappings \(F^{-}:Z \rightarrow2^{X} \), the inverse of F and \(F^{*}:Z\rightarrow2^{X}\) the dual of F, defined as \(F^{-}(z)=\{ x\in X: z \in F(x)\}\) and \(F^{*}(z)=X\setminus F^{-}(z)\).
Definition 2.1
([21])
- (i)
A point \(z\in A\) is called a minimal point of A if \(A \cap(z-S) =\{z\}\), and MinA denotes the set of all minimal points of A.
- (ii)
A point \(z\in A\) is called a weakly minimal point of A if \(A \cap(z- \operatorname {int}S) = \emptyset\), and \(\operatorname {Min}_{w} A\) denotes the set of all weakly minimal points of A.
- (iii)
A point \(z\in A\) is called a maximal point of A if \(A \cap (z+S) =\{z\}\), and MaxA denotes the set of all maximal points of A.
- (iv)
A point \(z\in A\) is called a weakly maximal point of A if \(A \cap(z+\operatorname {int}S) =\emptyset\), and \(\operatorname {Max}_{w} A\) denotes the set of all weakly maximal points of A.
For a nonempty compact subset \(A\subset Z\), it follows from [12] that \(\emptyset\neq \operatorname {Min}A \subset \operatorname {Min}_{w} A\); \(A \subset \operatorname {Min}A +S\) and \(\emptyset\neq \operatorname {Max}A \subset \operatorname {Max}_{w} A\); \(A\subset \operatorname {Max}A -S\). We note that, when \(Z=R\), MinA and MaxA are equivalent to \(\operatorname {Min}_{w} A\) and \(\operatorname {Max}_{w} A\), respectively.
Definition 2.2
([22])
- (i)
F is said to be upper semicontinuous (shortly, u.s.c.) at \(x_{0}\in X\), if for any neighborhood \(N(F(x_{0}))\) of \(F(x_{0})\), there exists a neighborhood \(N(x_{0})\) of \(x_{0}\) such that \(F(x)\subset N(F(x_{0}))\), \(\forall x\in N(x_{0})\). F is u.s.c. on X if F is u.s.c. at any \(x\in X\).
- (ii)
F is said to be lower semicontinuous (shortly, l.s.c.) at \(x_{0}\in X\), if for any open neighborhood N in Z satisfying \(F(x_{0})\cap N \neq\emptyset\), there exists a neighborhood \(N(x_{0})\) of \(x_{0}\) such that \(F(x)\cap N \neq\emptyset\), \(\forall x\in N(x_{0})\). F is l.s.c. on X if F is l.s.c. at any \(x\in X\).
- (iii)
F is said to be continuous at \(x_{0}\in X\), if F is both u.s.c. and l.s.c. at \(x_{0}\). F is continuous on X if F is continuous at any \(x\in X\).
- (iv)
F is said to be closed if the graph of F is closed subset of \(X\times Z\).
Definition 2.3
([17])
- (i)F is said to be S-concave (respectively, S-convex) on X, if for any \(x_{1}, x_{2} \in X\) and \(\lambda\in[0,1]\),$$\begin{aligned}& \lambda F(x_{1})+ (1-\lambda) F(x_{2}) \subset F\bigl( \lambda x_{1}+ (1-\lambda) x_{2}\bigr) -S \\& \bigl(\mbox{respectively, } F\bigl(\lambda x_{1}+ (1-\lambda) x_{2}\bigr)\subset\lambda F(x_{1})+ (1-\lambda) F(x_{2}) -S \bigr); \end{aligned}$$
- (ii)F is said to be properly S-quasiconcave (respectively, properly S-quasiconvex) on X, if for any \(x_{1}, x_{2} \in X\) and \(\lambda \in[0,1]\),$$\begin{aligned}& \mbox{either } F(x_{1})\subset F\bigl(\lambda x_{1}+(1- \lambda)x_{2}\bigr)-S \quad \mbox{or}\quad F(x_{2})\subset F \bigl(\lambda x_{1}+(1-\lambda)x_{2}\bigr)-S \\& \bigl(\mbox{respectively, either } F\bigl(\lambda x_{1}+(1- \lambda)x_{2}\bigr)\subset F(x_{1})-S \quad \mbox{or }\\& \quad F \bigl(\lambda x_{1}+(1-\lambda)x_{2}\bigr) \subset F(x_{2})-S\bigr); \end{aligned}$$
- (iii)F is said to be naturally S-quasiconcave (respectively, naturally S-quasiconvex) on X, if for any \(x_{1}, x_{2}\in X \) and \(\lambda\in[0,1]\)$$\begin{aligned}& \operatorname {co}\bigl(F(x_{1}) \cup F(x_{2})\bigr) \subset F\bigl( \lambda x_{1}+(1-\lambda)x_{2}\bigr) -S \\& \bigl(\mbox{respectively, } F\bigl(\lambda x_{1}+(1- \lambda)x_{2}\bigr)\subset \operatorname {co}\bigl(F(x_{1}) \cup F(x_{2})\bigr)-S\bigr). \end{aligned}$$
Remark 2.1
- (1)
Obviously, any S-concave (S-convex) mapping F is naturally S-quasiconcave (naturally S-quasiconvex); any properly S-quasiconcave (properly S-quasiconvex) mapping F is naturally S-quasiconcave (naturally S-quasiconvex).
- (2)
One should note that the S-concave (respectively, S-convex, properly S-quasiconcave, properly S-quasiconvex, naturally S-quasiconcave, naturally S-quasiconvex) mapping is defined as above S-concave (respectively, above S-convex, above properly S-quasiconcave, above properly S-quasiconvex, above naturally S-quasiconcave, above naturally S-quasiconvex) mapping in [18, 19].
Lemma 2.1
([22])
Let \(F: X\rightarrow2^{Z}\) be a set-valued mapping. If X is compact and F is u.s.c. with compact values, then \(F(X)=\bigcup_{x\in X}F(x)\) is compact.
Lemma 2.2
([17])
In the sequel we need the following alternative theorem which is a variant form of Balaj [20].
Lemma 2.3
([20])
- (i)
for each \(x\in X\), \(\operatorname {co}\mathcal{F}_{1}(x) \subset\mathcal{F}_{2}(x) \subset\mathcal{F}_{3}(x)\);
- (ii)
\(\mathcal{F}_{3}(\operatorname {co}A) \subset\mathcal{F}_{4}(A)\) for any finite subset \(A \subset X\);
- (iii)
\(\mathcal{F}_{1}\) and \(\mathcal{F}_{4}^{*}\) are u.s.c.;
- (iv)
\(\mathcal{F}_{2}\) and \(\mathcal{F}_{3}^{*}\) have compact values.
- (a)
There exists \(x_{0} \in X\) such that \(\mathcal{F}_{1}(x_{0}) =\emptyset\).
- (b)
\(\bigcap_{x\in X}\mathcal{F}_{4}(x) \neq\emptyset\).
3 Hierarchical minimax theorems for scalar set-valued mappings
In this section, we first establish the following hierarchical minimax theorems for scalar set-valued mappings.
Theorem 3.1
- (i)
\((x,y)\rightarrow F_{1}(x,y)\) is u.s.c. with nonempty closed values, and \((x,y)\rightarrow F_{4}(x,y)\) is l.s.c.
- (ii)
\(y \rightarrow F_{2}(x,y)\) is naturally \(R_{+}\)-quasiconcave on Y for each \(x\in X\), and \(x\rightarrow F_{3}(x,y)\) is naturally \(R_{+}\)-quasiconvex on X for each \(y\in Y\).
- (iii)
\(y\rightarrow F_{2}(x,y)\) is closed for all \(x\in X\), and \(x\rightarrow F_{3}(x,y)\) is l.s.c. for all \(y\in Y\).
Then either there is \(x_{0}\in X\) such that \(F_{1}(x_{0}, y)\subset(-\infty, \alpha)\) for all \(y\in Y\) or there is \(y_{0}\in Y\) such that \(F_{4}(x, y_{0}) \cap[\beta, +\infty) \neq\emptyset\) for all \(x\in X\).
- (iv)for each \(w\in Y\), there exists \(x_{w}\in X\) such that$$ \operatorname {Max}F_{4}(x_{w},w) \leq \operatorname {Max}\bigcup _{y\in Y} \operatorname {Min}\bigcup_{x\in X} F_{4}(x,y). $$(1)
Proof
For any \(x\in X\), we see that \(\mathcal{F}_{2}(x)\) is convex valued. In fact, for any \(y_{1}, y'_{1} \in\mathcal{F}_{2}(x)\), there exist \(f_{1}\in F_{2}(x, y_{1})\) and \(f'_{1}\in F_{2}(x, y'_{1})\) such that \(f_{1}\geq\alpha\) and \(f'_{1}\geq\alpha\). Since \(y \rightarrow F_{2}(x,y)\) is naturally \(R_{+}\)-quasiconcave, we have \(\lambda f_{1}+(1-\lambda)f'_{1} \in\lambda F_{2}(x,y_{1}) +(1-\lambda) F_{2}(x,y'_{1}) \subset \operatorname {co}(F_{2}(x,y_{1})\cup F_{2}(x,y'_{1})) \subset F_{2}(x, \lambda y_{1}+(1-\lambda)y'_{1})-R_{+}\), \(\forall \lambda\in[0,1]\). Then there exist \(f\in F_{2}(x, \lambda y_{1}+(1-\lambda )y'_{1}) \) and \(r\in R_{+}\) such that \(f\in\lambda f_{1}+(1-\lambda)f'_{1} +r \geq\alpha\). Therefore, \(\lambda y_{1}+(1-\lambda)y'_{1} \in\mathcal {F}_{2}(x)\), i.e. \(\mathcal{F}_{2}(x)\) is convex valued. Thus \(\operatorname {co}\mathcal{F}_{1}(x)\subset \operatorname {co}\mathcal{F}_{2}(x) = \mathcal{F}_{2}(x)\), \(\forall x\in X\).
Let \(y\in\mathcal{F}_{3}(\operatorname {co}A)\) for a finite subset \(A\subset X\). Without loss of generality, we suppose that \(y\in\mathcal{F}_{3}(\lambda x_{1}+(1-\lambda)x_{2})\) for some \(x_{1}, x_{2} \in A\) and \(\lambda\in[0,1]\). Then there exists \(f\in F_{3}(\lambda x_{1}+ (1-\lambda)x_{2}, y)\) such that \(f>\beta\). Since \(x\rightarrow F_{3}(x,y)\) is naturally \(R_{+}\)-quasiconvex for each \(y\in Y\), there exists \(f'\in \operatorname {co}(F_{3}(x_{1},y)\cup F_{3}(x_{2},y))\) such that \(f \in f'-R_{+}\). Therefore, there exist \(\mu\in[0,1]\) and \(f_{1}, f_{2} \in F_{3}(x_{1},y)\cup F_{3}(x_{2},y)\) and \(r\in R_{+}\) such that \(f=f'-r=\mu f_{1}+(1-\mu)f_{2} -r >\beta\). Then at least one of the assertions \(f_{1}>\beta\) and \(f_{2}>\beta\) holds. Hence, \(y \in(\mathcal {F}_{3}(x_{1})\cup\mathcal{F}_{3}(x_{2})) \subset\mathcal{F}_{3}(A)\). Therefore, \(\mathcal{F}_{3}(\operatorname {co}A)\subset\mathcal{F}_{3}(A)\subset\mathcal {F}_{4}(A)\).
For any sequence \((x_{n},y_{n}) \in \operatorname {graph}\mathcal{F}_{1}=\{(x,y): \exists f\in F_{1}(x,y), f\geq\alpha\}\) with \((x_{n}, y_{n}) \rightarrow(x,y)\), there exist \(f_{n} \in F_{1}(x_{n},y_{n})\) such that \(f_{n} \geq\alpha\). We can take subsequence \(\{f_{n_{k}} \}\) such that \(\lim_{k\rightarrow\infty} f_{n_{k}} = \liminf_{n\rightarrow\infty} f_{n} = f_{0}\). Then \(f_{0}\geq\alpha \). Since \(F_{1}\) is u.s.c. with closed values, Then \(F_{1}\) is closed. Thus \(f_{0} \in F(x_{0},y_{0})\), and so \((x_{0},y_{0}) \in \operatorname {graph}\mathcal{F}_{1}\). This implies that \(\mathcal{F}_{1}\) is closed. From compactness of Y it follows that \(\mathcal{F}_{1}\) is upper semicontinuous.
Now, we show that \(\operatorname {graph}\mathcal{F}_{4}^{*}=\{(x,y): \forall f\in F_{4}(x,y), f\leq\beta\}\) is closed. Let \((x_{n},y_{n}) \in \operatorname {graph}\mathcal {F}_{4}^{*}\) with \((x_{n},y_{n})\rightarrow(x_{0}, y_{0})\). From lower semicontinuity of \(F_{4}\), it follows that for any \(f_{0}\in F_{4}(x_{0},y_{0})\), there exists \(f_{n} \in F_{4}(x_{n},y_{n})\) such that \(f_{n} \rightarrow f_{0}\). Then \(f_{0} \leq\beta\). Therefore \(\operatorname {graph}\mathcal{F}_{4}^{*}\) is closed. Noticing the compactness of Y, we see that \(\mathcal{F}_{4}^{*}\) is upper semicontinuous.
Since \(y\rightarrow F_{2}(x,y)\) is closed for all \(x\in X\), \(\mathcal {F}_{2}\) is closed valued. In fact, for any sequence \({y_{n}}\subset \mathcal{F}_{2}(x)\) with \(y_{n} \rightarrow y_{0}\), there exists \(f_{n} \in F_{2}(x,y_{n})\) such that \(f_{n} \geq\alpha\). We can take subsequence \(\{ f_{n_{k}} \}\) such that \(\lim_{k\rightarrow\infty} f_{n_{k}} = \liminf_{n\rightarrow\infty} f_{n} = f_{0}\). Then \(f_{0}\geq\alpha\). It follows from the closedness of \(F(x,\cdot)\) that \(f_{0} \in F(x, y_{0})\), and so \(\mathcal{F}_{2}\) has closed values. Next, we claim that \(\mathcal {F}_{3}^{*}\) has closed values. For any sequence \({x_{n}}\subset\mathcal {F}_{3}^{*}(y)\) that converges to some point \(x_{0} \in X\), we see that \(y\notin\mathcal{F}_{3}(x_{n})\). Then \(f\leq\beta\) for any \(f\in F_{3}(x_{n}, y)\). Since \(x\rightarrow F_{3}(x,y)\) is lower semicontinuous for all \(y\in Y\), for any \(y_{0}\in F(x_{0},y)\) there exists \(f_{n} \in F_{3}(x_{n},y)\) such that \(f_{n} \rightarrow f_{0}\). Then \(f_{0} \leq\beta\) and hence \(x_{0} \in\mathcal {F}_{3}^{*}(y)\). This proves that \(\mathcal{F}_{3}^{*}\) has closed values. It follows from the compactness of X and Y that both \(\mathcal{F}_{2}\) and \(\mathcal{F}_{3}^{*}\) have compact values.
Then from Lemma 2.3, it follows that either there is \(x_{0}\in X\) such that \(\mathcal{F}_{1}(x_{0})=\emptyset\), or \(\bigcap_{x\in X} \mathcal {F}_{4}(x)\neq\emptyset\). That is, for any real numbers \(\alpha, \beta \in R\) with \(\alpha>\beta\), either there is \(x_{0}\in X\) such that \(F_{1}(x_{0}, y)\subset(-\infty, \alpha)\) for all \(y\in Y\) or there is \(y_{0}\in Y\) such that \(F_{4}(x, y_{0}) \cap[\beta, +\infty) \neq\emptyset\) for all \(x\in X\).
Example 3.1
When \(F_{1}(x,y)=F_{2}(x,y)=F(x,y)\) and \(F_{3}(x,y)=F_{4}(x,y)=G(x,y)\) in Theorem 3.1, we state the special case of Theorem 3.1 as follows.
Theorem 3.2
- (i)
\((x,y)\rightarrow F(x,y)\) is u.s.c. with nonempty closed values, and \((x,y)\rightarrow G(x,y)\) is l.s.c.
- (ii)
\(y \rightarrow F(x,y)\) is naturally \(R_{+}\)-quasiconcave on Y for each \(x\in X\), and \(x\rightarrow G(x,y)\) is naturally \(R_{+}\)-quasiconvex on X for each \(y\in Y\).
Then either there is \(x_{0}\in X\) such that \(F(x_{0}, y)\subset(-\infty, \alpha)\) for all \(y\in Y\) or there is \(y_{0}\in Y\) such that \(G(x, y_{0}) \cap[\beta, +\infty) \neq\emptyset\) for all \(x\in X\).
- (iii)for each \(w\in Y\), there exists \(x_{w}\in X\) such that$$\operatorname {Max}G(x_{w},w) \leq \max \bigcup_{y\in Y} \operatorname {Min}\bigcup_{x\in X} F_{4}(x,y). $$
Proof
Since F is u.s.c. with nonempty closed values, it follows that \(y\rightarrow F(x,y)\) is closed for all \(x\in X\) by Proposition 7 in [22], p. 110. From Theorem 3.1, it is easy to show the conclusion holds. □
Remark 3.1
It is obvious that \(F(x,y) \subset G(x,y)\) implies \(F(x,y)\subset G(x,y)-R_{+}\). So Theorem 3.2 generalizes Theorem 2.1 in [18].
It is well known that both sets \(\bigcup_{y\in Y}F(x,y)\) and \(\bigcup_{x\in X}G(x,y)\) are compact for any \(y\in Y\) and \(x\in X\) whenever the mappings F and G are upper semicontinuous with nonempty compact values. Hence we can deduce the following result.
Corollary 3.1
- (i)
\((x,y)\rightarrow F(x,y)\) is u.s.c., and \((x,y)\rightarrow G(x,y)\) is continuous.
- (ii)
\(y \rightarrow F(x,y)\) is naturally \(R_{+}\)-quasiconcave on Y for each \(x\in X\), and \(x\rightarrow G(x,y)\) is naturally \(R_{+}\)-quasiconvex on X for each \(y\in Y\).
- (iii)For each \(w\in Y\), there exists \(x_{w}\in X\) such that$$\operatorname {Max}G(x_{w},w) \leq \operatorname {Max}\bigcup_{y\in Y} \operatorname {Min}\bigcup_{x\in X} G(x,y). $$
4 Generalized hierarchical minimax theorem
In this section, we will discuss some generalized hierarchical minimax theorems for set-valued mappings valued in a complete locally convex Hausdorff topological vector space.
Lemma 4.1
Proof
Theorem 4.1
- (i)
\((x,y)\rightarrow F_{1}(x,y)\) is u.s.c., \(x\rightarrow F_{1}(x,y)\) is l.s.c. for each \(y\in Y\), and \((x,y)\rightarrow F_{4}(x,y)\) is continuous;
- (ii)
\(y \rightarrow F_{2}(x,y)\) is naturally S-quasiconcave on Y for each \(x\in X\), and \(x\rightarrow F_{3}(x,y)\) is naturally S-quasiconvex on X for each \(y\in Y\);
- (iii)
\(y\rightarrow F_{2}(x,y)\) is u.s.c. for all \(x\in X\), and \(x\rightarrow F_{3}(x,y)\) is l.s.c. for all \(y\in Y\);
- (iv)for each \(w\in Y\), there exists \(x_{w}\in X\) such that$$\operatorname {Max}\bigcup_{y\in Y} \operatorname {Min}_{w}\bigcup _{x\in X} F_{4}(x,y) - F_{4}(x_{w},w) \subset S ; $$
- (v)for each \(w\in Y\)$$\operatorname {Max}\bigcup_{y\in Y}\operatorname {Min}_{w} \bigcup _{x\in X} F_{4}(x,y) \subset \operatorname {Min}_{w} \bigcup _{x\in X} F_{4}(x,w) +S. $$
Proof
Example 4.1
When \(F_{1}(x,y)=F_{2}(x,y)=F(x,y)\) and \(F_{3}(x,y)=F_{4}(x,y)=G(x,y)\) in Theorem 4.1, we state the special case of Theorem 4.1 as follows.
Corollary 4.1
- (i)
\((x,y)\rightarrow F(x,y)\) is u.s.c., \(x\rightarrow F(x,y)\) is l.s.c. for each \(y\in Y\), and \((x,y)\rightarrow G(x,y)\) is continuous;
- (ii)
\(y \rightarrow F(x,y)\) is naturally S-quasiconcave on Y for each \(x\in X\), and \(x\rightarrow G(x,y)\) is naturally S-quasiconvex on X for each \(y\in Y\);
- (iii)for each \(w\in Y\), there exists \(x_{w}\in X\) such that$$\operatorname {Max}\bigcup_{y\in Y} \operatorname {Min}_{w}\bigcup _{x\in X} G(x,y) - G(x_{w},w)\subset S ; $$
- (iv)for each \(w\in Y\)$$\operatorname {Max}\bigcup_{y\in Y}\operatorname {Min}_{w} \bigcup _{x\in X} G(x,y) \subset \operatorname {Min}_{w} \bigcup _{x\in X} G(x,w) +S. $$
5 Concluding remarks
We have proven some hierarchical minimax theorems for scalar set-valued mappings and generalized hierarchical minimax theorems for set-valued mappings valued in a complete locally convex Hausdorff topological vector space. The imposed conditions involved four set-valued mappings. The main tools to prove our results have been an alternative principle and separation theorems. Some examples have been provided to illustrate our results.
Declarations
Acknowledgements
The author would like to thank the editor and anonymous reviewers for their valuable comments and suggestions which helped to improve the paper. This work was supported by ’Department of Mathematics, Taiyuan Normal University, 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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