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Generalized hierarchical minimax theorems for setvalued mappings
Journal of Inequalities and Applications volume 2016, Article number: 103 (2016)
Abstract
In this paper, we discuss generalized hierarchical minimax theorems with four setvalued 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.
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 twoperson nonzerosum 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 vectorvalued mappings (see [8–10]).
Since Kuroiwa [11] investigated minimax problems of setvalued mappings in 1996, many authors have devoted their efforts to the study of the minimax problems for setvalued mappings. Li et al. [12] proved some minimax theorems for setvalued by using section theorem and separation theorem. Some other minimax theorems for setvalued mappings can be found in [13–16]. Zhang et al. [17] established some minimax theorems for two setvalued mappings, which improved the corresponding results in [12, 13]. Lin et al. [18, 19] investigated some bilevel minimax theorems and hierarchical minimax theorems for setvalued mappings by using nonlinear scalarization function.
Recently, Balaj [20] proposed some minimax theorems for four realvalued functions by using some new alternative principles. Inspired by [17–20] we shall study some generalized hierarchical minimax theorems for setvalued mappings. The imposed conditions involve four setvalued mappings. In the second section, we introduce some notions and preliminary results. In the third section, we prove the hierarchical minimax theorem for scalar setvalued mappings. In the fourth section, we show some hierarchical minimax theorems for setvalued mappings in Hausdorff topological vector spaces by using the results obtained in the previous section.
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 setvalued 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])
Let \(A\subset Z\) be a nonempty subset.

(i)
A point \(z\in A\) is called a minimal point of A if \(A \cap(zS) =\{z\}\), 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])
Let \(F:X\rightarrow2^{Z}\) be a setvalued mapping with nonempty values.

(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])
Let X be a nonempty subset of a topological vector space, \(F: X\rightarrow2^{Z}\) be a setvalued mapping.

(i)
F is said to be Sconcave (respectively, Sconvex) 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 Squasiconcave (respectively, properly Squasiconvex) 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 Squasiconcave (respectively, naturally Squasiconvex) 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 Sconcave (Sconvex) mapping F is naturally Squasiconcave (naturally Squasiconvex); any properly Squasiconcave (properly Squasiconvex) mapping F is naturally Squasiconcave (naturally Squasiconvex).

(2)
One should note that the Sconcave (respectively, Sconvex, properly Squasiconcave, properly Squasiconvex, naturally Squasiconcave, naturally Squasiconvex) mapping is defined as above Sconcave (respectively, above Sconvex, above properly Squasiconcave, above properly Squasiconvex, above naturally Squasiconcave, above naturally Squasiconvex) mapping in [18, 19].
Lemma 2.1
([22])
Let \(F: X\rightarrow2^{Z}\) be a setvalued 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])
Let \(F: X\rightarrow2^{Z}\) be a continuous setvalued mapping with compact values. Then the setvalued mapping
is nonempty closed and upper semicontinuous.
In the sequel we need the following alternative theorem which is a variant form of Balaj [20].
Lemma 2.3
([20])
Let X, Y be two nonempty compact convex subsets in two local convex Hausdorff topological vector spaces. The setvalued mappings \(\mathcal{F}_{i}: X\rightarrow Z\), \(i=1,2, 3, 4\), satisfy the following conditions:

(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.
Then at least one of the following assertions holds:

(a)
There exists \(x_{0} \in X\) such that \(\mathcal{F}_{1}(x_{0}) =\emptyset\).

(b)
\(\bigcap_{x\in X}\mathcal{F}_{4}(x) \neq\emptyset\).
Hierarchical minimax theorems for scalar setvalued mappings
In this section, we first establish the following hierarchical minimax theorems for scalar setvalued mappings.
Theorem 3.1
Let X, Y be two nonempty compact convex subsets of local convex Hausdorff topological vector spaces, respectively. Let \(F_{i}: X\times Y \rightarrow2^{R}\), \(i=1, 2, 3, 4\) be setvalued mappings such that \(F_{i}(x,y) \subset F_{i+1}(x,y)R_{+}\). Assume that

(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 \(x\in X\).
Furthermore, assume that the sets \(\bigcup_{y\in Y}F_{1}(x,y)\) and \(\bigcup_{x\in X}F_{4}(x,y)\) are compact for all \(y\in Y\) and \(x\in X\), respectively. Assume the following condition holds:

(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)
Then
Proof
For any real numbers \(\alpha, \beta\in R\) with \(\alpha>\beta\), we define the mappings \(\mathcal{F}_{i}: X \rightarrow2^{Y}\), \(i=1, 2, 3, 4\) by
Then we can see that \(\mathcal{F}_{1}(x)\subset\mathcal{F}_{2}(x) \subset \mathcal{F}_{3}(x) \subset\mathcal{F}_{4}(x)\), \(\forall x\in X\). For any \(x\in X\), if \(y\in\mathcal{F}_{1}(x)\), there exists \(f_{1}\in F_{1}(x,y)\) such that \(f_{1}\geq\alpha\). Since \(F_{1}(x,y) \subset F_{2}(x,y)R_{+}\), there are \(f_{2} \in F_{2}(x,y)\) and \(r\in R_{+}\) such that \(f_{2}=f_{1}+r \geq \alpha\). Then \(y\in\mathcal{F}_{2}(x)\), and so \(\mathcal{F}_{1}(x) \subset\mathcal{F}_{2}(x)\). Noticing that \(\alpha> \beta\), one can show \(\mathcal{F}_{2}(x)\subset\mathcal{F}_{3}(x)\subset\mathcal{F}_{4}(x)\) by using similar deduction.
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 \(x\in X\).
Furthermore, the compactness of \(\bigcup_{x\in X} F_{4}(x,y)\) implies that \(\operatorname {Min}\bigcup_{x\in X} F_{4}(x,y)\) is nonempty for all \(y\in Y\). Since \((x,y) \rightarrow F_{4}(x,y)\) is lower semicontinuous, it follows that \(y\rightarrow\bigcup_{x\in X}F_{4}(x,y)\) is lower semicontinuous. By the compactness of Y and the proof of Lemma 3.2 [12], the set \(\bigcup_{y\in Y} \operatorname {Min}\bigcup_{x\in X}F_{4}(x,y)\) is nonempty and compact, and so \(\operatorname {Max}\bigcup_{y\in Y} \operatorname {Min}\bigcup_{x\in X}F_{4}(x,y) \neq \emptyset\). Set any real numbers \(\alpha, \beta\in R\) with \(\alpha> \beta> \operatorname {Max}\bigcup_{y\in Y}\operatorname {Min}\bigcup_{x\in X}F_{4}(x,y)\). From (iv), we see that, for each \(w\in Y\), there exists \(x_{w}\in X\) such that \(F_{4}(x_{w},w) \cap[\beta, +\infty)=\emptyset\). Therefore, there is \(x_{0}\in X\) such that \(F_{1}(x_{0}, y)\subset(\infty, \alpha)\) for all \(y\in Y\). Hence
By the arbitrariness of α and β, (2) holds. □
Example 3.1
Let \(X=Y=[0,1]\subset R\). Define four mappings \(F_{i}:X\times Y\rightarrow2^{R}\), \(i=1,2,3,4\), as
We can see that \(F_{i}(x,y)\subset F_{i+1}(x,y)R_{+}\) for all \((x,y) \in X\times Y\) and conditions (i)(iii) of Theorem 3.1 hold. It is obvious that \(\bigcup_{x\in X}F_{1}(x,y)\) and \(\bigcup_{y\in Y}F_{4}(x,y)\) are compact for all \(y\in Y\) and \(x\in X\), respectively. Now, we show condition (iv) of Theorem 3.1 is true. One can calculate that \(\operatorname {Min}\bigcup_{x\in X}F_{4}(x,y)=\{y\}\), \(\forall y\in Y\), and \(\operatorname {Max}\bigcup_{y\in Y} \operatorname {Min}\bigcup_{x\in X} F_{4}(x,y)=1\). Taking \(x=0\), we have
Then all of the conditions of Theorem 3.1 valid. So, the conclusion of Theorem 3.1 holds. In fact, \(\operatorname {Min}\bigcup_{x\in X} \operatorname {Max}\bigcup_{y\in Y} F_{1}(x,y)=0<1=\operatorname {Max}\bigcup_{y\in Y} \operatorname {Min}\bigcup_{x\in X} F_{4}(x,y)\).
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
Let X, Y be two nonempty compact convex subsets of local convex Hausdorff topological vector spaces, respectively. The setvalued mappings \(F, G: X\times Y \rightarrow2^{R}\) with \(F(x,y) \subset G(x,y)R_{+}\). Assume that

(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 \(x\in X\).
Furthermore, assume that the sets \(\bigcup_{y\in Y}F(x,y)\) and \(\bigcup_{x\in X}G(x,y)\) are compact for all \(y\in Y\) and \(x\in X\), respectively. Assume the following condition holds:

(iii)
for each \(w\in Y\), there exists \(x_{w}\in X\) such that
$$\operatorname {Max}G(x_{w},w) \leq \max \bigcup_{y\in Y} \operatorname {Min}\bigcup_{x\in X} F_{4}(x,y). $$
Then
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
Let X, Y be two nonempty compact convex subsets of local convex Hausdorff topological vector spaces, respectively. The setvalued mappings \(F, G: X\times Y \rightarrow2^{R}\) come with nonempty compact values and \(F(x,y) \subset G(x,y)R_{+}\). Assume that

(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
$$\operatorname {Max}G(x_{w},w) \leq \operatorname {Max}\bigcup_{y\in Y} \operatorname {Min}\bigcup_{x\in X} G(x,y). $$
Then
Remark 3.2
Corollary 3.1 generalizes Theorem 2.1 in [17] and weakens the continuity of \(F_{1}\) in Theorem 2.1 in [17]. It also generalizes Theorem 2.1 in [12] from one setvalued mapping to two setvalued mappings.
Generalized hierarchical minimax theorem
In this section, we will discuss some generalized hierarchical minimax theorems for setvalued mappings valued in a complete locally convex Hausdorff topological vector space.
Lemma 4.1
Let X, Y be two nonempty compact convex subsets of local convex Hausdorff topological vector spaces, respectively. The setvalued mapping \(F: X\times Y \rightarrow2^{Z}\) comes with nonempty compact values. If \((x,y)\rightarrow F(x,y)\) is u.s.c., and \(x\rightarrow F(x,y)\) is l.s.c. for each \(y\in Y\), then the setvalued mapping
is u.s.c. with nonempty compact values.
Proof
Define a setvalued mapping \(T: X \rightarrow 2^{Z}\) as
It follows from Lemma 2.4 in [16] that T is continuous. By Lemma 2.1 and compactness of Y, T is compactvalued. Then, by Lemma 2.2, we see that A is nonempty closed and u.s.c. on X. By compactness of X, it follows that \(A(x)\) is compact for each \(x\in X\). □
Theorem 4.1
Let X, Y be two nonempty compact convex subsets of local convex Hausdorff topological vector spaces, respectively, Z be a complete locally convex Hausdorff topological vector space. The setvalued mappings \(F_{i}: X\times Y \rightarrow2^{Z}\), \(i=1, 2, 3, 4\) come with nonempty compact values and \(F_{i}(x,y) \subset F_{i+1}(x,y)S\). Assume that

(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 Squasiconcave on Y for each \(x\in X\), and \(x\rightarrow F_{3}(x,y)\) is naturally Squasiconvex 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 \(w\in Y\)
$$\operatorname {Max}\bigcup_{y\in Y}\operatorname {Min}_{w} \bigcup _{x\in X} F_{4}(x,y) \subset \operatorname {Min}_{w} \bigcup _{x\in X} F_{4}(x,w) +S. $$
Then
Proof
Let \(L(x):=\operatorname {Max}_{w} \bigcup_{y\in Y} F_{1}(x,y) \). By Lemma 4.1, \(L(x)\) is u.s.c. with nonempty compact values. From Lemma 2.1, it follows that \(L(X)= \bigcup_{x\in X} L(x)\) is compact, and so is \(\operatorname {co}(L(X))\). Then \(\operatorname {co}(L(X))+S\) is a closed set with nonempty interior. Suppose that \(v\in Z\) and \(v\notin \operatorname {co}(L(X))+S\). By the separation theorem, there exist \(\xi\in Z^{*}\) and \(\alpha_{1}, \alpha_{2} \in R\) such that
By using a similar discussion to Theorem 3.1 in [17], we have \(\xi\in S^{*}\) and \(\xi(S)=R^{+}\). From assumptions (i) and (iii), it is easy to see that \((x,y)\rightarrow\xi(F_{1}(x,y))\) is u.s.c., \((x,y)\rightarrow\xi(F_{4}(x,y))\) is l.s.c., \(y\rightarrow\xi (F_{2}(x,y))\) is closed for all \(x\in X\), and \(x\rightarrow\xi (F_{3}(x,y))\) is l.s.c. for all \(y\in Y\). From condition (ii), applying Proposition 3.9 and Proposition 3.13 in [16], we see that \(y \rightarrow\xi(F_{2}(x,y))\) is naturally \(R^{+}\)quasiconcave on Y for each \(x\in X\), and \(x\rightarrow\xi(F_{3}(x,y))\) is naturally \(R^{+}\)quasiconvex on X for each \(y\in Y\). By the condition (iv), for each \(w\in Y\), there exists \(x_{w}\in X\) such that
Since \(F_{1}\) and \(F_{4}\) are u.s.c. and come with compact values, we see that \(\bigcup_{x\in X}\xi(F_{1}(x,y))\) and \(\bigcup_{y\in Y}\xi (F_{4}(x,y))\) are compact for all \(y\in Y\) and \(x\in X\), respectively. Then for setvalued mappings \(\xi(F_{i})\), \(i=1,2,3,4\), all conditions of Theorem 3.1 hold. Therefore we see that
Since Y is compact and \(F_{1}\) has nonempty compact values, for any \(x\in X\), there exist \(y_{x}\) and \(f(x,y_{x})\in F_{1}(x,y_{x})\) with \(f(x,y_{x}) \in L(x)\) such that
From (4), choosing \(s=0\) and \(u=f(x,y_{x})\), it follows that
for all \(x\in X\). Then
By (5),
Since Y is compact, there exists \(y'\in Y\) such that
From \(\xi(s)\geq0\) for all \(s\in S\), it follows that \(v \notin\bigcup_{x\in X}(F_{4}(x,y'))+S \), and then
Combined with the assumption (v), we have
That is, for any \(v\in \operatorname {Max}\bigcup_{y\in Y}\operatorname {Min}_{w}\bigcup_{x\in X}F_{4}(x,y) \),
Hence
Since \(\operatorname {co}(L(X))=\operatorname {co}(\bigcup_{x\in X}L(x))=\operatorname {co}(\bigcup_{x\in X}\operatorname {Max}_{w} \bigcup_{y\in Y}F_{1}(x,y))\) is compact, we have
Therefore, (3) holds. □
Example 4.1
Let \(X=Y=[0,1]\), \(Z=R^{2}\), and \(S=R^{2}_{+}\). Define setvalued mappings \(F_{i}:X\times Y\rightarrow2^{Z}\), \(i=1,2,3,4\), as
For all \((x,y)\in X\times Y\), we can see that the \(F_{i}(x,y)\), \(i=1,2,3,4\), are compact and
It is easy to show that the conditions (i)(iii) hold in Theorem 4.1. We explain conditions (iv) and (v) are valid. We can calculate that
For each \(w\in Y\), let \(x_{w}= 0\). Then
The condition (iv) holds. We can see that
Then all of the assumptions of Theorem 4.1 are valid. So, the conclusion of Theorem 4.1 holds. In fact,
Then
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
Let X, Y be two nonempty compact convex subsets of local convex Hausdorff topological vector spaces, respectively, Z be a complete locally convex Hausdorff topological space. The setvalued mappings \(F, G: X\times Y \rightarrow2^{Z}\) come with nonempty compact values and \(F(x,y) \subset G(x,y)S\). Assume that

(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 Squasiconcave on Y for each \(x\in X\), and \(x\rightarrow G(x,y)\) is naturally Squasiconvex 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 \(w\in Y\)
$$\operatorname {Max}\bigcup_{y\in Y}\operatorname {Min}_{w} \bigcup _{x\in X} G(x,y) \subset \operatorname {Min}_{w} \bigcup _{x\in X} G(x,w) +S. $$
Then
Remark 4.1
Concluding remarks
We have proven some hierarchical minimax theorems for scalar setvalued mappings and generalized hierarchical minimax theorems for setvalued mappings valued in a complete locally convex Hausdorff topological vector space. The imposed conditions involved four setvalued 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.
References
Kan, F: Minimax theorems. Proc. Natl. Acad. Sci. USA 39, 4247 (1953)
Ha, CW: Minimax and fixed point theorems. Math. Ann. 248, 7377 (1980)
Ha, CW: A minimax theorem. Acta Math. Hung. 101, 149154 (2003)
Zhang, QB, Cheng, CZ: Some fixedpoint theorems and minimax inequalities in FCspaces. J. Math. Anal. Appl. 328, 13691377 (2007)
Tuy, H: A new topological minimax theorem with application. J. Glob. Optim. 50, 371378 (2011)
Cheng, CZ, Lin, BL: A two functions, noncompact topological minimax theorems. Acta Math. Hung. 73, 6569 (1996)
Cheng, CZ: A general twofunction minimax theorem. Acta Math. Sin. Engl. Ser. 26, 595602 (2010)
Nieuwenhuis, JW: Some minimax theorems in vectorvalued functions. J. Optim. Theory Appl. 40, 463675 (1983)
Tanaka, T: Generalized quasiconvexities, cone saddle points and minimax theorems for vectorvalued functions. J. Optim. Theory Appl. 81, 355377 (1994)
Tan, KK, Yu, J, Yuan, XZ: Existence theorems for saddle points of vectorvalued maps. J. Optim. Theory Appl. 89, 731747 (1996)
Kuroiwa, D: Convexity for setvalued maps. Appl. Math. Lett. 9, 97101 (1996)
Li, SJ, Chen, GY, Lee, GM: Minimax theorems for setvalued mappings. J. Optim. Theory Appl. 106, 183199 (2000)
Li, SJ, Chen, GY, Teo, KL, Yang, XQ: Generalized minimax inequalities for setvalued mappings. J. Math. Anal. Appl. 281, 707723 (2003)
Zhang, Y, Li, SJ: Minimax theorems for scalar setvalued mappings with nonconvex domains and applications. J. Glob. Optim. 57, 15732916 (2013)
Zhang, Y, Li, SJ: Ky Fan minimax inequalities for setvalued mappings. Fixed Point Theory Appl. 2012, Article ID 64 (2012)
Lin, YC, Ansari, QH, Lai, HC: Minimax theorems for setvalued mappings under coneconvexities. Abstr. Appl. Anal. 2012, Article ID 310818 (2012)
Zhang, QB, Cheng, CZ, Li, XX: Generalized minimax theorems for two setvalued mappings. J. Ind. Manag. Optim. 9, 112 (2013)
Lin, YC: Bilevel minimax theorems for noncontinuous setvalued mappings. J. Inequal. Appl. 2014, Article ID 182 (2014)
Lin, YC, Pang, CT: The hierarchical minimax inequalities for setvalued mappings. Abstr. Appl. Anal. 2014, Article ID 190821 (2014)
Balaj, M: Alternative principles and their applications. J. Glob. Optim. 50, 537547 (2011)
John, J: Vector Optimization, Theory, Application, and Extensions. Springer, Berlin (2004)
Aubin, JP, Ekeland, I: Applied Nonlinear Analysis. Wiley, New York (1984)
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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.
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Wang, H. Generalized hierarchical minimax theorems for setvalued mappings. J Inequal Appl 2016, 103 (2016). https://doi.org/10.1186/s136600161050z
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DOI: https://doi.org/10.1186/s136600161050z
Keywords
 minimax theorems
 setvalued mappings
 coneconvexities