Generalized hierarchical minimax theorems for set-valued mappings

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.

1 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])

Let $A\subset Z$ be a nonempty subset.

• (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])

Let $F:X\rightarrow2^{Z}$ be a set-valued 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 set-valued mapping.

• (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])

Let $F: X\rightarrow2^{Z}$ be a continuous set-valued mapping with compact values. Then the set-valued mapping

$$\Gamma(x)= \operatorname {Max}_{w} F(x) $$

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 set-valued 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) $\bigcap_{x\in X}\mathcal{F}_{4}(x) \neq\emptyset$.

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

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 set-valued 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

$$ \operatorname {Min}\bigcup_{x\in X} \operatorname {Max}\bigcup _{y\in Y} F_{1}(x,y)\leq \operatorname {Max}\bigcup _{y\in Y} \operatorname {Min}\bigcup_{x\in X}F_{4}(x,y). $$

(2)

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

$$\begin{aligned}& \mathcal{F}_{1}(x)=\bigl\{ y\in Y: \exists f\in F_{1}(x,y), f\geq\alpha\bigr\} , \qquad \mathcal{F}_{2}(x)=\bigl\{ y\in Y: \exists f\in F_{2}(x,y), f\geq\alpha\bigr\} , \\& \mathcal{F}_{3}(x)=\bigl\{ y\in Y: \exists f\in F_{3}(x,y),f> \beta\bigr\} , \qquad \mathcal{F}_{4}(x)=\bigl\{ y\in Y: \exists f\in F_{4}(x,y), f> \alpha\bigr\} . \end{aligned}$$

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

$$\operatorname {Min}\bigcup_{x\in X} \operatorname {Max}\bigcup _{y\in Y} F_{1}(x,y)\leq \operatorname {Max}\bigcup _{y\in Y} F_{1}(x_{0},y)\leq\alpha. $$

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

$$\begin{aligned} &F_{1}(x,y)=\bigl[x^{2}-1+y,x \bigr]; \qquad F_{2}(x,y)=\biggl[x^{2}-\frac {1}{2}+y, x+ \frac{1}{2}\biggr]; \\ &F_{3}(x,y)=\bigl[x^{2}+y,x^{2}+1\bigr];\qquad F_{4}(x,y)=\bigl[x^{2}+y,x+1\bigr]. \end{aligned} $$

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

$$\operatorname {Max}F_{4}(0,y) \leq \operatorname {Max}\bigcup_{y\in Y} \operatorname {Min}\bigcup_{x\in X} F_{4}(x,y), \quad \forall y\in Y. $$

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 set-valued 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

$$\operatorname {Min}\bigcup_{x\in X} \operatorname {Max}\bigcup _{y\in Y} F(x,y)\leq \operatorname {Max}\bigcup_{y\in Y} \operatorname {Min}\bigcup_{x\in X}G(x,y). $$

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

Let X, Y be two nonempty compact convex subsets of local convex Hausdorff topological vector spaces, respectively. The set-valued 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

$$\operatorname {Min}\bigcup_{x\in X} \operatorname {Max}\bigcup _{y\in Y} F(x,y)\leq \operatorname {Max}\bigcup_{y\in Y} \operatorname {Min}\bigcup_{x\in X}G(x,y). $$

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 set-valued mapping to two set-valued mappings.

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

Let X, Y be two nonempty compact convex subsets of local convex Hausdorff topological vector spaces, respectively. The set-valued 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 set-valued mapping

$$A(x)=\operatorname {Max}_{w} \bigcup_{y\in Y} F(x,y) $$

is u.s.c. with nonempty compact values.

Proof

Define a set-valued mapping $T: X \rightarrow 2^{Z}$ as

$$T(x)=\bigcup_{y\in Y} F(x,y). $$

It follows from Lemma 2.4 in [16] that T is continuous. By Lemma 2.1 and compactness of Y, T is compact-valued. 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 set-valued 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 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 $w\in Y$

  $$\operatorname {Max}\bigcup_{y\in Y}\operatorname {Min}_{w} \bigcup _{x\in X} F_{4}(x,y) \subset \operatorname {Min}_{w} \bigcup _{x\in X} F_{4}(x,w) +S. $$

Then

$$ \operatorname {Max}\bigcup_{x\in X} \operatorname {Min}_{w} \bigcup _{y\in Y} F_{4}(x,y)\subset \operatorname {Min}\biggl(\operatorname {co}\bigcup _{y\in Y}\operatorname {Max}_{w} \bigcup _{x\in X}F_{1}(x,y) \biggr)+S. $$

(3)

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

$$ \xi(v) < \alpha_{1} < \alpha_{2} < \xi(u+s),\quad \forall u \in \operatorname {co}\bigl(L(X)\bigr), \forall s\in S. $$

(4)

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

$$\operatorname {Max}\xi\bigl(F_{4}(x_{w},w)\bigr) \leq \operatorname {Max}\bigcup _{y\in Y} \operatorname {Min}\bigcup_{x\in X} \xi\bigl(F_{4}(x,y)\bigr). $$

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 set-valued mappings $\xi(F_{i})$, $i=1,2,3,4$, all conditions of Theorem 3.1 hold. Therefore we see that

$$ \operatorname {Min}\bigcup_{x\in X} \operatorname {Max}\bigcup _{y\in Y} \xi \bigl(F_{1}(x,y)\bigr)\leq \operatorname {Max}\bigcup _{y\in Y} \operatorname {Min}\bigcup_{x\in X} \xi\bigl(F_{4}(x,y)\bigr). $$

(5)

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

$$\xi\bigl(F_{1}(x,y_{x})\bigr)=\operatorname {Max}\bigcup _{y\in Y}\xi\bigl(F_{1}(x,y)\bigr). $$

From (4), choosing $s=0$ and $u=f(x,y_{x})$, it follows that

$$\xi(v) < \xi\bigl(f(x,y_{x})\bigr)= \operatorname {Max}\bigcup _{y\in Y}\xi\bigl(F_{1}(x,y)\bigr) $$

for all $x\in X$. Then

$$\xi(v) < \operatorname {Min}\bigcup_{x\in X} \operatorname {Max}\bigcup _{y\in Y}\xi\bigl(F_{1}(x,y)\bigr). $$

By (5),

$$\xi(v) < \operatorname {Max}\bigcup_{y\in Y} \operatorname {Min}\bigcup _{x\in X}\xi\bigl(F_{4}(x,y)\bigr). $$

Since Y is compact, there exists $y'\in Y$ such that

$$\xi(v) < \operatorname {Min}\bigcup_{x\in X}\xi\bigl(F_{4} \bigl(x,y'\bigr)\bigr)=\operatorname {Max}\bigcup_{y\in Y} \operatorname {Min}\bigcup_{x\in X}\xi\bigl(F_{4}(x,y) \bigr). $$

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

$$v \notin \operatorname {Min}_{w}\bigcup_{x\in X} \bigl(F_{4}\bigl(x,y'\bigr)\bigr)+S. $$

Combined with the assumption (v), we have

$$v \notin \operatorname {Max}\bigcup_{y\in Y}\operatorname {Min}_{w}\bigcup _{x\in X}F_{4}(x,y). $$

That is, for any $v\in \operatorname {Max}\bigcup_{y\in Y}\operatorname {Min}_{w}\bigcup_{x\in X}F_{4}(x,y)$,

$$v\in \operatorname {co}\bigl(L(x)\bigr)+S. $$

Hence

$$\operatorname {Max}\bigcup_{y\in Y}\operatorname {Min}_{w}\bigcup _{x\in X}\bigl(F_{4}(x,y)\bigr) \subset \operatorname {co}\bigl(L(x) \bigr)+S. $$

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

$$\operatorname {co}\biggl(\bigcup_{x\in X}\operatorname {Max}_{w} \bigcup _{y\in Y}F_{1}(x,y)\biggr) \subset \operatorname {Min}\biggl( \operatorname {co}\biggl(\bigcup_{x\in X}\operatorname {Max}_{w} \bigcup_{y\in Y}F_{1}(x,y)\biggr)\biggr)+S. $$

Therefore, (3) holds. □

Example 4.1

Let $X=Y=[0,1]$, $Z=R^{2}$, and $S=R^{2}_{+}$. Define set-valued mappings $F_{i}:X\times Y\rightarrow2^{Z}$, $i=1,2,3,4$, as

$$\begin{aligned}&F_{1}(x,y)=\bigl[x^{2}-1+y,x\bigr] \times\{-1\}, \qquad F_{2}(x,y)=\biggl[x^{2}- \frac{1}{2}+y, x+\frac{1}{2}\biggr]\times\biggl\{ x^{2}- \frac{1}{2}\biggr\} , \\ &F_{3}(x,y)=\bigl[x^{2}+y,x+1\bigr]\times\biggl\{ y+ \frac{1}{2}\biggr\} ,\qquad F_{4}(x,y)=\{x+1\}\times[y+1,2]. \end{aligned} $$

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

$$F_{i}(x,y) \subset F_{i+1}(x,y)-S. $$

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

$$\begin{aligned} &\operatorname {Min}_{w} \bigcup_{x\in X} F_{4}(x,y)= \{1\}\times[y+1,2]\cup[1,2]\times\{ y+1\}, \\ &\operatorname {Max}\bigcup_{y\in Y}\operatorname {Min}_{w} \bigcup _{x\in X} F_{4}(x,y)= \bigl\{ (2,2)\bigr\} . \end{aligned} $$

For each $w\in Y$, let $x_{w}= 0$. Then

$$\operatorname {Max}\bigcup_{y\in Y}\operatorname {Min}_{w}\bigcup _{x\in X} F_{4}(x,y)-F_{4}(x_{w},w)= \bigl\{ (2,2)\bigr\} -\{1\}\times[y+1,2]\subset S. $$

The condition (iv) holds. We can see that

$$\begin{aligned} &\operatorname {Max}\bigcup_{y\in Y} \operatorname {Min}_{w}\bigcup_{x\in X} F_{4}(x,y)\\ &\quad = \bigl\{ (2,2)\bigr\} \subset\{1\}\times[y+1,2]\cup[1,2]\times\{ y+1\}+S \\ &\quad =\operatorname {Min}_{w} \bigcup_{x\in X} F_{4}(x,y)+S. \end{aligned} $$

Then all of the assumptions of Theorem 4.1 are valid. So, the conclusion of Theorem 4.1 holds. In fact,

$$\operatorname {Min}\biggl(\operatorname {co}\bigcup_{x\in X}\operatorname {Max}_{w} \bigcup_{y\in Y} F_{1}(x,y)\biggr)= \bigl\{ (0,-1)\bigr\} . $$

Then

$$\begin{aligned} &\operatorname {Max}\bigcup_{y\in Y} \operatorname {Min}_{w}\bigcup_{x\in X} F_{4}(x,y)\\ &\quad = \bigl\{ (2,2)\bigr\} \subset\bigl\{ (0,-1)\bigr\} +S \\ &\quad =\operatorname {Min}_{w} \operatorname {Min}\biggl(\operatorname {co}\biggl(\bigcup _{x\in X}\operatorname {Max}_{w} \bigcup_{y\in Y} F_{1}(x,y)\biggr)\biggr)+S. \end{aligned} $$

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 set-valued 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 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 $w\in Y$

  $$\operatorname {Max}\bigcup_{y\in Y}\operatorname {Min}_{w} \bigcup _{x\in X} G(x,y) \subset \operatorname {Min}_{w} \bigcup _{x\in X} G(x,w) +S. $$

Then

$$ \operatorname {Max}\bigcup_{x\in X} \operatorname {Min}_{w} \bigcup _{y\in Y} G(x,y)\subset \operatorname {Min}\biggl(\operatorname {co}\bigcup _{y\in Y}\operatorname {Max}_{w} \bigcup_{x\in X}F(x,y) \biggr)+S. $$

(6)

Remark 4.1

• (1) Corollary 4.1 generalizes Theorem 3.1 in [17] and weakens the continuity if F.

• (2) Corollary 4.1 also generalizes Theorem 3.1 in [18] since $F(x,y)\subset G(x,y)$ implies $F(x,y)\subset G(x,y)-S$.

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.