# Ky Fan minimax inequalities for set-valued mappings

- Yu Zhang
^{1}Email author and - Sheng-Jie Li
^{1, 2}

**2012**:64

https://doi.org/10.1186/1687-1812-2012-64

© Zhang and Li; licensee Springer. 2012

**Received: **21 June 2011

**Accepted: **19 April 2012

**Published: **19 April 2012

## Abstract

In this article, by virtue of the Kakutani-Fan-Glicksberg fixed point theorem, two types of Ky Fan minimax inequalities for set-valued mappings are obtained. Some examples are given to illustrate our results.

**Mathematics Subject Classification (2010)**: 49J35; 49K35; 90C47.

## Keywords

## 1 Introduction

It is well known that Ky Fan minimax inequalities play a very important role in many fields, such as variational inequalities, game theory, mathematical economics, control theory, and fixed point theory. Because of its wide applications, Ky Fan minimax inequalities have been generalized in various ways. Since 1960s, Ky Fan minimax theorems of the real-valued functions have been discussed, such as [1–4] and references therein.

*H*-spaces by using a generalized Fan's section theorem and a generalized Browder's fixed point theorem. Chang et al. [6] obtained a Ky Fan minimax inequality for vector-valued mappings on

*W*-spaces by applying a generalized section theorem and a generalized fixed point theorem. Li and Wang [7] established the following Ky Fan minimax inequalities for vector-valued mappings:

Luo [8] also obtained some generalized Ky Fan minimax inequalities for vector-valued mappings by applying the classical Browder fixed point theorem and the Kakutani-Fan-Glicksberg fixed point theorem.

There are also many articles to study the minimax theorems for vector-valued mappings.

*f*(

*x, y*) =

*x*+

*y*. Tanaka [10–12] obtained minimax theorems of the separated vector-valued function of the type

*f*(

*x, y*) =

*u*(

*x*) +

*v*(

*y*) and investigated some existence results of cone saddle points for general vector-valued functions. Furthermore, by using the existence results of cone saddle points for vector-valued mappings, he obtained the following result:

Shi and Ling [13] proved, respectively, a minimax theorem and a cone saddle point theorem for a class of vector-valued functions, which include the separated functions as its proper subset. Ferro [14, 15] studied minimax theorems for general vector-valued functions. Gong [16] obtained a strong minimax theorem and established an equivalent relationship between the strong minimax inequality and a strong cone saddle point theorem for vector-valued functions. Li et al. [17] investigated a minimax theorem and a saddle point theorem for vector-valued functions in the sense of lexicographic order, respectively.

To the best of authors' knowledge, there are few articles to investigate minimax problems for set-valued mappings. Li et al. [18] obtained some minimax inequalities for set-valued mappings by using a section theorem and a linear scalarization function. Li et al. [19] studied some generalized minimax theorems for set-valued mappings by using a nonlinear scalarization function. Zhang et al. [20] investigated some minimax problems for set-valued mappings by applying the Fan-Browder Fixed Point Theorem. Motivated by the study of [7, 10, 13, 18, 20], we obtain two types of Ky Fan minimax inequalities for set-valued mappings.

The rest of the article is organized as follows. In Section 2, we introduce notations and preliminary results. In Section 3, we obtain two types of Ky Fan minimax inequalities for set-valued mappings. We also give some examples to illustrate our results.

## 2 Preliminaries

Let *X* and *V* be real Hausdorff topological vector spaces. Assume that *S* is a pointed closed convex cone in *V* with its interior $\text{int}\phantom{\rule{1em}{0ex}}S\ne \mathrm{\varnothing}$. Some fundamental terminologies are presented as follows.

**Definition 2.1** [21] Let *A* ⊂ *V* be a nonempty subset.

**(i)** A point *z* ∈ *A* is said to be a minimal point of *A* iff *A*⋂(*z-S*) = {*z*}, and Min *A* denotes the set of all minimal points of *A*.

**(ii)** A point *z* ∈ *A* is said to be a weakly minimal point of *A* iff *A*⋂(*z -* int*S*) = Ø, and Min_{
w
}*A* denotes the set of all weakly minimal points of *A*.

**(iii)** A point *z* ∈ *A* is said to be a maximal point of *A* iff *A*⋂(*z* + *S*) = {*z*}, and Max*A* denotes the set of all maximal points of *A*.

**(iv)** A point *z* ∈ *A* is said to be a weakly maximal point of *A* iff *A* ⋂(*z* + int*S*) = Ø, and Max_{
w
}denotes the set of all weakly maximal points of *A*.

**Definition 2.2** [22] Let *F* : *X* → 2^{
V
}be a set-valued mapping with nonempty values.

**(i)**

*F*is said to be upper semicontinuous (u.s.c.) at

*x*

_{0}∈

*X*, iff for any neighborhood

*N*(

*F*(

*x*

_{0})) of

*F*(

*x*

_{0}), there exists a neighborhood

*N*(

*x*

_{0}) of

*x*

_{0}such that

**(ii)**

*F*is said to be lower semicontinuous (l.s.c.) at

*x*

_{0}∈

*X*, iff for any open neighborhood

*N*in

*V*satisfying

*F*(

*x*

_{0}) ⋂

*N*≠ Ø, there exists a neighborhood

*N*(

*x*

_{0}) of

*x*

_{0}such that

**(iii)** *F* is said to be continuous at *x*_{0} ∈ *X* iff *F* is both u.s.c. and l.s.c. at *x*_{0}.

**Remark 2.1** [22] The nonempty compact-valued mapping *F* is said to be u.s.c. at *x*_{0} ∈ *X*_{0} if and only if for any net {*x*_{
α
}} ⊂ *X* with *x*_{
α
}→ *x*_{0} and for any *y*_{
α
}∈ *F*(*x*_{
α
}), there exist *y*_{0} ∈ *F*(*x*_{0}) and a subnet {*y*_{
β
}} of {*y*_{
α
}}, such that *y*_{
β
}→ *y*_{0}.

**Definition 2.3** Let *X*_{0} be a nonempty convex subset of *X*, and let *F* : *X*_{0} → 2^{
V
}be a set-valued mapping with nonempty values.

**(i)**

*F*is said to be properly

*S*-quasiconvex on

*X*

_{0}, iff for any

*x*

_{1},

*x*

_{2}∈

*X*

_{0}and

*l*∈ [0, 1], either

*F* is said to be properly *S*-quasiconcave on *X*_{0}, iff - *F* is properly *S*-quasiconvex on *X*_{0}.

**(ii)**

*F*is said to be

*S*-quasiconvex [23] on

*X*

_{0}, iff for any point

*z*∈

*V*, the level set

is convex. *F* is said to be *S*-quasiconcave on *X*_{0}, iff - *F* is *S*-quasiconvex on *X*_{0}.

**Remark 2.2** If *F* is a vector-valued mapping, then properly *S*-quasiconvex reduces to the ordinary properly *S*-quasiconvex in [14].

**Lemma 2.1** *Let X*_{0} *be a compact subset of X. Suppose that F* : *X*_{0} × *X*_{0} → *2*^{
V
}*is a continuous set-valued mapping and for each* (*x, y*) ∈ *X*_{0} × *X*_{0}, *F*(*x, y*) *is a compact set. Then* $\Gamma \left(y\right)=\text{Mi}{\text{n}}_{w}{\bigcap}_{x\in {X}_{0}}F\left(x,y\right)$ *and* $\Phi \left(x\right)=\text{Ma}{\text{x}}_{w}{\bigcup}_{y\in {X}_{0}}F\left(x,y\right)$ *are u.s.c. and compact-valued on X*_{0}, *respectively*.

**Proof**. It follows from Lemma 2.2 in [18] that Γ and Φ are u.s.c. By the compactness of *X*_{0} and the closeness of weakly minimal (maximal) point sets, Γ and Φ are also compact-valued.

□

**Lemma 2.2** [22] *Let X*_{0} *be a nonempty subset of X, and let F* : *X*_{0} → 2^{
V
}*be a set-valued mapping with nonempty values. If X*_{0} *is compact and if F is u.s.c. and compact-valued, then* $F\left({X}_{0}\right)={\bigcup}_{x\in {X}_{0}}F\left(x\right)$ *is compact*.

**Lemma 2.3** [14] *Let A* ⊂ *V be a nonempty compact subset. Then* (*i*) Min *A* ≠ Ø; (*ii*) *A* ⊂ Min *A* + *S;* (*iii*) *A* ⊂ *Min*_{
w
}*A* + int*S*∪{0_{
V
}}; (*iv*) Max*A* ≠ Ø; (*v*) *A* ⊂ Max*A* - *S; and* (*vi*) *A* ⊂ Max_{
w
}*A* - int*S*∪{0_{
V
}}.

**Lemma 2.4** [24] *(Kakutani-Fan-Glicksberg fixed point theorem) Let X*_{0} *be a nonempty compact convex subset of X. If* $T:{X}_{0}\to {2}^{{X}_{0}}$ *is u.s.c, and for any x* ∈ *X*_{0}, *T*(*x*) *is a nonempty, closed and convex set, then T has a fixed point*.

## 3 Ky Fan minimax inequalities for set-valued mappings

First, we prove the following interesting lemma.

**Lemma 3.1** *Let X*_{0} *be a nonempty compact convex subset of X, and let F* : *X*_{0} × *X*_{0} → 2^{
V
}*be a continuous set-valued mapping with nonempty compact values*.

**(i)**

*If for each x*∈

*X*

_{0},

*F*(

*x*, ⋅)

*is properly S-quasiconcave on X*

_{0},

*then there exists*$\stackrel{\u0304}{x}\in {X}_{0}$

*such that*

**(ii)**

*If for each y*∈

*X*

_{0},

*F*(⋅,

*y*)

*is properly S-quasiconvex on X*

_{0},

*then there exists*$\mathit{\u0233}\in {X}_{0}$

*such that*

**Proof**. (i) We define a multifunction $T:{X}_{0}\to {2}^{{X}_{0}}$ by the formula

*T*(

*x*) ≠ Ø, for each

*x*∈

*X*

_{0}. Since

*F*(

*x*, ⋅) is u.s.c. with compact values and

*X*

_{0}is compact, by Lemma 2.2, ${\bigcup}_{y\in {X}_{0}}F\left(x,y\right)$ is a compact set for each

*x*∈

*X*

_{0}. By Lemma 2.3, $\text{Ma}{\text{x}}_{w}{\bigcup}_{y\in {X}_{0}}F\left(x,y\right)\ne \varnothing $. For each

*x*∈

*X*

_{0}, let ${z}_{x}\in \text{Ma}{\text{x}}_{w}{\bigcup}_{y\in {X}_{0}}F\left(x,y\right)$. Then, there exists

*y*

_{ x }∈

*X*

_{0}such that

*z*

_{ x }∈

*F*(

*x*,

*y*

_{ x }). Namely,

Hence, for each *x* ∈ *X*_{0}, *T*(*x*) ≠ Ø.

*T*(

*x*) is a closed set, for each

*x*∈

*X*

_{0}. Let a net {

*y*

_{ α }:

*α*∈

*I*} ⊂

*T*(

*x*), for each

*x*∈

*X*

_{0}and

*y*

_{ α }→

*y*

_{0}. By the definition of

*T*, there exists {

*z*

_{ α }} such that

*z*

_{ α }∈

*F*(

*x*,

*y*

_{ α }) and ${z}_{\alpha}\in \text{Ma}{\text{x}}_{w}{\bigcup}_{y\in {X}_{0}}F\left(x,y\right)$. Since

*F*(

*x*, ⋅) is u.s.c. with nonempty compact values, by Remark 2.1, there exist a subnet {

*z*

_{ β }} of {

*z*

_{ α }} and

*z*

_{0}∈

*F*(

*x, y*

_{0}) satisfying

*z*

_{ β }→

*z*

_{0}. By the closeness of the weakly maximal point set, ${z}_{0}\in \text{Ma}{\text{x}}_{w}{\bigcup}_{y\in {X}_{0}}F\left(x,y\right)$. Thus, we have that

and hence for each *x* ∈ *X*_{0}, *T*(*x*) is a closed set.

*T*(

*x*) is a convex set, for each

*x*∈

*X*

_{0}. For each

*x*∈

*X*

_{0}, let

*y*

_{1},

*y*

_{2}∈

*T*(

*x*) and

*l*∈ [0, 1]. Suppose that there exists

*l*

_{0}∈ [0, 1] such that

*F*(

*x*,

*l*

_{0}

*y*

_{1}+ (1 -

*l*

_{0})

*y*

_{2}) ⊂

*F*(

*x*,

*X*

_{0}), by Lemma 2.3,

*z*

_{1},

*z*

_{2}∈

*V*such that

*z*

_{1},

*z*

_{2}, there exist ${z}_{1}^{\prime},{z}_{2}^{\prime}\in \text{Ma}{\text{x}}_{w}{\bigcup}_{y\in {X}_{0}}F\left(x,y\right)$ such that either

Clearly, this is a contradiction. Therefore, (2) holds, which also contradicts the assumption about *y*_{1} and *y*_{2}. Hence, *T*(*x*) is a convex set, for each *x* ∈ *X*_{0}.

*T*is u.s.c. on

*X*

_{0}. Since

*X*

_{0}is compact, we only need to show that

*T*is a closed map (see [22]). Let a net

*x*

_{ α },

*y*

_{ α }) → (

*x*

_{0}.

*y*

_{0}). By the definition of

*T*, there exists {

*z*

_{ α }} satisfying

*z*

_{ α }∈

*F*(

*x*

_{ α },

*y*

_{ α }) and ${z}_{\alpha}\in \text{Ma}{\text{x}}_{w}{\bigcup}_{y\in {X}_{0}}F\left({x}_{\alpha},y\right)$. By assumptions and Lemma 2.2, {

*z*

_{ α }} must have a convergence subnet. For convenience, let the convergence subnet be itself. Since

*F*is u.s.c. with nonempty compact values, by Remark 2.1, there exist a subnet {

*z*

_{ β }} of {

*z*

_{ α }} and

*z*

_{0}∈

*F*(

*x*

_{0},

*y*

_{0}) satisfying

*z*

_{ γ }} of {

*z*

_{ α }} and ${z}_{0}^{\prime}\in {\text{Max}}_{w}{\bigcup}_{y\in {X}_{0}}F\left({x}_{0},y\right)$ satisfying

Clearly, ${z}_{0}={z}_{0}^{\prime}$. That is (*x*_{0}, *y*_{0}) ∈ Graph*T*. Hence, *T* is u.s.c. on *X*_{0}.

- (ii)We also define a multifunction $W:{X}_{0}\to {2}^{{X}_{0}}$ by the formula$W\left(y\right)=\left\{x\in {X}_{0}:F\left(y,y\right)\bigcap \text{Mi}{\text{n}}_{w}\bigcup _{x\in {X}_{0}}F\left(x,y\right)\ne \varnothing \right\},\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{2.77695pt}{0ex}}y\in {X}_{0}.$

□

**Remark 3.1** *When F is a real-valued function, Lemma 3.1 (i) reduces to Lemma 6 in* [7].

**Theorem 3.1**
*Let X*
_{0}
*be a nonempty compact convex subset of X. Suppose that the following conditions are satisfied:*

**(i)** *F* : *X*_{0} × *X*_{0} → 2^{
V
}*is a continuous set-valued mapping with nonempty compact values;*

**(ii)** *for each x* ∈ *X*_{0}, *F*(*x*, ⋅) *is properly S-quasiconcave on X*_{0}.

*Then,*

*such that*

**Proof**. By assumptions and Lemmas 2.1-2.3,

By Lemmas 2.1 and 2.2, ${\bigcup}_{x\in {X}_{0}}F\left(x,x\right)$ and ${\bigcup}_{x\in {X}_{0}}{\text{Max}}_{w}F\left(x,{X}_{0}\right)$ are two compact sets.

Particularly, taking *u* = *v*, we have *z*_{1} ∈ *z*_{2} + *S*. This completes the proof. □

**Corollary 3.1**
*Let X*
_{0}
*be a nonempty compact convex subset of X. Suppose that the following conditions are satisfied:*

**(i)** *f* : *X*_{0} × *X*_{0} → *V is a continuous vector-valued mapping;*

**(ii)** *for each x* ∈ *X*_{0}, *f*(*x*, ⋅) *is properly S-quasiconcave on X*_{0}.

*Then,*

*such that*

**Proof**. Since $\text{Min}{\bigcup}_{x\in {X}_{0}}\text{Ma}{\text{x}}_{w}f\left(x,{X}_{0}\right)\subset \phantom{\rule{2.77695pt}{0ex}}\text{Mi}{\text{n}}_{w}{\bigcup}_{x\in {X}_{0}}\text{Ma}{\text{x}}_{w}f\left(x,{X}_{0}\right)$, by the proof of Theorem 3.1, the conclusion follows readily. □

**Remark 3.2** Corollary 3.1 is different from Theorems 3 and 4 in [7] and Corollary 3.8 in [8]. The following example illustrates that when Theorem 3 in [7] and Corollary 3.8 in [8] are not applicable, Corollary 3.1 is applicable.

**Example 3.1**Let

*X*=

*R*,

*V*=

*R*

^{2},

*X*

_{0}= [0, 1] and

*S*= {(

*u*,

*v*)|

*u*≥ 0,

*v*≥ 0}. Let

*f*: [0, 1] × [0, 1] →

*R*

^{2}

*f*is continuous and

*f*(

*x*, ⋅) is properly

*S*-quasiconcave for each

*x*∈

*X*

_{0}. All conditions of Corollary 3.1 are satisfied. So, inclusion (4) holds. Indeed, by the definition of

*f*, we have

*x*∈

*X*

_{0},

*x*

_{0}≠ 1, we have

Namely, the condition (iii) of Theorem 3 in [7] and the condition (ii) of Corollary 3.8 in [8] do not hold. So, Theorem 3 in [7] and Corollary 3.8 in [8] are not applicable.

**Theorem 3.2**
*Let X*
_{0}
*be a nonempty compact convex subset of X. Suppose that the following conditions are satisfied:*

**(i)** *F* : *X*_{0} × *X*_{0} → 2^{
V
}*is a continuous set-valued mapping with nonempty compact values;*

**(ii)** *for each y* ∈ *X*_{0}, *F*(⋅,*y*) *is properly S-quasiconvex on X*_{0}.

*Then,*

*such that*

**Proof**. By assumptions and Lemmas 2.1-2.3,

Particularly, taking *u* = *v*, we have *z*_{1} ∈ *z*_{2} - *S*. This completes the proof. □

**Theorem 3.3**
*Let X*
_{0}
*be compact convex subset of X. Suppose that the following conditions are satisfied:*

**(i)** *F* : *X*_{0} × *X*_{0} → 2^{
V
}*is a continuous set-valued mapping with nonempty compact values;*

**(ii)**

*for each x*∈

*X*

_{0},

*and any*$z\in \text{Min}{\bigcup}_{x\in {X}_{0}}\text{Ma}{\text{x}}_{w}F\left(x,{X}_{0}\right)$,

*the level set*

*is convex*.

**(iii)**

*for any x*∈

*X*

_{0},

*Then,*

**Proof**. By assumptions and Lemmas 2.1-2.3,

Obviously, by the conditions (ii) and (iii), we have that *W*(*x*) is a nonempty convex set, for all *x* ∈ *X*_{0}.

*W*(

*x*) is a closed set, for any

*x*∈

*X*

_{0}. Let a net {

*y*

_{ α }:

*α*∈

*I*} ⊂

*W*(

*x*), for each

*x*∈

*X*

_{0}and

*y*

_{ α }→

*y*

_{0}. By the definition of

*W*, there exists {

*z*

_{ α }} such that

*z*

_{ α }∈

*F*(

*x*,

*y*

_{ α }) and

*z*

_{ α }∈

*β*+

*S*. Since

*F*(

*x*, ⋅) is u.s.c. with compact values, by Remark 2.1, there exist a subnet {

*z*

_{ β }} of {

*z*

_{ α }} and

*z*

_{0}∈

*F*(

*x*,

*y*

_{0}) satisfying

*z*

_{ β }→

*z*

_{0}. By the closeness of

*S*,

*z*

_{0}∈

*β*+

*S*. Thus, we have

and hence for each *x* ∈ *X*_{0}, *W*(*x*) is a closed set.

*W*is upper semicontinuous on

*X*

_{0}. Since

*X*

_{0}is compact, we only need to show

*W*is a closed map (see [22]). Let a net

and (*x*_{
α
}, *y*_{
α
}) → (*x*_{0}.*y*_{0}). By the definition of *W*, there exists {*z*_{
α
}} satisfying *z*_{
α
}∈ *F*(*x*_{
α
}, *y*_{
α
}) and *z*_{
α
}∈ *β* + *S*. Since *F* is u.s.c. with compact values, by Remark 2.1, there exist a subnet {*z*_{
γ
}} of {*z*_{
α
}} and *z*_{0} ∈ *F*(*x*_{0}, *y*_{0}) satisfying *z*_{
γ
}→ *z*_{0}. By the closeness of *S*, *z*_{0} ∈ *β* + *S*. That is (*x*_{0}*, y*_{0}) ∈ Graph*W*. Namely, *W* is upper semicontinuous on *X*_{0}.

Hence, inclusion (6) holds. This completes the proof. □

**Remark 3.3 (i)** The condition (ii) of Theorem 3.3 can be replaced by "for any *x* ∈ *X*_{0}, *F*(*x*, ⋅) is *S*-quasiconcave on *X*_{0}".

**(ii)** If *F* is a scalar set-valued mapping, the condition (iii) of Theorem 3.3 always holds.

**(iii)** When *F* is a vector-valued mapping, Theorem 3.3 reduces to corresponding ones in [7, 8].

**Theorem 3.4**
*Let X*
_{0}
*be a compact convex subset of X. Suppose that the following conditions are satisfied:*

**(i)** *F* : *X*_{0} × *X*_{0} → 2^{
V
}*is a continuous set-valued mapping with nonempty compact values;*

**(ii)**

*for each y*∈

*X*

_{0},

*and any*$z\in \text{Max}{\bigcup}_{y\in {X}_{0}}\text{Mi}{\text{n}}_{w}F\left({X}_{0},y\right)$,

*the level set*

*is convex*.

**(iii)**

*for any y*∈

*X*

_{0},

*Then,*

**Proof**. By assumptions and Lemmas 2.1-2.3,

From the proof process of Theorem 3.3, inclusion (8) holds. This completes the proof. □

**Remark 3.4 (i)** The condition (ii) of Theorem 3.4 can be replaced by "for any *y* ∈ *X*_{0}, *F*(⋅, *y*) is *S*-quasiconvex on *X*_{0}".

**(ii)** If *F* is a scalar set-valued mapping, the condition (iii) of Theorem 3.4 always holds.

Next, we give an example for explaining Theorem 3.4.

**Example 3.2**Let

*X*=

*R*,

*V*=

*R*

^{2},

*X*

_{0}= [0, 1] ⊂

*X*,

*S*= {(

*u*,

*v*)|

*u*≥ 0,

*v*≥ 0}, and

*M*= {(

*u*,

*v*)|0 ≤

*u*≤ 1,0 ≤

*v*≤ 1}. Let

*f*: [0, 1] × [0, 1] →

*R*

^{2}and $F:\left[0,1\right]\times \left[0,1\right]\to {2}^{{R}^{2}}$,

*F*is continuous with nonempty compact values and

*F*(⋅,

*y*) is

*S*-quasiconvex for every

*y*∈

*X*

_{0}. By the definition of

*F*,

*y*∈

*X*

_{0},

*y*∈

*X*

_{0},

## Declarations

### Acknowledgements

The authors would like to thank the anonymous referees for their valuable comments and suggestions, which helped to improve the article. This study was supported by the National Natural Science Foundation of China No. 11171362.

## Authors’ Affiliations

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