Minimax Theorems for Set-Valued Mappings under Cone-Convexities

Abstract

The aim of this paper is to study the minimax theorems for set-valued mappings with or without linear structure. We define several kinds of cone-convexities for set-valued mappings, give some examples of such set-valued mappings, and study the relationships among these cone-convexities. By using our minimax theorems, we derive some existence results for saddle points of set-valued mappings. Some examples to illustrate our results are also given.

1. Introduction

The minimax theorems for real-valued functions were introduced by Fan [1, 2] in the early fifties. Since then, these were extended and generalized in many different directions because of their applications in variational analysis, game theory, mathematical economics, fixed-point theory, and so forth (see, for example, [3–11] and the references therein). The minimax theorems for vector-valued functions have been studied in [4, 9, 10] with applications to vector saddle point problems. However, the minimax theorems for set-valued bifunctions have been studied only in few papers, namely, [4–8] and the references therein.

In this paper, we establish some new minimax theorems for set-valued mappings. Section 2 deals with preliminaries which will be used in rest of the paper. Section 3 denotes the cone-convexities of set-valued mappings. In Section 4, we establish some minimax theorems by using separation theorems, Fan-Browder fixed-point theorem. In the last section, we discuss some existence results for different kinds of saddle points for set-valued mappings.

2. Preliminaries

Throughout the paper, unless otherwise specified, we assume that 𝑋, 𝑌 are two nonempty subsets, and 𝒵 is a real Hausdorff topological vector space, 𝐶 is a closed convex pointed cone in 𝒵 with int𝐶≠∅. Let 𝒵^∗ be the topological dual space of 𝒵, and let 𝐶^∗=𝑔∈𝒵^∗∶𝑔(𝑐)≥0∀𝑐∈𝐶. (2.1)

We present some fundamental concepts which will be used in the sequel.

Definition 2.1 (see [3, 4, 8]). Let 𝐴 be a nonempty subset of 𝒵. A point 𝑧∈𝐴 is called a (a)minimal point of 𝐴 if 𝐴∩(𝑧−𝐶)={𝑧}; Min𝐴 denotes the set of all minimal points of 𝐴;(b)maximal point of 𝐴 if 𝐴∩(𝑧+𝐶)={𝑧}; Max𝐴 denotes the set of all maximal points of 𝐴;(c)weakly minimal point of 𝐴 if 𝐴∩(𝑧−int𝐶)=∅; Min_𝑤𝐴 denotes the set of all weakly minimal points of 𝐴;(d)weakly maximal point of 𝐴 if 𝐴∩(𝑧+int𝐶)=∅; Max_𝑤𝐴 denotes the set of all weakly maximal points of 𝐴.

It can be easily seen that Min𝐴⊂Min_𝑤𝐴 and Max𝐴⊂Max_𝑤𝐴.

Lemma 2.2 (see [3, 4]). Let 𝐴 be a nonempty compact subset of 𝒵. Then, (a)Min𝐴≠∅; (b)A⊂MinA+𝐶; (c)Max𝐴≠∅; (d)𝐴⊂Max𝐴−𝐶.

Following [6], we denote both Max and Max_𝑤 by max (both Min and Min_𝑤 by min) in ℝ since both Max and Max_𝑤 (both Min and Min_𝑤) are the same in ℝ.

Definition 2.3. Let 𝒳, 𝒴 be Hausdorff topological spaces. A set-valued map 𝐹∶𝒳⇉𝒴 with nonempty values is said to be (a)upper semicontinuous at 𝑥₀∈𝒳 if for every 𝑥₀∈𝒳 and for every open set 𝑁 containing 𝐹(𝑥₀), there exists a neighborhood 𝑀 of 𝑥₀ such that 𝐹(𝑀)⊂𝑁;(b)lower semi-continuous at 𝑥₀∈𝒳 if for any sequence {𝑥_𝑛}⊂𝒳 such that 𝑥_𝑛→𝑥₀ and any 𝑦₀∈𝐹(𝑥₀), there exists a sequence 𝑦_𝑛∈𝐹(𝑥_𝑛) such that 𝑦_𝑛→𝑦₀;(c)continuous at 𝑥₀∈𝒳 if 𝐹 is upper semi-continuous as well as lower semi-continuous at 𝑥₀.

We present the following fundamental lemmas which will be used in the sequel.

Lemma 2.4 (see [9, Lemma 3.1]). Let 𝒳, 𝒴, and 𝒵 be three topological spaces. Let 𝒴 be compact, 𝐹∶𝒳×𝒴⇉𝒵 a set-valued mapping, and the set-valued mapping 𝑇∶𝒳⇉𝒵 defined by 𝑇(𝑥)=_𝑦∈𝒴𝐹𝑥,𝑦,∀𝑥∈𝒳. (2.2)(a)If F is upper semi-continuous on 𝒳×𝒴, then T is upper semi-continuous on 𝒳. (b)If F is lower semi-continuous on 𝒳, so is T.

Lemma 2.5 (see [9, Lemma 3.2]). Let 𝒵 be a Hausdorff topological vector space, 𝐹∶𝒵⇉ℝ a set-valued mapping with nonempty compact values, and the functions 𝑝, 𝑞∶𝒵→ℝ defined by 𝑝(𝑧)=max𝐹(𝑧) and 𝑞(𝑧)=min𝐹(𝑧). (a)If 𝐹 is upper semi-continuous, so is 𝑝. (b)If 𝐹 is lower semi-continuous, so is 𝑝. (c)If 𝐹 is continuous, so are 𝑝 and 𝑞.

We shall use the following nonlinear scalarization function to establish our results.

Definition 2.6 (see [6, 10]). Let 𝑘∈int𝐶 and 𝑣∈𝒵. The Gerstewitz function 𝜉_𝑘𝑣∶𝒵→ℝ is defined by 𝜉_𝑘𝑣(𝑢)=min{𝑡∈ℝ∶𝑢∈𝑣+𝑡𝑘−𝐶}. (2.3)

We present some fundamental properties of the scalarization function.

Proposition 2.7 (see [6, 10]). Let 𝑘∈int𝐶 and 𝑣∈𝒵. The Gerstewitz function 𝜉_𝑘𝑣∶𝒵→ℝ has the following properties: (a)𝜉_𝑘𝑣(𝑢)<𝑟⇔𝑢∈𝑣+𝑟𝑘−int𝐶; (b)𝜉_𝑘𝑣(𝑢)≤𝑟⇔𝑢∈𝑣+𝑟𝑘−𝐶; (c)𝜉_𝑘𝑣(𝑢)=0⇔𝑢∈𝑣−𝜕𝐶, where 𝜕𝐶 is the topological boundary of 𝐶; (d)𝜉_𝑘𝑣(𝑢)>𝑟⇔𝑢∉𝑣+𝑟𝑘−𝐶; (e)𝜉_𝑘𝑣(𝑢)≥𝑟⇔𝑢∉𝑣+𝑟𝑘−int𝐶; (f)𝜉_𝑘𝑣(⋅) is a convex function; (g)𝜉_𝑘𝑣(⋅) is an increasing function, that is, 𝑢₂−𝑢₁∈int𝑆⇒𝜉_𝑘𝑣(𝑢₁)<𝜉_𝑘𝑣(𝑢₂); (h)𝜉_𝑘𝑣(⋅) is a continuous function.

Theorem 2.8 ( Fan-Browder fixed-point theorem (see [12])). Let 𝑋 be a nonempty compact convex subset of a Hausdorff topological vector space and let 𝑇∶𝑋⇉𝑋 be a set-valued mapping with nonempty convex values and open fibers, that is, 𝑇⁻¹(𝑦)={𝑥∈𝑋∶𝑦∈𝑇(𝑥)} is open for all 𝑦∈𝑋. Then, 𝑇 has a fixed point.

3. Cone-Convexities

In this section, we present different kinds of cone-convexities for set-valued mappings and give some relations among them. Some examples of such set-valued mappings are also given.

Definition 3.1. Let 𝑋 be a nonempty convex subset of a topological vector space 𝒲. A set-valued mapping 𝐹∶𝑋⇉𝒵 is said to be (a) above -𝐶-convex [4] (resp., above-𝐶-concave [5]) on 𝑋 if for all 𝑥₁,𝑥₂∈𝑋 and all 𝜆∈[0,1], 𝐹𝜆𝑥₁+(1−𝜆)𝑥₂⊂𝜆𝐹𝑥₁+(1−𝜆)𝐹𝑥₂𝑥−𝐶,resp.,𝜆𝐹₁+(1−𝜆)𝐹𝑥₂⊂𝐹𝜆𝑥₁+(1−𝜆)𝑥₂−𝐶; (3.1)(b)below- 𝐶-convex [13] (resp., below- 𝐶-concave [9, 13]) on 𝑋 if for all 𝑥₁,𝑥₂∈𝑋 and all 𝜆∈[0,1], 𝜆𝐹𝑥₁+(1−𝜆)𝐹𝑥₂⊂𝐹𝜆𝑥₁+(1−𝜆)𝑥₂+𝐶resp.,𝐹𝜆𝑥₁+(1−𝜆)𝑥₂⊂𝜆𝐹𝑥₁+(1−𝜆)𝐹𝑥₂+𝐶; (3.2)(c)above- 𝐶-quasi-convex (resp., below- 𝐶-quasiconcave) [7, Definition 2.3] on 𝑋 if the set Lev_𝐹≤(𝑧)∶={𝑥∈𝑋∶𝐹(𝑥)⊂𝑧−𝐶}resp.,Lev_𝐹≥(𝑧)∶={𝑥∈𝑋∶𝐹(𝑥)⊂𝑧+𝐶}, (3.3) is convex for all 𝑧∈𝒵;(d)above-properly 𝐶-quasiconvex (resp., above-properly 𝐶-quasiconcave [6]) on 𝑋 if for all 𝑥₁,𝑥₂∈𝑋 and all 𝜆∈[0,1], either 𝐹𝜆𝑥₁+(1−𝜆)𝑥₂⊂𝐹𝑥₁𝑥−𝐶resp.,𝐹₁⊂𝐹𝜆𝑥₁+(1−𝜆)𝑥₂−𝐶 (3.4) or 𝐹𝜆𝑥₁+(1−𝜆)𝑥₂⊂𝐹𝑥₂𝑥−𝐶resp.,𝐹₂⊂𝐹𝜆𝑥₁+(1−𝜆)𝑥₂−𝐶; (3.5)

(e)below-properly 𝐶-quasiconvex [7] (resp., below-properly 𝐶-quasiconcave) on 𝑋 if for all 𝑥₁,𝑥₂∈𝑋 and all 𝜆∈[0,1], either 𝐹𝑥₁⊂𝐹𝜆𝑥₁+(1−𝜆)𝑥₂+𝐶resp.,𝐹𝜆𝑥₁+(1−𝜆)𝑥₂⊂𝐹𝑥₁+𝐶 (3.6) or 𝐹𝑥₂⊂𝐹𝜆𝑥₁+(1−𝜆)𝑥₂+𝐶resp.,𝐹𝜆𝑥₁+(1−𝜆)𝑥₂⊂𝐹𝑥₂+𝐶; (3.7)(f) above-naturally 𝐶-quasiconvex [6] on 𝑋 if for all 𝑥₁,𝑥₂∈𝑋 and all 𝜆∈[0,1], 𝐹𝜆𝑥₁+(1−𝜆)𝑥₂⊂co𝐹𝑥₁∪𝐹𝑥₂−𝐶, (3.8) where co𝐴 denotes the convex hull of a set 𝐴;(g)above -𝐶-convex-like (resp., above- 𝐶-concave-like) on 𝑋 (𝑋 is not necessarily convex) if for all 𝑥₁,𝑥₂∈𝑋 and all 𝜆∈[0,1], there is an 𝑥^′∈𝑋 such that 𝐹𝑥^′⊂𝜆𝐹𝑥₁+(1−𝜆)𝐹𝑥₂𝑥−𝐶resp.,𝜆𝐹₁+(1−𝜆)𝐹𝑥₂⊂𝐹𝑥^′−𝐶; (3.9)(h)below -𝐶-convex-like [13] (resp., below -𝐶-concave-like) on 𝑋 (𝑋 is not necessarily convex) if for all 𝑥₁,𝑥₂∈𝑋 and all 𝜆∈[0,1], there is an 𝑥^′∈𝑋 such that 𝜆𝐹𝑥₁+(1−𝜆)𝐹𝑥₂⊂𝐹𝑥^′𝑥+𝐶resp.,𝐹^′⊂𝜆𝐹𝑥₁+(1−𝜆)𝐹𝑥₂+𝐶. (3.10)

It is obvious that every above-𝐶-convex set-valued mapping or above-properly 𝐶-quasi-convex set-valued mapping is an above-naturally 𝐶-quasi-convex set-valued mapping, and every above-𝐶-convex (above-𝐶-concave) set-valued mapping is an above-𝐶-convex-like (above-𝐶-concave-like) set-valued mapping. Similar relations hold for cases below.

Remark 3.2. The definition of above-properly 𝐶-quasi-convex (above-properly 𝐶-quasi-concave) set-valued mapping is different from the one mentioned in [7, Definition 2.3] or [5, 6]. The following Examples 3.3 and 3.4 illustrate the reason why they are different from the one mentioned in [5–7]. However, if 𝐹 is a vector-valued mapping or a single-valued mapping, both mappings reduce to the ordinary definition of a properly 𝐶-quasi-convex mapping for vector-valued functions [7]. The above-𝐶-convexity in Definition 3.1 is also different from the below-𝐶-convexity used in [5, 9].

Example 3.3. Consider 𝐶={(𝑠,𝑡)∈ℝ²∶𝑠≥0,𝑡≥0}. Let 𝐹∶[𝑥₁,𝑥₂]⊂ℝ⇉ℝ² be a set-valued mapping defined by 𝐹𝑥₁∶=(𝑠,𝑡)∈ℝ²∶(𝑠−2)²+(𝑡−4)²)},𝐹𝑥=1,2≤𝑠≤3,4≤𝑡≤5{(𝑠,5∶−1≤𝑠≤2₂∶=(𝑠,𝑡)∈ℝ²∶(𝑠−6)²+(𝑡+2)²=1,6≤𝑠≤7,−2≤𝑡≤−1, (3.11) and for all 𝜆∈(0,1), 𝐹𝜆𝑥₁+(1−𝜆)𝑥₂∶=(𝑠,𝑡)∈ℝ²∶(𝑠−2)²+(𝑡−2)²=4,0≤𝑠≤2,0≤𝑡≤2. (3.12) Then 𝐹 is an above-properly 𝐶-quasi-convex set-valued mapping, but it is not below-properly 𝐶-quasi-convex.

On the other hand, let 𝐺∶[𝑥₁,𝑥₂]⊂ℝ⇉ℝ² be a set-valued mapping defined by 𝐺𝑥₁∶=(𝑠,𝑡)∈ℝ²∶(𝑠−1)²+(𝑡−4)²,𝐺𝑥=1,1≤𝑠≤2,4≤𝑡≤5₂∶=(𝑠,𝑡)∈ℝ²∶(𝑠−6)²+(𝑡+2)²=1,6≤𝑠≤7,−2≤𝑡≤−1, (3.13) and for all 𝜆∈(0,1), 𝐺𝜆𝑥₁+(1−𝜆)𝑥₂∶=(𝑠,𝑡)∈ℝ²∶(𝑠−2)²+(𝑡−2)²)}.=4,0≤𝑠≤2,0≤𝑡≤2{(𝑠,0∶2≤𝑠≤3 (3.14) Then, 𝐺 is a below-properly 𝐶-quasi-convex set-valued mapping, but it is not above-properly 𝐶-quasi-convex.

Example 3.4. Let 𝐶={(𝑠,𝑡)∶𝑠≥0,𝑡≥0}. Define 𝐹∶[−1,1]⇉ℝ² by 𝐹(𝑥)=(𝑥,𝑡)∶1−𝑥²≤𝑡≤1,∀𝑥∈[−1,1]. (3.15) Then 𝐹 is continuous, above-𝐶-quasi-convex, below-𝐶-quasi-concave, above-properly 𝐶-quasi-convex, and above-properly 𝐶-quasi-concave, but it is not below-properly 𝐶-quasi-conconvex.

Proposition 3.5. Let 𝑋 be a nonempty set (not necessarily convex) and for a given set-valued mapping 𝐹∶𝑋⇉𝒵 with nonempty compact values, define a set-valued mapping 𝑀∶𝑋⇉𝒵 as 𝑀(𝑥)=Max_𝑤𝐹(𝑥),∀𝑥∈𝑋. (3.16)(a)If 𝐹is above-𝐶-convex-like, then 𝑀 is so. (b)If 𝑋 is a topological space and 𝐹 is a continuous mapping, then 𝑀 is upper semicontinuous with nonempty compact values on 𝑋.

Proof. (a) Let 𝐹 be above-𝐶-convex-like, and let 𝑥₁,𝑥₂∈𝑋 be arbitrary. Since 𝐹 is above-𝐶-convex-like, for any 𝛼∈[0,1], there exists 𝑥^′∈𝑋 such that 𝐹𝑥^′⊂𝛼𝐹𝑥₁+(1−𝛼)𝐹𝑥₂−𝐶. (3.17) By Lemma 2.2, Max_𝑤𝐹𝑥^′⊂𝛼Max_𝑤𝐹𝑥₁+(1−𝛼)Max_𝑤𝐹𝑥₂−𝐶. (3.18) Therefore, 𝑥↦Max_𝑤𝐹(𝑥) is above-C-convex-like.
(b) The upper semicontinuity of 𝑀 was deduced in [4, Lemma 2].

Proposition 3.6. Let 𝑋 be a nonempty convex set, and let 𝐹∶𝑋⇉𝒵 be a set-valued mapping with nonempty compact values. Then, the set-valued mapping 𝑀∶𝑋⇉𝒵 defined by 𝑀(𝑥)=Max_𝑤𝐹(𝑥),∀𝑥∈𝑋, (3.19) is above-𝐶-quasiconvex if 𝐹 is so.

The following result can be easily derived, and therefore, we omit the proof.

Proposition 3.7. Let 𝑋 be a nonempty convex set and 𝐹∶𝑋⇉ℝ be above-ℝ₊-concave. Then the set-valued mapping 𝑥↦max𝐹(𝑥) is above-ℝ₊-concave and below-ℝ₊-quasiconcave. Furthermore, if 𝐹∶𝑋⇉ℝ is above-properly ℝ₊-quasiconcave, then the set-valued mapping 𝑥↦max𝐹(𝑥) is also above-properly ℝ₊-quasiconcave and below-ℝ₊-quasiconcave.

Let 𝜉∈𝐶^∗ and 𝐹∶𝑋⇉𝒵 be a set-valued mapping. Then, the composition mapping 𝜉∘𝐹∶𝑋⇉ℝ is defined by (𝜉∘𝐹)(𝑥)=𝜉(𝐹(𝑥))=_{𝑦∈𝐹(𝑥)}𝜉𝑦,∀𝑥∈𝑋. (3.20) Clearly, the composition mapping 𝜉∘𝐹∶𝑋⇉ℝ is also a set-valued mapping.

Proposition 3.8. Let 𝑋 be a nonempty set, 𝐹∶𝑋⇉𝒵 a set-valued mapping, and 𝜉∈𝐶^∗. (a)If 𝐹 is above-𝐶-convex-like, then 𝜉∘𝐹 is above-ℝ₊-convex-like. (b)If 𝐹 is below-𝐶-concave-like, then 𝜉∘𝐹 is below-ℝ₊-concave-like. (c)If 𝑋 is a topological space and F is upper semi-continuous, then so is 𝜉∘𝐹.

Proof. (a) By the definition of above-𝐶-convex-like set-valued mapping 𝐹∶𝑋⇉𝒵, for any 𝑥₁,𝑥₂∈𝑋 and all 𝜆∈[0,1], there exists 𝑥^′∈𝑋 such that 𝐹(𝑥^′)⊂𝜆𝐹(𝑥₁)+(1−𝜆)𝐹(𝑥₂)−𝐶. For any 𝑦∈𝐹(𝑥^′), there exist 𝑦₁∈𝐹(𝑥₁), 𝑦₂∈𝐹(𝑥₂) such that 𝜆𝑦₁+(1−𝜆)𝑦₂∈𝑓−𝐶. (3.21) For any 𝜉∈C^∗, we have 𝜉(𝑦)≤𝜆𝜉(𝑦₁)+(1−𝜆)𝜉(𝑦₂). Hence, 𝜉(𝐹(𝑥^′))⊂𝜆𝜉(𝐹(𝑥₁))+(1−𝜆)𝜉(𝐹(𝑥₂))−ℝ₊. Thus, 𝜉∘𝐹 is above-ℝ₊-convex-like.
The proof of (b) and (c) is easy, and therefore, we omit it.

Proposition 3.9. Let 𝑋 be a nonempty convex set and 𝜉∈𝐶^∗. (a)If 𝐹∶𝑋⇉𝒵 is above-𝐶-concave (above-properly 𝐶-quasi-concave), then 𝜉∘𝐹∶𝑋⇉ℝ is above-ℝ₊-concave (above-properly ℝ₊-quasi-concave).(b)If F∶X⇉𝒵 is above-properly 𝐶-quasi-convex, then 𝜉∘𝐹∶𝑋⇉ℝ is above-ℝ₊-quasi-convex and above-properly ℝ₊-quasi-convex.(c)If 𝐹∶𝑋⇉𝒵 is above-𝐶-convex, then 𝜉∘𝐹∶𝑋⇉ℝ is above-ℝ₊-convex and above-ℝ₊-quasi-convex.

Lemma 3.10. Let 𝒵 be a real Hausdorff topological vector space and 𝐶 a closed convex pointed cone in 𝒵 with int𝐶≠∅. Let 𝑋 be a nonempty compact subset of a topological space 𝒳, and let 𝐹∶𝑋⇉𝒵 be an upper semi-continuous set-valued mapping with nonempty compact values. Then, for any 𝜉∈𝐶^∗, there exists 𝑦∈Max_𝑤𝐹(𝑋) such that 𝜉(𝑦)=max⋃_𝑥∈𝑋𝜉(𝐹(𝑥)).

Proof. For any given 𝜉∈𝐶^∗, the mapping 𝑥⇉𝜉(𝐹(𝑥)) is upper semi-continuous by Proposition 3.8 (c). By the compactness of 𝑋, there exist 𝑥₀∈𝑋 and 𝑦₀∈𝐹(𝑥₀) such that 𝜉(𝑦₀)=max⋃_𝑥∈𝑋𝜉(𝐹(𝑥)). By Lemma 2.2, there exists 𝑦∈Max_𝑤⋃_𝑥∈𝑋𝐹(𝑥) such that 𝑦₀−𝑦∈−𝐶, and hence 𝜉(𝑦)≥𝜉(𝑦₀). On the other hand, 𝑦∈Max_𝑤⋃_𝑥∈𝑋𝐹(𝑥)⊂𝐹(𝑋), we know that 𝜉(𝑦)∈𝜉(𝐹(𝑋)), and then 𝜉(𝑦)≤max⋃_𝑥∈𝑋𝜉(𝐹(𝑥))=𝜉(𝑦₀). Therefore, the conclusion holds.

Proposition 3.11. Let 𝑋 be a nonempty convex set. If 𝐹∶𝑋⇉𝒵 is above-properly 𝐶-quasi-convex, then it is above-𝐶-quasi-convex.

Proof. For any 𝑧∈𝒵, let 𝑥₁,𝑥₂∈Lev_𝐹≤(𝑧). Then, 𝐹(𝑥₁) and 𝐹(𝑥₂) are subsets of 𝑧−𝐶. Since 𝐹 is above-properly 𝐶-quasi-convex, for any 𝜆∈[0,1], 𝐹(𝜆𝑥₁+(1−𝜆)𝑥₂) is contained in either 𝐹(𝑥₁)−𝐶 or 𝐹(𝑥₂)−𝐶, and hence, in 𝑧−𝐶. Thus, the set Lev_𝐹≤(𝑧) is convex, and therefore, 𝐹 is above-𝐶-quasi-convex.

Proposition 3.12. Let 𝑋 be a nonempty convex set. If 𝐹∶𝑋⇉𝒵 is above-naturally 𝐶-quasi-convex, then it is above-𝐶-quasi-convex.

Proof. Let 𝑧, 𝑥₁, and 𝑥₂ be the same as given as in Proposition 3.11. Then, co{𝐹(𝑥₁)∪𝐹(𝑥₂)}⊂𝑧−𝐶 since 𝑧−𝐶 is convex. By the above-naturally 𝐶-quasi-convexity, 𝐹(𝜆𝑥₁+(1−𝜆)𝑥₂)}⊂𝑧−𝐶 for all 𝜆∈[0,1]. Thus, the set Lev_𝐹≤(𝑧) is convex, and therefore, 𝐹 is above-𝐶-quasi-convex.

Proposition 3.13. Let 𝑋 be a nonempty convex set. If 𝐹∶𝑋⇉𝒵 is above-naturally 𝐶-quasi-convex, then 𝜉∘𝐹 is above-naturally ℝ₊-quasi-convex for any 𝜉∈𝐶^∗.

Proof. Let 𝜉∈𝐶^∗ be given. From the above-naturally 𝐶-quasi-convexity of 𝐹, for any 𝑥₁,𝑥₂∈𝑋 and any 𝜆∈[0,1], 𝐹𝜆𝑥₁+(1−𝜆)𝑥₂⊂co𝐹𝑥₁∪𝐹𝑥₂−𝐶. (3.22) For any 𝑦∈𝐹(𝛼𝑥₁+(1−𝛼)𝑥₂), there is a 𝑤∈co{𝐹(𝑥₁)∪𝐹(𝑥₂)} such that 𝑦∈𝑤−𝐶. Then there exist 𝑦_𝑖∈𝐹(𝑥₁)∪𝐹(𝑥₂) and 𝜆_𝑖∈[0,1], 1≤𝑖≤𝑛 such that 𝑤=∑^𝑛_𝑖=1𝜆_𝑖𝑦_𝑖. Hence, 𝜉(𝑤)=∑^𝑛_𝑖=1𝜆_𝑖𝜉(𝑦_𝑖), and 𝜉𝑦∈𝜉(𝑤)−ℝ₊=^𝑛_𝑖=1𝜆_𝑖𝜉𝑦_𝑖−ℝ₊⊂co𝜉𝐹𝑥₁∪𝜉𝐹𝑥₂−ℝ₊. (3.23) Therefore, 𝜉∘𝐹 is a above-naturally ℝ₊-quasi-convex.

Proposition 3.14. Let 𝐹∶𝑋⇉𝒵 be a set-valued mapping with nonempty compact values. For any 𝜉∈𝐶^∗, (a)if 𝜉(𝑑)=min⋃_𝑥∈𝑋𝜉(𝐹(𝑥)) for some 𝑑∈𝒵, then 𝑑∈Min_𝑤⋃_𝑥∈𝑋𝐹(𝑥);(b)if 𝜉(𝑒)=max⋃_𝑥∈𝑋𝜉(𝐹(𝑥)) for some 𝑒∈𝒵, then 𝑒∈Max_𝑤⋃_𝑥∈𝑋𝐹(𝑥).

Proof. Let 𝜉(𝑑)=min⋃_𝑥∈𝑋𝜉(𝐹(𝑥)). Suppose that 𝑑∉Min_𝑤⋃_𝑥∈𝑋𝐹(𝑥). Then _𝑥∈𝑋𝐹(𝑥)(𝑑−int𝐶)≠∅. (3.24) Then, there exists 𝑤∈⋃_𝑥∈𝑋𝐹(𝑥) and 𝑤∈𝑑−int𝐶. Therefore, there exists 𝑠∈𝑋 such that 𝑤∈𝐹(𝑠) and 𝑑−𝑤∈int𝐶. Since 𝜉∈𝐶^∗, 𝜉(𝑑)>𝜉(𝑤) and 𝜉(𝑤)≥min⋃_𝑥∈𝑋𝜉(𝐹(𝑥)). This implies that 𝜉(𝑑)>min⋃_𝑥∈𝑋𝜉(𝐹(𝑥)), which is a contradiction. This proves (a).
Analogously, we can prove (b), so we omit it.

Remark 3.15. Propositions 3.8 and 3.9, Lemma 3.10, and Propositions 3.13 and 3.14 are always true except Proposition 3.8 (b) if we replace 𝜉 by any Gerstewitz function.

4. Minimax Theorems for Set-Valued Mappings

In this section, we establish some minimax theorems for set-valued mappings with or without linear structure.

Theorem 4.1. Let 𝑋, 𝑌 be two nonempty compact subsets (not necessarily convex) of real Hausdorff topological spaces 𝒳 and 𝒴, respectively. Let the set-valued mapping 𝐹∶𝑋×𝑌⇉ℝ be lower semi-continuous on 𝑋 and upper semi-continuous on 𝑌 such that for all (𝑥,𝑦)∈𝑋×𝑌, 𝐹(𝑥,𝑦) is nonempty compact and satisfies the following conditions: (i)for each 𝑥∈𝑋, y↦𝐹(𝑥,𝑦) is below-ℝ₊-concave-like on 𝑌; (ii)for each 𝑦∈𝑌, x↦𝐹(𝑥,𝑦) is above-ℝ₊-convex-like on 𝑋. Then, max_𝑦∈𝑌min_𝑥∈𝑋𝐹𝑥,𝑦=min_𝑥∈𝑋max_𝑦∈𝑌𝐹𝑥,𝑦. (4.1)

Proof. Since max_𝑦∈𝑌min_𝑥∈𝑋𝐹𝑥,𝑦≤min_𝑥∈𝑋max_𝑦∈𝑌𝐹𝑥,𝑦, (4.2) it is sufficient to prove that max_𝑦∈𝑌min_𝑥∈𝑋𝐹𝑥,𝑦≥min_𝑥∈𝑋max_𝑦∈𝑌𝐹𝑥,𝑦. (4.3) Choose any 𝛼∈ℝ such that 𝛼<min⋃_𝑥∈𝑋max⋃_𝑦∈𝑌𝐹(𝑥,𝑦). For any 𝑦∈𝑌, let Lev_𝐹≤𝑦;𝛼=𝑥∈𝑋∶𝐹𝑥,𝑦⊂𝛼−ℝ₊. (4.4) Then, by the lower semi-continuity of the set-valued mapping 𝑥↦𝐹(𝑥,𝑦), the set Lev_𝐹≤(𝑦;𝛼) is closed, hence it is compact for all 𝑦∈𝑌. By the choice of 𝛼, we have _𝑦∈𝑌Lev_𝐹≤𝑦;𝛼=∅. (4.5) Since 𝑋 is compact and the collection {𝑋⧵Lev_𝐹≤(𝑦;𝛼)∶𝑦∈𝑌} covers 𝑋, there exist finite number of points 𝑦₁,𝑦₂,…,𝑦_𝑚 in 𝑌 such that 𝑋⊂^𝑚_𝑖=1𝑋⧵Lev_𝐹≤𝑦_𝑖;𝛼 (4.6) or ^𝑚_𝑖=1Lev_𝐹≤𝑦_𝑖;𝛼=∅. (4.7) This implies that max^𝑚_𝑖=1𝐹𝑥,𝑦_𝑖>𝛼,∀𝑥∈𝑋, (4.8) and therefore, min_𝑥∈𝑋max^𝑚_𝑖=1𝐹𝑥,𝑦_𝑖>𝛼. (4.9) Following the idea of Borwein and Zhuang [14], let 𝔈∶=(𝐳,𝑟)∈ℝ^𝑚+1∶thereis𝑥∈𝑋,𝐹𝑥,𝑦_𝑖⊂𝑟+𝑧_𝑖−ℝ₊,𝑖=1,2,…,𝑚, (4.10) where 𝐳=(𝑧₁,𝑧₂,…,𝑧_𝑚). Then the set 𝔈 is convex, so is int𝔈. We note that the interior int𝔈 of 𝔈 is nonempty since 𝟎,1+max^𝑚_𝑖=1𝐹𝑥,𝑦_𝑖∈int𝔈,∀𝑥∈𝑋. (4.11) Since (𝟎,𝛼)∉𝔈, by separation hyperplane theorem [15, Theorem 14.2], there is a (Ξ,𝜖)≠𝟎×{0} such that ⟨(Ξ,𝜀),(𝐳,𝑟)⟩≥⟨(Ξ,𝜀),(𝟎,𝛼)⟩,∀(𝐳,𝑟)∈𝔈, (4.12) where Ξ=(𝜆₁,𝜆₂,…,𝜆_𝑚), that is, Ξ𝐳+𝜀r≥𝜀𝛼,∀(𝐳,𝑟)∈𝔈. (4.13) By (4.11), (4.13), and the choice of 𝛼, we have that 𝜀>0. Furthermore, from the fact ^𝑚_𝑖=1𝐹𝑥,𝑦_𝑖+𝑟×{−𝑟}⊂𝔈, (4.14) we have 𝜂_𝑥,1+𝑟,𝜂_𝑥,2+𝑟,…,𝜂_𝑥,𝑚+𝑟,−𝑟∈𝔈,∀𝜂_𝑥,𝑖∈𝐹𝑥,𝑦_𝑖. (4.15) Hence, by (4.13), we have ^𝑚_𝑖=1𝜆_𝑖𝜂_𝑥,𝑖+𝑟+𝜀(−𝑟)≥𝜀𝛼 (4.16) or ^𝑚_𝑖=1𝜆_𝑖𝜀𝜂_𝑥,𝑖+∑^𝑚_𝑖=1𝜆_𝑖𝜀−1𝑟≥𝛼,∀𝑥∈𝑋,𝑟∈ℝ. (4.17) Thus, we have ∑^𝑚_𝑖=1(𝜆_i/𝜀)=1. Hence, by (4.17), we have ^𝑚_𝑖=1𝜆_𝑖𝜀𝐹𝑥,𝑦_𝑖⊂𝛼+ℝ₊. (4.18) Since 𝐹(𝑥,𝑦) is below-ℝ₊-concave-like in 𝑦, there is 𝑦^′∈𝑌 such that 𝐹𝑥,𝑦^′⊂^𝑚_𝑖=1𝜆_𝑖𝜀𝐹𝑥,𝑦_𝑖+ℝ₊,∀𝑥∈𝑋. (4.19) Therefore, _𝑥∈𝑋𝐹𝑥,𝑦^′⊂𝛼+ℝ₊, (4.20) and hence, max_𝑦∈𝑌min_𝑥∈𝑋𝐹𝑥,𝑦≥𝛼. (4.21) This completes the proof.

Remark 4.2. Theorem 4.1 is a modification of [14, Theorem A]. If 𝐹 is a real-valued function, then Theorem 4.1 reduces to the well-known minimax theorem due to Fan [2].
We next establish a minimax theorem for set-valued mappings defined on the sets with linear structure.

Theorem 4.3. Let 𝑋, 𝑌 be two nonempty compact convex subsets of real Hausdorff topological vector spaces 𝒳 and 𝒴, respectively. Let the set-valued mapping 𝐹∶𝑋×𝑌⇉ℝ be lower semi-continuous on 𝑋 and upper semi-continuous on 𝑌 such that for all (𝑥,𝑦)∈𝑋×𝑌, 𝐹(𝑥,𝑦) is nonempty compact, and satisfies the following conditions: (i)for each 𝑦∈𝑌, 𝑥↦𝐹(𝑥,𝑦) is above-ℝ₊-quasi-convex on 𝑋; (ii)for each 𝑥∈𝑋, 𝑦↦𝐹(𝑥,𝑦) is above-ℝ₊-concave, or above-properly ℝ₊-quasi-concave on 𝑌; (iii)for each 𝑦∈𝑌, there is a 𝑥_𝑦∈𝑌 such that max𝐹𝑥_𝑦,𝑦≤max_𝑦∈𝑌min_𝑥∈𝑋𝐹𝑥,𝑦. (4.22)Then, min_𝑥∈𝑋max_𝑦∈𝑌𝐹𝑥,𝑦=max_𝑦∈𝑌min_𝑥∈𝑋𝐹𝑥,𝑦. (4.23)

Proof. We only need to prove that max_𝑦∈𝑌min_𝑥∈𝑋𝐹𝑥,𝑦<min_𝑥∈𝑋max_𝑦∈𝑌𝐹𝑥,𝑦 (4.24) is impossible, since it is always true that max_𝑦∈𝑌min_𝑥∈𝑋𝐹𝑥,𝑦≤min_𝑥∈𝑋max_𝑦∈𝑌𝐹𝑥,𝑦. (4.25) Suppose that there is an 𝛼∈ℝ such that max_𝑦∈𝑌min_𝑥∈𝑋𝐹𝑥,𝑦<𝛼<min_𝑥∈𝑋max_𝑦∈𝑌𝐹𝑥,𝑦. (4.26) Define 𝐺∶𝑋×𝑌⇉𝑋×𝑌 by 𝐺𝑥,𝑦=𝑠∈𝑋∶max𝐹𝑠,𝑦<𝛼×{𝑡∈𝑌∶max𝐹(𝑥,𝑡)>𝛼}. (4.27) For each 𝑥∈𝑋, max⋃_𝑦∈𝑌𝐹(𝑥,𝑦)≥min⋃_𝑥∈𝑋max⋃_𝑦∈𝑌𝐹(𝑥,𝑦)>𝛼. Since 𝑌 is compact and the set-valued mapping 𝑦↦max𝐹(𝑥,𝑦) is upper semi-continuous, there is a 𝑡∈𝑌 such that max𝐹(𝑥,𝑡)=max⋃_𝑦∈𝑌𝐹(𝑥,𝑦)>𝛼.
On the other hand, from the condition (iii), for each 𝑦∈𝑌, there is a 𝑥_𝑦∈𝑌 such that max𝐹(𝑥_𝑦,𝑦)<𝛼. Hence, for each (𝑥,𝑦)∈𝑋×𝑌, 𝐺(𝑥,𝑦)≠∅. By (i) and Proposition 3.6, the mapping 𝑥→max𝐹(𝑥,𝑦) is above-ℝ₊-quasi-convex on 𝑋. By (ii) and Proposition 3.7, the mapping 𝑦→max𝐹(𝑥,𝑦) is below-ℝ₊-quasi-concave on 𝑦. Hence, for each (𝑥,𝑦)∈𝑋×𝑌, the set𝐺(𝑥,𝑦) is convex. From the lower semi-continuities on 𝑋 and upper semi-continuity on 𝑌 of 𝐹, the set 𝐺⁻¹(𝑠,𝑡)={𝑥∈𝑋∶max𝐹(𝑥,𝑡)>𝛼}×𝑦∈𝑌∶max𝐹𝑠,𝑦<𝛼 (4.28) is open in 𝑋×𝑌. By Fan-Browder fixed-point Theorem 2.8, there exists (𝑥,𝑦)∈𝑋×𝑌 such that 𝑥,𝑦∈𝐺𝑥,𝑦, (4.29) that is, max𝐹𝑥,𝑦>𝛼>max𝐹𝑥,𝑦, (4.30) which is a contradiction. This completes the proof.

Remark 4.4. [5, Propositions 2.7 and 2.1] can be deduced from Theorem 4.3. Indeed, in [5, Proposition 2.1], the above-naturally 𝐶-quasi-convexity is used. By Proposition 3.12, the condition (i) of Theorem 4.3 holds. Hence the conclusion of Proposition 2.1 in [5] holds. We also note that, in Theorem 4.3, the mapping 𝐹 need not be continuous on 𝑋×𝑌. Hence Theorem 4.3 is a slight generalization of [7, Theorem 3.1].

Theorem 4.5. Let 𝑋 and 𝑌 be nonempty compact (not necessarily convex) subsets of real Hausdorff topological vector spaces 𝒳 and 𝒴, respectively. Let the mapping 𝐹∶𝑋×𝑌⇉𝒵 be upper semi-continuous with nonempty compact values and lower semi-continuous on 𝑋 such that (i)for each 𝑥∈𝑋, 𝑦→𝐹(𝑥,𝑦) is below-𝐶-concave-like on 𝑌; (ii)for each 𝑦∈𝑌, 𝑥→𝐹(𝑥,𝑦) is above-𝐶-convex-like on 𝑋;(iii)for every 𝑦∈𝑌, Max_𝑦∈𝑌Min_{𝑤𝑥∈𝑋}𝐹𝑥,𝑦⊂Min_{𝑤𝑥∈𝑋}𝐹𝑥,𝑦+𝐶. (4.31)Then for any 𝑧₁∈Max_𝑦∈𝑌Min_{𝑤𝑥∈𝑋}𝐹𝑥,𝑦, (4.32) there is a 𝑧₂∈Minco_𝑥∈𝑋Max_{𝑤𝑦∈𝑌}𝐹𝑥,𝑦 (4.33) such that 𝑧₁∈𝑧₂+𝐶, (4.34) that is, Max_𝑦∈𝑌Min_{𝑤𝑥∈𝑋}𝐹𝑥,𝑦⊂Minco_𝑥∈𝑋Max_{𝑤𝑦∈𝑌}𝐹𝑥,𝑦+𝐶. (4.35)

Proof. Let Γ(𝑥)∶=Max_𝑤⋃_𝑦∈𝑌𝐹(𝑥,𝑦) for all 𝑥∈𝑋. From Lemma 2.4 and Proposition 3.5, the set-valued mapping 𝑥↦Γ(𝑥) is upper semi-continuous with nonempty compact values on 𝑋. Hence the set Γ(𝑋) is compact, and so is co{Γ(𝑋)}. Then co{Γ(𝑋)}+𝐶 is a closed convex set with nonempty interior. Suppose that 𝑣∉co{Γ(𝑋)}+𝐶. By separation hyperplane theorem [15, Theorem 14.2], there exist 𝑘∈ℝ, 𝜀>0 and a nonzero continuous linear functional 𝜉∶𝑍→ℝ such that 𝜉(𝑣)≤𝑘−𝜀<𝑘≤𝜉(𝑢+𝑐),forevery𝑢∈co{Γ(𝑋)},𝑐∈𝐶. (4.36) Therefore, 𝜉(𝑐)>𝜉(𝑣−𝑢),forevery𝑢∈co{Γ(𝑋)},𝑐∈𝐶. (4.37) This implies that 𝜉∈𝐶^∗ and 𝜉(𝑣)<𝜉(𝑢) for all 𝑢∈co{Γ(𝑋)}.
Let 𝑔∶=𝜉𝐹∶𝑋×𝑌⇉ℝ. From Lemma 3.10, for each fixed 𝑥∈𝑋, there exist 𝑦^∗_𝑥∈𝑌 and 𝑓(𝑥,𝑦^∗_𝑥)∈𝐹(𝑥,𝑦^∗_𝑥) with 𝑓(𝑥,𝑦^∗_𝑥)∈Γ(𝑥) such that 𝜉(𝑓(𝑥,𝑦^∗_𝑥))=max⋃_𝑦∈𝑌𝜉(𝐹(𝑥,𝑦)). Choosing 𝑐=0 and 𝑢=𝑓(𝑥,𝑦^∗_𝑥) in (4.36), we have max_𝑦∈𝑋𝜉𝐹𝑥,𝑦=𝜉𝑓𝑥,𝑦^∗_𝑥≥𝑘>𝑘−𝜀≥𝜉(𝑣),∀𝑥∈𝑋. (4.38) Therefore, min_𝑥∈𝑋max_𝑦∈𝑌𝜉𝐹𝑥,𝑦>𝜉(𝑣). (4.39)
By the conditions (i), (ii) and Proposition 3.8, the set-valued mapping 𝑦↦𝜉(𝐹(𝑥,𝑦)) is below-ℝ₊-concave-like on 𝑌 for all 𝑥∈𝑋, and the set-valued mapping 𝑥↦𝜉(𝐹(𝑥,𝑦)) is above-ℝ₊-convex-like on 𝑋 for all𝑦∈𝑌. From Theorem 4.1, we have max_𝑦∈𝑌min_𝑥∈𝑋𝜉𝐹𝑥,𝑦>𝜉(𝑣). (4.40) Since 𝑌 is compact, there is an 𝑦^′∈𝑌 such that min⋃_𝑥∈𝑋𝜉(𝐹(𝑥,𝑦^′))>𝜉(𝑣). For any 𝑥∈𝑋 and all 𝑔(𝑥,𝑦^′)∈𝐹(𝑥,𝑦^′), we have 𝜉𝑔𝑥,𝑦^′>𝜉(𝑣), (4.41) that is, 𝜉𝑔𝑥,𝑦^′−𝑣>0,∀𝑥∈𝑋,𝑔𝑥,𝑦^′∈𝐹𝑥,𝑦^′. (4.42) Thus, 𝑣∉⋃_𝑥∈𝑋𝐹(𝑥,𝑦^′)+𝐶, and hence, 𝑣∉Min_{𝑤𝑥∈𝑋}𝐹𝑥,𝑦^′+𝐶. (4.43) If 𝑣∈Max⋃_𝑦∈𝑌Min_𝑤⋃_𝑥∈𝑋𝐹(𝑥,𝑦), by the condition (iii), 𝑣∈Min_𝑤⋃_𝑥∈𝑋𝐹(𝑥,𝑦^′)+𝐶 which contradicts (4.43). Hence, for every 𝑣∈Max⋃_𝑦∈𝑌Min_𝑤⋃_𝑥∈𝑋𝐹(𝑥,𝑦), 𝑣∈co_𝑥∈𝑋Max_{𝑤𝑦∈𝑌}𝐹𝑥,𝑦+𝐶, (4.44) that is, Max_𝑦∈𝑌Min_𝑤𝑥∈X𝐹𝑥,𝑦⊂co_𝑥∈𝑋Max_{𝑤𝑦∈𝑌}𝐹𝑥,𝑦+𝐶 (4.45) or Max_𝑦∈𝑌Min_{𝑤𝑥∈𝑋}𝐹𝑥,𝑦⊂Minco_𝑥∈𝑋Max_{𝑤𝑦∈𝑌}𝐹𝑥,𝑦+𝐶. (4.46)

The following examples illustrate Theorem 4.5.

Example 4.6. Let 𝑋=𝑌={0}∪{1/𝑛∶𝑛∈ℕ}, C=ℝ²₊ and 𝐹𝑥,𝑦=(𝑠,𝑡)∈ℝ²∶𝑠=𝑥²,𝑡=1−𝑦²,∀𝑥,𝑦∈𝑋×𝑌. (4.47) It is obviously that 𝐹 is below-ℝ²₊-concave-like on 𝑌 and above-ℝ²₊-convex-like on 𝑋. We now verify the condition (iii) of Theorem 4.5. Indeed, for any 𝑦∈𝑌, _𝑥∈𝑋𝐹={0}∪1𝑥,𝑦𝑛²∶𝑛∈ℕ×1−𝑦²,Min_{𝑤𝑥∈𝑋}𝐹={0}∪1𝑥,𝑦𝑛²∶𝑛∈ℕ×1−𝑦². (4.48) Then, _𝑦∈𝑌Min_{𝑤𝑥∈𝑋}𝐹={0}∪1𝑥,𝑦𝑛²×{1}∪1∶𝑛∈ℕ1−𝑛²∶𝑛∈ℕ,Max_𝑦∈𝑌Min_{𝑤𝑥∈𝑋}𝐹𝑥,𝑦={(1,1)}. (4.49) Thus, for every 𝑦∈𝑌, Max_𝑦∈𝑌Min_{𝑤𝑥∈𝑋}𝐹⊂{0}∪1𝑥,𝑦𝑛²∶𝑛∈ℕ×1−𝑦²+𝐶=Min_{𝑤𝑥∈𝑋}𝐹𝑥,𝑦+𝐶, (4.50) and the condition (iii) of Theorem 4.5 holds.
Furthermore, for any 𝑥∈𝑋, _𝑦∈𝑌𝐹𝑥,𝑦=𝑥²×{1}∪11−𝑛²∶𝑛∈ℕ,Max_{𝑤𝑦∈𝑌}𝐹𝑥,𝑦=𝑥²×{1}∪11−𝑛²∶𝑛∈ℕ. (4.51) Then, _𝑥∈𝑋Max_{𝑤𝑦∈𝑌}𝐹={0}∪1𝑥,𝑦𝑛²×{1}∪1∶𝑛∈ℕ1−𝑛²∶𝑛∈ℕ,co_𝑥∈𝑋Max_{𝑤𝑦∈𝑌}𝐹𝑥,𝑦=[0,1]×[0,1]. (4.52) Thus, Minco_𝑥∈𝑋Max_{𝑤𝑦∈𝑌}𝐹𝑥,𝑦={(0,0)},Max_𝑦∈𝑌Min_{𝑤𝑥∈𝑋}𝐹𝑥,𝑦={(1,1)}⊂Minco_𝑥∈𝑋Max_{𝑤𝑦∈𝑌}𝐹𝑥,𝑦+𝐶. (4.53) Hence, the conclusion of Theorem 4.5 holds.

Example 4.7. Let 𝑋=[0,1], 𝑌=[−1,0], 𝐶=ℝ²₊, and 𝐺∶𝑌⇉𝑌 be defined by 𝐺𝑦=[]{0}−1,0,𝑦=0,,𝑦≠0. (4.54) Let 𝐹(𝑥,𝑦)={𝑥²}×𝐺(𝑦) for all (𝑥,𝑦)∈𝑋×𝑌. Then 𝐺 is upper semi-continuous, but not lower semi-continuous on ℝ, and 𝐹 is not continuous but is upper semi-continuous on 𝑋×𝑌. Moreover, 𝐹 has nonempty compact values and is lower semi-continuous on 𝑋. It is easy to see that 𝐹 is below-𝐶-concave-like on 𝑌 and is above-𝐶-convex-like on 𝑋. We verify the condition (iii) of Theorem 4.5. Indeed, for all 𝑦∈𝑌, ⋃_𝑥∈𝑋𝐹(𝑥,𝑦)=[0,1]×𝐺(𝑦). Min_{𝑤𝑥∈𝑋}𝐹=[]×{0}0}×[∪]×{𝑥,𝑦0,1,𝑦≠0,({−1,0])([0,1−1}),𝑦=0. (4.55) Then, _𝑦∈𝑌Min_{𝑤𝑥∈𝑋}𝐹𝑥,𝑦=({0}×[−1,0])∪([0,1]×{−1})∪([0,1]×{0}),Max_𝑦∈𝑌Min_{𝑤𝑥∈𝑋}𝐹𝑥,𝑦={(1,0)}⊂Min_{𝑤𝑥∈𝑋}𝐹𝑥,𝑦+𝐶. (4.56) Therefore, the condition (iii) of Theorem 4.5 holds.
Since 𝐹=𝑥𝑥,𝑦²×[]𝑥−1,0,𝑦=0,₂×{0},𝑦≠0, (4.57) for all (𝑥,𝑦)∈𝑋×𝑌, and Max⋃_𝑦∈𝑌Min_𝑤⋃_𝑥∈𝑋𝐹(𝑥,𝑦)={(1,0)}, for each 𝑦∈𝑌, we can choose 𝑥_𝑦=0∈𝑋 such that Max_𝑦∈𝑌Min_{𝑤𝑥∈𝑋}𝐹𝑥,𝑦⊂𝐹𝑥_𝑦,𝑦+𝐶. (4.58) Furthermore, _𝑦∈𝑌𝐹𝑥,𝑦=𝑥²×_𝑦∈𝑌𝐺𝑦=𝑥²×]∪{0=𝑥([−1,0})₂×[−1,0_𝑥∈𝑋],Max_{𝑤𝑦∈𝑌}𝐹𝑥,𝑦=[0,1]×[−1,0]. (4.59) Therefore, Max_𝑦∈𝑌Min_{𝑤𝑥∈𝑋}𝐹𝑥,𝑦={(1,0)}⊂{(0,−1)}+𝐶=Min_𝑥∈𝑋Max_{𝑤𝑦∈𝑌}𝐹𝑥,𝑦+𝐶. (4.60) Hence, the conclusion of Theorem 4.5 holds.

Remark 4.8. Theorem 3.1 in [5] Theorem 3.1 in [6], or Theorem 4.2 in [7] cannot be applied to Examples 4.6 and 4.7 because of the following reasons: (i)the two sets 𝑋 and 𝑌 are not convex in Example 4.6; (ii)𝐹 is not continuous on 𝑋×𝑌 in Examples 4.6 and 4.7.

Theorem 4.9. Let 𝑋, 𝑌 be two nonempty compact convex subsets of real Hausdorff topological vector spaces 𝒳 and 𝒴, respectively. Suppose that the set-valued mapping 𝐹∶𝑋×𝑌⇉𝒵 has nonempty compact values, and it is continuous on 𝑌 and lower semi-continuous on 𝑋 such that (i)for each 𝑦∈𝑌, 𝑥↦𝐹(𝑥,𝑦) is above-naturally 𝐶-quasi-convex on 𝑋;(ii)for each 𝑥∈𝑋, 𝑦↦𝐹(𝑥,𝑦) is above-𝐶-concave or above-properly 𝐶-quasi-concave on 𝑌;(iii)for every 𝑦∈𝑌, Max_𝑦∈𝑌Min_{𝑤𝑥∈𝑋}𝐹𝑥,𝑦⊂Min_{𝑤𝑥∈𝑋}𝐹𝑥,𝑦+𝐶; (4.61)
(iv) for any continuous increasing function ℎ and for each 𝑦∈𝑌, there exists 𝑥_𝑦∈𝑋 such that maxℎ𝐹𝑥_𝑦,𝑦≤max_𝑦∈𝑌min_𝑥∈𝑋ℎ𝐹𝑥,𝑦. (4.62) Then, for any 𝑧₁∈Max⋃_𝑦∈𝑌Min_𝑤⋃_𝑥∈𝑋𝐹(𝑥,𝑦), there is a 𝑧₂∈Min_𝑥∈𝑋Max_{𝑤𝑦∈𝑌}𝐹𝑥,𝑦 (4.63) such that 𝑧₁∈𝑧₂+𝐶, that is, Max_𝑦∈𝑌Min_{𝑤𝑥∈𝑋}𝐹𝑥,𝑦⊂Min_𝑥∈𝑋Max_{𝑤𝑦∈𝑌}𝐹𝑥,𝑦+𝐶. (4.64)

Proof. Let Γ(𝑥) be defined as the same as in the proof of Theorem 4.5. Following the same perspective as in the proof of Theorem 4.5, suppose that 𝑣∉⋃_𝑥∈𝑋Max_𝑤⋃_𝑦∈𝑌𝐹(𝑥,𝑦)+𝐶. For any 𝑘∈int𝐶 and Gerstewitz function 𝜉_kv∶𝒵⇉ℝ. By Proposition 2.7(d), we have 𝜉_𝑘𝑣(𝑢)>0,forevery𝑢∈Γ(𝑋). (4.65) Let 𝑔∶=𝜉_𝑘𝑣∘𝐹∶𝑋×𝑌⇉ℝ. From Lemma 3.10, for the mapping 𝜉_kv and Remark 3.15, for each 𝑥∈𝑋, there exist 𝑦^∗_𝑥∈𝑌and 𝑓(𝑥,𝑦^∗_𝑥)∈𝐹(𝑥,𝑦^∗_𝑥) with 𝑓(𝑥,𝑦^∗_𝑥)∈Max_𝑤⋃_𝑦∈𝑌𝐹(𝑥,𝑦) such that 𝜉_𝑘𝑣𝑓(𝑥,𝑦^∗_𝑥)=max⋃_𝑦∈𝑌𝜉_𝑘𝑣(𝐹(𝑥,𝑦)). Choosing 𝑢=𝑓(𝑥,𝑦^∗_𝑥) in (4.65), we have max_𝑦∈𝑌𝜉_𝑘𝑣𝐹𝑥,𝑦>0,∀𝑥∈𝑋. (4.66) Therefore, min_𝑥∈𝑋max_𝑦∈𝑌𝜉_𝑘𝑣𝐹𝑥,𝑦>0. (4.67)
By conditions (i), (ii) and Remark 3.15, the set-valued mapping 𝑦↦𝜉_𝑘𝑣(𝐹(𝑥,𝑦)) is upper semi-continuous, and either above-ℝ₊-concave or above-properly ℝ₊-quasi-concave on 𝑌, and the set-valued mapping 𝑥↦𝜉_𝑘𝑣(𝐹(𝑥,𝑦)) is lower semi-continuous and above-ℝ₊-quasi-convex on 𝑋. From Theorem 4.3, we have max_𝑦∈𝑌min_𝑥∈𝑋𝜉_𝑘𝑣𝐹𝑥,𝑦>0. (4.68) Since the set-valued mapping 𝑦↦𝐹(𝑥,𝑦) is lower semi-continuous on 𝑌, by Lemma 2.4 (b) and Lemma 2.5 (b), the set-valued mapping 𝑦↦min⋃_𝑥∈𝑋𝜉_𝑘𝑣(𝐹(𝑥,𝑦)) is upper semi-continuous on 𝑌. By the compactness of 𝑌, there exists 𝑦^′∈𝑌 such that min⋃_𝑥∈𝑋𝜉_𝑘𝑣(𝐹(𝑥,𝑦^′))>0. For all 𝑥∈𝑋 and all 𝑔(𝑥,𝑦^′)∈𝐹(𝑥,𝑦^′), we have 𝜉_𝑘𝑣(𝑔(𝑥,𝑦^′))>0. Thus, 𝑣∉⋃_𝑥∈𝑋𝐹(𝑥,𝑦^′)+𝐶, and hence, 𝑣∉Min_{𝑤𝑥∈𝑋}𝐹𝑥,𝑦^′+C. (4.69) If 𝑣∈Max⋃_𝑦∈𝑌Min_𝑤⋃_𝑥∈𝑋𝐹(𝑥,𝑦), by the condition (iii), 𝑣∈Min_𝑤⋃_𝑥∈𝑋𝐹(𝑥,𝑦^′)+𝐶 which contradicts (4.69). Hence, for every 𝑣∈Max⋃_𝑦∈𝑌Min_𝑤⋃_𝑥∈𝑋𝐹(𝑥,𝑦), 𝑣∈Min_𝑥∈𝑋Max_{𝑤𝑦∈𝑌}𝐹𝑥,𝑦+𝐶, (4.70) that is, Max_𝑦∈𝑌Min_{𝑤𝑥∈𝑋}𝐹𝑥,𝑦⊂Min_𝑥∈𝑋Max_{𝑤𝑦∈𝑌}𝐹𝑥,𝑦+𝐶. (4.71) This completes the proof.

The following example illustrates Theorem 4.9.

Example 4.10. Let 𝑋=𝑌=[0,1], 𝐶=ℝ²₊ and 𝐺∶𝑋⇉𝑌 be a set-valued mapping defined as 𝐺(𝑥)=[]{0}0,1,𝑥≠0,,𝑥=0. (4.72) Let 𝐹(𝑥,𝑦)=𝐺(𝑥)×{−𝑦²} for all (𝑥,𝑦)∈𝑋×𝑌. Then 𝐺 is lower semi-continuous, but not upper semi-continuous on ℝ, and 𝐹 is continuous on 𝑌, and 𝐹 has nonempty compact values and is lower semi-continuous on 𝑋. It is easy to see that 𝐹 is above-C-concave or above-properly 𝐶-quasi-concave on 𝑌 and is above-naturally 𝐶-quasi-convex on𝑋.
We verify the condition (iii) of Theorem 4.9. Indeed, for all 𝑦∈𝑌, ⋃_𝑥∈𝑋𝐹(𝑥,𝑦)=[0,1]×{−𝑦²} and Min_𝑤⋃_𝑥∈𝑋𝐹(𝑥,𝑦)=[0,1]×{−𝑦²}. Hence, _𝑦∈𝑌Min_{𝑤𝑥∈𝑋}𝐹𝑥,𝑦=[0,1]×[−1,0],Max_𝑦∈𝑌Min_{𝑤𝑥∈𝑋}𝐹𝑥,𝑦={(1,0)}⊂Min_{𝑤𝑥∈𝑋}𝐹𝑥,𝑦+𝐶. (4.73) Therefore, the condition (iii) of Theorem 4.9 holds.
Since Max⋃_𝑦∈𝑌Min_𝑤⋃_𝑥∈𝑋𝐹(𝑥,𝑦)={(1,0)} for any 𝑦∈𝑌, we can choose 𝑥_𝑦=0∈𝑋 such that 𝐹𝑥_𝑦,𝑦=0,−𝑦²⊂Max_𝑦∈𝑌Min_{𝑤𝑥∈𝑋}𝐹𝑥,𝑦−𝐶. (4.74) For any continuous increasing function ℎ, the condition (iv) of Theorem 4.9 holds.
Furthermore, since for each 𝑥∈𝑋, _𝑦∈𝑌𝐹𝑥,𝑦=𝐺(𝑥)×[−1,0],Max_{𝑤𝑦∈𝑌}𝐹={0}×[]1}×[⋃]×{0𝑥,𝑦−1,0,𝑥=0,({−1,0])([0,1}),𝑥≠0, (4.75) we have _𝑥∈𝑋Max_{𝑤𝑦∈𝑌}𝐹𝑥,𝑦=({0}×[−1,0])([0,1]×{0})({1}×[−1,0]),Min_𝑥∈𝑋Max_{𝑤𝑦∈𝑌}𝐹𝑥,𝑦={(0,−1)}. (4.76) Thus, Max_𝑦∈𝑌Min_{𝑤𝑥∈𝑋}𝐹𝑥,𝑦={(1,0)}⊂{(0,−1)}+𝐶=Min_𝑥∈𝑋Max_{𝑤𝑦∈𝑌}𝐹𝑥,𝑦+𝐶. (4.77) Therefore, the conclusion of Theorem 4.9 holds.

Remark 4.11. Theorem 3.1 in [5], Theorem 3.1 in [6], or Theorem 4.2 in [7] cannot be applied to Example 4.10 as 𝐹 is not continuous on 𝑋×𝑌.
If we choose 𝑍=ℝ and 𝐶=ℝ₊ in Theorems 4.5 and 4.9, we always have 𝐶^∗=ℝ₊ and for every 𝑦∈𝑌, max_𝑦∈𝑌min_𝑥∈𝑋𝐹𝑥,𝑦≥min_𝑥∈𝑋𝐹𝑥,𝑦. (4.78) Hence, the condition (iii) of Theorem 4.5 holds. Thus, we have the following corollaries.

Corollary 4.12. Let 𝑋, 𝑌 be nonempty compact (not necessarily convex) subsets of real Hausdorff topological vector space 𝒳 and 𝒴, respectively. Suppose that the set-valued mapping 𝐹∶𝑋×𝑌⇉ℝ has nonempty compact values such that it is lower semi-continuous on 𝑋 and is upper semi-continuous on 𝑋×𝑌. Assume that the following conditions hold: (i)for each 𝑥∈𝑋, 𝑦↦𝐹(𝑥,𝑦) is below-ℝ₊-concave-like on 𝑌; (ii)for each 𝑦∈𝑌, 𝑥↦𝐹(𝑥,𝑦) is above-ℝ₊-convex-like on 𝑋; (iii)for every 𝑦∈𝑌, max_𝑦∈𝑌min_𝑥∈𝑋𝐹𝑥,𝑦≥min_𝑥∈𝑋𝐹𝑥,𝑦. (4.79)Then, for any 𝑧₁∈max_𝑦∈𝑌min_x∈X𝐹𝑥,𝑦, (4.80) there is a 𝑧₂∈minco_𝑥∈𝑋max_𝑦∈𝑌𝐹𝑥,𝑦 (4.81) such that 𝑧₁≥𝑧₂, (4.82) that is, max_𝑦∈𝑌min_𝑥∈𝑋𝐹𝑥,𝑦≥minco_𝑥∈𝑋max_𝑦∈𝑌𝐹𝑥,𝑦. (4.83)

Corollary 4.13. Under the framework of Corollary 4.12, in addition, let 𝑋, 𝑌 be two convex subsets, and let 𝐹 be upper semi-continuous on 𝑋×𝑌. Then, max_𝑦∈𝑌min_𝑥∈𝑋𝐹𝑥,𝑦=min_𝑥∈𝑋max_𝑦∈𝑌𝐹𝑥,𝑦. (4.84)

Proof. By Corollary 4.12, we have max_𝑦∈𝑌min_𝑥∈𝑋𝐹𝑥,𝑦≥minco_𝑥∈𝑋max_𝑦∈𝑌𝐹𝑥,𝑦. (4.85) Since the set-valued mapping 𝐹 is upper semi-continuous on 𝑋×𝑌 and 𝑌 is compact, by Lemmas 2.4 and 2.5, the set-valued mapping 𝑥↦max⋃_𝑦∈𝑌𝐹(𝑥,𝑦) is upper semi-continuous on 𝑋. Since 𝑋 is convex, it is connected. By [16, Theorem 3.1], _𝑥∈𝑋max_𝑦∈𝑌𝐹𝑥,𝑦 (4.86) is connected in ℝ, and hence, it is convex. From (4.85), max_𝑦∈𝑌min_𝑥∈𝑋𝐹𝑥,𝑦≥min_𝑥∈𝑋max_𝑦∈𝑌𝐹𝑥,𝑦. (4.87) This completes the proof.

When 𝑍=ℝ and 𝐶=ℝ₊, from Theorem 4.9, we deduce the following corollary.

Corollary 4.14. Let 𝑋, 𝑌 be two nonempty compact convex subsets in real Hausdorff topological vector spaces 𝒳 and 𝒴, respectively. Suppose that the set-valued mapping 𝐹∶𝑋×𝑌⇉ℝ has nonempty compact values such that it is continuous on 𝑌 and is lower semi-continuous on 𝑋. Assume that the following conditions hold: (i)for each 𝑦∈𝑌, 𝑥→𝐹(𝑥,𝑦) is above-naturally ℝ₊-quasi-convex on 𝑋;(ii)for each 𝑥∈𝑋, 𝑦→𝐹(𝑥,𝑦) is above-ℝ₊-concave or above-properly ℝ₊-quasi-concave on 𝑌;(iii)for each 𝑦∈𝑌, there exists 𝑥_𝑦∈𝑋 such that max𝐹𝑥_𝑦,𝑦≤max_𝑦∈𝑌min_𝑥∈𝑋𝐹𝑥,𝑦. (4.88)Then, max_𝑦∈𝑌min_𝑥∈𝑋𝐹𝑥,𝑦=min_𝑥∈𝑋max_𝑦∈𝑌𝐹𝑥,𝑦. (4.89)

Remark 4.15. Corollary 4.14 includes Proposition 2.1 in [5].

5. Saddle Points for Set-Valued Mappings

In this section, we discuss the existence of several kinds of saddle points for set-valued mappings including the 𝐶-loose saddle points, weak 𝐶-saddle points, ℝ₊-loose saddle points, and R₊-saddle points of 𝐹 on 𝑋×𝑌.

Definition 5.1. Let F∶X×Y⇉𝒵 be a set-valued mapping. A point (𝑥,𝑦)∈𝑋×𝑌 is said to be a (a)𝐶-loose saddle point [7] of 𝐹 on 𝑋×𝑌 if 𝐹𝑥,𝑦Max_𝑦∈𝑌𝐹𝐹𝑥,𝑦≠∅,𝑥,𝑦Min_𝑥∈𝑋𝐹𝑥,𝑦≠∅; (5.1)(b) weak 𝐶-saddle point [7] of 𝐹 on 𝑋×𝑌 if 𝐹𝑥,𝑦Max_{𝑤𝑦∈𝑌}𝐹𝑥,𝑦Min_{𝑤𝑥∈𝑋}𝐹𝑥,𝑦≠∅; (5.2)(c)ℝ₊-loose saddle point of 𝐹 on 𝑋×𝑌 if 𝑍=ℝ and 𝐹𝑥,𝑦=min_𝑥∈𝑋𝐹𝑥,𝑦,max_𝑦∈𝑌𝐹𝑥,𝑦; (5.3)(d)ℝ₊-saddle point of 𝐹 on 𝑋×𝑌 if Z=ℝ and max_𝑦∈𝑌𝐹𝑥,𝑦=min_𝑥∈𝑋𝐹𝑥,𝑦=𝐹𝑥,𝑦. (5.4)

It is obvious that every weak 𝐶-saddle point is a 𝐶-loose saddle point and every ℝ₊-saddle point is a ℝ₊-loose saddle point.

Theorem 5.2. Under the framework of Theorem 4.1, 𝐹 has ℝ₊-saddle point if the set-valued mapping 𝑦↦𝐹(𝑥,𝑦) is continuous.

Proof. By Lemmas 2.4 and 2.5, we attained the max and min in Theorem 4.1. By the compactness of 𝑋 and 𝑌 and the lower semi-continuity of 𝐹 on 𝑋 and 𝑌, respectively, there exists (𝑥,𝑦)∈𝑋×𝑌 such that max_𝑦∈𝑌min_𝑥∈𝑋𝐹𝑥,𝑦=min_𝑥∈𝑋𝐹𝑥,𝑦,min_𝑥∈𝑋max_𝑦∈𝑌𝐹𝑥,𝑦=max_𝑦∈𝑌𝐹𝑥,𝑦. (5.5) Combining this with Theorem 4.1, we have max_𝑦∈𝑌𝐹𝑥,𝑦=min_𝑥∈𝑋𝐹𝑥,𝑦=𝐹𝑥,𝑦, (5.6) and hence, 𝐹 has ℝ₊-saddle point.

Theorem 5.3. Under the framework of Theorem 4.3, 𝐹 has ℝ₊-saddle point if the set-valued mapping 𝑦↦𝐹(𝑥,𝑦) is continuous.

Theorem 5.4. Under the framework of Theorem 4.5 or Theorem 4.9, 𝐹 has weak 𝐶-saddle point if the set-valued mapping 𝑦↦𝐹(𝑥,𝑦) is continuous.

Proof. For any 𝜉∈𝐶^∗, the set-valued mapping 𝜉∘𝐹 satisfies all the conditions of Theorem 5.2 or Theorem 5.3. Hence, 𝜉∘𝐹 has ℝ₊-saddle point, that is, there exists (𝑥,𝑦)∈𝑋×𝑌 such that max_𝑦∈𝑌𝜉𝐹𝑥,𝑦=min_𝑥∈𝑋𝜉𝐹𝑥,𝑦=𝜉𝐹𝑥,𝑦. (5.7) Then, for any 𝑧∈𝐹(𝑥,𝑦), 𝜉(𝑧)∈min_𝑥∈𝑋𝜉𝐹𝑥,𝑦,𝜉(𝑧)∈max_𝑦∈𝑌𝜉𝐹𝑥,𝑦. (5.8) Thus, by Proposition 3.14, 𝑧∈Min_{𝑤𝑥∈𝑋}𝐹𝑥,𝑦Max_{𝑤𝑦∈𝑌}𝐹𝑥,𝑦, (5.9) and (𝑥,𝑦) is a weak 𝐶-saddle point of 𝐹.

Acknowledgments

In this paper, the first author was partially supported by Grant NSC101-2115-M-039-001- from the National Science Council of Taiwan. The research part of the second author was done during his visit to King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia.