# Minimax Theorems for Set-Valued Mappings under Cone-Convexities

Authors: Yen-Cherng Lin; Qamrul Hasan Ansari; Hang-Chin Lai
Journal: Abstract and Applied Analysis
 (2012)

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## 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.


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## Body

## 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. Section2 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 thatX, 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
(2.1)C*={g∈𝒵*:g(c)≥0∀c∈C}.We present some fundamental concepts which will be used in the sequel.Definition 2.1 (see [3, 4, 8]).
LetA 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)=∅; MinwA denotes the set of all weakly minimal points of A;(d)
weakly maximal point of A if A∩(z+intC)=∅; MaxwA denotes the set of all weakly maximal points of A.It can be easily seen thatMinA⊂MinwA and MaxA⊂MaxwA.Lemma 2.2 (see [3, 4]).
LetA 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 Maxw by max (both Min and Minw by min) in ℝ since both Max and Maxw (both Min and Minw) are the same in ℝ.Definition 2.3.
Let𝒳, 𝒴 be Hausdorff topological spaces. A set-valued map F:𝒳⇉𝒴 with nonempty values is said to be  (a)
upper semicontinuous atx0∈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 atx0∈𝒳 if for any sequence {xn}⊂𝒳 such that xn→x0 and any y0∈F(x0), there exists a sequence yn∈F(xn) such that yn→y0;(c)
continuous atx0∈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]).
Let𝒳, 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
(2.2)T(x)=⋃y∈𝒴‍F(x,y),∀x∈𝒳.(a)
IfF is upper semi-continuous on 𝒳×𝒴, then T is upper semi-continuous on 𝒳.(b)
IfF is lower semi-continuous on 𝒳, so is T.Lemma 2.5 (see [9, Lemma 3.2]).
Let𝒵 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)
IfF is upper semi-continuous, so is  p.(b)
IfF is lower semi-continuous, so is  p.(c)
IfF 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]).
Letk∈intC and v∈𝒵. The Gerstewitz functionξkv:𝒵→ℝ is defined by
(2.3)ξkv(u)=min{t∈ℝ:u∈v+tk-C}.We present some fundamental properties of the scalarization function.Proposition 2.7 (see [6, 10]).
Letk∈intC and v∈𝒵. The Gerstewitz function ξkv:𝒵→ℝ has the following properties:  (a)
ξkv(u)<r⇔u∈v+rk-intC;(b)
ξkv(u)≤r⇔u∈v+rk-C;(c)
ξkv(u)=0⇔u∈v-∂C, where ∂C is the topological boundary of C;(d)
ξkv(u)>r⇔u∉v+rk-C;(e)
ξkv(u)≥r⇔u∉v+rk-intC;(f)
ξkv(·) is a convex function;(g)
ξkv(·) is an increasing function, that is, u2-u1∈intS⇒ξkv(u1)<ξkv(u2);(h)
ξkv(·) is a continuous function.Theorem 2.8 ( Fan-Browder fixed-point theorem (see [12])).
LetX 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.
LetX 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],
(3.1)F(λx1+(1-λ)x2)⊂λF(x1)+(1-λ)F(x2)-C,(resp.,λF(x1)+(1-λ)F(x2)⊂F(λx1+(1-λ)x2)-C);(b)
below-C-convex [13] (resp.,  below-C-concave [9, 13]) on X if for all x1,x2∈X and all λ∈[0,1],
(3.2)λF(x1)+(1-λ)F(x2)⊂F(λx1+(1-λ)x2)+C(resp.,F(λx1+(1-λ)x2)⊂λF(x1)+(1-λ)F(x2)+C);(c)
above-C-quasi-convex (resp.,  below-C-quasiconcave) [7, Definition 2.3] on X if the set
(3.3)LevF≤(z)∶={x∈X:F(x)⊂z-C}(resp.,LevF≥(z):={x∈X:F(x)⊂z+C}),
is convex for all z∈𝒵;(d)
above-properlyC-quasiconvex (resp., above-properlyC-quasiconcave [6]) on X if for all x1,x2∈X and all λ∈[0,1], either
(3.4)F(λx1+(1-λ)x2)⊂F(x1)-C(resp.,F(x1)⊂F(λx1+(1-λ)x2)-C)
or
(3.5)F(λx1+(1-λ)x2)⊂F(x2)-C(resp.,F(x2)⊂F(λx1+(1-λ)x2)-C);(e)
below-properly
C
-quasiconvex [7] (resp.,  below-properlyC-quasiconcave) on X if for all x1,x2∈X and all λ∈[0,1], either
(3.6)F(x1)⊂F(λx1+(1-λ)x2)+C(resp.,F(λx1+(1-λ)x2)⊂F(x1)+C)
or
(3.7)F(x2)⊂F(λx1+(1-λ)x2)+C(resp.,F(λx1+(1-λ)x2)⊂F(x2)+C);
(f)
above-naturally
C
-quasiconvex [6] on X if for all x1,x2∈X and all λ∈[0,1],
(3.8)F(λx1+(1-λ)x2)⊂co{F(x1)∪F(x2)}-C,
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
(3.9)F(x′)⊂λF(x1)+(1-λ)F(x2)-C(resp.,λF(x1)+(1-λ)F(x2)⊂F(x′)-C);
(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
(3.10)λF(x1)+(1-λ)F(x2)⊂F(x′)+C(resp.,F(x′)⊂λF(x1)+(1-λ)F(x2)+C).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-properlyC-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.
ConsiderC={(s,t)∈ℝ2:s≥0,t≥0}. Let F:[x1,x2]⊂ℝ⇉ℝ2 be a set-valued mapping defined by
(3.11)F(x1)∶={(s,t)∈ℝ2:(s-2)2+(t-4)2=1,2≤s≤3,4≤t≤5}⋃‍{(s,5):-1≤s≤2},F(x2)∶={(s,t)∈ℝ2:(s-6)2+(t+2)2=1,6≤s≤7,-2≤t≤-1},
and for all λ∈(0,1),
(3.12)F(λx1+(1-λ)x2)∶={(s,t)∈ℝ2:(s-2)2+(t-2)2=4,0≤s≤2,0≤t≤2}.
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, letG:[x1,x2]⊂ℝ⇉ℝ2 be a set-valued mapping defined by
(3.13)G(x1)∶={(s,t)∈ℝ2:(s-1)2+(t-4)2=1,1≤s≤2,4≤t≤5},G(x2)∶={(s,t)∈ℝ2:(s-6)2+(t+2)2=1,6≤s≤7,-2≤t≤-1},
and for all λ∈(0,1),
(3.14)G(λx1+(1-λ)x2)∶={(s,t)∈ℝ2:(s-2)2+(t-2)2=4,0≤s≤2,0≤t≤2}⋃{(s,0):2≤s≤3}.
Then, G is a below-properly C-quasi-convex set-valued mapping, but it is not above-properly C-quasi-convex.Example 3.4.
LetC={(s,t):s≥0,t≥0}. Define F:[-1,1]⇉ℝ2 by
(3.15)F(x)={(x,t):1-x2≤t≤1},∀x∈[-1,1].
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.
LetX 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
(3.16)M(x)=MaxwF(x),∀x∈X.(a)
IfFis above-C-convex-like, then M is so.(b)
IfX is a topological space and F is a continuous mapping, then M is upper semicontinuous with nonempty compact values on X.Proof.
(a) LetF 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
(3.17)F(x′)⊂αF(x1)+(1-α)F(x2)-C.
By Lemma 2.2,
(3.18)MaxwF(x′)⊂αMaxwF(x1)+(1-α)MaxwF(x2)-C.
Therefore, x↦MaxwF(x) is above-C-convex-like.
(b) The upper semicontinuity ofM was deduced in [4, Lemma 2].Proposition 3.6.
LetX 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
(3.19)M(x)=MaxwF(x),∀x∈X,
is above-C-quasiconvex if F is so.The following result can be easily derived, and therefore, we omit the proof.Proposition 3.7.
LetX 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.Letξ∈C* and F:X⇉𝒵 be a set-valued mapping. Then, the composition mapping ξ∘F:X⇉ℝ is defined by
(3.20)(ξ∘F)(x)=ξ(F(x))=⋃y∈F(x)‍ξ(y),∀x∈X.
Clearly, the composition mapping ξ∘F:X⇉ℝ is also a set-valued mapping.Proposition 3.8.
LetX be a nonempty set, F:X⇉𝒵 a set-valued mapping, and ξ∈C*.  (a)
IfF is above-C-convex-like, then ξ∘F is above-ℝ+-convex-like.(b)
IfF is below-C-concave-like, then ξ∘F is below-ℝ+-concave-like.(c)
IfX 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
(3.21)λy1+(1-λ)y2∈f-C.
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.
LetX be a nonempty convex set and ξ∈C*.  (a)
IfF: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)
IfF: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)
IfF:X⇉𝒵 is above-C-convex, then ξ∘F:X⇉ℝ is above-ℝ+-convex and above-ℝ+-quasi-convex.Lemma 3.10.
Let𝒵 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∈MaxwF(X) such that ξ(y)=max⋃x∈X‍ξ(F(x)).Proof.
For any givenξ∈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∈Maxw⋃x∈X‍F(x) such that y0-y∈-C, and hence ξ(y)≥ξ(y0). On the other hand, y∈Maxw⋃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.
LetX be a nonempty convex set. If F:X⇉𝒵 is above-properly C-quasi-convex, then it is above-C-quasi-convex.Proof.
For anyz∈𝒵, let x1,x2∈LevF≤(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 LevF≤(z) is convex, and therefore, F is above-C-quasi-convex.Proposition 3.12.
LetX be a nonempty convex set. If F:X⇉𝒵 is above-naturally C-quasi-convex, then it is above-C-quasi-convex.Proof.
Letz, 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 LevF≤(z) is convex, and therefore, F is above-C-quasi-convex.Proposition 3.13.
LetX 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.
Letξ∈C* be given. From the above-naturally C-quasi-convexity of F, for any x1,x2∈X and any λ∈[0,1],
(3.22)F(λx1+(1-λ)x2)⊂co{F(x1)∪F(x2)}-C.
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 yi∈F(x1)∪F(x2) and λi∈[0,1], 1≤i≤n such that w=∑i=1n‍λiyi. Hence, ξ(w)=∑i=1n‍λiξ(yi), and
(3.23)ξ(y)∈ξ(w)-ℝ+=∑i=1n‍λiξ(yi)-ℝ+⊂co{ξ(F(x1))∪ξ(F(x2))}-ℝ+.
Therefore, ξ∘F is a above-naturally ℝ+-quasi-convex.Proposition 3.14.
LetF:X⇉𝒵 be a set-valued mapping with nonempty compact values. For any ξ∈C*,  (a)
ifξ(d)=min⋃x∈X‍ξ(F(x)) for some d∈𝒵, then d∈Minw⋃x∈X‍F(x);(b)
ifξ(e)=max⋃x∈X‍ξ(F(x)) for some e∈𝒵, then e∈Maxw⋃x∈X‍F(x).Proof.
Letξ(d)=min⋃x∈X‍ξ(F(x)). Suppose that d∉Minw⋃x∈X‍F(x). Then
(3.24)(⋃x∈X‍F(x))⋂‍(d-intC)≠∅.
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.
Propositions3.8 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.
LetX, 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 eachx∈X, y↦F(x,y) is below-ℝ+-concave-like on Y;(ii)
for eachy∈Y, x↦F(x,y) is above-ℝ+-convex-like on X. Then,
(4.1)max⋃y∈Y‍min⋃x∈X‍F(x,y)=min⋃x∈X‍max⋃y∈Y‍F(x,y).Proof.
Since(4.2)max⋃y∈Y‍min⋃x∈X‍F(x,y)≤min⋃x∈X‍max⋃y∈Y‍F(x,y),
it is sufficient to prove that
(4.3)max⋃y∈Y‍min⋃x∈X‍F(x,y)≥min⋃x∈X‍max⋃y∈Y‍F(x,y).
Choose any α∈ℝ such that α<min⋃x∈X‍max⋃y∈Y‍F(x,y). For any y∈Y, let
(4.4)LevF≤(y;α)={x∈X:F(x,y)⊂α-ℝ+}.
Then, by the lower semi-continuity of the set-valued mapping x↦F(x,y), the set LevF≤(y;α) is closed, hence it is compact for all y∈Y. By the choice of α, we have
(4.5)⋂y∈Y‍LevF≤(y;α)=∅.
Since X is compact and the collection {X∖LevF≤(y;α):y∈Y} covers X, there exist finite number of points y1,y2,…,ym in Y such that
(4.6)X⊂⋃i=1m‍(X∖LevF≤(yi;α))
or
(4.7)⋂i=1m‍LevF≤(yi;α)=∅.
This implies that
(4.8)max⋃i=1m‍F(x,yi)>α,∀x∈X,
and therefore,
(4.9)min⋃x∈X‍max⋃i=1m‍F(x,yi)>α.
Following the idea of  Borwein and Zhuang  [14], let
(4.10)𝔈∶={(z,r)∈ℝm+1:thereisx∈X,F(x,yi)⊂r+zi-ℝ+,i=1,2,…,m},
where z=(z1,z2,…,zm). Then the set 𝔈 is convex, so is int𝔈. We note that the interior int𝔈 of 𝔈 is nonempty since
(4.11)(0,1+max⋃i=1m‍F(x,yi))∈int𝔈,∀x∈X.
Since (0,α)∉𝔈, by separation hyperplane theorem [15, Theorem 14.2], there is a (Ξ,ϵ)≠0×{0} such that
(4.12)〈(Ξ,ε),(z,r)〉≥〈(Ξ,ε),(0,α)〉,∀(z,r)∈𝔈,
where Ξ=(λ1,λ2,…,λm), that is,
(4.13)Ξz+εr≥εα,∀(z,r)∈𝔈.
By (4.11), (4.13), and the choice of α, we have that ε>0. Furthermore, from the fact
(4.14)∏i=1m(F(x,yi)+r)×{-r}⊂𝔈,
we have
(4.15)(ηx,1+r,ηx,2+r,…,ηx,m+r,-r)∈𝔈,∀ηx,i∈F(x,yi).
Hence, by (4.13), we have
(4.16)∑i=1m‍λi(ηx,i+r)+ε(-r)≥εα
or
(4.17)∑i=1m‍(λiε)ηx,i+(∑i=1m‍λiε-1)r≥α,∀x∈X,r∈ℝ.
Thus, we have ∑i=1m‍(λi/ε)=1. Hence, by (4.17), we have
(4.18)∑i=1m‍(λiε)F(x,yi)⊂α+ℝ+.
Since F(x,y) is below-ℝ+-concave-like in y, there is y′∈Y such that
(4.19)F(x,y′)⊂∑i=1m‍(λiε)F(x,yi)+ℝ+,∀x∈X.
Therefore,
(4.20)⋃x∈X‍F(x,y′)⊂α+ℝ+,
and hence,
(4.21)max⋃y∈Y‍min⋃x∈X‍F(x,y)≥α.
This completes the proof.Remark 4.2.
Theorem4.1 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.
LetX, 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 eachy∈Y, x↦F(x,y) is above-ℝ+-quasi-convex on X;(ii)
for eachx∈X, y↦F(x,y) is above-ℝ+-concave, or above-properly ℝ+-quasi-concave on Y;(iii)
for eachy∈Y, there is a  xy∈Y such that
(4.22)maxF(xy,y)≤max⋃y∈Y‍min⋃x∈X‍F(x,y).Then,
(4.23)min⋃x∈X‍max⋃y∈Y‍F(x,y)=max⋃y∈Y‍min⋃x∈X‍F(x,y).Proof.
We only need to prove that(4.24)max⋃y∈Y‍min⋃x∈X‍F(x,y)<min⋃x∈X‍max⋃y∈Y‍F(x,y)
is impossible, since it is always true that
(4.25)max⋃y∈Y‍min⋃x∈X‍F(x,y)≤min⋃x∈X‍max⋃y∈Y‍F(x,y).
Suppose that there is an α∈ℝ such that
(4.26)max⋃y∈Y‍min⋃x∈X‍F(x,y)<α<min⋃x∈X‍max⋃y∈Y‍F(x,y).
Define G:X×Y⇉X×Y by
(4.27)G(x,y)={s∈X:maxF(s,y)<α}×{t∈Y:maxF(x,t)>α}.
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 eachy∈Y, there is a xy∈Y such that maxF(xy,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
(4.28)G-1(s,t)={x∈X:maxF(x,t)>α}×{y∈Y:maxF(s,y)<α}
is open in X×Y. By Fan-Browder fixed-point Theorem 2.8, there exists (x-,y-)∈X×Y such that
(4.29)(x-,y-)∈G(x-,y-),
that is,
(4.30)maxF(x-,y-)>α>maxF(x-,y-),
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.
LetX 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 eachx∈X, y→F(x,y) is below-C-concave-like on Y;(ii)
for eachy∈Y, x→F(x,y) is above-C-convex-like on X;(iii)
for everyy∈Y,
(4.31)Max⋃y∈Y‍Minw⋃x∈X‍F(x,y)⊂Minw⋃x∈X‍F(x,y)+C.Then for any
(4.32)z1∈Max⋃y∈Y‍Minw⋃x∈X‍F(x,y),
there is a
(4.33)z2∈Min(co{⋃x∈X‍Maxw⋃y∈Y‍F(x,y)})
such that
(4.34)z1∈z2+C,
that is,
(4.35)Max⋃y∈Y‍Minw⋃x∈X‍F(x,y)⊂Min(co{⋃x∈X‍Maxw⋃y∈Y‍F(x,y)})+C.Proof.
LetΓ(x):=Maxw⋃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
(4.36)ξ(v)≤k-ε<k≤ξ(u+c),foreveryu∈co{Γ(X)},c∈C.
Therefore,
(4.37)ξ(c)>ξ(v-u),foreveryu∈co{Γ(X)},c∈C.
This implies that ξ∈C* and ξ(v)<ξ(u) for all u∈co{Γ(X)}.
Letg:=ξF:X×Y⇉ℝ. From Lemma 3.10, for each fixed x∈X, there exist yx*∈Y and f(x,yx*)∈F(x,yx*) with f(x,yx*)∈Γ(x) such that ξ(f(x,yx*))=max⋃y∈Y‍ξ(F(x,y)). Choosing c=0 and u=f(x,yx*) in (4.36), we have
(4.38)max⋃y∈X‍ξ(F(x,y))=ξf(x,yx*)≥k>k-ε≥ξ(v),∀x∈X.
Therefore,
(4.39)min⋃x∈X‍max⋃y∈Y‍ξ(F(x,y))>ξ(v).
By the conditions (i), (ii) and Proposition3.8, 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
(4.40)max⋃y∈Y‍min⋃x∈X‍ξ(F(x,y))>ξ(v).
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
(4.41)ξ(g(x,y′))>ξ(v),
that is,
(4.42)ξ(g(x,y′)-v)>0,∀x∈X,g(x,y′)∈F(x,y′).
Thus, v∉⋃x∈X‍F(x,y′)+C, and hence,
(4.43)v∉Minw⋃x∈X‍F(x,y′)+C.
If v∈Max⋃y∈Y‍Minw⋃x∈X‍F(x,y), by the condition (iii), v∈Minw⋃x∈X‍F(x,y′)+C which contradicts (4.43). Hence, for every v∈Max⋃y∈Y‍Minw⋃x∈X‍F(x,y),
(4.44)v∈co{⋃x∈X‍Maxw⋃y∈Y‍F(x,y)}+C,
that is,
(4.45)Max⋃y∈Y‍Minw⋃x∈X‍F(x,y)⊂co{⋃x∈X‍Maxw⋃y∈Y‍F(x,y)}+C
or
(4.46)Max⋃y∈Y‍Minw⋃x∈X‍F(x,y)⊂Min(co{⋃x∈X‍Maxw⋃y∈Y‍F(x,y)})+C.The following examples illustrate Theorem4.5.Example 4.6.
LetX=Y={0}∪{1/n:n∈ℕ}, C=ℝ+2 and
(4.47)F(x,y)={(s,t)∈ℝ2:s=x2,t=1-y2},∀(x,y)∈X×Y.
It is obviously that F is below-ℝ+2-concave-like on Y and above-ℝ+2-convex-like on X. We now verify the condition (iii) of Theorem 4.5. Indeed, for any y∈Y,
(4.48)⋃x∈X‍F(x,y)=({0}∪{1n2:n∈ℕ})×{1-y2},Minw⋃x∈X‍F(x,y)=({0}∪{1n2:n∈ℕ})×{1-y2}.
Then,
(4.49)⋃y∈Y‍Minw⋃x∈X‍F(x,y)=({0}∪{1n2:n∈ℕ})×({1}∪{1-1n2:n∈ℕ}),Max⋃y∈Y‍Minw⋃x∈X‍F(x,y)={(1,1)}.
Thus, for every y∈Y,
(4.50)Max⋃y∈Y‍Minw⋃x∈X‍F(x,y)⊂({0}∪{1n2:n∈ℕ})×{1-y2}+C=Minw⋃x∈X‍F(x,y)+C,
and the condition (iii) of Theorem 4.5 holds.
Furthermore, for anyx∈X,
(4.51)⋃y∈Y‍F(x,y)={x2}×({1}∪{1-1n2:n∈ℕ}),Maxw⋃y∈Y‍F(x,y)={x2}×({1}∪{1-1n2:n∈ℕ}).
Then,
(4.52)⋃x∈X‍Maxw⋃y∈Y‍F(x,y)=({0}∪{1n2:n∈ℕ})×({1}∪{1-1n2:n∈ℕ}),co{⋃x∈X‍Maxw⋃y∈Y‍F(x,y)}=[0,1]×[0,1].
Thus,
(4.53)Min(co{⋃x∈X‍Maxw⋃y∈Y‍F(x,y)})={(0,0)},Max⋃y∈Y‍Minw⋃x∈X‍F(x,y)={(1,1)}⊂Min(co{⋃x∈X‍Maxw⋃y∈Y‍F(x,y)})+C.
Hence, the conclusion of Theorem 4.5 holds.Example 4.7.
LetX=[0,1], Y=[-1,0], C=ℝ+2, and G:Y⇉Y be defined by
(4.54)G(y)={[-1,0],y=0,{0},y≠0.
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).
(4.55)Minw⋃x∈X‍F(x,y)={[0,1]×{0},y≠0,({0}×[-1,0])∪([0,1]×{-1}),y=0.
Then,
(4.56)⋃y∈Y‍Minw⋃x∈X‍F(x,y)=({0}×[-1,0])∪([0,1]×{-1})∪([0,1]×{0}),Max⋃y∈Y‍Minw⋃x∈X‍F(x,y)={(1,0)}⊂Minw⋃x∈X‍F(x,y)+C.
Therefore, the condition (iii) of Theorem 4.5 holds.
Since(4.57)F(x,y)={{x2}×[-1,0],y=0,{x2}×{0},y≠0,
for all (x,y)∈X×Y, and Max⋃y∈Y‍Minw⋃x∈X‍F(x,y)={(1,0)}, for each y∈Y, we can choose xy=0∈X such that
(4.58)Max⋃y∈Y‍Minw⋃x∈X‍F(x,y)⊂F(xy,y)+C.
Furthermore,
(4.59)⋃y∈Y‍F(x,y)={x2}×(⋃y∈Y‍G(y))={x2}×([-1,0]∪{0})={x2}×[-1,0],⋃x∈X‍Maxw⋃y∈Y‍F(x,y)=[0,1]×[-1,0].
Therefore,
(4.60)Max⋃y∈Y‍Minw⋃x∈X‍F(x,y)={(1,0)}⊂{(0,-1)}+C=Min⋃x∈X‍Maxw⋃y∈Y‍F(x,y)+C.
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 setsX 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.
LetX, 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 eachy∈Y, x↦F(x,y) is above-naturally C-quasi-convex on X;(ii)
for eachx∈X, y↦F(x,y) is above-C-concave or above-properly C-quasi-concave on Y;(iii)
for everyy∈Y,
(4.61)Max⋃y∈Y‍Minw⋃x∈X‍F(x,y)⊂Minw⋃x∈X‍F(x,y)+C;
(iv) for any continuous increasing functionh and for each y∈Y, there exists xy∈X such that
(4.62)maxh(F(xy,y))≤max⋃y∈Y‍min⋃x∈X‍h(F(x,y)).
Then, for any z1∈Max⋃y∈Y‍Minw⋃x∈X‍F(x,y), there is a
(4.63)z2∈Min⋃x∈X‍Maxw⋃y∈Y‍F(x,y)
such that z1∈z2+C, that is,
(4.64)Max⋃y∈Y‍Minw⋃x∈X‍F(x,y)⊂Min⋃x∈X‍Maxw⋃y∈Y‍F(x,y)+C.Proof.
LetΓ(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∈X‍Maxw⋃y∈Y‍F(x,y)+C. For any k∈intC and Gerstewitz function ξkv:𝒵⇉ℝ. By Proposition 2.7(d), we have
(4.65)ξkv(u)>0,foreveryu∈Γ(X).
Let g:=ξkv∘F:X×Y⇉ℝ. From Lemma 3.10, for the mapping ξkv and Remark 3.15, for each x∈X, there exist yx*∈Yand f(x,yx*)∈F(x,yx*) with f(x,yx*)∈Maxw⋃y∈Y‍F(x,y) such that ξkvf(x,yx*)=max⋃y∈Y‍ξkv(F(x,y)). Choosing u=f(x,yx*) in (4.65), we have
(4.66)max⋃y∈Y‍ξkv(F(x,y))>0,∀x∈X.
Therefore,
(4.67)min⋃x∈X‍max⋃y∈Y‍ξkv(F(x,y))>0.
By conditions (i), (ii) and Remark3.15, the set-valued mapping y↦ξkv(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↦ξkv(F(x,y)) is lower semi-continuous and above-ℝ+-quasi-convex on X. From Theorem 4.3, we have
(4.68)max⋃y∈Y‍min⋃x∈X‍ξkv(F(x,y))>0.
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∈X‍ξkv(F(x,y)) is upper semi-continuous on Y. By the compactness of Y, there exists y′∈Y such that min⋃x∈X‍ξkv(F(x,y′))>0. For all x∈X and all g(x,y′)∈F(x,y′), we have ξkv(g(x,y′))>0. Thus, v∉⋃x∈X‍F(x,y′)+C, and hence,
(4.69)v∉Minw⋃x∈X‍F(x,y′)+C.
If v∈Max⋃y∈Y‍Minw⋃x∈X‍F(x,y), by the condition (iii), v∈Minw⋃x∈X‍F(x,y′)+C which contradicts (4.69). Hence, for every v∈Max⋃y∈Y‍Minw⋃x∈X‍F(x,y),
(4.70)v∈Min⋃x∈X‍Maxw⋃y∈Y‍F(x,y)+C,
that is,
(4.71)Max⋃y∈Y‍Minw⋃x∈X‍F(x,y)⊂Min⋃x∈X‍Maxw⋃y∈Y‍F(x,y)+C.
This completes the proof.The following example illustrates Theorem4.9.Example 4.10.
LetX=Y=[0,1], C=ℝ+2 and G:X⇉Y be a set-valued mapping defined as
(4.72)G(x)={[0,1],x≠0,{0},x=0.
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 Theorem4.9. Indeed, for all y∈Y, ⋃x∈X‍F(x,y)=[0,1]×{-y2} and Minw⋃x∈X‍F(x,y)=[0,1]×{-y2}. Hence,
(4.73)⋃y∈Y‍Minw⋃x∈X‍F(x,y)=[0,1]×[-1,0],Max⋃y∈Y‍Minw⋃x∈X‍F(x,y)={(1,0)}⊂Minw⋃x∈X‍F(x,y)+C.
Therefore, the condition (iii) of Theorem 4.9 holds.
SinceMax⋃y∈Y‍Minw⋃x∈X‍F(x,y)={(1,0)} for any y∈Y, we can choose xy=0∈X such that
(4.74)F(xy,y)={(0,-y2)}⊂Max⋃y∈Y‍Minw⋃x∈X‍F(x,y)-C.
For any continuous increasing function h, the condition (iv) of Theorem 4.9 holds.
Furthermore, since for eachx∈X,
(4.75)⋃y∈Y‍F(x,y)=G(x)×[-1,0],Maxw⋃y∈Y‍F(x,y)={{0}×[-1,0],x=0,({1}×[-1,0])⋃‍([0,1]×{0}),x≠0,
we have
(4.76)⋃x∈X‍Maxw⋃y∈Y‍F(x,y)=({0}×[-1,0])⋃‍([0,1]×{0})⋃‍({1}×[-1,0]),Min⋃x∈X‍Maxw⋃y∈Y‍F(x,y)={(0,-1)}.
Thus,
(4.77)Max⋃y∈Y‍Minw⋃x∈X‍F(x,y)={(1,0)}⊂{(0,-1)}+C=Min⋃x∈X‍Maxw⋃y∈Y‍F(x,y)+C.
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 chooseZ=ℝ and C=ℝ+ in Theorems 4.5 and 4.9, we always have C*=ℝ+ and for every y∈Y,
(4.78)max⋃y∈Y‍min⋃x∈X‍F(x,y)≥min⋃x∈X‍F(x,y).
Hence, the condition (iii) of Theorem 4.5 holds. Thus, we have the following corollaries.Corollary 4.12.
LetX, 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 eachx∈X,  y↦F(x,y) is below-ℝ+-concave-like on Y;(ii)
for eachy∈Y, x↦F(x,y) is above-ℝ+-convex-like on X;(iii)
for everyy∈Y,
(4.79)max⋃y∈Y‍min⋃x∈X‍F(x,y)≥min⋃x∈X‍F(x,y).Then, for any
(4.80)z1∈max⋃y∈Y‍min⋃x∈X‍F(x,y),
there is a
(4.81)z2∈min(co{⋃x∈X‍max⋃y∈Y‍F(x,y)})
such that
(4.82)z1≥z2,
that is,
(4.83)max⋃y∈Y‍min⋃x∈X‍F(x,y)≥min(co{⋃x∈X‍max⋃y∈Y‍F(x,y)}).Corollary 4.13.
Under the framework of Corollary4.12, in addition, let X, Y be two convex subsets, and let F be upper semi-continuous on X×Y. Then,
(4.84)max⋃y∈Y‍min⋃x∈X‍F(x,y)=min⋃x∈X‍max⋃y∈Y‍F(x,y).Proof.
By Corollary4.12, we have
(4.85)max⋃y∈Y‍min⋃x∈X‍F(x,y)≥min(co{⋃x∈X‍max⋃y∈Y‍F(x,y)}).
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],
(4.86)⋃x∈X‍max⋃y∈Y‍F(x,y)
is connected in ℝ, and hence, it is convex. From (4.85),
(4.87)max⋃y∈Y‍min⋃x∈X‍F(x,y)≥min(⋃x∈X‍max⋃y∈Y‍F(x,y)).
This completes the proof.WhenZ=ℝ and C=ℝ+, from Theorem 4.9, we deduce the following corollary.Corollary 4.14.
LetX, 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 eachy∈Y, x→F(x,y) is above-naturally ℝ+-quasi-convex on X;(ii)
for eachx∈X, y→F(x,y) is above-ℝ+-concave or above-properly ℝ+-quasi-concave on Y;(iii)
for eachy∈Y, there exists xy∈X such that
(4.88)maxF(xy,y)≤max⋃y∈Y‍min⋃x∈X‍F(x,y).Then,
(4.89)max⋃y∈Y‍min⋃x∈X‍F(x,y)=min⋃x∈X‍max⋃y∈Y‍F(x,y).Remark 4.15.
Corollary4.14 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 theC-loose saddle points, weak C-saddle points, ℝ+-loose saddle points, and R+-saddle points of F on X×Y.Definition 5.1.
LetF: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
(5.1)F(x-,y-)⋂‍(Max⋃y∈Y‍F(x-,y))≠∅,F(x-,y-)⋂‍(Min⋃x∈X‍F(x,y-))≠∅;(b)
weakC-saddle point [7] of F on X×Y if
(5.2)F(x-,y-)⋂‍(Maxw⋃y∈Y‍F(x-,y))⋂‍(Minw⋃x∈X‍F(x,y-))≠∅;(c)
ℝ+-loose saddle point of F on X×Y if Z=ℝ and
(5.3)F(x-,y-)=[min⋃x∈X‍F(x,y-),max⋃y∈Y‍F(x-,y)];(d)
ℝ+-saddle point of F on X×Y if Z=ℝ and
(5.4)max⋃y∈Y‍F(x-,y)=min⋃x∈X‍F(x,y-)=F(x-,y-).It is obvious that every weakC-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 Theorem4.1, F has ℝ+-saddle point if the set-valued mapping y↦F(x,y) is continuous.Proof.
By Lemmas2.4 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
(5.5)max⋃y∈Y‍min⋃x∈X‍F(x,y)=min⋃x∈X‍F(x,y-),min⋃x∈X‍max⋃y∈Y‍F(x,y)=max⋃y∈Y‍F(x-,y).
Combining this with Theorem 4.1, we have
(5.6)max⋃y∈Y‍F(x-,y)=min⋃x∈X‍F(x,y-)=F(x-,y-),
and hence, F has ℝ+-saddle point.Theorem 5.3.
Under the framework of Theorem4.3, F has ℝ+-saddle point if the set-valued mapping y↦F(x,y) is continuous.Theorem 5.4.
Under the framework of Theorem4.5 or Theorem 4.9, F has weak C-saddle point if the set-valued mapping y↦F(x,y) is continuous.Proof.
For anyξ∈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
(5.7)max⋃y∈Y‍ξ(F(x-,y))=min⋃x∈X‍ξ(F(x,y-))=ξ(F(x-,y-)).
Then, for any z∈F(x-,y-),
(5.8)ξ(z)∈min⋃x∈X‍ξ(F(x,y-)),ξ(z)∈max⋃y∈Y‍ξ(F(x-,y)).
Thus, by Proposition 3.14,
(5.9)z∈Minw⋃x∈X‍F(x,y-)⋂‍Maxw⋃y∈Y‍F(x-,y),
and (x-,y-) is a weak C-saddle point of F.



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*Source: 310818-2012-12-26.xml*