For instance, let the set A = {(x, y) ∈ [0, 1] × [0, 1]; 0 ≤ y ≤ 1 2 x}\{(0, 0)}, and g be a real-valued function on [0, 1] × [0, 1], defined by

4 Minmax theorems under the region of the graph of a multifunction

On some feasible region, the two functions version of minimax theorems does not

hold again. However, by restricting to a proper range, the minimax theorem is down.

For instance, let the set A = {(x, y) ∈ [0, 1] × [0, 1]; 0 ≤ y ≤ ¹ 2 x}\{(0, 0)}, and g be a real-valued function on [0, 1] × [0, 1], defined by

g(x, y) =

 0 if (x, y) ∈ A ∪ {(0, 1)}

1 otherwise .

It is easy to see that inf

y∈Y sup

x∈X

g(x, y) = 1 > 0 = sup

x∈X

inf

y∈Y g(x, y).

If T is a multifunction on [0, 1] defined by T (x) =

  x, x + ¹ ₂ 

if x < ¹ ₂ [x, 1] if x ≥ ¹ ₂ , then we have (see Example 4.4)

inf

y∈Y sup

x∈T

(y)

g(x, y) = 1 = sup

x∈X

inf

y∈T (x) g(x, y).

This motivates us to define X-quasiconcave sets below. Recall that for a multifunction T : X −→ Y , the set H ^T = {g : X × Y → R} is said to be a X-quasiconcave of T if for all g ∈ H ^T , and x, x ₁ , x ₂ ∈ X, there exists x ³ ∈ X, h ∈ H ^T such that

h(x 3 , y) ≥ max{g(x ¹ , y), g(x 2 , y)}, ∀ y ∈ T (x).

Theorem 4.1. Let X be a nonempty compact convex set of a Hausdorff topological vector space, and Y be a nonempty set. Let T : X → Y be a multifunction having nonempty images, H _T a X-quasiconcave set of T , and f be a real-valued function defined on X × Y satisfied the following properties:

(0) sup _X f (x, y) ≤ sup X g(x, y) for all y ∈ Y and g ∈ H ^T ; (i) T is upper semicontinuous on X;

(ii) For each x ∈ X, y ∈ T (X) and g ∈ H ^T , g(x, ·) is lower semicontinous on T (X) and g(·, y) is quasiconcave on X.

Then for any λ ∈ R, there exists some g ∈ H ^T satisfying we have the following alternative:

Either sup _g∈H

T

sup _x∈X inf _{y∈T (x)} g(x, y) ≥ λ,

or there exists y ₀ ∈ T (X) such that f(x, y ⁰ ) ≤ λ for all x ∈ T ⁻¹ (y ₀ ).

Proof. For each λ ∈ R, let

U _g (y) = {x ∈ X; g(x, y) > λ}, and

V _g (x) = {y ∈ Y ; g(x, y) > λ}.

Fixed λ ∈ R. We may assume that

∀ y ∈ T (X) ∃ x ⁰ ∈ T ⁻¹ (y) s.t. f (x ₀ , y) > λ. (17)

For g ∈ H ^T , let S _g : X → X be defined as S ^g (x) = ∩ y∈T (x) U _g (y).

We want to show that

(α) S _g (x) 6= ∅ for some g ∈ H ^T . Fixed x ∈ X, let

A = {V ^ϕ (z) ∩ T (x); z ∈ X, ϕ ∈ H ^T }.

Obviously, the set A is a nonempty partially ordered set with the inclusion relation of subsets of Y . It is easy to see that any totally ordered subset of A has an upper bound.

By Zorn’s Lemma, there exists a maximal element V _g (ˆ x) ∩ T (x) of A, that is, ˆ x ∈ X and g ∈ H ^T satisfied that for any z ∈ X, ϕ ∈ H ^T

if V _g (ˆ x) ∩ T (x) ⊂ V ^ϕ (z) ∩ T (x) then V _ϕ (z) ∩ T (x) = V ^g (ˆ x) ∩ T (x). (18) Suppose V g (ˆ x)∩T (x) 6= T (x). Let y ∈ T (x) \ V ^g (ˆ x), by (17) we have some x 0 ∈ T ⁻¹ (y) such that f (x ₀ , y) > λ. By (0), we have sup _X g(x, y) ≥ sup X f (x, y) ≥ f(x ⁰ , y) > λ. This implies some x ₁ ∈ X such that g(x ¹ , y) > λ. Since H _T is X-quasiconcave, for g, ˆ x and x 1 there exists h ∈ H ^T , x 3 ∈ X such that

h(x ₃ , y) ≥ max{g(ˆ x, y), g(x ₁ , y)} , ∀ y ∈ T (x).

We have V _g (ˆ x) ∩T (x) V ^h (x ₃ ) ∩T (x), and get a contradiction from (18). This shows that V _g (ˆ x) ∩ T (x) = T (x) and hence T (x) ⊂ V ^g (ˆ x). In other words, for all y ∈ T (x), g(ˆ x, y) >

λ. It follows that ˆ x ∈ ∩ y∈T (x) U g (y). Hence, S g (x) 6= ∅ for some g ∈ H ^T . (β) S _g (x) is convex.

Since g(·, y) is quasiconcave by (ii) for each y ∈ T (X), U ^g (y) is convex. Hence S _g (x) is also convex.

(γ) for any z ∈ X, S g ⁻¹ (z) = {x ∈ X; g(z, y) > λ, ∀ y ∈ T (x)} is open.

Since g(x, ·) is l.s.c, U ^g (x) is open. It follows from upper semicontinuity of T that S _g ⁻¹ (z) = {x ∈ X; T (x) ⊂ U ^g (z)} = T ⁺ (U _g (z))

is open for each z ∈ X.

We already have some g ∈ H ^T satisfying (α), (β) and (γ). By Browder’s Fixed Point Theorem, there exists some x ₁ ∈ X such that x ¹ ∈ S ^g (x ₁ ). That is,

g(x ₁ , y) > λ ∀ y ∈ T (x ¹ ) =⇒ inf

y∈T (x

) g(x ₁ , y) > λ =⇒ sup

x∈X

inf

y∈T (x) g(x, y) ≥ λ.

We conclude that

sup

g∈H

sup

x∈X

inf

y∈T (x) g(x, y) ≥ λ.

Theorem 4.2. Let X be a nonempty compact convex set of a Hausdorff topological vector space, and Y be a nonempty set. Let T : X → Y be a multifunction having nonempty images, H _T a X-quasiconcave set for T , and f be a real-valued function defined on X × Y satisfied the following properties:

(0) sup _X f (x, y) ≤ sup X g(x, y) for all y ∈ Y and g ∈ H ^T ; (i) T is upper semicontinuous on X;

(ii) For each x ∈ X, y ∈ T (X) and g ∈ H ^T , g(x, ·) is lower semicontinous on T (X) and g(·, y) is quasiconcave on X.

Then there exists some g ∈ H ^T such that inf

y∈T (X) sup

x∈T

(y) f (x, y) ≤ sup

g∈H

sup

x∈X

inf

y∈T (x) g(x, y).

Proof. Let λ ∈ R and with

sup

g∈H

sup

x∈X

inf

y∈T (x) g(x, y) < λ.

Thus, by Theorem 4.1, there exists y ₀ ∈ T (X) such that f(x, y ⁰ ) ≤ λ for all x ∈ T ⁻¹ (y ₀ ).

This implies that sup _x∈T

(y

) f (x, y ₀ ) ≤ λ and hence inf y∈T (X) sup _x∈T

(y) f (x, y) ≤ λ. It follows that

inf

y∈T (X) sup

x∈T

(y) f (x, y) ≤ sup

g∈H

sup

x∈X

inf

y∈T (x) g(x, y).

Corollary 4.3. Let X be a nonempty compact convex set of a Hausdorff topological vector space, and Y be a nonempty set. Let T : X → Y be a multifunction having nonempty images, and f , g be two real-valued functions defined on X × Y satisfied the following properties:

(0) sup _X f (x, y) ≤ sup X g(x, y) for all y ∈ Y ; (i) T is upper semicontinuous on X;

(ii) For each x ∈ X, y ∈ T (X) and g(x, ·) is lower semicontinous on T (X) and g(·, y) is quasiconcave on X;

(iii) For all x, x ₁ , x ₂ ∈ X, there exists x ³ ∈ X such that

g(x 3 , y) ≥ max{g(x ¹ , y), g(x 2 , y)} ∀ y ∈ T (x).

Then

inf

y∈T (X) sup

x∈T

(y) f (x, y) ≤ sup

x∈X

inf

y∈T (x) g(x, y).

Proof. Let H _T = {g}, then by (iii) H ^T is X-quasiconcave of T . By Theorem 4.1 we have sup _g∈H

sup _x∈X inf _{y∈T (x)} g(x, y) = sup _x∈X inf _{y∈T (x)} g(x, y). Thus,

inf

y∈T (X)

sup

x∈T

(y) f (x, y) ≤ sup

x∈X

inf

y∈T (x)

g(x, y).

Example 4.4. Let the set A = {(x, y) ∈ [0, 1] × [0, 1]; 0 ≤ y ≤ ¹ 2 x}\{(0, 0)}, and g be a real-valued function defined on [0, 1] × [0, 1] by

g(x, y) =

 0 if (x, y) ∈ A ∪ {(0, 1)}

1 otherwise .

If T is a multifunction on [0, 1] defined by T (x) =

  x, x + ¹ ₂ 

if x < ¹ ₂ [x, 1] if x ≥ ¹ 2

figure 2 Then we have

y∈Y inf sup

x∈X

g(x, y) = 1 > 0 = sup

x∈X

y∈Y inf g(x, y).

We will check the conditions of Corollary 4.3.

(i) Clearly, T is u.s.c on X.

(ii)Let L ^β (x) = {y; g(x, y) ≤ β},

β ≥ 1, L ^β (x) = [0, 1].

when 0 ≤ β < 1, L ^β (x) =

  0, ¹ ₂ x 

, if 0 < x ≤ 1 {(0, 1)}, if x = 0 . β < 0, L ^β (x) = ∅.

Hence, g(x, ·) is lower semicontinuous on Y . (iii)Let U ^β (y) = {x; g(x, y) ≥ β}. Then

β > 1, U ^β (y) = ∅.

when 0 < β ≤ 1, U ^β (y) =

 

 

 



{(0, 0)}, if y = 0 [0, 2y) , if 0 < y ≤ ¹ 2

[0, 1] , if ¹ ₂ < y < 1 (0, 1] , if y = 1

.

β ≤ 0, U ^β (y) = [0, 1].

Hence, g(·, y) is quasi-concave on X.

(iv)For all x, x ₁ , x ₂ ∈ X, we have (by taking x ³ = x)

g(x, y) = 1 ≥ max{g(x ¹ , y), g(x ₂ , y)}, ∀y ∈ T ⁻¹ (x).

By Corollary 4.3, we have

y∈Y inf sup

x∈T

(y) g(x, y) ≤ sup

x∈X

inf

y∈T (x) g(x, y).

In fact, inf _y∈Y sup _x∈T

(y) g(x, y) = sup _x∈X inf _{y∈T (x)} g(x, y) = 0.