The Hierarchical Minimax Inequalities for Set-Valued Mappings

Definition 1 (see [1], [3].)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}; Min⁡A denotes the set of all minimal points of A;

• (b)

  maximal point of A if A∩(z + C) = {z}; Max⁡A denotes the set of all maximal points of A;

• (c)

  weakly minimal point of A if A∩(z − int⁡C) = ∅; Min_wA denotes the set of all weakly minimal points of A;

• (d)

  weakly maximal point of A if A∩(z + int⁡C) = ∅; Max_wA denotes the set of all weakly maximal points of A.

Definition 2 (see [5], [6].)Let U, V be Hausdorff topological spaces. A set-valued map F : U⇉V with nonempty values is said to be

• (a)

  upper semicontinuous at x₀ ∈ U if for every x₀ ∈ U and for every open set N containing F(x₀) there exists a neighborhood M of x₀ such that F(M) ⊂ N;

• (b)

  lower semicontinuous at x₀ ∈ U if for any net {x_ν} ⊂ U, x_ν → x₀, y₀ ∈ T(x₀) implies that there exists net y_ν ∈ T(x_ν) such that y_ν → y₀;

• (c)

  continuous at x₀ ∈ U if F is upper semicontinuous as well as lower semicontinuous at x₀.

Definition 3 (see [3], [7].)Let k ∈ int⁡C and v ∈ Z. The Gerstewitz function ξ_kv : Z → ℝ is defined by

Proposition 4 (see [3], [7].)Let k ∈ int⁡C and v ∈ Z. The Gerstewitz function ξ_kv : Z → ℝ has the following properties:

• (a)

  ξ_kv(u) > r⇔u ∉ v + rk − C;

• (b)

  ξ_kv(u) ≥ r⇔u ∉ v + rk − int⁡C;

• (c)

  ξ_kv(·) is a convex, continuous, and increasing function.

Definition 5 (see [1].)Let X be a nonempty convex subset of a topological vector space. A set-valued mapping F : X⇉Z is said to be

• (a)

  above-C-convex (resp., above-C-concave) on X if, for all x₁, x₂ ∈ X and all λ ∈ [0,1],

  $$ (2)

• (b)

  above-naturally C-quasiconvex on X if, for all x₁, x₂ ∈ X and all λ ∈ [0,1],

  $$ (3)
  where co⁡A denotes the convex hull of a set A;

• (c)

  above-C-convex-like (resp., above-C-concave-like) on X (X is not necessary convex) if, for all x₁, x₂ ∈ X and all λ ∈ [0,1], there is an x^′ ∈ X such that

  $$ (4)

Theorem 6. Let X be a nonempty compact convex subset of real Hausdorff topological vector space. Let the set-valued mappings F, G, H : X × X⇉ℝ such that F(x, y) ⊂ G(x, y) ⊂ H(x, y) for all (x, y) ∈ X × X; ⋃_y∈X F(x, y) and ⋃_x∈X H(x, y) are compact for each x ∈ X and for each y ∈ X and satisfy the following conditions:

• (i)

  x ↦ F(x, y) is lower semicontinuous on X for each y ∈ X and y ↦ F(x, y) is above-ℝ₊-concave on X for each x ∈ X;

• (ii)

  x ↦ G(x, y) is above-naturally ℝ₊-quasiconvex for each y ∈ X, and y ↦ G(x, y) is lower semicontinuous on X for each x ∈ X;

• (iii)

  x ↦ H(x, y) is lower semicontinuous on X for each y ∈ X, y ↦ H(x, y) is above-ℝ₊-concave on X for each x ∈ X, and y ↦ H(x, y) is lower semicontinuous for each x ∈ X.

Then one has

Lemma 7. Let F : X⇉ℝ be such that max⁡⋃_x∈X F(x), max⁡⋃_x∈X max⁡F(x), and max⁡F(x) exist for all x ∈ X. Then

Proof. By using the similar technique of Lemma 3.3 [9], we can show that the conclusion is valid.

Theorem 8. Let X be a nonempty compact (not necessarily convex) subset of a real Hausdorff topological space. Let the set-valued mappings F, S, T, G : X × X⇉ℝ with nonempty compact values such that

• (i)

  (x, y) ↦ F(x, y) and (x, y) ↦ G(x, y) are upper semicontinuous on X × X;

• (ii)

  x ↦ max⁡S(x, y) is convex-like for each y ∈ X, and y ↦ max⁡T(x, y) is concave-like on Y for each x ∈ X;

• (iii)

  for all (x, y) ∈ X × X, max⁡F(x, y) ≤ max⁡S(x, y) ≤ max⁡T(x, y) ≤ max⁡G(x, y).

Then the relation (s-Hi) holds.

Proof. From (i), we know that both sides of (s-Hi) exist. For any r ∈ ℝ,

Define M : X⇉X by for all x ∈ X. By (i), the set M(x) is closed for all x ∈ X. We claim that the whole intersection is empty. Indeed, if not, there exists y₀ ∈ ⋂_x∈X M(x) such that, for all x ∈ X, max⁡F(x, y₀) ≥ r. In particular, we choose x = y₀; then max⁡F(y₀, y₀) ≥ r which, with the aid of condition (iii), contradicts the choice of r. Hence, by the compactness of X, there exist x₁, x₂, …, x_m ∈ X such that Let for all x ∈ X. Then, by (iii), we have This implies that, for each y ∈ X, there is $$ such that Define two sets as follows: By the concave-like property of T, we can see that these two sets are disjoint. For each y ∈ X, by the separation theorem, there exists nonzero vector (τ₁, τ₂, …, τ_m) ∈ ℝ^m such that for all (z₁, z₂, …, z_m) ∈ L₂. Then, $$ and τ_i > 0 for all i = 1,2, …, m. Let $$ for all i = 1,2, …, m. Then we have For each i = 1,2, …, m, by taking z_i = r and noting max⁡S(x_i, y) ≤ max⁡T(x_i, y), we have for all y ∈ X. Since the mapping x ↦ max⁡S(x, y) is convex-like for each y ∈ Y, there is x₀ ∈ X such that Since max⁡F(x, y) ≤ max⁡S(x, y) for all (x, y) ∈ X × X, we have for all y ∈ X. By Lemma 7, we know that Therefore, the relation (s-Hi) holds.

Theorem 9. Let X be a nonempty compact convex subset of a real Hausdorff topological vector space. Let the set-valued mappings F, G : X × X⇉ℝ with nonempty compact values such that

• (i)

  (x, y) ↦ F(x, y) and (x, y) ↦ G(x, y) are upper semicontinuous on X × X;

• (ii)

  y ↦ max⁡F(x, y) is quasiconcave for each x ∈ X; that is, for each x ∈ X, the set {y ∈ X : max⁡F(x, y) ≥ r} is convex in X;

• (iii)

  for all (x, y) ∈ X × X, max⁡F(x, y) ≤ max⁡G(x, y).

Then the relation (s-Hi) holds.

Proof. By (i), we know that both sides of (s-Hi) exist. Choose any r ∈ ℝ satisfies

Define W : X⇉X by for all x ∈ X. By (ii), the set W(x) is convex for all x ∈ X. By (iii), we have Hence, for all x ∈ X. By the upper semicontinuity of F, we know that the mapping x ↦ max⁡F(x, y) is upper semicontinuous for each x ∈ X. Thus, for each x ∈ X, W(x) is closed; hence it is compact. In order to claim that the mapping x ↦ W(x) is upper semicontinuous on X, we only need to show that the mapping x ↦ W(x) has a closed graph. Since, for any net {(x_α, y_α)} ⊂ Graph⁡(W) we have the net {(x_α, y_α)}. converges to some point (x₀, y₀). Then, for each α, max⁡F(x_α, y_α) ≥ r. Since the mapping (x, y) ↦ max⁡F(x, y) is upper semicontinuous, we have Thus, (x₀, y₀) ∈ Graph⁡(W). Suppose that W(x) ≠ ∅ for all x ∈ X. Then, by Kakutani fixed point theorem, the mapping W has a fixed point which is a contradiction to (24). Hence, there is an x₀ ∈ X such that W(x₀) = ∅. From this, we know that This implies that the relation (s-Hi) holds.

Example 10. Let X = {0}∪{1/n : n ∈ ℕ} and f(x) = x², g(y) = 1 − y² for all x, y ∈ X. Define F, S, T, G : X × X⇉ℝ by F(x, y) = [0, f(x)g(y)], S(x, y) = [−1, f(x)g(y) + 1], T(x, y) = [2, f(x)g(y) + 2], and G(x, y) = [3, f(x)g(y) + 3]. Obviously, all conditions of Theorem 8 hold. Hence the relation (s-Hi) holds. Indeed, by simple calculation, we can see that

Example 11. Let X = [0,1]. The mappings f, g, F, and G are the same as in Example 10. Then, all conditions of Theorem 9 hold. We can see that both values of min⁡⋃_x∈X max⁡⋃_y∈X F(x, y) and max⁡⋃_x∈X G(x, x) are the same as those in Example 10. Hence the relation (s-Hi) holds.

Theorem 12. Let X be a nonempty compact convex subset of a real Hausdorff topological vector space. Let the set-valued mappings F, G, H : X × X⇉Z with nonempty compact values such that F(x, y) ⊂ G(x, y) ⊂ H(x, y) for all (x, y) ∈ X × Y satisfy the following conditions:

• (i)

  (x, y) ↦ F(x, y) is upper semicontinuous, y ↦ F(x, y) is above-C-concave on Y for each x ∈ X, and x ↦ F(x, y) is lower semicontinuous on X for each y ∈ Y;

• (ii)

  x ↦ G(x, y) is above-naturally C-quasiconvex for each y ∈ Y, and y ↦ G(x, y) is lower semicontinuous on Y for each x ∈ X;

• (iii)

  y ↦ H(x, y) is lower semicontinuous and above-C-concave on Y for each x ∈ X, and x ↦ H(x, y) is lower semicontinuous on X for each y ∈ Y;

• (iv)

  for each y ∈ Y,

  $$ (28)

Then the relation (Hi-1) is valid.

Proof. Let Γ(x): = Max_w⋃_y∈Y F(x, y) for all x ∈ X. From Lemma 2.4 and Proposition  3.5 in [1], the mapping x ↦ Γ(x) is upper semicontinuous with nonempty compact values on X. Hence ⋃_x∈X Γ(x) is compact and so is co⁡(⋃_x∈X Γ(x)). Then co⁡(⋃_x∈X Γ(x)) + C is a closed convex set with nonempty interior. Suppose that v ∉ co⁡(⋃_x∈X Γ(x)) + C. By separation theorem, there is a k ∈ ℝ, ϵ > 0, and a nonzero continuous linear functional ξ : Z ↦ ℝ such that

for all u ∈ co(⋃_x∈X Γ(x)) and c ∈ C. From this we can see that ξ ∈ C^⋆, where C^⋆ = {g : Z ↦ ℝ : g(c) ≥ 0  ∀ c ∈ C}, and ξ(v) < ξ(u) for all u ∈ co⁡(⋃_x∈X Γ(x)). By Proposition 3.14 of [1], for any x ∈ X, there is a $$ and $$ with $$ such that Let us choose c = 0 and $$ in (29); we have for all x ∈ X. Therefore, From conditions (i)–(iii), by applying Proposition 3.9 and Proposition  3.13 in [1], all conditions of Theorem 6 hold. Hence, we have Since Y is compact, there is y^′ ∈ Y such that Thus, and, hence, Therefore, By taking into account condition (iv), we know that Hence, the relation (Hi-1) is valid.

Example 13. Let X = [0,1], $$, and f : X⇉ℝ define

and F, G, H : X × X⇉ℝ²define for all (x, y) ∈ X × X.

Theorem 14. Let X be a nonempty compact convex subset of real Hausdorff topological vector space. Let the set-valued mappings F, G, H : X × X⇉Z such that F(x, y) ⊂ G(x, y) ⊂ H(x, y) for all (x, y) ∈ X × X and satisfy the following conditions:

• (i)

  (x, y) ↦ F(x, y) is continuous with nonempty compact values, and y ↦ ξ_kvF(x, y) is above-ℝ₊-concave on X for each x ∈ X and any Gerstewitz function ξ_kv;

• (ii)

  x ↦ G(x, y) is above-naturally C-quasiconvex for each y ∈ X, and y ↦ G(x, y) is lower semicontinuous on X for each x ∈ X;

• (iii)

  (x, y) ↦ H(x, y) is upper semicontinuous with nonempty compact values, y ↦ ξ_kvH(x, y) is above-ℝ₊-concave on X for each x ∈ X, and x ↦ H(x, y) is lower semicontinuous on X for each y ∈ X and any Gerstewitz function ξ_kv;

• (iv)

  for each y ∈ Y,

  $$ (46)

Then the relation (Hi-2) is valid.

Proof. Let Γ(x) be defined the same as that in Theorem 12 for all x ∈ X. From the process in the proof of Theorem 12, we know that the set ⋃_x∈X Γ(x) is nonempty compact. Suppose that v ∉ ⋃_x∈X Γ(x) + C. For any k ∈ int⁡C, there is a Gerstewitz function ξ_kv : Z ↦ ℝ such that

for all u ∈ ⋃_x∈X Γ(x). Then, for each x ∈ X, there is $$ and $$ with $$ such that Choosing $$ in (47), we have for all x ∈ X. Therefore,

By conditions (i)–(iii), we know that all conditions of Theorem 6 hold for the mappings ξ_kvF(x, y), ξ_kvG(x, y), and ξ_kvH(x, y), and, hence, we have

Since X is compact, there is a y^′ ∈ X such that Thus, and, hence, If v ∈ Max⁡⋃_y∈X Min_w⋃_x∈X F(x, y), then, by (iv), we have which contradicts (54). From this, we can deduce that the relation (Hi-2) is valid.

Theorem 15. Let Z be a finite dimensional space and the set-valued mappings F and T are two upper semicontinuous mappings with nonempty compact values such that,

• (i)

  for each x ∈ K, there is s ∈ T(x) such that F(s, x, x)⊄− int C;

• (ii)

  for each x ∈ K, the sets {(s, y) ∈ TK × K : F(s, x, y)⊂− int C} and T(x) are convex.

Then (SEP)_F has a weak solution.

Proof. Define Ω : K⇉K by

for all y ∈ K. By (i), y ∈ Ω(y) for all y ∈ K. Hence the set Ω(y) is nonempty for all y ∈ K. Next, we claim that the set Ω(y) is closed for all y ∈ K. Let a net {x_α} ⊂ Ω converge to some point x₀. Then there are s_α ∈ Tx_α and z_α ∈ F(x_α, x_α, y) such that z_α ∈ Z∖(−int⁡C). By the upper semicontinuities of F and T, the sets TK and F(TK × K × K) are compact. Hence, there is a convergent subnet $$ of {z_α} that converges to some point z₀. Furthermore, the net $$ has a convergent subnet $$ which converges to some point s₀. Again, by the upper semicontinuities of F and T, we have z₀ ∈ T(x₀) and z₀ ∈ F(s₀, z₀, y). Since the set Z∖(−int⁡C) is closed, z₀ ∈ Z∖(−int⁡C). Hence, x₀ ∈ Ω(y), and, thus, Ω(y) is closed for all y ∈ K. We next claim that the mapping Ω : K⇉K is a KKM mapping. Indeed, if not, there exist x₁, x₂, …, x_n ∈ K and x₀ such that Then there is $$ where $$ and λ_i ≥ 0 for all i = 1,2, …, n.

Since $$, for all i ∈ {1,2, …, n}, choose any s_i ∈ T(x₀); we have

By (ii), This implies that which contradicts (i). Thus, the mapping Ω : K⇉K is a KKM mapping. By the Fan lemma, the whole intersection is nonempty. Any point in the whole intersection is a weak solution for (SEP) _F.

Theorem 16. Under the framework of Theorem 15, in addition, the mappings A, B, G : TK × K × K⇉Z with nonempty compact values such that

• (i)

  the mapping s ↦ G(s, x, y) is upper semicontinuous mappings for each x, y ∈ K;

• (ii)

  both sets ⋃_s∈T(x) F(s, x, y) and ⋃_y∈K G(s, x, y) are compact for x, y ∈ K, s ∈ T(x);

• (iii)

  the mapping s ↦ max⁡B(s, x, y) is concave-like for each x, y ∈ K, and the mapping y ↦ max⁡A(s, x, y) is convex-like for each x, y ∈ K, s ∈ T(x);

• (iv)

  for each x, y ∈ K, s ∈ T(x),   max⁡F(s, x, y) ≤ max⁡A(s, x, y) ≤ max⁡B(s, x, y) ≤ max⁡G(s, x, y);

• (v)

  for each y ∈ K, there is an $$ with $$ such that

  $$ (64)

Then (SEP)_G has a strong solution.

Proof. According to Theorem 15, we know that (SEP)_F has a weak solution. That is, there is an $$ such that

for all x ∈ K and for some $$. For any k ∈ int⁡C, from Proposition 4, the Gerstewitz function ξ_k0 satisfies Hence, there is $$ such that, for each x ∈ K, Thus, we have By conditions (i)–(v), all conditions of Theorem 6 hold; hence we have Since $$ is compact, there is $$ such that This implies that or for all x ∈ K. Therefore, (SEP) _G has a strong solution.

Example 17. Let K = [1,2], C = ℝ₊, Z = ℝ, and T : K⇉ℝ be defined by T(x) = [0,2x] for all x ∈ K. Then we define F, A, B, G : TK × K × K⇉ℝ which are defined by

for all (x, y) ∈ K × K.