Existence of Solutions for Generalized Vector Quasi-Equilibrium Problems by Scalarization Method with Applications

Definition 1. Let B ⊂ Z be nonempty convex subset. G : B → 2^Y is called generalized D-quasiconvex, if for any y ∈ Y, the set {u ∈ B : G(u) ⊂ y − D} is convex (here, ∅ is regarded as a convex set).

Example 2. Let Y = ℝ²,   Z = B = ℝ, and $$ and define G₁, G₂ : B → 2^Y as

Then G₁ is generalized D-quasiconvex but not D-quasiconvex, while G₂ is just the reverse. Indeed, for any fixed y = (x₁, x₂) ∈ Y, both the sets are convex. But the set is not convex and neither is the set

Lemma 3 (see [15].)Let X and Y be topological spaces and S : X → 2^Y a set-valued mapping.

• (1)

  If S is usc with closed values, then S is closed.

• (2)

  If X is compact and S is usc with compact values, then S(X) is compact.

Lemma 4 (see [13].)Let X and Y be Hausdorff topological spaces and E ⊂ X a nonempty compact set and let h : E × Y → ℝ be a function and G : E → 2^Y a set-valued mapping. If h is continuous on E × Y and G is continuous with compact values, then the marginal function V(u) = sup _y∈G(u)h(u, y) is continuous and the marginal set-valued mapping δ(u) = {y ∈ G(u) : V(u) = h(u, y)} is usc.

Lemma 5 (Kakutani, see [13, Theorem 13 in Section 4 Chapter 6]). Let E be a nonempty compact and convex subset of a locally convex TVS X. If Φ : E → 2^E is usc and Φ(x) is a nonempty, convex, and closed subset for any x ∈ E, then there exists $$ such that $$.

Definition 6. The general nonlinear scalarization function ξ_G : E × F → ℝ, of G is defined as

Remark 7. If X = Y = Z = E = F and G(u) = {u},   for  all  u ∈ F, then the general nonlinear scalarization function ξ_G of G becomes the nonlinear scalarization function defined in [2].

Example 8. Let X = Z = ℝ,   Y = ℝ², and E = F = [0, +∞) ⊂ X and let G, C : E → 2^Y,   e : E → Y define as

respectively. We can see that ξ_G(x, u) = (1 + (2/(1 + 2x)))u.

Definition 9. Let D ⊂ Y be a cone, H : F → 2^Y, and ς : F → ℝ. ς is called monotone (resp., strictly monotone) with respect to (wrt for brevity) H, if for any u₁, u₂ ∈ F, H(u₁) − H(u₂) ⊂ int D implies that ς(u₂) ≤ ς(u₁)  (resp., ς(u₂) < ς(u₁)).

Example 10. Let Y = Z = ℝ,  F = [0,10], and D = ℝ₊ and let

Then H(u₁) − H(u₂) ⊂ int D implies that ς(u₁) > ς(u₂). Thus ς is strictly monotone wrt H, while ς is not monotone in the normal sense.

Example 11. Let Y = Z = ℝ, F = [0,5], and D = ℝ₊ and let ς(u) = u, for  all  u ∈ F, and H(u) = [−u − 1, −u], for  all  u ∈ F. Then ς is strictly monotone in the normal sense, but ς is not monotone wrt H. As the case stands, H(u₁) − H(u₂) ⊂ int D is equivalent to u₁, u₂ ∈ F and u₂ − u₁ > 1, which just results in ς(u₁) < ς(u₂).

Now some main properties of the general nonlinear scalarization function are established. First, according to [16, Proposition 3.1], we have the following.

Theorem 12. For each λ ∈ ℝ, x ∈ E and u ∈ F, the following assertions hold.

• (1)

  ξ_G(x, u) < λ⇔G(u) ⊂ λe(x) − int C(x).

• (2)

  ξ_G(x, u) ≤ λ⇔G(u) ⊂ λe(x) − C(x).

Theorem 13. ξ_G is strictly monotone wrt G in the second variable.

Proof. Letting u₁, u₂ ∈ F such that G(u₁) − G(u₂) ⊂ int C(x) and λ = ξ_G(x, u₁), we have

It is from this assertion that ξ_G(x, u₂) < λ = ξ_G(x, u₁) by Theorem 12 (1).

Theorem 14. Suppose that e is continuous.

• (1)

  If W and G are usc on E and F, respectively, where

  $$ ()

•  

  then ξ_G is usc on E × F.

• (2)

  If C is usc on E and G is lsc on F, then ξ_G is lsc on E × F.

Proof. (1) It is sufficient to attest the fact that for each λ ∈ ℝ, the set

is closed. As a matter of fact, for any sequence {(x_k, u_k)} in A such that (x_k, u_k)→(x₀, u₀) as k → ∞, ξ_G(x_k, u_k) ≥ λ, that is, G(u_k)⊄λe(x_k) − int C(x_k), by Theorem 12 (1). Obviously, $$ is compact and so is G(F₀) by Lemma 3 (2). Then there exists y_k ∈ G(u_k) ⊂ G(F₀) such that y_k ∉ λe(x_k) − int C(x_k) and a subsequence $$ of {y_k}, such that $$ as i → ∞ and Obviously, G is usc with closed values, which implies that G is closed by Lemma 3 (1). Hence, y₀ ∈ G(u₀). Similarly, W is closed. Letting i → ∞ in (12) and applying the continuity of e and the closeness of W, we obtain that λe(x₀) − y₀ ∈ W(x₀). Namely, y₀ ∉ λe(x₀) − int C(x₀). Thus G(u₀)⊄λe(x₀) − int C(x₀), which is equivalent to ξ_G(x₀, u₀) ≥ λ by Theorem 12 (1). Therefore, λ ∈ A and A is closed.

(2) It′s enough to argue that for each λ ∈ ℝ, the set

is closed. In fact, for any sequence {(x_k, u_k)} in B such that (x_k, u_k)→(x₀, u₀) as k → ∞, by Theorem 12 (2),   ξ_G(x_k, u_k) ≤ λ, equivalently, G(u_k) ⊂ λe(x_k) − C(x_k). For any y₀ ∈ G(u₀), there exists y_k ∈ G(u_k) such that y_k → y₀ as k → ∞ according to the equivalent definition of lower semicontinuity of G (see [13], page 108). Also, Linking the continuity of e with the closeness of C inferred from Lemma 3 (1) and letting k → ∞ in (14), we get λe(x₀) − y₀ ∈ C(x₀) and G(u₀) ⊂ λe(x₀) − C(x₀). Consequently, Theorem 12 (2) leads to ξ_G(x₀, u₀) ≤ λ which amounts to (x₀, u₀) ∈ B. B is closed.

If X = Y = Z = E = F (F is not required to be compact) and G(u) = {u}, for  all  u ∈ F, then Theorem 14 becomes [2, Theorem 2.1]. Actually, [2, Theorem 2.1] can be regarded as the case where G is continuous in Theorem 14. In addition, Example 2.1 (resp., Example 2.2) in [2] shows that if C (resp., W) is not usc, maybe the general nonlinear scalarization function fails to be lsc (resp., usc) under all the other assumptions. Now the following example demonstrates that the assumption of the upper semicontinuity (resp., lower semicontinuity) of G is necessary in Theorem 14 (1) (resp., (2)) even if C is continuous.

Example 15. Let X = Z = ℝ, Y = ℝ², E = ℝ₊, and F = [−10,10] ⊂ Z, and let

(1) Define

Evidently, G is not usc on F. After simply calculating, ξ_G is not usc on E × F due to the fact that {(x, u) ∈ E × F : ξ(x, u) ≥ 1} = ℝ₊ × (0,10] is not closed.

(2) Consider the following mapping:

Obviously, G is not lsc on F. Also, ξ_G fails to be lsc on E × F, where

Lemma 16. Let e : E → Y be a vector-valued mapping such that e(x) ∈ int C(x) for all x ∈ E and f : E × F → 2^Y a set-valued mapping and define f_z(x) = f(x, z) for each z ∈ F. If for each z ∈ F, f_z is generalized C(g(z))-quasiconvex with compact values, then x ↦ h(x, z) is ℝ₊-quasiconvex, where $$ and $$ is the general nonlinear scalarization function of f_z.

Proof. It′s sufficient to testify that L_z(λ) ⊂ X is convex, where

for each z ∈ F and λ ∈ ℝ. Indeed, for any x₁, x₂ ∈ L_z(λ), Clearly, for each z ∈ E,   x₁, x₂ ∈ M, where M is convex by reason of the generalized C(g(z))-quasiconvexity of f_z and so $$ for all 0 ≤ t ≤ 1. Hence, which is equal to $$ by Theorem 12 (2). Thus, $$ and L_z(λ) is convex.

Theorem 17. Let E and F be compact and convex subsets and P : E → 2^E,   Q : E → 2^F, and f : E × F → 2^Y set-valued mappings. For each z ∈ F, define f_z(x) = f(x, z). Suppose that the following conditions are fulfilled:

• (a)

  x ↦ int C(x) has a continuous select e : E → Y;

• (b)

  both C and W are usc on E, where

  $$ ()

• (c)

  g is continuous, f and P are strict and continuous, and Q is strict and usc;

• (d)

  for each z ∈ F, f_z is generalized C(g(z))-quasiconvex;

• (e)

  for each (x, z) ∈ E × F, f(x, z) is compact and for each x ∈ E, both P(x) and Q(x) are closed and convex.

Then the GVQEP1 has a solution $$; that is, there exist $$ and $$ such that Further assume that f(x, z) ⊂ C(g(z)) for each x ∈ E and z ∈ Q(x). Then

Proof. Denote K = E × F and w = (x, z) where x ∈ E and z ∈ F, and define $$ as

Obviously, $$ is continuous on K. The general nonlinear scalarization function of f is continuous on E × K by Theorem 14. Set $$, where $$ is the general nonlinear scalarization function of f_z. Since h is continuous on K by virtue of the continuity of ξ_f and $$. Define $$ as Obviously, $$ has compact values and is continuous. Define a set-valued mapping δ : E × F → 2^E×F as By Lemma 4, δ(x, z) is usc. Let Then δ(x, z) = η(x, z)×{z}. The upper semicontinuity of η is obvious. Now we show that for each (x, z) ∈ E × F,   η(x, z) is convex and closed. Detailedly,

• (i)

  Define

  $$ ()

Then for any u₁, u₂ ∈ η(x, z), h(u_i, z) = λ, i = 1,2. Clearly, $$ for all t ∈ [0,1] since P(x) is convex. In view of condition (d) and Lemma 16, x ↦ h(x, z) is ℝ₊-quasiconvex, which deduces that for each z ∈ F, the set is convex. Thus, $$. It follows from the definition of λ that $$, which results in $$ and the convexity of η(x, z).

• (ii)

  For each (x, z) ∈ E × F and for any sequence {u_k} ⊂ η(x, z) such that u_k → u₀ as k → ∞,

  $$ ()

Since h is continuous, Thus u₀ ∈ η(x, z) and η(x, z) is closed.

Now consider a set-valued mapping Φ : E × F → 2^E×F prescribed as

It′s easy to check that Φ is usc on E × F and for each (x, z) ∈ E × F,   Φ(x, z) is convex and closed. By Lemma 5, Φ exists a fixed point $$, that is to say, Hence, for each $$, $$. In other words, which deduces that $$, and by Theorem 13. The first conclusion is proved. Further assume that f(x, z) ⊂ C(g(z)) for each x ∈ E and z ∈ Q(x). If the second conclusion is not true; namely, there exists $$ such that $$, then This is absurd.

Corollary 18. Let E and F be compact and convex subsets, P : E → 2^E and Q : E → 2^F set-valued mappings and f : E × F → Y a vector-valued mapping. For each z ∈ F, define f_z(x) = f(x, z). Suppose that the following conditions are in force:

• (a)

  x ↦ int C(x) has a continuous select e : E → Y;

• (b)

  both C and W are usc on E, where

  $$ ()

• (c)

  f and g are continuous, P is strict and continuous, and Q is strict and usc;

• (d)

  for each z ∈ F,   f_z is C(g(z))-quasiconvex;

• (e)

  for each x ∈ E, both P(x) and Q(x) are closed and convex.

Then there exist $$ and $$ such that Further suppose that f(x, z) ∈ C(g(z)) for each x ∈ E and z ∈ Q(x). Then

Theorem 19. Let E and F be compact and convex subsets and P : E → 2^E,   Q : E → 2^F, and f : E × E × F → 2^Y set-valued mappings. For each (x, z) ∈ E × F, define f_xz(u) = f(u, x, z). Suppose that the following conditions hold:

• (a)

  x ↦ int C(x) has a continuous select e : E → Y;

• (b)

  both C and W are usc on E, where

  $$ ()

• (c)

  g is continuous, f and P are strict and continuous, and Q is strict and usc;

• (d)

  for each (x, z) ∈ E × F, f_xz is generalized C(g(z))-quasiconvex;

• (e)

  for each (u, x, z) ∈ E × E × F, f(u, x, z) is compact and for each x ∈ E, both P(x) and Q(x) are closed and convex.

Then the GVQEP2 exists a solution $$; namely, there exist $$ and $$ such that Furthermore, if f(x, x, z) ⊂ C(g(z)) for each x ∈ E and z ∈ Q(x), then

Proof. Denoting K = E × F, we see that K is compact. Define $$ and S : E → 2^K as $$ and S(x) = {x} × Q(x),   for  all  x ∈ E, respectively. Then $$ is continuous on K and for each w ∈ K, f_w is generalized $$-quasiconvex according to condition (d). Since Q is strict and usc with compact values, so is S. Replacing F, g, and Q in Theorem 17 by $$, and S, respectively, we see that these conclusions are true.

Corollary 20 (see [2].)Let E and F be compact and convex subsets, P : E → 2^E and Q : E → 2^F set-valued mappings, and f : E × E × F → Y a vector-valued mapping. For each (x, z) ∈ E × F, define f_xz(u) = f(u, x, z). The following assumptions are in operation:

• (a)

  x ↦ int C(x) has a continuous select e : E → Y;

• (b)

  both C and W are usc on E, where

  $$ ()

• (c)

  f and g are continuous, P is strict and continuous, and Q is strict and usc;

• (d)

  for each (x, z) ∈ E × F, f_xz is C(g(z))-quasiconvex;

• (e)

  for each x ∈ E, both P(x) and Q(x) are closed and convex.

Then there exist $$ and $$ such that Additionally, on the assumption that f(x, x, z) ∈ C(g(z)) for each x ∈ E and z ∈ Q(x),