The Convergent Behavior for Parametric Generalized Vector Equilibrium Problems

Theorem 2.1. Let X, Y, Z, C, K, 𝒦, T, and f be given as in Section 1, the parametric spaces Δ₁, Δ₂ be two Hausdorff topological vector spaces. Let the mapping f : Δ₁ × Z × 𝒦 × 𝒦 → Y be such that (ξ, s, x, y) → f(ξ, s, x, y) is continuous and y → f(ξ, s, x, y) is C(x)-convex for every (ξ, s, x) ∈ Δ₁ × Z × 𝒦, the mapping T : 𝒦 → 2^Z be an upper semicontinuous with nonempty compact values, and the mapping K : Δ₂ → 2^X is continuous with nonempty compact and convex values. Suppose that the following conditions hold the following:

• (a)

  for any ξ ∈ Δ₁, x ∈ 𝒦, there is an s ∈ Tx, such that f(ξ, s, x, x) ∉ (−int C(x));

• (b)

  the mapping x → Y∖(−int C(x)) is closed [33] on 𝒦.

Then, we have

• (1)

  for every (ξ, η) ∈ Δ₁ × Δ₂, the weak efficient solutions for (PGVEP) exist, that is, the set Γ_w(ξ, η) is nonempty, where $$ for some $$ for all y ∈ K(η)}.

• (2)

  Γ_w : Δ₁ × Δ₂ → 2^X is upper semicontinuous on Δ₁ × Δ₂ with nonempty compact values.

Proof. (1) For any fixed (ξ, η) ∈ Δ₁ × Δ₂, we can easy check that the mappings (s, x) → f(ξ, s, x, y), y → f(ξ, s, x, y) satisfy all conditions of Corollary 2.2 in [15] with K = 𝒦 and D = conv(𝒦). Hence, from this corollary, we know that Γ_w(ξ, η) is nonempty.

(2) For any fixed (ξ, η) ∈ Δ₁ × Δ₂, we first claim that Γ_w(ξ, η) is closed in K(η), hence it is compact. Indeed, let a net {x_α} ⊂ Γ_w(ξ, η) and x_α → p for some p ∈ X. Then, x_α ∈ K(η) and f(ξ, s_αy, x_αy, y) ∉ −int C(x_α) for all y ∈ K(η) and for some s_αy ∈ T(x_α). Since K(η) is compact, p ∈ K(η). For each α and for each y ∈ K(η), there exists an s_αy ∈ T(x_α) such that f(ξ, s_αy, x_α, y) ∈ Y∖(−int C(x_α)). Since T is upper semicontinuous with nonempty compact values, and the set {x_α}∪{p} is compact, T({x_α}∪{p}) is compact. Therefore, without loss of generality, we may assume that the net {s_αy} converges to some s_y. Then s_y ∈ T(p). Since the mapping (s, x) → f(ξ, s, x, y) is continuous, we have

Since f(ξ, s_αy, x_α, y) ∈ Y∖(−int C(x_α)), x_α → p and the mapping x → Y∖(−int C(x)) is closed, we have This proves that p ∈ Γ_w(ξ, η), and hence Γ_w(ξ, η) is closed. Since K(η) is compact, so is Γ_w(ξ, η).

We next prove that the mapping Γ_w : Δ₁ × Δ₂ → 2^K(η) is upper semicontinuous. That is, for any (ξ, η) ∈ Δ₁ × Δ₂, if there is a net {(ξ_β, η_β)} converges to (ξ, η) and some x_β ∈ Γ_w(ξ_β, η_β), we need to claim that there is a p ∈ Γ_w(ξ, η) and a subnet $$ of {x_β} such that $$. Indeed, since x_β ∈ K(η_β) and K : Δ₂ → 2^X are upper semicontinuous with nonempty compact values, there is a p ∈ K(η) and a subnet $$ of {x_β} such that $$.

If we can claim that p ∈ Γ_w(ξ, η), then we can see that Γ_w : Δ₁ × Δ₂ → 2^X is upper semicontinuous on Δ₁ × Δ₂, and complete our proof. Indeed, if not, there is a y ∈ K(η) such that for every s ∈ T(p) we have

Since K is lower semicontinuous, there is a net $$ with $$ and $$. Since $$, we have $$ and, for each $$,

for some $$.

Since T is upper semicontinuous and the net $$, without loss of generality, we may assume that $$ for some s ∈ T(x). Since the mapping (ξ, s, x, y) → f(ξ, s, x, y) is continuous, we have

From (2.4) and the closedness of the mapping x → Y∖(−int C(x)), we have which contradicts (2.3). Hence, we have p ∈ Γ_w(ξ, η).

Theorem 3.1. Under the framework of Theorem 2.1, for each (ξ, η) ∈ Δ₁  ×  Δ₂, there is an $$ with $$. In addition, if $$ is convex, the mapping $$ is properly quasi $$-convex (Definition 1.1 of [28]) on $$ for each (ξ, y) ∈ Δ₁ × K(η). Assume that the mapping $$ satisfies the following conditions:

• (i)
  $$ ()

•  

    for every $$;

• (ii)

  for any fixed x ∈ K(η), if $$ and δ cannot be comparable with $$ which does not equal to δ, then $$;

• (iii)

  if $$, there exists an $$ such that $$.

Then, we have

• (a)

  for every (ξ, η) ∈ Δ₁ × Δ₂, the efficient solutions exists, that is, the set Γ(ξ, η) is nonempty, furthermore, it is compact;

• (b)

  the mapping Γ : Δ₁ × Δ₂ → 2^X is upper semicontinuous on Δ₁ × Δ₂ with nonempty compact values;

• (c)

  for each (ξ, η) ∈ Δ₁ × Δ₂, the set Γ(ξ, η) is connected if C : K(η) → 2^Y is constant, and for any (ξ, η) ∈ Δ₁ × Δ₂, x ∈ K(η) and s ∈ T(K(η)), f(ξ, s, x, K(η)) + C is convex.

Proof. (a) Fixed any (ξ, η) ∈ Δ₁ × Δ₂, we can easy see that all conditions of Theorem 2.3 of [15] hold, hence from Theorem 2.3 of [15], we know that Γ(ξ, η) is nonempty and compact.

(b) Let {(ξ_α, η_α)} ⊂ Δ₁ × Δ₂ be a net such that (ξ_α, η_α)→(ξ, η) and {x_α} be a net with x_α ∈ Γ(ξ_α, η_α). Since x_α ∈ K(η_α) and K : Δ₂ → 2^X are upper semicontinuous with nonempty compact values, there are an x ∈ K(η) and a subnet $$ of {x_α} such that $$. Since T : K → 2^Z is upper semicontinuous with nonempty compact values, $$ is compact. Since $$, there is an s ∈ T(x) such that a subnet of $$ converges to s. Without loss of generality, we still denote the subnet by $$, and hence $$.

If x ∉ Γ(ξ, η), then there is a y ∈ K(η) such that

Since K(η) is compact, there is a net, say $$, in K(η) converges to y. Since the mapping (ξ, s, x, y) → f(ξ, s, x, y) is continuous, and the mapping x → Y∖(−int C(x)) is closed, we have which contracts (3.2). Thus, x ∈ Γ(ξ, η).

In order to prove (c), we introduce Lemmas 3.2–3.4 as follows.

Let Y^⋆ be the topological dual space of Y. For each x ∈ 𝒦,

Let C^⋆ = ∩_x∈𝒦C^⋆(x), then C^⋆ is nonempty and connected. If $$ is a constant mapping, then C^⋆(x) = C^⋆ for all x ∈ 𝒦. In the sequel, we suppose that C^⋆ is not a singleton. That is, C^⋆∖{0} ≠ ∅, and hence it is connected. For each g ∈ C^⋆∖{0}, let us denote the set of g-efficient solutions to (PGVEP) by Lemma  3.2.   Under the framework of Theorem 3.1, for every g ∈ C^⋆∖{0}.

Proof. From (a) of Theorem 3.1, we know that, for each (ξ, η) ∈ Δ₁ × Δ₂, there is an $$ with $$ such that

for all y ∈ K(η) and for all g ∈ C^⋆∖{0}. Thus, $$ for every g ∈ C^⋆∖{0}. Hence, S^ξ,η(g) ≠ ∅ for every g ∈ C^⋆∖{0}.

Lemma  3.3.  Suppose that for any (ξ, η) ∈ Δ₁ × Δ₂ and y ∈ K(η), f(ξ, T(K(η)), K(η), y) are bounded. Then, the mapping S^ξ,η : C^⋆∖{0} → 2^K(η) is upper semicontinuous with compact values.

Proof. Fixed any (ξ, η) ∈ Δ₁ × Δ₂. We first claim that the mapping S^ξ,η : C^⋆∖{0} → 2^K(η) is closed. Let x_ν ∈ S^ξ,η(g_ν), x_ν → x and g_ν → g with respect to the strong topology σ(Y^⋆, Y) in Y^⋆.

Since x_ν ∈ S^ξ,η(g_ν), there is an s_ν ∈ T(x_ν) such that g(f(ξ, s_ν, x_ν, y)) ≥ 0 for all y ∈ K(η). Since T is upper semicontinuous with nonempty compact values, by a similar argument in the proof of Theorem 3.1(b), there is an s ∈ T(x) such that a subnet of {s_ν} converges to s. Without loss of generality, we still denote the subnet by {s_ν}.

For each y ∈ K(η), we define P_{f(ξ,T(K(η)),K(η),y)}(g) = sup _{z∈f(ξ,T(K(η)),K(η),y)} | g(z)| for all g ∈ Y^⋆. We note that the set f(ξ, T(K(η)), K(η), y) is bounded by assumption, hence P_{f(ξ,T(K(η)),K(η),y)}(g) is well defined and is a seminorm of Y^⋆. For any ε > 0, let 𝒰_ε = {g ∈ Y^⋆ : P_{f(ξ,T(K(η)),K(η),y)}(g) < ε} be a neighborhood of 0 with respect to σ(Y^⋆, Y). Since g_ν → g, there is a α₀ ∈ Λ such that g_ν − g ∈ 𝒰_ε for every ν ≥ ν₀. That is, P_{f(ξ,T(K(η)),K(η),y)}(g_ν − g) = sup _{z∈f(ξ,T(K(η)),K(η),y)} | (g_ν − g)(z)| < ε/2 for every ν ≥ ν₀. This implies that

for all ν ≥ ν₀. Since the mapping (s, x) → f(ξ, s, x, y) is continuous and (s_ν, x_ν) → (s, x), we have By the continuity of g, we have for some ν₁ and all ν ≥ ν₁. Let us choose ν₂ = max {ν₀, ν₁}. Combining (3.8) and (3.10), we know that, for all ν ≥ ν₂, That is g_ν(f(ξ, s_ν, x_ν, y)) → g(f(ξ, s, x, y)). Since g_ν(f(ξ, s_ν, x_ν, y)) ≥ 0, there is an s ∈ T(x) such that g(f(ξ, s, x, y)) ≥ 0, which proves that x ∈ S^ξ,η(g). Therefore, the mapping S^ξ,η : C^⋆∖{0} → 2^K(η) is closed. By the compactness and Corollary 7 in [33, page 112], the mapping S^ξ,η is upper semicontinuous with compact values.

Lemma  3.4.  Suppose that for any (ξ, η) ∈ Δ₁ × Δ₂, x ∈ K(η) and s ∈ T(K(η)), f(ξ, s, x, K(η)) + C(x) is convex. Then

Furthermore, if C : K(η) → 2^Y is constant, then we have

Proof. We first claim that $$.

If $$, there is a g ∈ C^⋆∖{0} such that x ∈ S^ξ,η(g). Then, there is a g ∈ C^⋆∖{0} such that

for all y ∈ K(η). This implies that f(ξ, s, x, y) ∉ −int C(x) for all y ∈ K(η). Indeed, if there is a $$ such that $$. Since g ∈ C^⋆∖{0}, we have g(f(ξ, s, x, y)) < 0 which contracts (3.14). Thus, x ∈ Γ(ξ, η). This proves (3.12) holds.

Second, if C : K(η) → 2^Y is constant, we claim that $$.

If x ∈ Γ(ξ, η), then x ∈ K(η) with s ∈ T(x) and f(ξ, s, x, y) ∉ −int C for all y ∈ K(η), that is, f(ξ, s, x, K(η))∩(−int C) = ∅. Hence,

Since f(ξ, s, x, K(η)) + C is convex, by Eidelheit separation theorem, there is a g ∈ Y^⋆∖{0} and ρ ∈ ℝ such that

for all y ∈ K(η), w ∈ C, w^′ ∈ −int C. Then, for all y ∈ K(η), w ∈ C, w′ ∈ −int C.

Without loss of generality, we denote g − ρ by g, then

for all y ∈ K(η), w ∈ C, w^′ ∈ −int C. By the left-hand side inequality of (3.18) and the linearity of g, we have g(m) > 0 for all m ∈ int C. Since C is closed, for any m in the boundary of C, there is a net {m_ν} ⊂ int C such that m_ν → m. By the continuity of g, g(m) = g(lim _νm_ν) = lim _νg(m_ν) ≥ 0. Hence, for all w ∈ C, g(w) ≥ 0, that is g ∈ C^⋆∖{0}.

By the right-hand side inequality of (3.18), for all w ∈ C, there is an s ∈ T(x) such that g(f(ξ, s, x, y) + w) ≥ 0 for all y ∈ K(η). This implies that g(f(ξ, s, x, y)) ≥ 0 for all y ∈ K(η) if we choose w = 0. Hence, sup _s∈T(x)g(f(ξ, s, x, y)) ≥ 0 for all y ∈ K(η). Thus, x ∈ S^ξ,η(g). Therefore, $$, and hence

Combining this with (3.12), we have

Proof of Theorem 3.1(c). From Lemmas 3.2 and 3.3, the mapping S^ξ,η : C^⋆∖{0} → 2^K(η) is upper semicontinuous with nonempty compact values. From Lemma 3.4 and Theorem 3.1 [29], we know that for each (ξ, η) ∈ Δ₁ × Δ₂, the set Γ(ξ, η) is connected.

Example 3.5   3.5. Let Δ₁ = Δ₂ = X = Y = Z = ℝ, K(η) = [0,1] for all η ∈ Δ₂, $$, C(x) = [0, ∞) for all x ∈ 𝒦. Choose T : 𝒦 → 2^Z by T(x) = {x, x/2} for all x ∈ 𝒦. Define f(ξ, s, x, y) = s  −  y  +  ξ² for all (ξ, x, y) ∈ Δ₁  ×  X  ×  Y. Then, all the conditions of Theorem 2.1 hold, and Γ_w(ξ, η) = [1 − ξ², 1]∩[0,1] for all (ξ, η) ∈ Δ₁ × Δ₂. Indeed, since there are two choices for s, one is x, and the other is x/2. If the nonnegative number ξ² is less than 1, for any y in [0,1], and we always choose s = x/2, then for this case, the set Γ_w(ξ, η) will contain all elements of the set [2(1 − ξ²), 1]. Furthermore, if we always choose s = x, then the set Γ_w(ξ, η) will contain all elements of the set [(1 − ξ²), 1]. If the nonnegative number ξ² is greater than or equal to 1, then the set Γ_w(ξ, η) will contain all elements of the set [0,1]. Hence,

Here, we note that we cannot apply Theorem 3.1 since T(x) is not convex.

Example 3.6. Following Example 3.5, let T(x) = [x/2, x] for all x ∈ 𝒦 = [0,1]. By Theorem 2.1, the set Γ_w(ξ, η) ≠ ∅. We choose any $$, and we can see the mapping $$ is properly quasi $$-convex on $$ for any (ξ, y) ∈ Δ₁ × K(η). Since $$ for all $$. So, condition (i) of Theorem 3.1 holds. Obviously, the condition (ii) also holds, since no such δ exists in this example. Now, we can see condition (iii) holds. Indeed, from the facts

we know that if $$, then we can choose $$ such that s − 1 + ξ² ≥ 0. Hence, we can apply Theorem 3.1, and we know that Γ(ξ, η) is nonempty compact and connected. Let us compute the set Γ(ξ, η) for any (ξ, η) ∈ Δ₁ × Δ₂. If we choose any $$ for some t ∈ [1/2,1], we can see that all the points in the set [(1 − ξ²)/t, 1] are efficient solutions for (PGVEP). Hence, for any (ξ, η) ∈ Δ₁ × Δ₂.

Example 3.7 (see [32].)The perturbed nonlinear program (Pμ) described successfully that the optimal solutions set Γ(μ) of (Pμ) will approach the optimal solutions set Γ of linear program (P), where (P) and (Pμ) are as follows

where μ > 0.

We further note that, such convergent behavior will be described by upper semicontinuity by Theorems 2.1 and 3.1. That is,

Furthermore, the correspondent solution sets will preserve some kinds of topological properties, such as compactness and connectedness, under the convergent process.

We would like to point out an open question that naturally raises from Theorems 2.1 and 3.1. Under what conditions the mappings Γ_w and Γ will be lower semicontinuous?