Continuity of the Solution Maps for Generalized Parametric Set-Valued Ky Fan Inequality Problems

Special Case

• (i)

  If for any λ ∈ Λ, x, y ∈ A(λ), F(x, y, λ)∶ = φ(x, y, λ) + ψ(y, λ) − ψ(x, λ), where φ : A(μ) × A(μ) × Λ → 2^Y is a set-valued mapping and ψ : A(μ) × Λ → Y is a single-valued mapping, the (PKFI) reduces to the weak parametric vector equilibrium problem ((W)PVEP) considered in [15].

• (ii)

  When F is a vector-valued mapping, that is, F : B × B × Λ ⊂ X × X × Z → Y, the (PKFI) reduces to the parametric Ky Fan inequality in [14].

• (iii)

  If for any λ ∈ Λ, x, y ∈ A(λ), F(x, y, λ)∶ = φ(x, y, λ) + ψ(y, λ) − ψ(x, λ), where φ : A(μ) × A(μ) × Λ → Y and ψ : A(μ) × Λ → Y are two vector-valued maps, the (PKFI) reduces to the parametric (weak) vector equilibrium problem (PVEP) considered in [10, 11, 13, 16].

Definition 2.1. Let F : X × X × Z⇉Y∖{∅} be a trifunction.

• (i)

  F(x, ·, λ) is called C-convex function on A(λ), if and only if for every x₁, x₂ ∈ A(λ) and t ∈ [0,1], tF(x, x₁, λ) + (1 − t)F(x, x₂, λ) ⊂ F(x, tx₁ + (1 − t)x₂, λ) + C.

• (ii)

  F(x, ·, λ) is called C-like-convex function on A(λ), if and only if for any x₁, x₂ ∈ A(λ) and any t ∈ [0,1], there exists x₃ ∈ A(λ) such that tF(x, x₁, λ)+(1 − t)F(x, x₂, λ) ⊂ F(x, x₃, λ) + C.

• (iii)

  F(·, ·, ·) is called C-monotone on A(Λ) × A(Λ) × Λ, if and only if for any λ ∈ Λ and x, y ∈ A(λ), F(x, y, λ) + F(y, x, λ)⊂−C. The mapping F is called C-strictly monotone (or called C-strongly monotone in [10]) on A(Λ) × A(Λ) × Λ if F is C-monotone and if for any given λ ∈ Λ, for all x, y ∈ A(λ) and x ≠ y, s.t. F(x, y, λ) + F(y, x, λ) ⊂ −int C.

Definition 2.2 (see [18].)Let X and Y be topological spaces, T : X⇉Y∖{∅} be a set-valued mapping.

• (i)

  T is said to be upper semicontinuous (u.s.c., for short) at x₀ ∈ X if and only if for any open set V containing T(x₀), there exists an open set U containing x₀ such that T(x)⊆V for all x ∈ U.

• (ii)

  T is said to be lower semicontinuous (l.s.c., for short) at x₀ ∈ X if and only if for any open set V with T(x₀)∩V ≠ ∅, there exists an open set U containing x₀ such that T(x)∩V ≠ ∅ for all x ∈ U.

• (iii)

  T is said to be continuous at x₀ ∈ X, if it is both l.s.c. and u.s.c. at x₀ ∈ X. T is said to be l.s.c. (resp. u.s.c.) on X, if and only if it is l.s.c. (resp., u.s.c.) at each x ∈ X.

Proposition 2.3. Let X and Y be topological spaces, let T : X⇉Y∖{∅} be a set-valued mapping.

• (i)

  T is l.s.c. at x₀ ∈ X if and only if for any net {x_α} ⊂ X with x_α → x₀ and any y₀ ∈ T(x₀), there exists y_α ∈ T(x_α) such that y_α → y₀.

• (ii)

  If T has compact values (i.e., T(x) is a compact set for each x ∈ X), then T is u.s.c. at x₀ if and only if for any net {x_α} ⊂ X with x_α → x₀ and for any y_α ∈ T(x_α), there exist y₀ ∈ T(x₀) and a subnet {y_β} of {y_α}, such that y_β → y₀.

Theorem 3.1. For the problem (PKFI), suppose that the following conditions are satisfied:

• (i)

  A(·) is continuous with nonempty compact value on Λ;

• (ii)

  F(·, ·, ·) is l.s.c. on B × B × Λ.

Then, V(F, ·) is u.s.c. and closed on Λ.

Proof. (i) Firstly, we prove V(F, ·) is u.s.c. on Λ. Suppose to the contrary, there exists some μ₀ ∈ Λ such that V(F, ·) is not u.s.c. at μ₀. Then, there exist an open set V satisfying V(F, μ₀) ⊂ V and sequences μ_n → μ₀ and x_n ∈ V(F, μ_n), such that

Since x_n ∈ A(μ_n) and A(·) are u.s.c. at μ₀ with compact values by Proposition 2.3, there is an x₀ ∈ A(μ₀) such that x_n → x₀ (here, we can take a subsequence $$ of {x_n} if necessary).

Now, we need to show that x₀ ∈ V(F, μ₀). By contradiction, assume that x₀ ∉ V(F, μ₀). Then, there exists y₀ ∈ A(μ₀) such that

that is,

By the lower semicontinuity of A(·) at μ₀, for y₀ ∈ A(μ₀), there exists y_n ∈ A(μ_n) such that y_n → y₀.

It follows from x_n ∈ V(F, μ_n) and y_n ∈ A(μ_n) that

Since F(·, ·, ·) is l.s.c. at (x₀, y₀, λ₀), for z₀ ∈ F(x₀, y₀, μ₀), there exists z_n ∈ F(x_n, y_n, μ_n) such that

From (3.3), (3.5), and the openness of int C, there exists a positive integer N sufficiently large such that for all n ≥ N, which contradicts (3.4). So, we have Since x_n → x₀ (here we can take a subsequence $$ of {x_n} if necessary), we can find (3.7) contradicts (3.1). Consequently, V(F, ·) is u.s.c. on Λ.

(ii) In a similar way to the proof of (i), we can easily obtain the closeness of V(F, ·) on Λ. This completes the proof.

Remark 3.2. Theorem 3.1 generalizes and improves the corresponding results of Gong [10, Theorem 3.1] in the following four aspects:

• (i)

  the condition that A(·) is convex values is removed;

• (ii)

  the vector-valued mapping F(·, ·, ·) is extended to set-valued mapping, and the condition that C-monotone of mapping is removed;

• (iii)

  the assumption (iii) of Theorem  3.1 in [10] is removed;

• (iv)

  the condition that A(·) is uniformly compact near μ ∈ Λ is not required.

Moreover, we also can see that the obtained result extends Theorem 2.1 of [15].

Example 3.3. Let   X = Z = Y = ℝ, C = ℝ₊, Λ = [0, 2^1/2] be a subset of Z. Let F : X × X × Λ⇉Y be a set-valued mapping defined by F(x, y, λ) = [(y + 1)(λ² + 1)(x − λ), 10 + λ²] and let A : Λ⇉X defined by A(λ) = [λ², 2].

It follows from direct computation that

Then, we can verify that all assumptions of Theorem 3.1 are satisfied. By Theorem 3.1, V(F, ·) is u.s.c. and closed on Λ. Therefore, Theorem 3.1 is applicable.

Corollary 3.4. For the problem (PKFI), suppose that F : X × X × Z → Y is a vector-valued mapping and the following conditions are satisfied:

• (i)

  A(·) is continuous with nonempty compact value on Λ;

• (ii)

  F(·, ·, ·) is continuous on B × B × Λ.

Then, V(F, ·) is u.s.c. and closed on Λ.

Theorem 3.5. Let f ∈ C^*∖{0}. Suppose that the following conditions are satisfied:

• (i)

  A(·) is continuous with nonempty compact value on Λ;

• (ii)

  F(·, ·, ·) is u.s.c. with nonempty compact values on B × B × Λ;

• (iii)

  for each λ ∈ Λ, x ∈ A(λ)∖V_f(F, λ), there exists y ∈ V_f(F, λ), such that

  $$ (3.9)
  where γ > 0 is a positive constant.

Proof. By the contrary, assume that there exists λ₀ ∈ Λ, such that V_f(F, ·) is not l.s.c. at λ₀. Then, there exist λ_α with λ_α → λ₀ and x₀ ∈ V_f(F, λ₀), such that for any x_α ∈ V_f(F, λ_α) with x_α↛x₀.

Since x₀ ∈ A(λ₀) and A(·) are l.s.c. at λ₀, there exists $$ such that $$. We claim that $$. If not, for $$, it follows from above-mentioned assumption that $$, which is a contradiction.

By (iii), there exists y_α ∈ V_f(F, λ_α), such that

For y_α ∈ V_f(F, λ_α) ⊂ A(λ_α), because A(·) is u.s.c. at λ₀ with compact values by Proposition 2.3, there exist y₀ ∈ A(λ₀) and a subsequence $$ of {y_α} such that $$. In particular, for (3.10), we have

Then, there exist $$ and $$ such that Since F(·, ·, ·) is u.s.c. with compact values on B × B × Λ by Proposition 2.3, there exist $$ and $$ such that $$, $$. From $$, $$, the continuity of d(·, ·), and the closedness of C, we have It follows from x₀ ∈ V_f(F, λ₀) and y₀ ∈ A(λ₀) that $$. Particularly, we have On the other hand, since $$ and $$, we have $$. Also, we have $$. It follows from the continuity of f that we have By (3.14), (3.15), and the linearity of f, we get

For the above x₀ and y₀, we consider two cases:

Remark 3.6. Theorem 3.5 generalizes and improves the corresponding results of [14, Lemma 3.1] in the following three aspects:

• (i)

  the condition that A(·) is convex values is removed;

• (ii)

  the vector-valued mapping F(·, ·, ·) is extended to set-valued map;

• (iii)

  the constant γ can be any positive constant (γ > 0) in Theorem 3.5, while it should be strictly restricted to γ = 1 in Lemma 3.1 of [14].

Moreover, we also can see that the obtained result extends the ones of Gong and Yao [11, Theorem 2.1], where a strong assumption that C-strict/strong monotonicity of the mappings is required.

Example 3.7. Let X = Y = ℝ, C = ℝ₊. Let Λ = [3,5] be a subset of Z. For each λ ∈ Λ, x, y ∈ X, let A(λ) = [λ − 3,2] and F : X × X × Λ⇉Y∖{∅} be a set-valued mapping defined by

Obviously, assumptions (i) and (ii) of Theorem 3.5 are satisfied, and A(λ) = [0,2],   for  all  λ ∈ Λ. For any given λ ∈ Λ, let f(F(x, y, λ)) = z/3,   for  all  z ∈ F(x, y, λ). Then, it follows from a direct computation that

However, V_f(F, λ) is even not l.c.s. at λ = 3. The reason is that the assumption (iii) is violated. Indeed, if x = 0 ∈ V_f(F, λ), for λ = 3 and for all γ > 0, there exist y = 1/2 ∈ A(λ)∖V_f(F, λ) = (0,2), such that

if x = 2 ∈ V_f(F, λ), for λ = 3 and for all γ > 0, there exist y = 1/2 ∈ A(λ)∖V_f(F, λ), using a similar method, we have F(x, y, λ) + F(y, x, λ) + B(0, d^γ(x, y))⊈−C. Therefore, (iii) is violated.

Now, we show that V_f(F, ·) is not l.s.c. at λ = 3. Indeed, there exists 0 ∈ V_f(F, 3) and there exists a neighborhood (−2/9, 2/9) of 0, for any neighborhood N(3) of 3, there exists $$ such that $$ and

Thus, By Definition 2.2 (or page 108 in [18]), we know that V_f(F, ·) is not l.c.s. at λ = 3. So, the assumption (iii) of Theorem 3.5 is essential.

Proposition 3.8. Suppose that for each λ ∈ Λ and x ∈ A(λ), F(x, A(λ), λ) + C is a convex set, then

Theorem 3.9. For the problem (PKFI), suppose that the following conditions are satisfied:

• (i)

  A(·) is continuous with nonempty compact convex value on Λ;

• (ii)

  F(·, ·, ·) is continuous with nonempty compact values on B × B × Λ;

• (iii)

  for each λ ∈ Λ, x ∈ A(λ)∖V_f(F, λ), there exists y ∈ V_f(F, λ), such that

  $$ (3.25)
  where γ > 0 is a positive constant.

• (iv)

  for each λ ∈ Λ and for each x ∈ A(λ), F(x, ·, λ) is C-like-convex on A(λ).

Proof. From Theorem 3.1, it is easy to see that V(F, ·) is u.s.c. and closed on Λ. Now, we will only prove that V(F, ·) is l.s.c. on Λ. For each λ ∈ Λ and for each x ∈ A(λ), since F(x, ·, λ) is C-like-convex on A(λ), F(x, A(λ), λ) + C is convex. Thus, by virtue of Proposition 3.8, for each λ ∈ Λ, it holds

By Theorem 3.5, for each f ∈ C^*∖{0}, V_f(F, ·) is l.s.c. on Λ. Therefore, in view of Theorem 2 in [20, page 114], we have V(F, ·) is l.s.c. on Λ. This completes the proof.

Remark 3.10. Theorem 3.9 generalizes and improves the work in [15, Theorems  3.4-3.5]. Our approach on the (semi)continuity of the solution mapping V(F, ·) is totally different from the ones by Chen and Gong [15]. In [15], the V_f(F, λ) is strictly to be a singleton, while it may be a set-valued one in our paper. In addition, the assumption that C-strictly monotonicity of the mapping F is not required and the C-convexity of F is generalized to the C-like-convexity.

Corollary 3.11. For the problem (PKFI), suppose that F : X × X × Z → Y is a vector-valued mapping and the following conditions are satisfied:

• (i)

  A(·) is continuous with nonempty compact convex value on Λ;

• (ii)

  F(·, ·, ·) is continuous on B × B × Λ;

• (iii)

  for each λ ∈ Λ, x ∈ A(λ)∖V_f(F, λ), there exists y ∈ V_f(F, λ), such that

  $$ (3.27)
  where γ > 0 is a positive constant.

• (iv)

  for each λ ∈ Λ and for each x ∈ A(λ), F(x, ·, λ) is C-like-convex on A(λ).

Then, V(F, ·) is closed and continuous (i.e., both l.s.c. and u.s.c.) on Λ.

Remark 3.12. Corollary 3.11 generalizes and improves [10, Theorem 4.2] and [13, Theorem 4.2], respectively, because the assumption that C-strict monotonicity of the mapping F is not required.

Next,we give the following example to illustrate the case.

Example 3.13. Let $$, Λ = [−1,1] be a subset of Z. Let F : X × X × Λ → Y be a mapping defined by

and define A : Λ → 2^Y by A(λ) = [−1,1].

Obviously, A(·) is a continuous set-valued mapping from Λ to R with nonempty compact convex values, and conditions (ii) and (iv) of Corollary 3.11 are satisfied.

Let f = (0,2) ∈ C^*∖{0}, it follows from a direct computation that V_f(F, λ) = [0,1] for any λ ∈ Λ. Hence, for any x ∈ A(λ)∖V_f(F, λ), there exists y = 0 ∈ V_f(F, λ), such that,

Thus, the condition (iii) of Corollary 3.11 is also satisfied. By Corollary 3.11, V(F, ·) is closed and continuous (i.e., both l.s.c. and u.s.c.) on Λ.

However, the condition that F is a C-strictly monotone mapping is violated. Indeed, for any λ ∈ Λ = [−1,1] and x ∈ A(λ)∖V_f(F, ·), there exist y = −x ∈ V_f(F, ·) with y = −x, such that

which implies that F(·, ·, ·) is not $$-strictly monotone on A(Λ) × A(Λ) × Λ. Then, Theorem  4.2 in [10] and Theorem  4.2 in [13] are not applicable, and the corresponding results in references (e.g., [11, Lemma 2.2, Theorem 2.1]) are also not applicable.