Continuity of solutions for parametric generalized quasi-variational relation problems

In this article, we establish sufficient conditions for the solution sets of parametric generalized quasi-variational relation problems with the stability properties such as the upper semicontinuity, lower semi-continuity, the Hausdorff lower semicontinuity, continuity, Hausdorff continuity, and closedness. Our results improve recent existing ones in the literature.

Mathematics Subject Classification 2010: 90C31; 49J53; 49J40; 49J45.

Let X, Y be Hausdorff topological vector spaces and Λ, Γ, M be topological spaces. Let A ⊆ X and B ⊆ Y be nonempty sets. Let K₁: A × Λ → 2^A, K₂: A × Λ → 2^A, T : A × A × Γ → 2^B be multifunctions and R(x, t, y, μ) be a relation linking x ∈ A, t ∈ B, y ∈ A and μ ∈ M.

For the sake of simplicity, we adopt the following notations (see [1,2]). Letters w, m, and s are used for weak, middle, and strong, respectively, kinds of considered problems. For subsets U and V under consideration we adopt the notations

Let α ∈ {w, m, s}, ρ ∈ {ρ₁, ρ₂}, and . We consider the following for parametric generalized quasi-variational relation problem (in short, (QVR_α)):

(QVR_α): Find such that satisfying

For each λ ∈ Λ, γ ∈ Γ, μ ∈ M, we let E(λ) := {x ∈ A|x ∈ K₁(x, λ)} and let S_α : Λ × Γ × M → 2^A be a set-valued mapping such that S_α(λ, γ, μ) is the solution set of (QVR_α).

Throughout the article, we assume that S_α(λ, γ, μ) ≠ ∅ for each (λ, γ, μ) in the neighborhoods (λ₀, γ₀, μ₀) ∈ Λ × Γ × M.

The parametric generalized quasi-variational relation problems are more general than many following problems.

(a) The parametric variational relation problem (VR):

Let A, B, X, Y, M = Γ = Λ, K₁, K₂, T, α = s as in (QVR_α). Then, (QVR_α) becomes (VR) is studied in [3]:

Find such that

(b) The parametric generalized quasi-variational inclusion problem (QGVIP_α):

Let A, B, X, Y, M, Γ, Λ, K₁, K₂, T as in (QVR_α) and let Z be a Hausdorff topological vector space. Given a mapping F : A × B × A × M → 2^Z , the relation R is defined as follows

Then, (QVR_α) becomes (QGVIP_α)

Find such that satisfying

(c) The parametric quasi-variational inclusion problem (P_αρ):

Let A, B, X, Y, M, Γ, Λ, K₁, K₂, T, R as in (QVR_α) and let Z be a Hausdorff topological vector space. Let F : A × B × A × M → 2^Z and G : A × B × A × M → 2^Z be multivalued mappings. The relation R is defined as follows

Then, (QVR_α) becomes (P_αρ) is studied in [1,2]:

Find such that satisfying

(d) The parametric vector quasi-equilibrium problems:

Let A, X, M, Γ, Λ, K₁ ≡ K₂ ≡ K, T as in (QVR_α) and let Y be a Hausdorff topological vector space. Given a mapping F : A × A × M → 2^Y and C ⊆ Y be a closed subset with nonempty interior, the relation R is defined as follows

Then, (QVR_α) becomes the parametric vector quasi-equilibrium problems is studied in [4]. Find such that satisfying

(e) The parametric multivalued vector quasi-equilibrium problems:

Let A = B, X = Y, M = Γ, Λ, K₁ = clK, K₂ = K, T = {t} as in (QVR_α) and let Z be a Hausdorff topological vector space. Given a mapping F : A × A × M → 2^Z and C ⊆ Z be a closed subset with nonempty interior, the relation R is defined as follows

Then, (QVR_α) becomes the parametric multivalued vector quasi-equilibrium problems is studied in [5]. Find such that

(f) The parametric generalized vector quasi-equilibrium problems (QEP_αρ):

Let A, B, X, Y, M, Γ, Λ, K₁, K₂, T as in (QVR_α) and let Z be a Hausdorff topological vector space. Given a mapping F : A × B × A × M → 2^Z and C ⊆ Z be a closed subset with nonempty interior, the relation R is defined as follows

Then, (QVR_α) becomes (QEP_αρ)

Find such that satisfying

Stability properties of solution sets for parametric generalized quasi-variational relation problem is an important topic in optimization theory and applications. Recently, the continuity, especially the upper semicontinuity, the lower semicontinuity and the Hausdorff lower semicontinuity of the solution sets have been investigated in models as equilibrium problems [1,2,4-13], variational inequality problems [14-19], and the references therein.

The structure of this article is as follows. In the remaining part of this section, we recall definitions for later uses. Section "Main results" is devoted to the upper semicontinuity, the lower semicontinuity, and the Hausdorff lower semicontinuity of solutions for problem (QVR_α). Applications to the parametric vector quasi-equilibrium problem are presented in Section "Applications".

Now we recall some notions see [5,6,20,21]. Let X and Y be as above and G : X → 2^Y be a multifunction. G is said to be lower semicontinuous (lsc) at x₀ if G(x₀) ∩ U ≠ ∅ for some open set U ⊆ Y implies the existence of a neighborhood N of x₀ such that, for all x ∈ N, G(x) ∩ U ≠ ∅. An equivalent formulation is that: G is lsc at x₀ if . G is called upper semicontinuous (usc) at x₀ if for each open set U ⊇ G(x₀), there is a neighborhood N of x₀ such that U ⊇ G(x), for all x ∈ N. G is said to be Hausdorff upper semicontinuous (H-usc in short; Hausdorff lower semicontinuous, H-lsc, respectively) at x₀ if for each neighborhood B of the origin in Y, there exists a neighborhood N of x₀ such that, G(x) ⊆ G(x₀) + B, ∀x ∈ N (G(x₀) ⊆ G(x) + B, ∀x ∈ N). G is said to be continuous at x₀ if it is both lsc and usc at x₀ and to be H-continuous at x₀ if it is both H-lsc and H-usc at x₀. G is called closed at x₀ if for each net {(x_α, z_α )} graphG := {(x, z) | z ∈ G(x)}, (x_α, z_α) → (x₀, z₀), z₀ must belong to G(x₀). We say that G satisfies a certain property in a subset A ⊆ X if G satisfies it at every points of A. If A = X we omit "in X" in the statement.

Let A and Y be as above and G : A → 2^Y be a multifunction.

(i) If G is usc at x₀, then G is H-usc at x₀. Conversely if G is H-usc at x₀ and if G(x₀) is compact, then G is usc at x₀;

(ii) If G is H-lsc at x₀, then G is lsc at x₀. The converse is true if G(x₀) is compact;

(iii) If Y is compact and G is closed at x₀, then G is usc at x₀;

(iv) If G is usc at x₀ and G(x₀) is closed, then G is closed at x₀;

(v) If G has compact values, then G is usc at x₀ if and only if, for each net {x_α} ⊆ A which converges to x₀ and for each net {y_α} ⊆ G(x_α), there are y₀ ∈ G(x₀) and a subnet {y_β} of {y_α} such that y_β → y₀.

Now we let A, B, X, Y, M, Γ, Λ, R as in (QVR_α), we use the following notations for level sets of R

In this section, we discuss the upper semicontinuity, the lower semicontinuity, the Hausdorff lower semicontinuity, continuity, and H-continuity of solution sets for parametric quasi-variational relation problem (QVR_α).

Theorem 1 Assume for problem (QVR_α) that

(i) E is usc at λ₀ and E(λ₀) is compact, and K₂ is lsc in K₁(A, Λ) × {λ₀};

(ii) in K₁(A, Λ) × K₂(K₁(A, Λ), Λ) × {γ₀}, T is usc and compact-valued if α = w (or α = m), and lsc if α = s;

(iii) in K₁(A, Λ) × T(K₁(A, Λ), K₂(K₁(A, Λ), Λ), Γ) × K₂(K₁(A, Λ), Λ) × {μ₀}, lev_upperR is closed.

Then S_α is both usc and closed at (λ₀, γ₀, μ₀).

Proof. Since α = {w, m, s}, we have in fact three cases. However, the proof techniques are similar. We consider only the cases α = w. We first prove that S_w is upper semicontinuous at (λ₀, γ₀, μ₀). Indeed, we suppose to the contrary that S_w is not upper semicontinuous at (λ₀, γ₀, μ₀), i.e., there is an open subset U of S_w(λ₀, γ₀, μ₀) such that for all nets {(λ_n, γ_n, μ_n)} convergent to (λ₀, γ₀, μ₀), there exists x_n ∈ S_w(λ_n, γ_n, μ_n), x_n ∉ U, ∀n. By the upper semicontinuity of E and the compactness of E(λ₀), one can assume that x_n → x₀ for some x₀ ∈ E(λ₀). If x₀ ∉ S_w(λ₀, γ₀, μ₀), then ∃y₀ ∈ K₂(x₀, λ₀), ∀t₀ ∈ T(x₀, y₀, γ₀) such that

By the lower semicontinuity of K₂ at (x₀, λ₀), there exists y_n ∈ K₂(x_n, λ_n) such that y_n → y₀. Since x_n ∈ S_w(λ_n, γ_n, μ_n), ∃t_n ∈ T (x_n, y_n, γ_n) such that

Since T is usc at (x₀, y₀, γ₀) and T (x₀, y₀, γ₀) is compact, there exists t₀ ∈ T (x₀, y₀, γ₀) such that t_n → t₀ (can take a subnet if necessary). By the condition (iii) and (2), we have

we see a contradiction between (1) and (3). Thus, x₀ ∈ S_w(λ₀, γ₀, μ₀) ⊆ U, this contradicts to the fact x_n ∉ U, ∀n. Hence, S_w is upper semicontinuous at (λ₀, γ₀, μ₀).

Now we prove that S_w is closed at (λ₀, γ₀, μ₀). Indeed, we supposed that S_w is not closed at (λ₀, γ₀, μ₀), i.e., there is a net {(x_n, λ_n, γ_n, μ_n)} → (x₀, λ₀, γ₀, μ₀) with x_n ∈ S_w(λ_n, γ_n, μ_n) but x₀ ∉ S_w(λ₀, γ₀, μ₀). The further argument is the same as above. And so we have S_w is closed at (λ₀, γ₀, μ₀). □

The following example shows that the upper semicontinuity and the compactness of E are essential.

Example 2 Let A = B = X = Y = ℝ, Λ = Γ = M = [0, 1], λ₀ = 0, F(x, t, y, λ) = 2^λ+1, K₁(x, λ) = (−λ − 1, λ], K₂(x, λ) = {-1} and . We let relation R be defined by R(x, t, y, λ) holds iff F (x, t, y, λ) ⊆ ℝ₊. Then, we have E(0) = (-1, 0] and E(λ) = (−λ − 1, λ], ∀λ ∈ (0, 1]. We show that K₂ is lsc and assumptions (ii) and (iii) of Theorem 1 are fulfilled. But S_α is neither usc nor closed at (0, 0, 0). The reason is that E is not usc at 0 and E(0) is not compact. In fact, S_α(0, 0, 0) = (-1, 0] and S_α(λ, γ, μ) = (−λ −1, λ], ∀λ ∈ (0, 1].

The following example shows that the lower semicontinuity of K₂ is essential.

Example 3 Let X, Y, Λ, Γ, M, λ₀ as in Example 2 and let A = B = [-3, 3], F (x, t, y, λ) = x + y + λ, K₁(x, λ) = [0, 3], T (x, y, λ) = {t}. Let relation R be defined by R(x, t, y, λ) holds iff F(x, t, y, λ) ⊆ ℝ₊ and

We have E(λ) = [0, 3], ∀λ ∈ [0, 1]. Hence E is usc at 0 and E(0) is compact and the conditions (ii) and (iii) of Theorem 1 are easily seen to be fulfilled. But S_α is not upper semicontinuous at (0, 0, 0). The reason is that K₂ is not lower semicontinuous. In fact,

The following example shows that the condition (iii) of Theorem 1 is essential.

Example 4 Let Λ, Γ, M, T, λ₀ as in Example 3 and let X = Y = A = B = [0, 1]. Let relation R be defined by R(x, t, y, λ) holds iff F(x, t, y, λ) ⊆ ℝ₊, K₁(x, λ) = K₂(x, λ) = [0, 1] and , , . We show that the assumptions (i) and (ii) of Theorem 1 are easily seen to be fulfilled and

But S_α is not usc at (0, 0, 0). The reason is that assumption (iii) is violated. Indeed, taking x_n = 0, t_n = 0, , as n → ∞, then and , but .

The following example shows that all assumptions of Theorem 1 are fulfilled. But Theorem 3.2 in [5] cannot be applied.

Example 5 Let A, B, X, Y, Λ, Γ, M, λ₀ as in Example 2 and let K₁(x, λ) = K₂(x, λ) = [0, 1], and

Let relation R be defined by R(x, t, y, λ) holds iff F(x, t, y, λ) ⊆ ℝ₊. We show that the assumptions (i), (ii), and (iii) of Theorem 1 are easily seen to be fulfilled and

Hence, S_α is usc at (0, 0, 0). But Theorem 3.2 in [5] cannot be applied. The reason is that F is not usc at (x, t, y, 0).

The following example shows that all assumptions of Theorem 1 are fulfilled. But Theorem 3.4 in [5] cannot be applied.

Example 6 Let A, B, X, Y, Λ, Γ, M, λ₀, as in, Example, 5 and, let, K₁(x, λ), = K₂(x, λ) = [0, 3], T (x, y, γ) = {t} and

Let relation R be defined by R(x, t, y, λ) holds iff F(x, t, y, λ) ⊆ ℝ₊. We show that the assumption (i), (ii,) and (iii) of Theorem 1 are easily seen to be fulfilled

Hence, S_α is usc at (0, 0, 0). But Theorem 3.4 in [5] cannot be applied. The reason is that F is not usc (x, t, y, 0).

Assumptions in Theorem 1, we have K₂ is lsc in K₁(A, Λ) × {λ₀} (which is not imposed in this Theorem 4.1 of [10]). The Example 3 shows that the lower semicontinuity of K₂ needs to be added to Theorem 4.1 of [10].

Remark 7 (i) In the special case, if T (x, y, γ) = {t}, Λ = Γ = M, A = B, X = Y, K₁ = K₂ = K and the variational relation R is defined as follows holds iff F(x, y, λ) ⊄ -intC(x, λ) (or F (x, y, λ) ∩−intC(x, λ) = ∅), where F : A × A × Λ → 2^Y and C : A × Λ → 2^Y be multifunctions, with C(x, λ) being a convex cone. Then, (QVR_α) becomes (PGQVEP) and (PEQVEP) in [10].

(ii) In the special case as in Remark 7 (i). Then, Theorem 1 reduces to Theorem 4.1 in [10]. However the proof of the Theorem 4.1 in a different way. Its assumptions (i)-(iv) derive (i) Theorem 1, assumptions (v) and (vi) coincide with (iii) of Theorem 1.

The following example shows a case where the assumed compactness in Theorem 4.1 of [10] is violated but the assumptions of Theorem 1 are fulfilled.

Example 8 Let X, Y, Λ, Γ, M, T, λ₀, as in Example 6 and we let A = B = [0, 3), F(x, y, λ) = x − y and K₁(x, λ) = K₂(x, λ) = [1, 2]. Let relation R be defined by R(x, t, y, λ) holds iff F(x, t, y, λ) ⊆ ℝ₊. We show that the assumptions of Theorem 1 are easily seen to be fulfilled and so S_α is usc and closed at (0, 0, 0), although A is not compact. In fact, S_α (λ, γ, μ) = {2},∀λ ∈ [0, 1].

Theorem 9 Assume for problem (QVR_α) that

(i) E is lsc at λ₀, K₂ is usc and compact-valued in K₁(A, Λ) × {λ₀};

(ii) in K₁(A, Λ) × K₂(K₁(A, Λ), Λ) × {γ₀}, T is usc and compact-valued if α = s, and lsc if α = w (or α = m);

(iii) in K₁(A, Λ) × T(K₁(A, Λ), K₂(K₁(A, Λ), Λ), Γ) × K₂(K₁(A, Λ), Λ) × {μ₀}, lev_lowerR is closed.

Then S_α is lower semicontinuous at (λ₀, γ₀, μ₀).

Proof. Since α = {w, m, s}, we have in fact three cases. However, the proof techniques are similar. We consider only the cases α = s. Suppose to the contrary that S_s is not lsc at (λ₀, γ₀, μ₀), i.e., there are x₀ ∈ S_s(λ₀, γ₀, μ₀) and net {(λ_n, γ_n, μ_n)}, (λ_n, γ_n, μ_n) → (λ₀, γ₀, μ₀) such that ∀x_n ∈ S_s(λ_n, γ_n, μ_n), x_n → x₀. Since E is lsc at λ₀, there is with . By the above contradiction assumption, there must be a subnet of such that, , i.e., , such that

As K₂, is usc at (x₀, λ₀) and K₂(x₀, λ₀) is compact, one has y₀ ∈ K₂(x₀, λ₀) such that y_m → y₀ (taking a subnet if necessary). By the upper semicontinuity of T at (x₀, y₀, γ₀), one has t₀ ∈ T(x₀, y₀, γ₀) such that t_m → t₀.

Since and by the condition (iii) and (4), yields that

which is impossible since x₀ ∈ S_s(λ₀, γ₀, μ₀). Therefore, S_s is lsc at (λ₀, γ₀, μ₀). □

The following example shows that the lower semicontinuity of E is essential

Example 10 Let A, B, X, Y, Λ, Γ, M, λ₀ as in Example 2 and let F(x, t, y, λ) = 2^λ, T (x, y, λ) = {t}, K₂(x, λ) = [0, 1]. Let relation R be defined by R(x, t, y, λ) holds iff F(x, t, y, λ) ⊆ (0, +∞) and

We have E(0) = [-1, 1], E(λ) = [−λ − 1, 0], ∀λ ∈ (0, 1]. Hence K₂ is usc and the conditions (ii) and (iii) of Theorem 9 are easily seen to be fulfilled. But S is not lower semicontinuous at (0, 0, 0). The reason is that E is not lower semicontinuous at 0. In fact, S_α (0, 0, 0) = [-1, 1] and S_α(λ, γ, μ) = [−λ − 1, 0], ∀λ ∈ (0, 1].

The following example shows that all assumptions of Theorem 9 are fulfilled. But Theorems 2.1 and 2.3 in [5] and Theorem 2.2 in [4] are not fulfilled.

Example 11 Let A, B; X, Y, T, Λ, Γ, M, λ₀ as in Example 10, let and

and we let relation R be defined by R(x, t, y, μ) holds iff F(x, y, λ) ⊆ (0, +∞). We show that the assumptions (i), (ii) and (iii) of Theorem 9 are satisfied and , . Theorems 2.1 and 2.3 in [5] and Theorem 2.2 in [4] are not fulfilled. The reason is that F is neither usc nor lsc at (x, y, 0).

Theorem 12 Impose the assumption of Theorem 9 and the following additional conditions:

(iv) K₂(., λ₀) is lsc in K₁(A, Λ) and E(λ₀) is compact;

(v) in K₁(A, Λ) × K₂(K₁(A, Λ), Λ), T (., ., γ₀) is usc and compact-valued if α = w (or α = m), and lsc if α = s;

(vi) in K₁(A, Λ) × T(K₁(A, Λ), K₂(K₁(A, Λ), Λ), Γ) × K₂(K₁(A, Λ), Λ), lev_upperR(., ., μ₀) is closed ;

Then S_α is Hausdorff lower semicontinuous at (λ₀, γ₀, μ₀).

Proof. We consider only for the cases α = s. We first prove that S_s(λ₀, γ₀, μ₀) is closed. Indeed, we let x_n ∈ S_s(λ₀, γ₀, μ₀) such that x_n → x₀. If x₀ ∉ S_s(λ₀, γ₀, μ₀), then ∃y₀ ∈ K₂(x₀, λ₀), ∃t₀ ∈ T (x₀, y₀, γ₀) such that

By the lower semicontinuity of K₂(., λ₀) at x₀, one has y_n ∈ K₂(x_n, λ₀) such that y_n → y₀. By the lower semicontinuity of T (., ., γ₀) at (x₀, y₀), one has t_n ∈ T(x_n, y_n, γ₀) such that t_n → t₀. Since x_n ∈ S_s(λ₀, γ₀, μ₀), we have

Since (x_n, t_n, y_n) → (x₀, t₀, y₀) and by the condition (vi) and (6) yields that

we see a contradiction between (5) and (7). Therefore, S_s(λ₀, γ₀, μ₀) is closed.

On the other hand, since S_s(λ₀, γ₀, μ₀) ⊆ E(λ₀) is compact by E(λ₀) compact. Since S_s is lower semicontinuous at (λ₀, γ₀, μ₀) and S_s(λ₀, γ₀, μ₀) compact. Hence S_s is Hausdorff lower, semicontinuous at (λ₀, γ₀, μ₀). And so we complete the proof. □

The following example shows that the assumed compactness in (iv) is essential

Example 13 Let X = A = B = ℝ², Y = ℝ, Λ = Γ = M = [0, 1], λ₀ = 0, and , , and

Let relation R be defined by R(x, t, y, λ)^:holds iff F(x, t, y, λ) ⊆ (0, +∞). We have E(0) =, {x ∈ ℝ² | x² = 0} and , ∀λ ∈ (0, 1]. We show that the assumptions, of Theorem 12 are satisfied, but the compactness of E(0) is not satisfied. Hence, S_α is not, Hausdorff lower semicontinuous at (0, 0, 0). In fact, S_α(0, 0, 0) = {(x₁, x₂) ∈ ℝ²|x₂ = 0} and , ∀λ ∈ (0, 1].

Corollary 14 Suppose that all conditions in Theorems 1 and 9 are satisfied. Then, we have Sα is both continuous and closed at (λ₀, γ₀, μ₀).

Corollary 15 Suppose that all conditions in Theorems 1 and 12 are satisfied. Then, we have S_α is Hausdorff continuous and closed at (λ₀, γ₀, μ₀).

Since our generalized quasi-variational relation problems include many rather general problems as particular cases as mentioned in Section "Introduction". The results of Section "Main results" can derive corresponding to results of these special cases. In Section "Applications" we discuss only some corollaries for generalized vector quasi-equilibrium problems as example.

In this section, we discuss the upper semicontinuity, the lower semicontinuity, the Hausdorff lower semicontinuity, continuity, H-continuity of solution sets for generalized parametric vector quasi-equilibrium problems (QEP_αρ).

For each λ ∈ Λ, γ ∈ Γ, μ ∈ M, let Ψ_αρ : Λ × Γ × M → 2^A e a set-valued mapping such that Ψ_αρ(λ, γ, μ) is the solution set of (QEP_αρ).

Throughout the article, we assume that Ψ_αρ(λ, γ, μ) ≠ ∅ for each (λ, γ, μ) in the neighborhoods (λ₀, γ₀, μ₀) ∈ Λ × Γ × M.

Corollary 16 Assume for problem (QEP_αρ) that

(i) E is usc at λ₀ and E(λ₀) is compact, and K₂ is lsc in K₁(A_, Λ) × {λ₀};

(ii) in K₁(A, Λ) × K₂(K₁(A, Λ), Λ) × {γ₀}, T is usc and compact-valued if α = w (or α = m), and lsc if α = s;

(iii) in K₁(A, Λ) × T(K₁(A, Λ), K₂(K₁(A, Λ), Λ), Γ) × K₂(K₁(A, Λ), Λ) × {μ₀}, the set is closed.

Then Ψ_αρ is both usc and closed at (λ₀, γ₀, μ₀).

Proof. Since α = {w, m, s}, ρ = {ρ₁, ρ₂}, we have in fact six cases. However, the proof techniques are similar. We consider only the cases α = w, ρ = ρ₁. Let relation R be defined by R(x, t, y, μ) holds iff To apply Theorem 1, we need to check only that in K₁(A, Λ) × T(K₁(A, Λ), K₂(K₁(A, Λ), Λ), Γ) × K₂(K₁(A, Λ), Λ) × {μ₀}, the set is closed.

Indeed, for all nets {(x_n, t_n, y_n, _n)} → (x₀, t₀, y₀, μ₀) such that

By assumption (iii), we have

   □

Corollary 17 Assume for problem (QEP_αρ) that

(i) E is lsc at λ₀, K₂ is usc and compact-valued in K₁(A, Λ) × {λ₀};

(ii) in K₁(A, Λ) × K₂(K₁(A, Λ), Λ) × {γ₀}, T is usc and compact-valued if α = s, and lsc if α = w (or α = m);

(iii) in K₁(A, Λ) × T(K₁(A, Λ), K₂(K₁(A, Λ), Λ), Γ) × K₂(K₁(A, Λ), Λ) × {μ₀}, the set is closed.

Then Ψ_αρ is lower semicontinuous at (λ₀, γ₀, μ₀).

Proof. Since α = {w, m, s}, ρ = {ρ₁, ρ₂}, we have in fact six cases. However, the proof techniques are similar. We consider only the cases α = s, ρ = ρ₁. Let relation R be defined by R(x, t, y, μ) holds iff F(x, t, y, μ) ⊆ C. To apply Theorem 9, we need to check only that in K₁(A, Λ) × T (K₁(A, Λ), K₂(K₁(A, Λ), Λ), Γ) × K₂(K₁(A, Λ), Λ) × {μ₀}, the set {(x, t, y, μ) ∈ A × B × A × M | F(x, t, y, μ) ⊆ C)} is closed.

Indeed, for all nets {(x_n, t_n, y_n, μ_n)} → (x₀, t₀, y₀, μ₀) such that

By assumption (iii), we have

   □

Corollary 18 Impose the assumption of Corollary 17 and the following additional conditions:

(iv) K₂(., λ₀) is lsc in K₁(A, Λ) and E(λ₀) is compact;

(v) in K₁(A, Λ) × K₂(K₁(A, Λ), Λ), T (., ., γ₀) is usc and compact-valued if α = w (or α = m), and lsc if α = s;

(vi) in K₁(A, Λ) × T(K₁(A, Λ), K₂(K₁(A, Λ), Λ), Γ) × K₂(K₁(A, Λ), Λ), the set {(x, t, y) ∈ A × B × A | ρ(F(x, t, y, μ₀); C)} is closed.

Then Ψ_αρ is Hausdorff lower semicontinuous at (λ₀, γ₀, μ₀).

Proof. Since α = {w, m, s}, ρ = {ρ₁, ρ₂}, we have in fact six cases. However, the proof techniques are similar. We consider only the cases α = s, ρ = ρ₁. Let relation R be defined by R(x, t, y, μ) holds iff F(x, t, y, μ) ⊆ C. To apply Theorem 12, we need to check only that in K₁(A, Λ) × T (K₁(A, Λ), K₂(K₁(A, Λ), Λ), Γ) × K₂(K₁(A, Λ), Λ), the set {(x, t, y) ∈ A × B × A | F(x, t, y, μ₀) ⊆ C)} is closed. Indeed, for all nets {(x_n, t_n, y_n)} → (x₀, t₀, y₀) such that R(x_n, t_n, y_n, μ₀) holds. By assumption (vi), we have F(x₀, t₀, y₀, μ₀) ⊆ C. □

Remark 19 (i) Suppose that all conditions in Corollaries 16 and 17 are satisfied. Then, we have Ψ_α is both continuous and closed at (λ₀, γ₀, μ₀).

(ii) Suppose that all conditions in Corollaries 16 and 18 are satisfied. Then, we have Ψ_αρ is Hausdorff continuous and closed at (λ₀, γ₀, μ₀).

The author declares that they have no competing interests.

The author is grateful to Prof. Phan Quoc Khanh and Dr. Lam Quoc Anh for their encour-agements in research. The author also thanks to the two anonymous referees for their valuable remarks and suggestions, which helped them to improve considerably the article.

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