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 K ₁: A × Λ → 2 ^A , K ₂: 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

( u , v ) w U × V means ∀ u ∈ U , ∃ v ∈ V , ( u , v ) m U × V means ∃ v ∈ V , ∀ u ∈ U , ( u , v ) s U × V means ∀ u ∈ U , ∀ v ∈ V , ρ 1 ( U , V ) means U ⊆ V , ρ 2 ( U , V ) means U ∩ V ≠ ∅ , ( u , v ) w ̄ U × V means ∃ u ∈ U , ∀ v ∈ V and similarly for m ̄ , s ̄ , ρ ̄ 1 ( U , V ) means U ⊈ V and similarly for ρ ̄ 2 .

Let α ∈ {w, m, s}, α ̄ ∈ { w ̄ , m ̄ , s ̄ } , ρ ∈ {ρ ₁, ρ ₂}, and ρ ̄ ∈ { ρ ̄ 1 , ρ ̄ 2 } . We consider the following for parametric generalized quasi-variational relation problem (in short, (QVR _α )):

(QVR _α ): Find x ̄ ∈ K 1 ( x ̄ , λ ) such that ( y , t ) α K 2 ( x ̄ , λ ) × T ( x ̄ , y , γ ) satisfying

R ( x ̄ , t , y , μ ) holds .

For each λ ∈ Λ, γ ∈ Γ, μ ∈ M, we let E(λ) := {x ∈ A|x ∈ K ₁(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 (λ ₀, γ ₀, μ ₀) ∈ 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, K ₁, K ₂, T, α = s as in (QVR _α ). Then, (QVR _α ) becomes (VR) is studied in 3 :

Find x ̄ ∈ K 1 ( x ̄ , λ ) such that

R ( x ̄ , t , y , λ ) holds , ∀ t ∈ T ( x ̄ , y , λ ) , ∀ y ∈ K 2 ( x ̄ , λ ) .

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

Let A, B, X, Y, M, Γ, Λ, K ₁, K ₂, 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

R ( x , t , y , μ ) holds iff 0 ∈ F ( x , t , y , μ ) .

Then, (QVR _α ) becomes (QGVIP _α )

Find x ̄ ∈ K 1 ( x ̄ , λ ) such that ( y , t ) α K 2 ( x ̄ , λ ) × T ( x ̄ , y , γ ) satisfying

0 ∈ F ( x ̄ , t , y , μ ) .

(c) The parametric quasi-variational inclusion problem (P _αρ ):

Let A, B, X, Y, M, Γ, Λ, K ₁, K ₂, 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

R ( x , t , y , μ ) holds iff ρ ( F ( x , t , y , μ ) , G ( x , t , x , μ ) ) .

Then, (QVR _α ) becomes (P _αρ ) is studied in 1 2 :

Find x ̄ ∈ K 1 ( x ̄ , λ ) such that ( y , t ) α K 2 ( x ̄ , λ ) × T ( x ̄ , y , γ ) satisfying

ρ ( F ( x ̄ , t , y , μ ) , G ( x ̄ , t , x ̄ , μ ) ) .

(d) The parametric vector quasi-equilibrium problems:

Let A, X, M, Γ, Λ, K ₁ ≡ K ₂ ≡ 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

R ( x , t , y , μ ) holds iff ρ ( F ( t , y , μ ) , ( Y \ - intC ) ) .

Then, (QVR _α ) becomes the parametric vector quasi-equilibrium problems is studied in 4 . Find x ̄ ∈ clK ( x ̄ , λ ) such that ( y , t ) α K ( x ̄ , λ ) × T ( x ̄ , y , γ ) satisfying

ρ ( F ( t , y , μ ) , ( Y \ - intC ) ) .

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

Let A = B, X = Y, M = Γ, Λ, K ₁ = 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

R ( x , t , y , μ ) holds iff ρ ( F ( x , y , μ ) , ( Z \ - intC ) ) .

Then, (QVR _α ) becomes the parametric multivalued vector quasi-equilibrium problems is studied in 5 . Find x ̄ ∈ clK ( x ̄ , λ ) such that

ρ ( F ( x ̄ , y , μ ) , ( Z \ - intC ) ) , ∀ y ∈ K ( x ̄ , λ ) .

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

Let A, B, X, Y, M, Γ, Λ, K ₁, K ₂, 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

R ( x , t , y , μ ) holds iff ρ ( F ( x , t , y , μ ) , C ) .

Then, (QVR _α ) becomes (QEP _αρ )

Find x ̄ ∈ K 1 ( x ̄ , λ ) such that ( y , t ) α K 2 ( x ̄ , λ ) × T ( x ̄ , y , γ ) satisfying

ρ ( F ( x ̄ , t , y , μ ) , C ) .

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 5 6 7 8 9 10 11 12 13 , variational inequality problems 14 15 16 17 18 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 x ₀ if G(x ₀) ∩ U ≠ ∅ for some open set U ⊆ Y implies the existence of a neighborhood N of x ₀ such that, for all x ∈ N, G(x) ∩ U ≠ ∅. An equivalent formulation is that: G is lsc at x ₀ if ∀ x α → x 0 , ∀ z 0 ∈ G ( x 0 ) , ∃ z α ∈ G ( x α ) , z α → z 0 . G is called upper semicontinuous (usc) at x ₀ if for each open set U ⊇ G(x ₀), there is a neighborhood N of x ₀ 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 x ₀ if for each neighborhood B of the origin in Y, there exists a neighborhood N of x ₀ such that, G(x) ⊆ G(x ₀) + B, ∀x ∈ N (G(x ₀) ⊆ G(x) + B, ∀x ∈ N). G is said to be continuous at x ₀ if it is both lsc and usc at x ₀ and to be H-continuous at x ₀ if it is both H-lsc and H-usc at x ₀. G is called closed at x ₀ if for each net {(x_α , z_α )} graphG := {(x, z) | z ∈ G(x)}, (x_α , z_α ) → (x ₀, z ₀), z ₀ must belong to G(x ₀). 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 x ₀, then G is H-usc at x ₀. Conversely if G is H-usc at x ₀ and if G(x ₀) is compact, then G is usc at x ₀;

(ii) If G is H-lsc at x ₀, then G is lsc at x ₀. The converse is true if G(x ₀) is compact;

(iii) If Y is compact and G is closed at x ₀, then G is usc at x ₀;

(iv) If G is usc at x ₀ and G(x ₀) is closed, then G is closed at x ₀;

(v) If G has compact values, then G is usc at x ₀ if and only if, for each net {x_α } ⊆ A which converges to x ₀ and for each net {y_α } ⊆ G(x_α ), there are y ₀ ∈ G(x ₀) and a subnet {y_β } of {y_α } such that y_β → y ₀.

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

1 e v upper R : = { ( x , t , y , μ ) | R ( x , t , y , μ ) holds } . 1 e v upper R ( . , . , . , μ 0 ) : = { ( x , t , y ) | R ( x , t , y , μ 0 ) holds } . 1 e v lower R : = { ( x , t , y , μ ) | R ( x , t , y , μ ) does not hold } .

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 λ ₀ and E(λ ₀) is compact, and K ₂ is lsc in K ₁(A, Λ) × {λ ₀};

(ii) in K ₁(A, Λ) × K ₂(K ₁(A, Λ), Λ) × {γ ₀}, T is usc and compact-valued if α = w (or α = m), and lsc if α = s;

(iii) in K ₁(A, Λ) × T(K ₁(A, Λ), K ₂(K ₁(A, Λ), Λ), Γ) × K ₂(K ₁(A, Λ), Λ) × {μ ₀}, lev_upperR is closed.

Then S_α is both usc and closed at (λ ₀, γ ₀, μ ₀).

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 (λ ₀, γ ₀, μ ₀). Indeed, we suppose to the contrary that S_w is not upper semicontinuous at (λ ₀, γ ₀, μ ₀), i.e., there is an open subset U of S_w (λ ₀, γ ₀, μ ₀) such that for all nets {(λ_n , γ_n , μ_n )} convergent to (λ ₀, γ ₀, μ ₀), 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(λ ₀), one can assume that x_n → x ₀ for some x ₀ ∈ E(λ ₀). If x ₀ ∉ S_w (λ ₀, γ ₀, μ ₀), then ∃y ₀ ∈ K ₂(x ₀, λ ₀), ∀t ₀ ∈ T(x ₀, y ₀, γ ₀) such that

R ( x 0 , t 0 , y 0 , μ 0 ) does not hold .

By the lower semicontinuity of K ₂ at (x ₀, λ ₀), there exists y_n ∈ K ₂(x_n , λ_n ) such that y_n → y ₀. Since x_n ∈ S_w (λ_n, γ_n , μ_n ), ∃t_n ∈ T (x_n , y_n , γ_n ) such that

R ( x n , t n , y n , μ n ) holds .

Since T is usc at (x ₀, y ₀, γ ₀) and T (x ₀, y ₀, γ ₀) is compact, there exists t ₀ ∈ T (x ₀, y ₀, γ ₀) such that t_n → t ₀ (can take a subnet if necessary). By the condition (iii) and (2), we have

R ( x 0 , t 0 , y 0 , μ 0 ) holds ,

we see a contradiction between (1) and (3). Thus, x ₀ ∈ S_w (λ₀, γ ₀, μ ₀) ⊆ U, this contradicts to the fact x_n ∉ U, ∀n. Hence, S_w is upper semicontinuous at (λ ₀, γ ₀, μ ₀).

Now we prove that S_w is closed at (λ ₀, γ ₀, μ ₀). Indeed, we supposed that S_w is not closed at (λ ₀, γ ₀, μ ₀), i.e., there is a net {(x_n , λ_n , γ_n , μ_n )} → (x ₀, λ ₀, γ ₀, μ ₀) with x_n ∈ S_w (λ_n , γ_n , μ_n ) but x ₀ ∉ S_w (λ ₀, γ ₀, μ ₀). The further argument is the same as above. And so we have S_w is closed at (λ ₀, γ ₀, μ ₀). □

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], λ ₀ = 0, F(x, t, y, λ) = 2 ^λ+1, K ₁(x, λ) = (−λ − 1, λ], K ₂(x, λ) = {-1} and T x , y , λ = [ 0 , e 2 λ + cos λ ] . 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 K ₂ 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 K ₂ is essential.

Example 3 Let X, Y, Λ, Γ, M, λ ₀ as in Example 2 and let A = B = [-3, 3], F (x, t, y, λ) = x + y + λ, K ₁(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

K 2 ( x , λ ) = { - 3 , 0 , 3 } if λ = 0 , { 0 , 3 } otherwise .

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 K ₂ is not lower semicontinuous. In fact,

S α ( λ , γ , μ ) = { 3 } if λ = 0 , [ 0 , 3 ] if λ ∈ ( 0 , 1 ] .

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

Example 4 Let Λ, Γ, M, T, λ ₀ 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₊, K ₁(x, λ) = K ₂(x, λ) = [0, 1] and F ( x , t , y , 0 ) = x 2 - y 2 , F ( x , t , y , λ ) = y 2 - x 3 , ∀ λ ∈ ( 0 , 1 ] . We show that the assumptions (i) and (ii) of Theorem 1 are easily seen to be fulfilled and

S α ( λ , γ , μ ) = { 0 } if λ ∈ ( 0 , 1 ] , { 1 } if λ = 0 .

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, y n = 1 2 , λ n = 1 n → 0 as n → ∞, then { ( x n , t n , y n , λ n ) } → ( 0 , 0 , 1 2 , 0 ) and F ( x n , t n , y n , λ n ) = F ( 0 , 0 , 1 2 , 1 n ) = 1 4 > 0 , but F ( 0 , 0 , 1 2 , 0 ) = - 1 4 < 0 .

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, λ ₀ as in Example 2 and let K₁(x, λ) = K₂(x, λ) = [0, 1], T ( x , y , γ ) = [ 0 , 2 cos 6 x + sin 4 x + 2 ] and

F ( x , t , y , λ ) = { 0 } if λ = 0 , 3 sin 4 x + cos 2 x + 2 otherwise .

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

S α ( λ , γ , μ ) = [ 0 , 1 ] , ∀ λ ∈ [ 0 , 1 ]

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, λ ₀, as in, Example, 5 and, let, K ₁(x, λ), = K ₂(x, λ) = [0, 3], T (x, y, γ) = {t} and

F ( x , t , y , λ ) = { 0 } if λ = 0 , e cos 2 λ + 1 otherwise .

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 K ₂ is lsc in K ₁(A, Λ) × {λ ₀} (which is not imposed in this Theorem 4.1 of 10 ). The Example 3 shows that the lower semicontinuity of K ₂ 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, K ₁ = K ₂ = K and the variational relation R is defined as follows R ( x , t , y , λ ) 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, λ ₀, as in Example 6 and we let A = B = [0, 3), F(x, y, λ) = x − y and K ₁(x, λ) = K ₂(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 , K ₂ is usc and compact-valued in K ₁(A, Λ) × {λ₀};

(ii) in K ₁(A, Λ) × K ₂(K ₁(A, Λ), Λ) × {γ ₀}, T is usc and compact-valued if α = s, and lsc if α = w (or α = m);

(iii) in K ₁(A, Λ) × T(K ₁(A, Λ), K ₂(K ₁(A, Λ), Λ), Γ) × K ₂(K ₁(A, Λ), Λ) × {μ ₀}, lev_lowerR is closed.

Then S_α is lower semicontinuous at (λ ₀, γ ₀, μ ₀).

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 (λ ₀, γ ₀, μ ₀), i.e., there are x ₀ ∈ S_s (λ ₀, γ ₀, μ ₀) and net {(λ_n , γ_n , μ_n )}, (λ_n , γ_n , μ_n ) → (λ ₀, γ ₀, μ ₀) such that ∀x_n ∈ S_s (λ_n , γ_n , μ_n ), x_n → x ₀. Since E is lsc at λ ₀, there is x n ′ ∈ E ( λ n ) with x n ′ → x 0 . By the above contradiction assumption, there must be a subnet x ′ m of x ′ n such that, ∀ m , x m ′ ∉ S s ( λ m , γ m , μ m ) , i.e., ∃ y m ∈ K 2 ( x m ′ , λ m ) , ∃ t m ∈ T ( x m ′ , y m , γ m ) such that

R ( x m ′ , t m , y m , μ m ) does not hold .

As K ₂, is usc at (x ₀, λ ₀) and K ₂(x ₀, λ ₀) is compact, one has y ₀ ∈ K ₂(x ₀, λ ₀) such that y_m → y ₀ (taking a subnet if necessary). By the upper semicontinuity of T at (x ₀, y ₀, γ ₀), one has t ₀ ∈ T(x ₀, y ₀, γ ₀) such that t_m → t ₀.

Since ( x m ′ , t m , y m , λ m , γ m , μ m ) → ( x 0 , t 0 , y 0 , λ 0 , γ 0 , μ 0 ) and by the condition (iii) and (4), yields that

R ( x 0 , t 0 , y 0 , μ 0 ) does not hold ,

which is impossible since x ₀ ∈ S_s (λ₀, γ ₀, μ ₀). Therefore, S_s is lsc at (λ ₀, γ ₀, μ ₀). □

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

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

K 1 ( x , λ ) = [ - 1 , 1 ] if λ = 0 , [ - λ - 1 , 0 ] otherwise .

We have E(0) = [-1, 1], E(λ) = [−λ − 1, 0], ∀λ ∈ (0, 1]. Hence K ₂ 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, λ ₀ as in Example 10, let K 1 ( x , λ ) = K 2 ( x , λ ) = [ 0 , 1 2 ] and

F ( x , y , λ ) = [ 1 2 , 1 ] if λ = 0 , [ 2 , 3 λ + 2 ] otherwise .

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 S α ( λ , γ , μ ) ) = [ 0 , 1 2 ] , ∀ λ ∈ [ 0 , 1 ] . 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) K ₂(., λ ₀) is lsc in K ₁(A, Λ) and E(λ ₀) is compact;

(v) in K ₁(A, Λ) × K ₂(K ₁(A, Λ), Λ), T (., ., γ ₀) is usc and compact-valued if α = w (or α = m), and lsc if α = s;

(vi) in K ₁(A, Λ) × T(K ₁(A, Λ), K ₂(K ₁(A, Λ), Λ), Γ) × K ₂(K ₁(A, Λ), Λ), lev_upperR(., ., μ ₀) is closed ;

Then S_α is Hausdorff lower semicontinuous at (λ ₀, γ ₀, μ ₀).

Proof. We consider only for the cases α = s. We first prove that S_s (λ ₀, γ ₀, μ ₀) is closed. Indeed, we let x_n ∈ S_s (λ ₀, γ ₀, μ ₀) such that x_n → x ₀. If x ₀ ∉ S_s (λ ₀, γ ₀, μ ₀), then ∃y ₀ ∈ K ₂(x ₀, λ ₀), ∃t ₀ ∈ T (x ₀, y ₀, γ ₀) such that

R ( x 0 , t 0 , y 0 , μ 0 ) does not hold .

By the lower semicontinuity of K ₂(., λ ₀) at x ₀, one has y_n ∈ K ₂(x_n , λ ₀) such that y_n → y ₀. By the lower semicontinuity of T (., ., γ ₀) at (x ₀, y ₀), one has t_n ∈ T(x_n , y_n , γ ₀) such that t_n → t ₀. Since x_n ∈ S_s (λ ₀, γ ₀, μ ₀), we have

R ( x n , t n , y n , μ 0 ) holds .

Since (x_n , t_n , y_n ) → (x ₀, t ₀, y ₀) and by the condition (vi) and (6) yields that

R ( x 0 , t 0 , y 0 , μ 0 ) holds ,

we see a contradiction between (5) and (7). Therefore, S_s (λ ₀, γ ₀, μ ₀) is closed.

On the other hand, since S_s (λ ₀, γ ₀, μ ₀) ⊆ E(λ ₀) is compact by E(λ ₀) compact. Since S_s is lower semicontinuous at (λ ₀, γ ₀, μ ₀) and S_s (λ ₀, γ ₀, μ ₀) compact. Hence S_s is Hausdorff lower, semicontinuous at (λ ₀, γ ₀, μ ₀). 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], λ ₀ = 0, and x = ( x 1 , x 2 ) ∈ ℝ 2 , K 1 ( x , λ ) = K 2 ( x , λ ) = { ( x 1 , λ x 1 4 ) } , T ( x , y , λ ) = [ 0 , 3 sin 4 x + sin 2 x + 1 ] and

F ( x , t , y , λ ) = 1 2 if λ = 0 , 1 2 + λ 2 λ + 1 otherwise .

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 | x ² = 0} and E ( λ ) = { x ∈ ℝ 2 | x 2 = λ x 1 4 ) } , ∀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) = {(x ₁, x ₂) ∈ ℝ²|x ₂ = 0} and S α ( λ , γ , μ ) = { ( x 1 , x 2 ) ∈ ℝ 2 | x 2 = λ x 1 4 ) } , ∀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 (λ ₀, γ ₀, μ ₀).

Corollary 15 Suppose that all conditions in Theorems 1 and 12 are satisfied. Then, we have S_α is Hausdorff continuous and closed at (λ ₀, γ ₀, μ ₀).

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 (λ ₀, γ ₀, μ ₀) ∈ Λ × Γ × M.

Corollary 16 Assume for problem (QEP _αρ ) that

(i) E is usc at λ ₀ and E(λ ₀) is compact, and K ₂ is lsc in K ₁(A _, Λ) × {λ₀};

(ii) in K ₁(A, Λ) × K ₂(K ₁(A, Λ), Λ) × {γ ₀}, T is usc and compact-valued if α = w (or α = m), and lsc if α = s;

(iii) in K ₁(A, Λ) × T(K ₁(A, Λ), K ₂(K ₁(A, Λ), Λ), Γ) × K ₂(K ₁(A, Λ), Λ) × {μ ₀}, the set { ( x , t , y , μ ) ∈ A × B × A × M | ρ ( F ( x ̄ , t , y , μ ) ; C ) } is closed.

Then Ψ _αρ is both usc and closed at (λ ₀, γ ₀, μ ₀).

Proof. Since α = {w, m, s}, ρ = {ρ ₁, ρ ₂}, we have in fact six cases. However, the proof techniques are similar. We consider only the cases α = w, ρ = ρ ₁. Let relation R be defined by R(x, t, y, μ) holds iff F ( x ̄ , t , y , μ ) ⊆ C . To apply Theorem 1, we need to check only that in K ₁(A, Λ) × T(K ₁(A, Λ), K ₂(K ₁(A, Λ), Λ), Γ) × K ₂(K ₁(A, Λ), Λ) × {μ ₀}, 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 , _n )} → (x ₀, t ₀, y ₀, μ ₀) such that

R ( x n , t n , y n , μ n ) holds .

By assumption (iii), we have

F ( x 0 , t 0 , y 0 , μ 0 ) ⊆ C .

   □

Corollary 17 Assume for problem (QEP _αρ ) that

(i) E is lsc at λ ₀, K ₂ is usc and compact-valued in K ₁(A, Λ) × {λ ₀};

(ii) in K ₁(A, Λ) × K ₂(K ₁(A, Λ), Λ) × {γ ₀}, T is usc and compact-valued if α = s, and lsc if α = w (or α = m);

(iii) in K ₁(A, Λ) × T(K ₁(A, Λ), K ₂(K ₁(A, Λ), Λ), Γ) × K ₂(K ₁(A, Λ), Λ) × {μ ₀}, the set { ( x , t , y , μ ) ∈ A × B × A × M | ρ ̄ ( F ( x , t , y , μ ) ; C ) } is closed.

Then Ψ _αρ is lower semicontinuous at (λ ₀, γ ₀, μ ₀).

Proof. Since α = {w, m, s}, ρ = {ρ ₁, ρ ₂}, we have in fact six cases. However, the proof techniques are similar. We consider only the cases α = s, ρ = ρ ₁. 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 K ₁(A, Λ) × T (K ₁(A, Λ), K ₂(K ₁(A, Λ), Λ), Γ) × K ₂(K ₁(A, Λ), Λ) × {μ ₀}, 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 )} → (x ₀, t ₀, y ₀, μ ₀) such that

R ( x n , t n , y n , μ n ) does not hold .

By assumption (iii), we have

F ( x 0 , t 0 , y 0 , μ 0 ) ⊄ C .

   □

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

(iv) K ₂(., λ ₀) is lsc in K ₁(A, Λ) and E(λ ₀) is compact;

(v) in K ₁(A, Λ) × K ₂(K ₁(A, Λ), Λ), T (., ., γ ₀) is usc and compact-valued if α = w (or α = m), and lsc if α = s;

(vi) in K ₁(A, Λ) × T(K ₁(A, Λ), K ₂(K ₁(A, Λ), Λ), Γ) × K ₂(K ₁(A, Λ), Λ), the set {(x, t, y) ∈ A × B × A | ρ(F(x, t, y, μ ₀); C)} is closed.

Then Ψ _αρ is Hausdorff lower semicontinuous at (λ ₀, γ ₀, μ ₀).

Proof. Since α = {w, m, s}, ρ = {ρ ₁, ρ ₂}, we have in fact six cases. However, the proof techniques are similar. We consider only the cases α = s, ρ = ρ ₁. 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 K ₁(A, Λ) × T (K ₁(A, Λ), K ₂(K ₁(A, Λ), Λ), Γ) × K ₂(K ₁(A, Λ), Λ), the set {(x, t, y) ∈ A × B × A | F(x, t, y, μ ₀) ⊆ C)} is closed. Indeed, for all nets {(x_n , t_n , y_n )} → (x ₀, t ₀, y ₀) such that R(x_n , t_n , y_n , μ ₀) holds. By assumption (vi), we have F(x ₀, t ₀, y ₀, μ ₀) ⊆ 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 (λ ₀, γ ₀, μ ₀).

(ii) Suppose that all conditions in Corollaries 16 and 18 are satisfied. Then, we have Ψ _αρ is Hausdorff continuous and closed at (λ ₀, γ ₀, μ ₀).

The author declares that they have no competing interests.