Research

Continuity of solutions for parametric generalized quasi-variational relation problems

Nguyen Van Hung

Author Affiliations

Department of Mathematics, Dong Thap University, 783 Pham Huu Lau Street, Ward 6, Cao Lanh City, Vietnam

Fixed Point Theory and Applications 2012, 2012:102 doi:10.1186/1687-1812-2012-102

 Received: 1 January 2012 Accepted: 21 June 2012 Published: 21 June 2012

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.

Keywords:
quasi-variational relation problems; upper semicontinuity; lower semicontinuity; Hausdorff lower semicontinuity; upper semicontinuity; continuity; H-continuity; closedness

Introduction and preliminaries

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 → 2A, K2: A × Λ → 2A, T : A × A × Γ → 2B 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, (QVRα)):

(QVRα): Find such that satisfying

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

Throughout the article, we assume that Sα(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 (QVRα). Then, (QVRα) becomes (VR) is studied in [3]:

Find such that

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

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

Then, (QVRα) becomes (QGVIPα)

Find such that satisfying

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

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

Then, (QVRα) becomes (Pαρ) 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 (QVRα) and let Y be a Hausdorff topological vector space. Given a mapping F : A × A × M → 2Y and C Y be a closed subset with nonempty interior, the relation R is defined as follows

Then, (QVRα) 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 (QVRα) and let Z be a Hausdorff topological vector space. Given a mapping F : A × A × M → 2Z and C Z be a closed subset with nonempty interior, the relation R is defined as follows

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

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

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

Then, (QVRα) becomes (QEPαρ)

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 (QVRα). 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 → 2Y 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 {(xα, zα )} graphG := {(x, z) | z G(x)}, (xα, zα) → (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 → 2Y 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 {xα} ⊆ A which converges to x0 and for each net {yα} ⊆ G(xα), there are y0 G(x0) and a subnet {yβ} of {yα} such that yβ y0.

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

Main results

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 (QVRα).

Theorem 1 Assume for problem (QVRα) 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}, levupperR is closed.

Then Sα 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 Sw is upper semicontinuous at (λ0, γ0, μ0). Indeed, we suppose to the contrary that Sw is not upper semicontinuous at (λ0, γ0, μ0), i.e., there is an open subset U of Sw(λ0, γ0, μ0) such that for all nets {(λn, γn, μn)} convergent to (λ0, γ0, μ0), there exists xn Sw(λn, γn, μn), xn U, ∀n. By the upper semicontinuity of E and the compactness of E(λ0), one can assume that xn x0 for some x0 E(λ0). If x0 Sw(λ0, γ0, μ0), then ∃y0 K2(x0, λ0), ∀t0 T(x0, y0, γ0) such that

(1)

By the lower semicontinuity of K2 at (x0, λ0), there exists yn K2(xn, λn) such that yn y0. Since xn Sw(λn, γn, μn), ∃tn T (xn, yn, γn) such that

(2)

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

(3)

we see a contradiction between (1) and (3). Thus, x0 Sw0, γ0, μ0) ⊆ U, this contradicts to the fact xn U, ∀n. Hence, Sw is upper semicontinuous at (λ0, γ0, μ0).

Now we prove that Sw is closed at (λ0, γ0, μ0). Indeed, we supposed that Sw is not closed at (λ0, γ0, μ0), i.e., there is a net {(xn, λn, γn, μn)} → (x0, λ0, γ0, μ0) with xn Sw(λn, γn, μn) but x0 Sw(λ0, γ0, μ0). The further argument is the same as above. And so we have Sw 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) = 2λ+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 Sα 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, Sα(0, 0, 0) = (-1, 0] and Sα(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 Sα 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 Sα is not usc at (0, 0, 0). The reason is that assumption (iii) is violated. Indeed, taking xn = 0, tn = 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, Sα 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, Sα 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 → 2Y and C : A × Λ → 2Y be multifunctions, with C(x, λ) being a convex cone. Then, (QVRα) 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 Sα is usc and closed at (0, 0, 0), although A is not compact. In fact, Sα (λ, γ, μ) = {2},∀λ ∈ [0, 1].

Theorem 9 Assume for problem (QVRα) 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}, levlowerR is closed.

Then Sα 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 Ss is not lsc at (λ0, γ0, μ0), i.e., there are x0 Ss(λ0, γ0, μ0) and net {(λn, γn, μn)}, (λn, γn, μn) → (λ0, γ0, μ0) such that ∀xn Ss(λn, γn, μn), xn 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

(4)

As K2, is usc at (x0, λ0) and K2(x0, λ0) is compact, one has y0 K2(x0, λ0) such that ym 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 tm t0.

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

which is impossible since x0 Ss0, γ0, μ0). Therefore, Ss 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) = 2λ, 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, Sα (0, 0, 0) = [-1, 1] and Sα(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), levupperR(., ., μ0) is closed ;

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

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

(5)

By the lower semicontinuity of K2(., λ0) at x0, one has yn K2(xn, λ0) such that yn y0. By the lower semicontinuity of T (., ., γ0) at (x0, y0), one has tn T(xn, yn, γ0) such that tn t0. Since xn Ss(λ0, γ0, μ0), we have

(6)

Since (xn, tn, yn) → (x0, t0, y0) and by the condition (vi) and (6) yields that

(7)

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

On the other hand, since Ss(λ0, γ0, μ0) ⊆ E(λ0) is compact by E(λ0) compact. Since Ss is lower semicontinuous at (λ0, γ0, μ0) and Ss(λ0, γ0, μ0) compact. Hence Ss 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, Sα is not, Hausdorff lower semicontinuous at (0, 0, 0). In fact, Sα(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 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 Sα is Hausdorff continuous and closed at (λ0, γ0, μ0).

Applications

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 (QEPαρ).

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

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

Corollary 16 Assume for problem (QEPαρ) 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 Ψαρ 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 {(xn, tn, yn, n)} → (x0, t0, y0, μ0) such that

By assumption (iii), we have

□

Corollary 17 Assume for problem (QEPαρ) 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 Ψαρ 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 {(xn, tn, yn, μn)} → (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 Ψαρ 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 {(xn, tn, yn)} (x0, t0, y0) such that R(xn, tn, yn, 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 Ψα is both continuous and closed at (λ0, γ0, μ0).

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

Competing interests

The author declares that they have no competing interests.

Acknowledgements

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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