On the lower semicontinuity of the solution mapping for parametric vector mixed quasivariational inequality problem of the Minty type

 Thu Dau Mot University Journal of Science - Volume 2 - Issue 2-2020
On the lower semicontinuity of the solution map-
ping for parametric vector mixed quasivariational
inequality problem of the Minty type
 by Nguyen Van Hung (Posts and Telecommunications Institute of Tech-
nology Ho Chi Minh City), Vo Viet Tri (Thu Dau Mot university)
Article Info: Received 10 April. 2020, Accepted 7 May. 2020, Available online 30 May.
2020
 Corresponding author: trivv@tdmu.edu.vn (Vo Viet Tri)
 https://doi.org/10.37550/tdmu.EJS/2020.02.041
 ABSTRACT
 In this paper, we study a class of parametric vector mixed quasivariational
 inequality problem of the Minty type (in short, (MQVIP)). Afterward, we
 establish some sufficient conditions for the stability properties such as the
 inner-openness, lower semicontinuity and Hausdorff lower semicontinuity of
 the solution mapping for this problem. The results presented in this paper
 is new and wide to the corresponding results in the literature
 Keywords: problem of the minty type, inner-openness, lower semicontinu-
 ity, Hausdorff lower semicontinuity
1 Introduction
In 1980, Giannessi [3] was introduced a vector variational inequality in a finite-
dimensional Euclidean space. After that, there are many papers on the property
of solution sets for vector variational inequality problems in abstract spaces, see
[1,8,4] and the references therein. On the other hand, stability of solutions for
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Nguyen Van Hung, Vo Viet Tri - Volume 2 - Issue 2-2020, p.150-157
parametric vector mixed quasivariational inequality problem is an important topic
in optimization theory and applications. Recently, the continuity, especially the
upper semicontinuity, lower semicontinuity and Hausdorff lower semicontinuity of
the solution sets for parametric optimization problems, parametric vector variational
inequality problems and parametric vector quasiequilibrium problems have been
studied in the literature, see [1,4,5,6, 10].
 Motivated and inspired by the works mentioned above, in this paper we introduce
and study the parametric vector mixed quasivariational inequality problem. Let
X, Y be two Hausdorff topological vector spaces, P be a closed pointed convex in
Y with intP 6= ∅, A be a nonempty subset of X and Λ be a topological space.
The space of all linear continuous operators from X into Y denoted by L(X, Y ).
Let K : A × Λ ⇒ X, T : A × Λ ⇒ L(X, Y ) be set-valued mappings, and let
g : A × A × Λ → X, f : A × A × Λ → Y be two continuous single-valued mappings.
Denoted hz, xi by the value of a linear operator z ∈ L(X, Y ) at x ∈ X. We always
assume that h., .i is continuous.
 For γ ∈ Λ, we consider the following parametric vector mixed quasivariational
inequality problems (in short, (MQVIP-j), j=1,2).
 (MQVIP-1) Findx ¯ ∈ A such that
  x¯ ∈ K(¯x, γ)
 (1.1)
 hz, g(y, x,¯ γ)i + f(y, x,¯ γ) ∈/ −intP for all y ∈ K(¯x, γ), z ∈ T (y, γ).
 (MQVIP-2) Findx ¯ ∈ A such that
  x¯ ∈ K(¯x, γ)
 (1.2)
 hz, g(y, x,¯ γ)i + f(y, x,¯ γ) ∈/ −P for all y ∈ K(¯x, γ), z ∈ T (y, γ).
 For each γ ∈ Λ we let E(γ) := {x ∈ X|x ∈ K(x, γ)} and Ψj :Λ ⇒ X be a set-
valued mapping such that Ψj(γ) is the solution set of (MQVIP-j), j=1,2. Throughout
this paper, we always assume that Ψj(γ) 6= ∅ for each γ in the neighborhood of
γ0 ∈ Λ.
 The structure of our paper is as follows. In the first part of this article, we
introduce the model parametric vector mixed quasivariational inequality problems
of the Minty type. Section 2, we recall definitions for later uses. In Section 3, we
establish the lower semicontinuity, inner-openness and Hausdorff lower semicontinu-
ity of the solutions for parametric vector mixed quasivariational inequality problem
of the Minty type.
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 Thu Dau Mot University Journal of Science - Volume 2 - Issue 2-2020
2 Preliminaries
In this section, we recall some basic definitions and their some properties.
 We first recall a new limit in [7,9], the inferior open limit. Let X and Y be two
topological vector spaces.
Definition 2.1 (see [7,9]) Let A be a nonempty subset of X and G : A ⇒ Y be a
multifunction.
 (i) For x0 ∈ A, denote liminfox→x0 G(x) := {z ∈ Y : there are open neighborhoods
 U of x0 and V of z such that V ⊂ G(x) for all x ∈ U\{x0}}.
 (ii) Limsupx→x0 G(x) := {y ∈ Y : there exists a net {(xα, yα)} ⊂ A × Y converging
 to (x0, y) with yα ∈ G(xα)}
 (iii) G is said that inner-open at x0 if liminfox→x0 G(x) ⊃ G(x0). G is said to be
 inner-open on A if it is inner-open at every a ∈ A.
 From the Item (iii) of Lemma 2.1 in [7], we deduce that liminfox→x0 G(x) =
 c c c
 lim supx→x0 G (x) , where G (x) = Y \ G(x).
Definition 2.2 ([2]). Let X and Y be two topological vector spaces and G : A ⊂
X ⇒ Y be a multifunction.
 (i) G is said to be lower semicontinuous (lsc) at x0 ∈ A if for every open subset
 U of Y with G(x0) ∩ U 6= ∅, there exists a neighborhood N of x0 such that
 G(x) ∩ U 6= ∅, for all x ∈ N ∩ A.
 (ii) G is said to be upper semicontinuous (usc) at x0 ∈ A if for each open set
 U ⊃ G(x0), there is a neighborhood N of x0 such that U ⊃ G(x) for all
 x ∈ N ∩ A.
 (iii) G is said to be Hausdorff lower semicontinuous (H-lsc) at x0 ∈ A if for each
 neighborhood B of the origin in Y , there exists a neighborhood N of x0 such
 that G(x0) ⊂ G(x) + B for all x ∈ N ∩ A.
 (iv) G is said to be closed at x0 ∈ A if and only if for every net {(xα, yα)} ⊂ A × Y
 converging (x0, y0) with yα ∈ G(xα), we have y0 ∈ G(x0).
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 (v) G is said to be lower semicontinuous (upper semicontinuous, Hausdorff lower
 semicontinous, closed, respectively) on A if it is lower semicontinuous (upper
 semicontinuous, Hausdorff lower semicontinous, closed, respectively) at each
 x0 ∈ A.
Lemma 2.3 ([2]). Let X and Y be two topological vector spaces and G : A ⊂ X ⇒
Y be a multifunction. Then, the following assertions hold.
 (i) G is lsc at x0 ∈ A iff for every y ∈ G(x0) and each neighborhood V of y, there
 exists a neighborhood U of x0 such that G(x) ∩ V 6= ∅ for all x ∈ U ∩ A.
 (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 G has compact values, then G is usc at x0 if and only if, for each net
 {xα} ⊂ X which converges to x0 and for each net {yα} with yα ∈ G(xα), there
 are y ∈ G(x0) and a subnet {yβ} of {yα} such that yβ → y.
3 Main Results
In this section, with the contexts and hypothesis of the problem (MQVIP-j) de-
scribing in the Section 1 we establish the inner-openness, lower semicontinuity and
Hausdorff lower semicontinuity of the solution set of (MQVIP-j).
Theorem 3.1 Suppose the following conditions hold
 (i) E is inner-open on Λ,
 (ii) K is upper semicontinuous on A × Λ with compact values, and
 (iii) T is upper semicontinuous on A × Λ with compact values.
Then Ψ2 is inner-open on Λ.
Proof. Suppose to the contrary that Ψ2 is not inner-open at γ0. Then, there exists
x0 ∈ Ψ2(γ0), x0 6∈ liminfoγ→γ0 Ψ2(γ). Since
  c c
 liminfoγ→γ0 Ψ2(γ) = lim sup(Ψ2) (γ) ,
 γ→γ0
 c
we have x0 ∈ lim supγ→γ0 (Ψ2) (γ). Thus, there exists a net {(γα, xα)} converging
 c
to (γ0, x0) with xα ∈ (Ψ2) (γα). Since E is inner-open at γ0 and x0 ∈ E(γ0) which
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 Thu Dau Mot University Journal of Science - Volume 2 - Issue 2-2020
implies x0 ∈ liminfoγ→γ0 E(γ). There exist neighborhoods U of γ0, N of x0 such
that x0 ∈ N ⊂ E(γ) for all γ ∈ U. Since (xα, γα) → (x0, γ0) and by the above
contradiction assumption, there must be a subnet {(xβ, γβ)} of net {(xα, γα)} such
that for all β, xβ 6∈ Ψ2(γβ), i.e., there exist yβ ∈ K(xβ, γβ) and zβ ∈ T (yβ, γβ) such
that
 hzβ, g(yβ, xβ, γβ)i + f(yβ, xβ, γβ) ∈ −P. (3.1)
 Since K is upper semicontinuity on A × Λ and K(x0, γ0) is compact, one has
y0 ∈ K(x0, γ0) such that yβ → y0 (taking a subnet if necessary) by Lemma 2.3.
Also, by the upper semicontinuity of T at (y0, µ0) and T (y0, γ0) is compact, one has
 0 0
zβ ∈ T (yβ, γβ) such that zβ → z0. Since g, f and h., .i are continuous, we have
 0
 hzβ, g(yβ, xβ, γα)i + f(yβ, xβ, γβ) → hz0, g(y0, x0, γ0)i + f(y0, x0, γ0). (3.2)
From P is closed, (3.1) and (3.2), it follows that
 hz0, g(y0, x0, γ0)i + f(y0, x0, γ0) ∈ −P,
which is impossible since x0 ∈ Ψ2(γ0). The roof is complete. 
 The following example shows that the inner-openness of E is essential.
Example 3.1 Let X = Y = R,A = [0, 1], Λ = [0, 1], P = R+, and for every
(x, y, γ) ∈ A × A × Λ we define f(y, x, γ) = 0;
 (
 (0, 1] if γ ∈ (0, 1],
 K(x, γ) =
 [−1, 0] if γ = 0;
 T (y, γ) = [1, 21+cos2 γ] and
 g(y, x, γ) = {2 + 3γ2}.
It follows that E(γ) = (0, 1] for all γ ∈ (0, 1] and E(0) = [−1, 0]. It is not hard to
see that the assumptions (ii) and (iii) in Theorem 3.1 are satisfied. But E is not
inner-open at 0. Hence, Ψ2 also not inner-open at 0. Thus, Theorem 3.1 cannot be
applied. In fact, Ψ2(0) = [−1, 0] and Ψ2(γ) = (0, 1] for all γ ∈ (0, 1].
 Next, we establish the lower semicontinuity of the solutions for parametric vector
mixed quasivariational inequality problem of the Minty type (MQVIP-1). We denote
 Er(Ψ1) = {(x, γ) ∈ A × Γ: +f(y, x, γ) ∈/ −intP
 ∀y ∈ K(x, γ) and z ∈ T (y, γ)},
 Graph(Ψ1) = {(x, γ) ∈ A × Γ: x ∈ Ψ1(γ)}.
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Nguyen Van Hung, Vo Viet Tri - Volume 2 - Issue 2-2020, p.150-157
Theorem 3.2 Suppose the following conditions hold:
 (i) E is lower semicontinuous on Λ;
 (ii) A net (xα, γα) converging to some element in Graph(Ψ1) implies that there
 exists α satisfying (xα, γα) ∈ Er(Ψ1).
Then, Ψ1 is lower semicontinuous on Λ.
Proof. Suppose to the contrary that Ψ1 is not lower semicontinuous at γ0, by Lemma
2.3 there exist x0 ∈ Ψ1(γ0) and a neighborhood V of x0 such that
 for every neighborhood U of γ0, we have Ψ1(γ0) ∩ V = ∅ for all γ ∈ U. (3.3)
This, we can find a net γα such that γα → γ0
 Ψ1(γα) ∩ V = ∅ ∀α. (3.4)
Since E is lower semicontinuous at γ0 ∈ Λ, there is a net {xα} with xα ∈ E(γα),
xα → x0. Without loss of generality we can assume xα ∈ V for al α (taking a subnet
if necessary). By the condition (ii), there exists α satisfying (xα, γα) ∈ gr(Ψ1), thus
(xα, γα) ∈ Graph(Ψ1) which contradicts (3.4). Therefore, Ψ1 is lower semicontinous.

 The following example ensures that the lower semicontinuity of E in Theorem
3.2 is essential.
Example 3.2 Let X = Y = R,A = [−1, 1],P = R+, Λ = [0, 1], for (y, x, γ) ∈
A × A × Λ we defined
 2
 T (y, γ) = [0, 1], f(y, x, γ) = g(y, x, γ) = {1 + 2sin (γ)},
 (
 {−1, 0}, if γ = 0,
 K(x, γ) =
 {0, 1}, if γ ∈ (0, 1].
It is clear to see that E(0) = {−1, 0},E(γ) = {0, 1}, ∀γ ∈ (0, 1] and the assumption
(ii) of Theorem 3.2 is fulfilled. However, Ψ1 is not lower semicontinuous at 0. The
reason is that E is not lower semicontinuous at 0.
 Finally, we study Hausdorff lower semicontinuity of the solutions for parametric
vector mixed quasivariational inequality problem of the Minty type (MQVIP-1).
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 Thu Dau Mot University Journal of Science - Volume 2 - Issue 2-2020
Theorem 3.3 Impose the assumption of Theorem 3.2 and the following additional
condition:
 (iii) K(., γ0) and T (., γ0) are lower semicontinuous on X and E(γ0) is compact.
Then Ψ1 is Hausdorff lower semicontinuous on Λ.
Proof. We first prove that Ψ1(γ0) is closed. Indeed, we let {xα} ⊂ Ψ1(γ0) such that
xα → x0. If x0 6∈ Ψ1(γ0), then there exist y0 ∈ K(x0, γ0) and z0 ∈ T (y0, γ0) such
that
 hz0, g(y0, x0, γ0)i + f(y0, x0, γ0) ∈ −intP. (3.5)
From the lower semicontinuity of K(., γ0) at x0, by the Lemma 2.3 one has a net
{yα} converging to y0 with yα ∈ K(xα, γ0). Also, by the lower semicontinuity of
T (., γ0) at y0, one has a net zα → z0 with zα ∈ T (yα, γ0). Since {xα} ⊂ Ψ1(γ0), we
have
 hzα, g(yα, xα, γ0)i + f(yα, xα, γ0) 6∈ −intP for all α. (3.6)
Since (xα, zα, yα) → (x0, z0, y0) and from the continuity of g, f, h., .i and (3.6), we
have
 hz0, g(y0, x0, γ0)i + f(y0, x0, γ0) 6∈ −intP, (3.7)
we see a contradiction between (3.5) and (3.7). Hence, Ψ1(γ0) is a closed set.
 On the other hand, Ψ1(γ0) is closed subset of the compact E(γ0), therfore, it is
compact. Since Ψ1 is lower semicontinuous in Λ (by Theorem 3.2) and Ψ1(γ0) is
compact, by Lemma 2.3, it follows that Ψ1 is Hausdorff lower semicontinuous in Λ.

 m
Remark 3.1 If A = X = R ,Y = R,P = R+, g(y, x, γ) = y − x, f(y, x, γ) = 0,
then the problem (MQVIP-1) is reduced to the following quasivariational inequality
problem of the Minty type (in short, (MVI)):
 (MVI) Findx ¯ ∈ X withx ¯ ∈ K(¯x, γ) such that
 hz,¯ y − x¯i ≥ 0, ∀y ∈ K(¯x, γ), z ∈ T (y, γ).
This problem has been studied in [8]. Then the lower semicontinuity of the solution
set in Theorem 4.1 in [8] is a particular case of our Theorem 3.2, while our Theorem
3.3 is completely different Theorem 4.4 in [8]. Note that, our Theorem 3.1 is new
for vector mixed quasivariational inequality problem of the Minty type.
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Nguyen Van Hung, Vo Viet Tri - Volume 2 - Issue 2-2020, p.150-157
4 Acknowledgements
The authors wish to thank the anonymous referees for their valuable comments.
This research is funded by Thu Dau Mot University, Binh Duong province, Viet
Nam.
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