In this section, we investigate the stability of solutions of (PGVQVLIP), that is, the upper and lower semicontinuity of the solution mapping S(λ, μ) for (PGVQVLIP) corresponding to a pair (λ, μ) of parameters in Hausdorff topological vector spaces.
Theorem 3.1. Let T : X × ∨ → 2^{L(X,Y)}be a setvalued mapping with nonempty values, C : X → 2^{Y}be a setvalued mapping such that, for each x ∈ X, C(x) is a pointed closed and convex cone in Y and \mathsf{\text{int}}\phantom{\rule{1em}{0ex}}C\left(x\right)\ne \varnothing ,\eta :X\times X\to X and ϕ : X × X → Y be two vectorvalued mappings. Assume that the following conditions are satisfied:

(a)
η (x, x) = 0 and ϕ(x, x) = 0 for all x ∈ X;

(b)
η (x, y) is continuous and affine with respect to the first argument;

(c)
ϕ (x, y) is continuous and C(x)convex with respect to the first argument;

(d)
T (x, μ) is weakly (η, ϕ, C(x))pseudomapping with respect to the first argument and Bu.s.c with compactvalues on X × ∨;

(e)
there is a continuous selection t of T on X × ∨;

(f)
the mapping W (⋅) = Y\ intC(⋅) such that the graph Gr(W) of W is weakly closed in X × Y;

(g)
K : ∧ → 2^{X}is Bu.s.c and Bl.s.c with weakly compact and convexvalues.
Then the following hold:

(1)
The solution mapping S(⋅,⋅) is nonempty and closed on ∧ × ∨;

(2)
The solution mapping S(⋅,⋅) is Bu.s.c on ∧ × ∨.
Proof. For any (λ, μ) ∈ ∧ × ∨, we first show that S(λ, μ) is nonempty. Since T has a continuous selection t and T(x, μ) is weakly (η, ϕ, C(x))pseudomapping with respect to the first argument on X × ∨, we know that t(x, μ) is also weakly (η, ϕ, C(x))pseudomapping with respect to the first argument on X × ∨.
Now, we define two setvalued mappings ϒ_{1}, ϒ_{2} : K(λ) → 2^{K(λ)}as follows: for all y ∈ K(λ),
{\Upsilon}_{1}\left(y\right)=\left\{x\in K\left(\lambda \right):\u27e8t\left(x,\mu \right),\eta \left(y,x\right)\u27e9+\varphi \left(y,x\right)\notin \mathsf{\text{int}}\phantom{\rule{1em}{0ex}}C\left(x\right)\right\}
and
{\Upsilon}_{2}\left(y\right)=\left\{x\in K\left(\lambda \right):\u27e8t\left(y,\mu \right),\eta \left(y,x\right)\u27e9+\varphi \left(y,x\right)\notin \mathsf{\text{int}}\phantom{\rule{1em}{0ex}}C\left(x\right)\right\}.
Since η(x, x) = 0 and ϕ (x, x) = 0 for all x ∈ X, we have y ∈ ϒ_{1}(y) and y ∈ ϒ_{2}(y) and so ϒ_{1}(y) and ϒ_{2}(y) are nonempty for any y ∈ K(λ). By virtue of the weakly (η, ϕ, C(x))pseudomapping of t(x, μ) with respect to the first argument, we have
{\Upsilon}_{1}\left(y\right)\subseteq {\Upsilon}_{2}\left(y\right),\phantom{\rule{1em}{0ex}}\forall y\in K\left(\lambda \right).
(3.1)
First, we assert that ϒ_{1} is a KKM mapping. Suppose that there exists a finite subset {y_{1}, y_{2},...,y_{
m
}} ⊆ K(λ) such that
\mathsf{\text{co}}\left\{{y}_{1},{y}_{2},...,{y}_{m}\right\}\u2288\bigcup _{i=1}^{m}{\Upsilon}_{1}\left({y}_{i}\right).
Then there exists \u0233\in \mathsf{\text{co}}\left\{{y}_{1},{y}_{2},...,{y}_{m}\right\}, i.e., \u0233={\sum}_{i=1}^{m}{\iota}_{i}{y}_{i}\in K\left(\lambda \right) for some nonnegative real number ι_{
i
}with 1 ≤ i ≤ m and {\sum}_{i=1}^{m}{\iota}_{i}=1 such that \u0233\notin {\bigcap}_{i=1}^{m}{\Upsilon}_{1}\left({y}_{i}\right). Moreover, \u0233\notin {\Upsilon}_{1}\left({y}_{i}\right) for 1 ≤ i ≤ m. This yields that
\u27e8t\left(\u0233,\mu \right),\eta \left({y}_{i},\u0233\right)\u27e9+\varphi \left({y}_{i},\u0233\right)\in \mathsf{\text{int}}\phantom{\rule{1em}{0ex}}C\left(\u0233\right)
and so
\sum _{i=1}^{n}{\iota}_{i}\left(\u27e8t\left(\u0233,\mu \right),\eta \left({y}_{i},\u0233\right)\u27e9+\varphi \left({y}_{i},\u0233\right)\right)\in \mathsf{\text{int}}\phantom{\rule{1em}{0ex}}C\left(\u0233\right).
Taking into account (b) and (c) that
\u27e8t\left(\u0233,\mu \right),\eta \left(\sum _{i=1}^{m}{\iota}_{i}{y}_{i},\u0233\right)\u27e9+\varphi \left(\sum _{i=1}^{m}{\iota}_{i}{y}_{i},\u0233\right)=\u27e8t\left(\u0233,\mu \right),\eta \left(\u0233,\u0233\right)\u27e9+\varphi \left(\u0233,\u0233\right)\in \mathsf{\text{int}}\phantom{\rule{1em}{0ex}}C\left(\u0233\right).
(3.2)
Again, from (a) together with (3.2), we have 0\in \mathsf{\text{int}}\phantom{\rule{1em}{0ex}}C\left(\u0233\right), which is a contradiction. Hence ϒ_{1} is a KKM mapping. It follows from (3.1) that ϒ_{2} is also a KKM mapping.
Second, we show that {\bigcap}_{y\in K\left(\lambda \right)}{\Upsilon}_{2}\left(y\right)\ne \varnothing. Taking any net {x_{
β
}} of ϒ_{2}(y) such that {x_{
β
}} is weakly convergent to a point \stackrel{\u0303}{x}\in K\left(\lambda \right). Then, for each β, one has
\u27e8t\left(y,\mu \right),\eta \left(y,{x}_{\beta}\right)\u27e9+\varphi \left(y,{x}_{\beta}\right)\notin \mathsf{\text{int}}\phantom{\rule{1em}{0ex}}C\left({x}_{\beta}\right).
From (b)(e), it follows that
\left({x}_{\beta},\u27e8t\left(y,\mu \right),\eta \left(y,{x}_{\beta}\right)\u27e9+\varphi \left(y,{x}_{\beta}\right)\right)\to \left(\stackrel{\u0303}{x},\u27e8t\left(y,\mu \right),\eta \left(y,\stackrel{\u0303}{x}\right)\u27e9+\varphi \left(y,\stackrel{\u0303}{x}\right)\right)\in \mathsf{\text{Gr}}\left(W\right).
Consequently, we get
\u27e8t\left(y,\mu \right),\eta \left(y,\stackrel{\u0303}{x}\right)\u27e9+\varphi \left(y,\stackrel{\u0303}{x}\right)\in Y\backslash \left(\mathsf{\text{int}}\phantom{\rule{1em}{0ex}}C\left(\stackrel{\u0303}{x}\right)\right),
that is,
\u27e8t\left(y,\mu \right),\eta \left(y,\stackrel{\u0303}{x}\right)\u27e9+\varphi \left(y,\stackrel{\u0303}{x}\right)\notin \mathsf{\text{int}}\phantom{\rule{1em}{0ex}}C\left(\stackrel{\u0303}{x}\right).
Therefore, \stackrel{\u0303}{x}\in {\Upsilon}_{2}\left(y\right) and so ϒ_{2}(y) is weakly closed set for any y ∈ K(λ). By the compactness of K(λ), ϒ_{2}(y) is weakly compact subset of K(λ). From Lemma 2.1, it follows that
\bigcap _{y\in K\left(\lambda \right)}{\Upsilon}_{2}\left(y\right)\ne \varnothing ,
i.e., there exists \stackrel{\u0304}{x}\in K\left(\lambda \right) such that
\u27e8t\left(y,\mu \right),\eta \left(y,\stackrel{\u0304}{x}\right)\u27e9+\varphi \left(y,\stackrel{\u0304}{x}\right)\notin \mathsf{\text{int}}\phantom{\rule{1em}{0ex}}C\left(\stackrel{\u0304}{x}\right),\phantom{\rule{1em}{0ex}}\forall y\in K\left(\lambda \right).
(3.3)
Third, we prove that \stackrel{\u0304}{x}\in {\bigcap}_{y\in K\left(\lambda \right)}{\Upsilon}_{1}\left(y\right). For any y ∈ K(λ), set {x}_{r}=\left(1r\right)\stackrel{\u0304}{x}+ry for all r ∈ (0,1).
Then x_{
r
}∈ K(λ). So, from (3.3), we have
\u27e8t\left({x}_{r},\mu \right),\eta \left({x}_{r},\stackrel{\u0304}{x}\right)\u27e9+\varphi \left({x}_{r},\stackrel{\u0304}{x}\right)\notin \mathsf{\text{int}}\phantom{\rule{1em}{0ex}}C\left(\stackrel{\u0304}{x}\right),\phantom{\rule{1em}{0ex}}\forall y\in K\left(\lambda \right).
(3.4)
Note that
\begin{array}{l}\u27e8t\left({x}_{r},\mu \right),\eta \left({x}_{r},\stackrel{\u0304}{x}\right)\u27e9+\varphi \left({x}_{r},\stackrel{\u0304}{x}\right)r\left(\u27e8t\left({x}_{r},\mu \right),\eta \left(y,\stackrel{\u0304}{x}\right)\u27e9+\varphi \left(y,\stackrel{\u0304}{x}\right)\right)\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}=\u27e8t\left({x}_{r},\mu \right),\eta \left({x}_{r},\stackrel{\u0304}{x}\right)\u27e9+\varphi \left({x}_{r},\stackrel{\u0304}{x}\right)r\left(\u27e8t\left({x}_{r},\mu \right),\eta \left(y,\stackrel{\u0304}{x}\right)\u27e9+\varphi \left(y,\stackrel{\u0304}{x}\right)\right)\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\left(1r\right)\left(\u27e8t\left({x}_{r},\mu \right),\eta \left(\stackrel{\u0304}{x},\stackrel{\u0304}{x}\right)\u27e9+\varphi \left(\stackrel{\u0304}{x},\stackrel{\u0304}{x}\right)\right)\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}\in \mathsf{\text{int}}\phantom{\rule{1em}{0ex}}C\left(\stackrel{\u0304}{x}\right).\phantom{\rule{2em}{0ex}}\end{array}
It follows from (3.4) that
r\left(\u27e8t\left({x}_{r},\mu \right),\eta \left(y,\stackrel{\u0304}{x}\right)\u27e9+\varphi \left(y,\stackrel{\u0304}{x}\right)\right)\notin \mathsf{\text{int}}\phantom{\rule{1em}{0ex}}C\left(\stackrel{\u0304}{x}\right),\phantom{\rule{1em}{0ex}}\forall y\in K\left(\lambda \right),
and so
\u27e8t\left({x}_{r},\mu \right),\eta \left(y,\stackrel{\u0304}{x}\right)\u27e9+\varphi \left(y,\stackrel{\u0304}{x}\right)\notin \mathsf{\text{int}}\phantom{\rule{1em}{0ex}}C\left(\stackrel{\u0304}{x}\right),\phantom{\rule{1em}{0ex}}\forall y\in K\left(\lambda \right).
Since t is continuous, we have
\left({x}_{r},\u27e8t\left({x}_{r},\mu \right),\eta \left(y,\stackrel{\u0304}{x}\right)\u27e9+\varphi \left(y,\stackrel{\u0304}{x}\right)\right)\to \left(\stackrel{\u0304}{x},\u27e8t\left(\stackrel{\u0304}{x},\mu \right),\eta \left(y,\stackrel{\u0304}{x}\right)\u27e9+\varphi \left(y,\stackrel{\u0304}{x}\right)\right)\in \mathsf{\text{Gr}}\left(W\right)
as r → 0. Therefore, by the weak closedness of Gr (W), we have
\u27e8t\left(\stackrel{\u0304}{x},\mu \right),\eta \left(y,\stackrel{\u0304}{x}\right)\u27e9+\varphi \left(y,\stackrel{\u0304}{x}\right)\in Y\backslash \left(\mathsf{\text{int}}\phantom{\rule{1em}{0ex}}C\left(\stackrel{\u0304}{x}\right)\right),
that is,
\u27e8t\left(\stackrel{\u0304}{x},\mu \right),\eta \left(y,\stackrel{\u0304}{x}\right)\u27e9+\varphi \left(y,\stackrel{\u0304}{x}\right)\notin \mathsf{\text{int}}\phantom{\rule{1em}{0ex}}C\left(\stackrel{\u0304}{x}\right),\phantom{\rule{1em}{0ex}}\forall y\in K\left(\lambda \right).
(3.5)
By the condition (e) and (3.5), there exist \stackrel{\u0304}{x}\in K\left(\lambda \right) and \xi \in T\left(\stackrel{\u0304}{x},\mu \right) such that
\u27e8\xi ,\eta \left(y,\stackrel{\u0304}{x}\right)\u27e9+\varphi \left(y,\stackrel{\u0304}{x}\right)\notin \mathsf{\text{int}}\phantom{\rule{1em}{0ex}}C\left(\stackrel{\u0304}{x}\right),\phantom{\rule{1em}{0ex}}\forall y\in K\left(\lambda \right)
(3.6)
and so S(λ, μ) is nonempty for any (λ, μ) ∈ ∧ × ∨.
Fourth, we show that the solution mapping S(⋅,⋅) is Bu.s.c on ∧ × ∨. Suppose that there exist (λ_{0}, μ_{0}) ∈ ∧ × ∨ such that S(⋅,⋅) is not Bu.s.c at (λ_{0}, μ_{0}). Then there exist an open set V with S(λ_{0}, μ_{0}) ⊂ V, a net {(λ_{α}, μ_{α})} and x_{α} ∈ S (λ_{α}, μ_{α}) such that (λ_{α}, μ_{α}) → (λ_{0}, μ_{0}) and x_{α} ∉ V for all α. Since x_{α} ∈ S(λ_{α}, μ_{α}), it follows that x_{α} ∈ K(λ_{α}). By the condition (g), K(⋅) is Bu.s.c with compactvalues at λ_{0}. Then there exists x_{0} ∈ K(λ_{0}) such that x_{α} → x_{0} (here we may take a subnet {x_{
β
}} of {x_{
α
}} if necessary). Suppose that x_{0} ∉ S(λ_{0}, μ_{0}), that is, for any \stackrel{\u0304}{\xi}\in T\left({x}_{0},{\mu}_{0}\right), there exists \u0233\in K\left({\lambda}_{0}\right) such that
\u27e8\stackrel{\u0304}{\xi},\eta \left(\u0233,{x}_{0}\right)\u27e9+\varphi \left(\u0233,{x}_{0}\right)\in \mathsf{\text{int}}\phantom{\rule{1em}{0ex}}C\left({x}_{0}\right).
(3.7)
Since x_{α} ∈ S (λ_{
α
}, μ_{
α
}), there exist {\xi}_{\alpha}^{\prime}\in T\left({x}_{\alpha},{\mu}_{\alpha}\right) such that
\u27e8{\xi}_{\alpha}^{\prime},\eta \left({z}_{\alpha},{x}_{\alpha}\right)\u27e9+\varphi \left({z}_{\alpha},{x}_{\alpha}\right)\notin \mathsf{\text{int}}\phantom{\rule{1em}{0ex}}C\left({x}_{\alpha}\right),\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{1em}{0ex}}\forall {z}_{\alpha}\in K\left({\lambda}_{\alpha}\right).
Since K is Bl.s.c at λ_{0}, it follows that, for any net {λ_{
α
}} ⊆ ∧ with λ_{
α
}→ λ_{0} and z_{0} ∈ K(λ_{0}), there exists z_{
α
}∈ K(λ_{
α
}) such that z_{
α
}→ z_{0}. Again, from the condition (d), T is Bu.s.c with compactvalues at (x_{0}, μ_{0}) and, for any net {(x_{
α
}, μ_{
α
})} ⊆ X × ∨ with (x_{
α
}, μ_{
α
}) → (x_{0}, μ_{0}), there exists ξ_{0} ∈ T(z_{0}, μ_{0}) such that {\xi}_{\alpha}^{\prime}\to {\xi}_{0}.
Therefore, from (b), (c) and (f), we have
\left({x}_{\alpha},\u27e8{\xi}_{\alpha}^{\prime},\eta \left({z}_{\alpha},{x}_{\alpha}\right)\u27e9+\varphi \left({z}_{\alpha},{x}_{\alpha}\right)\right)\to \left({x}_{0},\u27e8{\xi}_{0},\eta \left({z}_{0},{x}_{0}\right)\u27e9+\varphi \left({z}_{0},{x}_{0}\right)\right)\in \mathsf{\text{Gr}}\left(W\right).
Furthermore, we have
\u27e8{\xi}_{0},\eta \left({z}_{0},{x}_{0}\right)\u27e9+\varphi \left({z}_{0},{x}_{0}\right)\notin \mathsf{\text{int}}\phantom{\rule{1em}{0ex}}C\left({x}_{0}\right),\phantom{\rule{1em}{0ex}}\forall {z}_{0}\in K\left({\lambda}_{0}\right),
which contradicts (3.7). So, x_{0} ∈ S (λ_{0}, μ_{0}) ⊂ V, which is a contradiction. Since x_{
α
}∉ V for all α, it follows that x_{
α
}→ x_{0} and V is open. Consequently, the solution mapping S(⋅,⋅) is Bu.s.c at any (λ_{0}, μ_{0}) ∈ ∧ × ∨.
Finally, we show that S(⋅,⋅) is closed at any (λ_{0}, μ_{0}) ∈ ∧ × ∨. Taking x_{
α
}∈ S(λ_{
α
}, μ_{
α
}) with (λ_{
α
}, μ_{
α
}) → (λ_{0}, μ_{0}) and x_{
α
}→ x_{0}. Then x_{
α
}∈ K(λ_{
α
}). By (g), x_{0} ∈ K(λ_{0}). By the same proof as above, we have x_{0} ∈ S (λ_{0}, μ_{0}), which implies that the solution mapping S(⋅,⋅) is closed on ⋀ × M. This completes the proof.
Remark 3.1. From Lemma 2.4, we know that, if all the conditions of Theorem 3.1 are satisfied, then the solution mapping S(⋅,⋅) is Hu.s.c on ∧ × ∨.
From Theorem 3.1, we can conclude the following:
Corollary 3.2. Let (λ_{0}, μ_{0}) ∈ ∧ × ∨ be a point, K(λ_{0}) be a compact set, T : X × ∨ → 2^{L(X,Y)}be a setvalued mapping with nonempty values, C : X → 2^{Y}be a setvalued mapping such that, for each x ∈ X, C(x) is a pointed closed and convex cone in Y and \mathsf{\text{int}}\phantom{\rule{1em}{0ex}}C\left(x\right)\ne \varnothing ,\eta :X\times X\to X and ϕ :X × X → Y be two vectorvalued mappings. Assume that the conditions (a)(c) and (f) in Theorem 3.1 and the following conditions are satisfied:
(d)' T(x, μ) is weakly (η, ϕ, C(x))pseudomapping with respect to the first argument and Bu.s.c with compactvalues on X × {μ_{0}};
(e)' there is a continuous selection t of T on X × {μ_{0}}.
Then the following hold:

(1)
The solution mapping S(⋅,⋅) is nonempty and weakly compact at (λ_{0}, μ_{0});

(2)
The solution mapping S(⋅,⋅) is Bu.s.c at (λ_{0}, μ_{0}).
If the setvalued mapping K : ∧ → 2^{X}is unboundedvalues, then we have the following:
Theorem 3.3. Let T : X × ∨ → 2^{L(X,Y)}be a setvalued mapping with nonempty values, C : X → 2^{Y}be a setvalued mapping such that, for each x ∈ X, C(x) is a pointed closed and convex cone in Y and \mathsf{\text{int}}\phantom{\rule{1em}{0ex}}C\left(x\right)\ne \varnothing ,\eta :X\times X\to X and ϕ : X × X → Y be two vectorvalued mappings. Assume that conditions (a)(f) in Theorem 3.1 and the following conditions are satisfied:
(g)' K : ∧ → 2^{X}is Bu.s.c and Bl.s.c with closed and convexvalues;

(h)
for any (λ,μ) ∈ ∧ × ∨, there exists a weakly compact subset Δ(λ) of X and z_{0} ∈ Δ(λ) ⋂ K(λ) such that
\u27e8t\left({z}_{0},\mu \right),\eta \left({z}_{0},x\right)\u27e9+\varphi \left({z}_{0},x\right)\in \mathsf{\text{int}}\phantom{\rule{1em}{0ex}}C\left(x\right),\phantom{\rule{1em}{0ex}}\forall x\in K\left(\lambda \right)\backslash \Delta \left(\lambda \right).
Then the following hold:

(1)
The solution mapping S(⋅,⋅) is nonempty and closed on ∧ × ∨;

(2)
The solution mapping S(⋅,⋅) is Bu.s.c on ∧ × ∨.
Proof. By the proof of Theorem 3.1, we only need to prove that ϒ_{2}(z_{0}) is weakly compact. Since ϒ_{2}(z_{0}) ⊆ Δ(λ) and ϒ_{2}(z_{0}) is closed, it follows that ϒ_{2}(z_{0}) is weakly compact and S(λ, μ) ⊆ Δ(λ) for each (λ, μ) ∈ ∧ × ∨. This completes the proof.
Remark 3.2. In Theorems 3.1 and 3.3, if the condition (d) is replaced by the condition that T(x, μ) is (strictly) (η, ϕ, C(x))pseudomapping with respect to the first argument and Bu.s.c with compactvalues on X × ∨, then Theorems 3.1 and 3.3 still hold.
Inspired the results in Chen et al. [34], we introduce the following function by the nonlinear scalarization function ξ_{
e
}. Suppose that K(λ) is a compact set for any λ ∈ ∧, T(x, μ) is also a compact set for any (x, μ) ∈ X × ∨, η (x, x) = ϕ (x, x) = 0 for all x ∈ X, V(⋅) =: Y\ intC(⋅) and C(⋅) are Bu.s.c on X. We define a function g: X × ∧ × ∨ → R as follows:
g\left(x,\lambda ,\mu \right)=:\underset{\zeta \in T\left(x,\mu \right)}{\text{max}}\underset{y\in K\left(\lambda \right)}{\text{min}}{\xi}_{e}\left(x,\u27e8\zeta ,\eta \left(y,x\right)\u27e9+\varphi \left(y,x\right)\right),\phantom{\rule{1em}{0ex}}\forall x\in K\left(\lambda \right).
(3.8)
Since K(λ) and T(x, μ) are compact sets and ξ_{
e
}(⋅,⋅) is continuous, g(x, λ, μ) is welldefined. Forward, we use the function g(x, λ, μ) to discuss the continuity of the solution mapping of (PGVQVLIP).
First, we discuss the relations between g(⋅,⋅,⋅) and the solution mapping S(⋅,⋅).
Lemma 3.1. (1) g(x_{0}, λ_{0}, μ_{0}) = 0 if and only if x_{0} ∈ S (λ_{0}, μ_{0});

(2)
g(x, λ, μ) ≤ 0 for all x ∈ K(λ).
Proof. The proof is similar to the proof of Proposition 4.1 [34] and so the proof is omitted.
Remark 3.3. We say that the function g is a parametric gap function for (PGVQVLIP) if and only if (1) and (2) of Lemma 3.1 are satisfied. In fact, the gap functions are widely applied in optimization problems, equation problems, variational inequalities problems and others. The minimization of the gap function is an effectively approach for solving variational inequalities. Many authors have investigated the gap functions and applied to construct some algorithms for variational inequalities and equilibrium problems (see, for instance, [8, 14, 41]).
Lemma 3.2. Let K(λ) be nonempty compact for any λ ∈ ∧. Assume that the following conditions are satisfied:

(a)
T(⋅,⋅) is Bl.s.c with compactvalues on X × ∨;

(b)
C(⋅) is Bu.s.c on X, and e(⋅) ∈ intC(⋅) is continuous on X.
Then g(⋅,⋅,⋅) is a lower semicontinuous function.
Proof. The proof is similar to the proof of Lemma 4.2 [34] and so the proof is omitted.
If the conditions of Lemma 3.2 are strengthened, then we can get the continuity of g.
Lemma 3.3. Let K(λ) be nonempty compact for any λ ∈ ∧. Assume that the following conditions are satisfied:

(a)
T(⋅,⋅) is Bcontinuous with compactvalues on X × ∨;

(b)
C(⋅) and V(⋅) = Y\ intC(⋅) are Bu.s.c on X and e(⋅) ∈ intC(⋅) is continuous on X.
Then g(⋅,⋅,⋅) is continuous.
Proof. By Lemma 3.2, we only need to prove that g is upper semicontinuous. We can show that g is lower semicontinuous. The proof method of the lower semicontinuity of g is similar to that of the upper semicontinuity of g and so the proof is omitted.
Motivated by the hypothesis (H_{1}) of [32, 33], (Hg) of [21, 34] and (Hg)' of [35], by virtue of the parametric gap function g, we also introduce the following key assumption:
(Hg)" For any (λ_{0}, μ_{0}) ∈ ∧ × ∨ and ϵ > 0, there exist ϱ > 0 and δ > 0 such that, for any (λ, μ) ∈ B((λ_{0}, μ_{0}), δ) and x ∈ Δ(λ, μ, ϵ) = K(λ) \ U(S(λ, μ), ϵ),
g\left(\lambda ,\mu ,x\right)\le \varrho .
Remark 3.4. It is easy to see that, if ∧ and ∨ are the same spaces and ∧ is a metric space, C(x) ≡ C for all x ∈ X and μ = λ, then the hypothesis (Hg)" is reduced to the hypothesis (Hg)' of [35].
Remark 3.5. As pointed in [21, 32, 34, 35], the hypothesis (Hg)" can be explained by the geometric properties that, for any small positive number ϵ, one can take two small positive real number ϱ and δ such that, for all problems in the δneighborhood of a pair parameters (λ_{0}, μ_{0}), if a feasible point x is away from the solution set by a distance of at least ϵ, then a "gap" by an amount of at least ϱ will be generated. As mentioned out in [32], the above hypothesis (Hg)" is characterized by a common theme used in mathematical analysis. Such a theme interprets a proposition associated with a set in terms of other propositions related with the complement set. Instead of looking for restrictions within the solution set, the hypothesis (Hg)" puts restrictions on the behavior of the parametric gap function on the complement of solution set. As showed in [34], the hypothesis (Hg)" seems to be reasonable in establishing the Hausdorff continuity of S(⋅,⋅) because of the complexity of the problem structure.
Theorem 3.4. Assume that (Hg)" and all the conditions of Theorem 3.1 holds and the following conditions are satisfied:

(a)
T(⋅,⋅) is Bcontinuous mapping with compactvalues on X × ∨;

(b)
C(⋅) is Bu.s.c on X and e(⋅) ∈ intC(⋅) is continuous on X.
Then the following hold:

(1)
The solution mapping S(⋅,⋅) is nonempty and closed on ∧ × ∨;

(2)
The solution mapping S(⋅,⋅) is Hcontinuous on ∧ × ∨.
Proof. By Theorem 3.1 and Lemma 2.2, we know that the solution mapping S(⋅,⋅) is nonempty closed and Hu.s.c on ∧ × ∨.
Now, we only need to prove that the solution mapping S(⋅,⋅) is Hl.s.c on ∧ × ∨. Suppose that there exists (λ_{0}, μ_{0}) ∈ ∧ × ∨ such that the solution mapping S is not Hl.s.c at (λ_{0}, μ_{0}). Then there exist a neighborhood V of 0_{
X
}, nets {(λ_{
α
}, μ_{
α
})} ⊂ ∧ × ∨ with (λ_{
α
}, μ_{
α
}) → (λ_{0}, μ_{0}) and {x_{
α
}} such that
{x}_{\alpha}\in S\left({\lambda}_{0},{\mu}_{0}\right)\backslash \left(S\left({\lambda}_{\alpha},{\mu}_{\alpha}\right)+V\right).
(3.9)
By Corollary 3.2, S(λ_{0}, μ_{0}) is a compact set. Without loss of generality, assume that x_{
α
}→ x_{0} ∈ S(λ_{0}, μ_{0}). For V and any ϵ > 0, there exists a balanced open neighborhood V(ϵ) of 0_{
X
}such that V (ϵ)+ V (ϵ)+ V (ϵ) ⊂ V. It is easy to see that, for all ϵ > 0,
\left({x}_{0}+V\left(\epsilon \right)\right)\cap K\left({\lambda}_{0}\right)\ne \varnothing .
Since K (⋅) is Bl.s.c at λ_{0}, there exists β_{1} such that
\left({x}_{0}+V\left(\epsilon \right)\right)\cap K\left({\lambda}_{\beta}\right)\ne \varnothing ,\phantom{\rule{1em}{0ex}}\forall \beta \ge {\beta}_{1}.
For any ϵ ∈ (0,1], assume that y_{
β
}∈ (x_{0} + V(ϵ))⋂K(λ_{
β
}). Then y_{
β
}→ x_{0}. We assert that y_{
β
}∉ S(λ_{
β
}, μ_{
β
})+V(ϵ). Suppose that y_{
β
}∈ S(λ_{
β
}, μ_{
β
}) + V(ϵ). Then there exists z_{
β
}∈ S(λ_{
β
}, μ_{
β
}) such that y_{
β
} z_{
β
}∈ V(ϵ). Note that x_{
α
}→ x_{0} ∈ S(λ_{0}, μ_{0}). Without loss of generality, we may assume that x_{
β
} x_{0} ∈ V(ϵ). Therefore, one has
{x}_{\beta}{z}_{\beta}=\left({x}_{\beta}{x}_{0}\right)+\left({x}_{0}{y}_{\beta}\right)+\left({y}_{\beta}{z}_{\beta}\right)\in V\left(\epsilon \right)+V\left(\epsilon \right)+V\left(\epsilon \right)\subset V.
This yields that x_{
β
}∈ S(λ_{
β
}, μ_{
β
}) + V, which contradicts (3.9). Thus y_{
β
}∉ S(λ_{
β
}, μ_{
β
}) + V(ϵ). In the light of (Hg)", there exist two positive real numbers ϱ > 0 and δ > 0 such that, for any (λ_{
β
}, μ_{
β
}) ∈ B((λ_{0}, μ_{0}), δ) and y_{
β
}∉ S(λ_{
β
}, μ_{
β
}) + V(ϵ),
g\left({y}_{\beta},{\lambda}_{\beta},{\mu}_{\beta}\right)\le \varrho .
(3.10)
By Lemma 3.2, g is lower semicontinuous. So, it follows that, for any real number σ > 0,
g\left({y}_{\beta},{\lambda}_{\beta},{\mu}_{\beta}\right)\ge g\left({x}_{0},{\lambda}_{0},{\mu}_{0}\right)\sigma .
(3.11)
Without loss of generality, assume that σ < ϱ. Then, from (3.10) and (3.11), it follows that
g\left({x}_{0},{\lambda}_{0},{\mu}_{0}\right)\le \sigma \varrho <0,
that is,
g\left({x}_{0},{\lambda}_{0},{\mu}_{0}\right)=\underset{\zeta \in T\left({x}_{0},{\mu}_{0}\right)}{\text{max}}\underset{y\in K\left({\lambda}_{0}\right)}{\text{min}}{\xi}_{e}\left({x}_{0},\u27e8\zeta ,\eta \left(y,{x}_{0}\right)\u27e9+\varphi \left(y,{x}_{0}\right)\right)<0.
Hence there exist {\u0177}_{0}\in K\left({\lambda}_{0}\right) and ζ_{0} ∈ T(x_{0}, μ_{0}) such that
{\xi}_{e}\left({x}_{0},\u27e8{\zeta}_{0},\eta \left({\u0177}_{0},{x}_{0}\right)\u27e9+\varphi \left({\u0177}_{0},{x}_{0}\right)\right)<0.
From Proposition 2.1, it follows that
\u27e8{\zeta}_{0},\eta \left({\u0177}_{0},{x}_{0}\right)\u27e9+\varphi \left({\u0177}_{0},{x}_{0}\right)\in \mathsf{\text{int}}\phantom{\rule{1em}{0ex}}C\left({y}_{0}\right),
which contradicts x_{0} ∈ S(λ_{0}, μ_{0}). Therefore, the solution mapping S is Hl.s.c on ∧ × ∨. This completes the proof.
Now, we give two examples to validate Theorems 3.1 and 3.4.
Example 3.1. Let ∧ = ∨ = (1,1),X = R,Y = R^{2} and let C\left(x\right)={R}_{+}^{2} and e\left(x\right)={\left(1,1\right)}^{T}\in \mathsf{\text{int}}\phantom{\rule{1em}{0ex}}{R}_{+}^{2} for all x ∈ X. Define the setvalued mappings K : ∧ → 2^{X}and T : X × ∨ → 2^{Y}as follows: for any x ∈ X, μ ∈ ∨ and λ ∈ ∧,
K\left(\lambda \right)=:\left[\frac{\lambda}{2},\left\lambda \right\right],\phantom{\rule{1em}{0ex}}T\left(x,\mu \right)=:\left\{{\left(0,\ell \right)}^{T}:1\le \ell \le +{\mu}^{2}\right\}.
It is easy to see that the conditions (a)(g) of Theorem 3.1 and the conditions (a) and (b) of Theorem 3.4 are satisfied. From simple computation, we get S\left(\lambda ,\mu \right)=K\left(\lambda \right)=\left[\frac{\lambda}{2},\left\lambda \right\right] for all (λ, μ) ∈ ∧ × ∨. Therefore, S(⋅,⋅) is Hcontinuous on ∧ × ∨.
The following example illustrate the assumption (Hg)" in Theorem 3.4 is essential.
Example 3.2. Let ∧ = ∨ = [0,1], X = R, Y = R^{2} and let η (y, x) = y  x, ϕ (y, x) = 0 and C\left(x\right)={R}_{+}^{2} for all x, y ∈ X. Define the setvalued mappings K : ∧ → 2^{X}and T : X × ∨ → 2^{Y}by
K\left(\lambda \right)=:\left[1,1\right],\phantom{\rule{1em}{0ex}}T\left(x,\lambda \right)=:\left\{{\left(4,{x}^{2}+\lambda \right)}^{T}\right\},\phantom{\rule{1em}{0ex}}\forall x\in X,\lambda \in \wedge .
It is easy to see that the conditions (a)(g) of Theorem 3.1 and the conditions (a) and (b) of Theorem 3.4 are satisfied. From simple computation, one has
S\left(\lambda \right)=\left\{\begin{array}{cc}\left\{1,0\right\},\hfill & \mathsf{\text{if}}\phantom{\rule{2.77695pt}{0ex}}\lambda =0,\hfill \\ \left\{1\right\},\hfill & \mathsf{\text{otherwise}}\mathsf{\text{.}}\hfill \end{array}\right.
Therefore, S(⋅,⋅) is not Hcontinuous at λ = 0. Let us show that the assumption (Hg)" is not satisfied at 0. Put e\left(x\right)={\left(1,1\right)}^{T}\in \mathsf{\text{int}}\phantom{\rule{1em}{0ex}}{R}_{+}^{2}. Then, from Example 2.1,
\begin{array}{ll}\hfill g\left(x,\lambda \right)& =\underset{\zeta \in T\left(x,\lambda \right)}{\text{max}}\underset{y\in K\left(\lambda \right)}{\text{min}}{\xi}_{e}\left(x,\u27e8\zeta ,\eta \left(y,x\right)\u27e9+\varphi \left(y,x\right)\right)\phantom{\rule{2em}{0ex}}\\ =\underset{y\in K\left(\lambda \right)}{\text{min}}\text{max}\left\{4\left(yx\right),\left({x}^{2}+\lambda \right)\left(yx\right)\right\}\phantom{\rule{2em}{0ex}}\\ =\left({x}^{2}+\lambda \right)\left(1x\right).\phantom{\rule{2em}{0ex}}\end{array}
It is easy to see that g is a parametric gap function for (PGVQVLIP). Take ϵ ∈ (0,1) and, for any ϱ > 0, set λ_{
n
}→ 0 with 0 < λ_{
n
}< ϱ and x_{
n
}= 0 ∈ Δ(λ_{
n
}, ϵ) = K(λ_{
n
}) \ U(S(λ_{
n
}),ϵ) for all n ≥ 1. Then we have
g\left({x}_{n},{\lambda}_{n}\right)={\lambda}_{n}>\varrho .
Hence the assumption (Hg)" fails to hold at 0.
From Lemma 2.4, Remark 3.1 and Theorems 3.1 and 3.4, we can get the following:
Corollary 3.5. Assume that all the conditions of Theorem 3.4 are satisfied. Then the solution mapping S(⋅,⋅) is Bcontinuous.