In this section, we investigate the stability of solutions of (PGVQVLIP), that is, the upper and lower semi-continuity 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 × ∨ → 2L(X,Y)be a set-valued mapping with nonempty values, C : X → 2Ybe a set-valued mapping such that, for each x ∈ X, C(x) is a pointed closed and convex cone in Y and and ϕ : X × X → Y be two vector-valued 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))-pseudo-mapping with respect to the first argument and B-u.s.c with compact-values 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 : ∧ → 2Xis B-u.s.c and B-l.s.c with weakly compact and convex-values.
Then the following hold:
-
(1)
The solution mapping S(⋅,⋅) is nonempty and closed on ∧ × ∨;
-
(2)
The solution mapping S(⋅,⋅) is B-u.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))-pseudo-mapping with respect to the first argument on X × ∨, we know that t(x, μ) is also weakly (η, ϕ, C(x))-pseudo-mapping with respect to the first argument on X × ∨.
Now, we define two set-valued mappings ϒ1, ϒ2 : K(λ) → 2K(λ)as follows: for all y ∈ K(λ),
and
Since η(x, x) = 0 and ϕ (x, x) = 0 for all x ∈ X, we have y ∈ ϒ1(y) and y ∈ ϒ2(y) and so ϒ1(y) and ϒ2(y) are nonempty for any y ∈ K(λ). By virtue of the weakly (η, ϕ, C(x))-pseudo-mapping of t(x, μ) with respect to the first argument, we have
(3.1)
First, we assert that ϒ1 is a KKM mapping. Suppose that there exists a finite subset {y1, y2,...,y
m
} ⊆ K(λ) such that
Then there exists , i.e., for some nonnegative real number ι
i
with 1 ≤ i ≤ m and such that . Moreover, for 1 ≤ i ≤ m. This yields that
and so
Taking into account (b) and (c) that
(3.2)
Again, from (a) together with (3.2), we have , which is a contradiction. Hence ϒ1 is a KKM mapping. It follows from (3.1) that ϒ2 is also a KKM mapping.
Second, we show that . Taking any net {x
β
} of ϒ2(y) such that {x
β
} is weakly convergent to a point . Then, for each β, one has
From (b)-(e), it follows that
Consequently, we get
that is,
Therefore, and so ϒ2(y) is weakly closed set for any y ∈ K(λ). By the compactness of K(λ), ϒ2(y) is weakly compact subset of K(λ). From Lemma 2.1, it follows that
i.e., there exists such that
(3.3)
Third, we prove that . For any y ∈ K(λ), set for all r ∈ (0,1).
Then x
r
∈ K(λ). So, from (3.3), we have
(3.4)
Note that
It follows from (3.4) that
and so
Since t is continuous, we have
as r → 0. Therefore, by the weak closedness of Gr (W), we have
that is,
(3.5)
By the condition (e) and (3.5), there exist and such that
(3.6)
and so S(λ, μ) is nonempty for any (λ, μ) ∈ ∧ × ∨.
Fourth, we show that the solution mapping S(⋅,⋅) is B-u.s.c on ∧ × ∨. Suppose that there exist (λ0, μ0) ∈ ∧ × ∨ such that S(⋅,⋅) is not B-u.s.c at (λ0, μ0). Then there exist an open set V with S(λ0, μ0) ⊂ V, a net {(λα, μα)} and xα ∈ S (λα, μα) such that (λα, μα) → (λ0, μ0) and xα ∉ V for all α. Since xα ∈ S(λα, μα), it follows that xα ∈ K(λα). By the condition (g), K(⋅) is B-u.s.c with compact-values at λ0. Then there exists x0 ∈ K(λ0) such that xα → x0 (here we may take a subnet {x
β
} of {x
α
} if necessary). Suppose that x0 ∉ S(λ0, μ0), that is, for any , there exists such that
(3.7)
Since xα ∈ S (λ
α
, μ
α
), there exist such that
Since K is B-l.s.c at λ0, it follows that, for any net {λ
α
} ⊆ ∧ with λ
α
→ λ0 and z0 ∈ K(λ0), there exists z
α
∈ K(λ
α
) such that z
α
→ z0. Again, from the condition (d), T is B-u.s.c with compact-values at (x0, μ0) and, for any net {(x
α
, μ
α
)} ⊆ X × ∨ with (x
α
, μ
α
) → (x0, μ0), there exists ξ0 ∈ T(z0, μ0) such that .
Therefore, from (b), (c) and (f), we have
Furthermore, we have
which contradicts (3.7). So, x0 ∈ S (λ0, μ0) ⊂ V, which is a contradiction. Since x
α
∉ V for all α, it follows that x
α
→ x0 and V is open. Consequently, the solution mapping S(⋅,⋅) is B-u.s.c at any (λ0, μ0) ∈ ∧ × ∨.
Finally, we show that S(⋅,⋅) is closed at any (λ0, μ0) ∈ ∧ × ∨. Taking x
α
∈ S(λ
α
, μ
α
) with (λ
α
, μ
α
) → (λ0, μ0) and x
α
→ x0. Then x
α
∈ K(λ
α
). By (g), x0 ∈ K(λ0). By the same proof as above, we have x0 ∈ S (λ0, μ0), 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 H-u.s.c on ∧ × ∨.
From Theorem 3.1, we can conclude the following:
Corollary 3.2. Let (λ0, μ0) ∈ ∧ × ∨ be a point, K(λ0) be a compact set, T : X × ∨ → 2L(X,Y)be a set-valued mapping with nonempty values, C : X → 2Ybe a set-valued mapping such that, for each x ∈ X, C(x) is a pointed closed and convex cone in Y and and ϕ :X × X → Y be two vector-valued 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))-pseudo-mapping with respect to the first argument and B-u.s.c with compact-values on X × {μ0};
(e)' there is a continuous selection t of T on X × {μ0}.
Then the following hold:
-
(1)
The solution mapping S(⋅,⋅) is nonempty and weakly compact at (λ0, μ0);
-
(2)
The solution mapping S(⋅,⋅) is B-u.s.c at (λ0, μ0).
If the set-valued mapping K : ∧ → 2Xis unbounded-values, then we have the following:
Theorem 3.3. Let T : X × ∨ → 2L(X,Y)be a set-valued mapping with nonempty values, C : X → 2Ybe a set-valued mapping such that, for each x ∈ X, C(x) is a pointed closed and convex cone in Y and and ϕ : X × X → Y be two vector-valued mappings. Assume that conditions (a)-(f) in Theorem 3.1 and the following conditions are satisfied:
(g)' K : ∧ → 2Xis B-u.s.c and B-l.s.c with closed and convex-values;
-
(h)
for any (λ,μ) ∈ ∧ × ∨, there exists a weakly compact subset Δ(λ) of X and z0 ∈ Δ(λ) ⋂ K(λ) such that
Then the following hold:
-
(1)
The solution mapping S(⋅,⋅) is nonempty and closed on ∧ × ∨;
-
(2)
The solution mapping S(⋅,⋅) is B-u.s.c on ∧ × ∨.
Proof. By the proof of Theorem 3.1, we only need to prove that ϒ2(z0) is weakly compact. Since ϒ2(z0) ⊆ Δ(λ) and ϒ2(z0) is closed, it follows that ϒ2(z0) 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))-pseudo-mapping with respect to the first argument and B-u.s.c with compact-values 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 B-u.s.c on X. We define a function g: X × ∧ × ∨ → R as follows:
(3.8)
Since K(λ) and T(x, μ) are compact sets and ξ
e
(⋅,⋅) is continuous, g(x, λ, μ) is well-defined. 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(x0, λ0, μ0) = 0 if and only if x0 ∈ S (λ0, μ0);
-
(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 B-l.s.c with compact-values on X × ∨;
-
(b)
C(⋅) is B-u.s.c on X, and e(⋅) ∈ intC(⋅) is continuous on X.
Then g(⋅,⋅,⋅) is a lower semi-continuous 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 B-continuous with compact-values on X × ∨;
-
(b)
C(⋅) and V(⋅) = Y\ intC(⋅) are B-u.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 semi-continuous. We can show that -g is lower semi-continuous. The proof method of the lower semi-continuity of -g is similar to that of the upper semi-continuity of g and so the proof is omitted.
Motivated by the hypothesis (H1) 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 (λ0, μ0) ∈ ∧ × ∨ and ϵ > 0, there exist ϱ > 0 and δ > 0 such that, for any (λ, μ) ∈ B((λ0, μ0), δ) and x ∈ Δ(λ, μ, ϵ) = K(λ) \ U(S(λ, μ), ϵ),
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 δ-neighborhood of a pair parameters (λ0, μ0), 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 B-continuous mapping with compact-values on X × ∨;
-
(b)
C(⋅) is B-u.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 H-continuous on ∧ × ∨.
Proof. By Theorem 3.1 and Lemma 2.2, we know that the solution mapping S(⋅,⋅) is nonempty closed and H-u.s.c on ∧ × ∨.
Now, we only need to prove that the solution mapping S(⋅,⋅) is H-l.s.c on ∧ × ∨. Suppose that there exists (λ0, μ0) ∈ ∧ × ∨ such that the solution mapping S is not H-l.s.c at (λ0, μ0). Then there exist a neighborhood V of 0
X
, nets {(λ
α
, μ
α
)} ⊂ ∧ × ∨ with (λ
α
, μ
α
) → (λ0, μ0) and {x
α
} such that
(3.9)
By Corollary 3.2, S(λ0, μ0) is a compact set. Without loss of generality, assume that x
α
→ x0 ∈ S(λ0, μ0). 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,
Since K (⋅) is B-l.s.c at λ0, there exists β1 such that
For any ϵ ∈ (0,1], assume that y
β
∈ (x0 + V(ϵ))⋂K(λ
β
). Then y
β
→ x0. 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
→ x0 ∈ S(λ0, μ0). Without loss of generality, we may assume that x
β
- x0 ∈ V(ϵ). Therefore, one has
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((λ0, μ0), δ) and y
β
∉ S(λ
β
, μ
β
) + V(ϵ),
(3.10)
By Lemma 3.2, g is lower semi-continuous. So, it follows that, for any real number σ > 0,
(3.11)
Without loss of generality, assume that σ < ϱ. Then, from (3.10) and (3.11), it follows that
that is,
Hence there exist and ζ0 ∈ T(x0, μ0) such that
From Proposition 2.1, it follows that
which contradicts x0 ∈ S(λ0, μ0). Therefore, the solution mapping S is H-l.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 = R2 and let and for all x ∈ X. Define the set-valued mappings K : ∧ → 2Xand T : X × ∨ → 2Yas follows: for any x ∈ X, μ ∈ ∨ and λ ∈ ∧,
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 for all (λ, μ) ∈ ∧ × ∨. Therefore, S(⋅,⋅) is H-continuous on ∧ × ∨.
The following example illustrate the assumption (Hg)" in Theorem 3.4 is essential.
Example 3.2. Let ∧ = ∨ = [0,1], X = R, Y = R2 and let η (y, x) = y - x, ϕ (y, x) = 0 and for all x, y ∈ X. Define the set-valued mappings K : ∧ → 2Xand T : X × ∨ → 2Yby
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
Therefore, S(⋅,⋅) is not H-continuous at λ = 0. Let us show that the assumption (Hg)" is not satisfied at 0. Put . Then, from Example 2.1,
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
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 B-continuous.