First, we investigate relations between solution set of the WGVEP (SGVEP) and solution set of the DWGVEP (DSGVEP) when K is bounded.
Theorem 3.1 Let K ⊆ X be a nonempty and convex closed bounded set. If F : D × K × K → 2Ysatisfies the followings:
(i) F (u, x, x) ⊆ C, ∀x ∈ K, u ∈ T (x);
(ii) the set {(u, x), u ∈ T (x), x ∈ K : F (u, x, y) ⊈ -int C} is closed for any y ∈ K;
(iii) F is weakly C-pseudomonotone;
(iv) the set {y ∈ K | F(u, x, y) ⊈ int C} is closed and F (u, x, .) is C-convex for any x ∈ K, u ∈ T(x).
Then the WGVEP has a nonempty solution set and x* ∈ K is a solution of the WGVEP if and only if
Proof. Set Γ: D × K → 2Kby
We claim that Γ is a KKM map. Suppose on the contrary, it does not hold, then there exists a finite set {x1, . . ., x
n
} ⊆ K and z ∈ co{x1, . . ., x
n
} such that Thus, there exists v
i
∈ T(x
i
) such that F(v
i
, x
i
, z) ⊆ int C, ∀i = 1, . . ., n. It follows from the weak C-pseudomonotonity of F that
(3.1)
Taking into account that int C is convex, we obtain
where and For the above t
i
, due to the convexity of F(u, x,.), one has
which contradicts (3.1). By the condition (iv), we derive that the Γ is closed valued. Hence Γ is a KKM map. By the KKM Theorem, there exists x*∈ K such that x* ∈ ⋂v ∈ T(y), y ∈ KΓ(v, y). That is, F (v, y, x*) ⊈ int C,∀y ∈ K, v ∈ T(y).
Let us verify Take any x*∈ K, obviously
(3.2)
For every y ∈ K, consider x
t
= x* + t(y - x*), ∀t ∈ (0, 1). Clearly, x
t
∈ K. The C-convexity of F (u, x
t
,.) implies that
Let us show tF (u, x
t
, y) ⊈-int C by contradiction. Suppose on the contrary, then tF (u, x
t
, y) ⊆ -int C. For any p ∈ tF (u, x
t
, y), it holds
So F (u, x
t
, x*) ⊆ int C, which contradicts (3.2). Noting that -int C is convex cone, we deduce
(3.3)
Letting t → 0 in (3.3), we obtain by assumption (ii) and Lemma 2.4 that there exists u* ∈ T(x*) such that
On the other hand, by the weak C-pseudomonotonity of F, we have .
Hence, .
Theorem 3.2 Let K ⊆ X be a nonempty and convex closed bounded set. If F : D × K × K → 2Ysatisfies the followings:
(i) F (u, x, x) ⊆ C,∀x ∈ K, u ∈ T(x);
(ii) the set {(u, x), u ∈ T(x), x ∈ K | F (u, x, y) ∩ -int C = ∅} is closed for all y ∈ K;
(iii) F is strongly C-pseudomonotone;
(iv) the set {y ∈ K | F (u, x, y) ∩ int C = ∅} is closed and F (u, x, .) is C-convex for any x ∈ K, u ∈ T(x).
Then the SGVEP has a nonempty solution set and x* ∈ K is a solution of the SGVEP if and only if
Proof. Set Γ: D × K → 2Kby
Following the similar arguments in the proof of Theorem 3.1, we can obtain the desired result.
In following sequel, we shall present some sufficient conditions for the nonemptiness and boundedness of the solution set of the WGVEP provided that it is strictly feasible in the strong sense.
Theorem 3.3 Let K ⊆ X be a nonempty, closed, convex and well-positioned set. If F : D × K × K → 2Ysatisfies the followings:
(i) F (u, x, x) ⊆ C, ∀x ∈ K, u ∈ T(x);
(ii) the set {(u, x), u ∈ T (x), x ∈ K | F (u, x, y) ⊈-int C} is closed for any y ∈ K;
(iii) F is weakly C-pseudomonotone;
(iv) F (u, x, .) is C-convex and weakly lower semicontinuous for x ∈ K, u ∈ T(x).
Then the WGVEP has a nonempty bounded solution set whenever it is generalized strictly feasible in the strong sense.
Proof. Suppose that the WGVEP is generalized strictly feasible in the strong sense. Then there exists x0 ∈K such that x0 ∈ F
s
+, i.e.,
Set
By assumptions (i) and (iv), x0 ∈ D and D is weakly closed. We assert that D is bounded. Suppose on the contrary it does not holds, then there exists a sequence {x
n
} ⊆ M with ||x
n
|| → +∞ as n → +∞. Since X is a reflexive Banach space, without loss of generality,
we may take a subsequence of {x
n
} such that
By Lemma 2.3, z ≠ 0 since K is well-positioned. It follows from x0 ∈ F
s
+ that
(3.4)
Noting that F (u, x, .) is C-convex, we have
That is,
We claim that Suppose on the contrary, we observe
which contradicts Taking into account the condition (iv), we obtain
This is a contradiction to (3.4). Thus, D is bounded and it is weakly compact. For each p ∈ K, set
Then D
p
≠ ∅. Indeed, given p ∈ K, v ∈ T (p), set K0 = conv (D ⋃ p) ⊆ K, where conv means the convex hull of a set. Then K0 is nonempty, convex, and weakly compact. By Theorem 3.1, there exists such that
Then implies and implies We obtain D
p
≠ ∅. Obviously, D
p
is nonempty and weakly compact.
Next we prove that {D
p
| p ∈ K} has the finite intersection property. For any finite set {p
i
| i = 1, 2, . . ., n} ⊆ K, let K1 = conv{D ⋃ {p1, p2, . . ., p
n
}}. Then K1 is weakly compact. By Theorem 3.1, there exists such that
In particular, it holds
This means that Thus {D
p
| p ∈ K} has the finite intersection property. Since D is weakly compact and D
p
⊆ D is weakly closed for all p ∈ K, v ∈ T (p), It follows that
Let x* ∈ ⋂p ∈ KD
p
It follows that
By Theorem 3.1, x* is a solution of the WGVEP. As for the boundedness of the solution set of the WGVEP, it follows from Theorem 3.1 that the solution set of the WGVEP is a subset of D.
Theorem 3.4 Let K ⊆ X be a nonempty, closed, convex, and well-positioned set. If F : D × K × K → 2Ysatisfies the followings:
(i) F (u, x, x) ⊆ C, ∀x ∈ K, u ∈ T (x);
(ii) the set {(u, x), u ∈ T(x), x ∈ K | F(u, x, y) ⋂ - int C = ∅} is closed for all y ∈ K;
(iii) F is strongly C-pseudomonotone;
(iv) F (u, x, .) is C-convex and weakly lower semicontinuous for x ∈ K, u ∈ T(x);
(v) F is positively homogeneous with degree α > 0, i.e., there exists α > 0 such that
Then the SGVEP has a nonempty bounded solution set whenever it is generalized strictly feasible in the weak sense.
Proof. Suppose that the SGVEP is generalized strictly feasible in the weak sense. Then there exists x0 ∈ K such that , i.e.,
Set
By assumptions (i) and (iv), x0 ∈ D and D is weakly closed. We claim that D is bounded. Suppose on the contrary it does not holds, then there exists a sequence {x
n
} ⊆ M with ||x
n
|| → +∞ as n → +∞. Since X is a reflexive Banach space, without loss of generality, we may take a subsequence of {x
n
} such that
By Lemma 2.3, z ≠ 0 since K is well-positioned. It follows from x0 ∈ F
w
+ that
(3.5)
Since and F is positively homogenous with degree α > 0, it holds
Taking into account the condition (iv), we obtain
This is a contradiction to (3.5). Thus, D is bounded and it is weakly compact. Following the similar arguments in the proof of Theorem 3.3, we can prove the Theorem 3.4.
Remark 3.1 Assumption (v) of Theorem 3.4 is not new. Clearly, if F(x, y) = 〈u, y -x〉, ∀u ∈ T(x), then F is positively homogeneous with degree = 1.
Remark 3.2 Since SS
K
⊆ WS
K
, conditions for the solution set of the SGVEP to be nonempty and bounded are stronger than the WGVEP. Compared with Theorem 3.3, the condition that F is positively homogeneous in Theorem 3.4 is not dropped for the SGVEP.
The following example shows that the converse of Theorem 3.3 or 3.4 is not true in general.
Example 3.1 Let X = R, K = R, D = [0, 1], Y = R, and
Let F : D × K × K → 2Ybe defined by
It is easily to see that K is well-positioned and F satisfies assumptions of Theorems 3.3 and 3.4. It can be verified that the WGVEP and the SGVEP have the same solution set {0}. On the other hand, it is easy to verify that
For general generalized vector equilibrium problem, the following example shows WS
K
≠ ∅, but SS
K
= ∅.
Example 3.2 Let X = R, K = R, D = [-1, 1], Y = R, C = R+ and
It is obvious that the WGVEP has solution set WS
K
= R, but solution set of the SGVEP SS
K
= ∅.