Well-posedness for bilevel vector equilibrium problems with variable domination structures

Definition 2.1

[38] Let X and Y be two topological spaces. A set-valued mapping T : X → 2 Y is said to be

1. upper semi-continuous (for short, u.s.c.) at x 0 ∈ X if, for each open set V ⊆ Y with T ( x 0 ) ⊆ V , there exists a neighborhood U of x 0 such that T ( x ) ⊆ V for all x ∈ U ;

2. lower semi-continuous (for short, l.s.c.) at x 0 ∈ X if, for each open set V ⊆ Y with T ( x 0 ) ∩ V ≠ ∅ , there exists a neighborhood U of x 0 such that T ( x ) ∩ V ≠ ∅ for all x ∈ U ;

3. Hausdorff upper semicontinuous (for short, H -u.s.c.) at x 0 ∈ X if, for each neighborhood V 0 of the origin in Y , there exists a neighborhood U of x 0 such that T ( x ) ⊆ T ( x 0 ) + V 0 for all x ∈ U ;

4. Hausdorff lower semicontinuous (for short, H -l.s.c.) at x 0 ∈ X if, for each neighborhood V 0 of the origin in Y , there exists a neighborhood U of x 0 such that T ( x 0 ) ⊆ T ( x ) + V 0 for all x ∈ U ;

5. u.s.c. (resp. l.s.c., H -u.s.c., H -l.s.c.) on X if it is u.s.c. (resp. l.s.c., H -u.s.c., H -l.s.c.) at every point x ∈ X ;

6. continuous (resp. Hausdorff continuous) on X if it is both u.s.c. and l.s.c. (resp. H -u.s.c. and H -l.s.c.) on X ;

7. closed if the graph of T is closed, i.e., the set graph ( T ) = { ( x , y ) ∈ X × Y : y ∈ T ( x ) } is closed in X × Y .

Lemma 2.1

[38] Let X and Y be two topological spaces and T : X → 2 Y be a set-valued mapping.

1. If Y is Hausdorff, T is u.s.c. and compact-valued, then T is closed.

2. If both X and Y are Hausdorff and Y is compact and the mapping T has nonempty closed values, then T is u.s.c. on X if and only if it is closed.

3. For any given x 0 ∈ X , if T ( x 0 ) is compact, then T is u.s.c. at x 0 if and only if, for any net { x α } ⊆ X with x α → x 0 and for any net { y α } with y α ∈ T ( x α ) , there exists a subset { y β } of { y α } such that y β → y 0 for some y 0 ∈ T ( x 0 ) .

4. T is l.s.c. at x 0 ∈ X if and only if, for any net { x α } with x α → x 0 and for any y 0 ∈ T ( x 0 ) , there exists a net { y α } with y α ∈ T ( x α ) such that y α → y 0 .

Definition 2.2

[39] Let X , Y be Hausdorff topological spaces and Z be a real Hausdorff topological vector space. Let C : X → 2 Z be a cone-valued mapping. A vector-valued mapping f : X × Y → Z is said to be

1. C -continuous at ( x 0 , y 0 ) ∈ X × Y if, for any neighborhood V of the origin in Z , there exist neighborhoods U x 0 and U y 0 of x 0 and y 0 , respectively, such that

   f ( x , y ) ∈ f ( x 0 , y 0 ) + V + C ( x 0 ) , ∀ ( x , y ) ∈ U x 0 × U y 0 ;

2. C -continuous on X × Y if it is C -continuous at every point ( x , y ) ∈ X × Y .

Definition 2.3

[32] Let X be a topological space, Y be a real normed vector space, and cl ( B 1 ) be the closed unit ball in Y . A cone-valued mapping C : X → 2 Y is said to be

1. cosmically upper continuous (for short, c.u.c.) at x 0 ∈ X if the set-valued mapping x → C ( x ) ∩ cl ( B 1 ) is u.s.c. at x 0 ;

2. cosmically lower continuous (for short, c.l.c.) at x 0 ∈ X if the set-valued mapping x → C ( x ) ∩ cl ( B 1 ) is l.s.c. at x 0 ;

3. c.u.c. (c.l.c.) on X if it is c.u.c. (c.l.c.) at every point x ∈ X .

Remark 2.1

It is easy to see that if a cone-valued mapping C is c.u.c. (c.l.c.) at x 0 ∈ X , then the cone-valued mapping − C is also c.u.c. (c.l.c.) at x 0 ∈ X ; and vice versa.

Lemma 2.2

[32] Let X be a topological space and Y be a real normed vector space. Let C : A → 2 Z be a cone-valued mapping.

1. C is c.u.c. at x 0 ∈ X if and only if, for every bounded closed set M ⊆ Y , the set-valued mapping x → C ( x ) ∩ M is u.s.c. at x 0 ;

2. C is c.l.c. at x 0 ∈ X if and only if it is l.s.c. at x 0 .

Definition 2.4

[40] Let Y be a real norm vector space and P be a closed, convex, and pointed cone of Y . A nonempty convex subset B ⊆ P is said to be a base of P if 0 ∉ cl ( B ) and P = cone ( B ) .

Lemma 2.3

[36] Let X be a nonempty closed convex subset of a Hausdorff topological vector space. Let Y be a real normed vector space. Let C : X → 2 Y be a cone-valued mapping such that, for each x ∈ X , C ( x ) ⊆ Y is a closed, convex and pointed cone. For any given x 0 ∈ X , if C ( x 0 ) has a bounded closed base and the mapping C is c.u.c. at x 0 , then, for any 0 < ε < l , there exists a neighborhood U of x 0 such that

Definition 2.5

[41] Let A and B be nonempty subsets of a metric space ( E , d ) . The Hausdorff distance ℋ ( ⋅ , ⋅ ) between A and B is defined by

where e ( A , B ) ≔ sup a ∈ A d ( a , B ) with d ( a , B ) = inf b ∈ B d ( a , b ) . Let { A n } be a sequence of nonempty subsets of E . We say that A n converges to A in the sense of Hausdorff metric if ℋ ( A n , A ) → 0 . It is easy to see that e ( A n , A ) → 0 if and only if d ( a n , A ) → 0 for each selection a n ∈ A n . For more details on this topic, we refer the readers to [41].

Definition 3.1

A sequence { x n ∗ } = { ( x n , λ n ) } ⊆ A × Λ is called an LP approximating solution sequence for (BEP) if there exists a sequence { ε n } ⊆ R + converging to 0 such that, for any n ∈ N ,

Definition 3.2

A sequence { x n ∗ } = { ( x n , λ n ) } ⊆ A × Λ is called an LP approximating solution sequence for (WBEP) if there exists a sequence { ε n } ⊆ R + converging to 0 such that, for any n ∈ N ,

Definition 3.3

The problem (BEP) (resp. (WBEP)) is said to be LP well-posed if

1. Ψ 1 (resp. Ψ 2 ) is singleton;

2. every LP approximating solution sequence { x n ∗ } for (BEP) (resp. (WBEP)) converges to its unique solution.

Definition 3.4

The problem (BEP) (resp. (WBEP)) is said to be LP well-posed in the generalized sense if

1. Ψ 1 (resp. Ψ 2 ) is nonempty;

2. every LP approximating solution sequence { x n ∗ } for (BEP) (resp. (WBEP)) has a subsequence converging to some point of Ψ 1 (resp. Ψ 2 ).

Remark 3.1

To deal with well-posedness for (BEP), Anh and Hung [25] also introduced a definition of approximating solution set, in which an approximating solution x ∗ = ( x , λ ) ∈ A × Λ need to satisfy not only all the requirements appeared in the aforementioned set Ψ ˜ 1 ( ε ) but also the following additional one x ∗ ∈ graph S 1 − 1 . However, if this additional requirement x ∗ ∈ graph S 1 − 1 is satisfied, then we can conclude from the definition of S 1 that x ∈ K 1 ( x , λ ) . It follows d ( x , K 1 ( x , λ ) ) = 0 ≤ ε for all ε ∈ R + . This indicates that the constraint d ( x , K 1 ( x , λ ) ) ≤ ε existing in the definition of approximating solution set holds trivially and so is redundant. Thus, in this article, we do not need this additional requirement x ∗ ∈ graph S 1 − 1 in the definitions of approximating solution sets for (BEP) and (WBEP).

Theorem 3.1

Let A and Λ be nonempty compact subsets of X and W, respectively. Assume that

1. C 1 and C 2 are c.u.c.;

2. K 1 is u.s.c. and compact-valued, K 2 is l.s.c.;

3. f is continuous on A × A × Λ ;

4. for each y ∗ ∈ graph S 1 − 1 , h ( ⋅ , y ∗ ) is continuous on B.

Proof

The proof is divided into two steps:

(I) Ψ ˜ 1 is u.s.c. and compact-valued at 0.

Suppose to the contrary that Ψ ˜ 1 is not u.s.c. at 0. Then, there exist an open subset V with Ψ ˜ 1 ( 0 ) = Ψ 1 ⊆ V and sequences { ε n } ⊆ R + and { x n ∗ } ⊆ A × Λ such that ε n → 0 ( n → ∞ ) and x n ∗ = ( x n , λ n ) ∈ Ψ ˜ 1 ( ε n ) ⧹ V for all n . Notice that, A and Λ are compact. Without loss of generality, we may assume that the sequence { x n ∗ } converges to some point x 0 ∗ = ( x 0 , λ 0 ) ∈ A × Λ . We assert that x 0 ∗ ∈ Ψ ˜ 1 ( 0 ) . In fact, for each n ∈ N , since K 1 is compact-valued, there must exist some u n ∈ K 1 ( x n , λ n ) such that d ( x n , u n ) = d ( x n , K 1 ( x n , λ n ) ) . In addition, as K 1 is u.s.c., by Lemma 2.1 (iii), we can assume that { u n } converges to some u 0 ∈ K 1 ( x 0 , λ 0 ) . It follows ( x n , u n , ε n ) → ( x 0 , u 0 , 0 ) and d ( x n , u n ) ≤ ε n . Then, by the continuity of d , we can conclude d ( x 0 , u 0 ) = 0 . So x 0 ∈ K 1 ( x 0 , λ 0 ) as the set K 1 ( x 0 , λ 0 ) is closed.

Next, we shall prove that f ( x 0 , y , λ 0 ) ∈ C 1 ( x 0 ) for all y ∈ K 2 ( x 0 , λ 0 ) . Indeed, for any y 0 ∈ K 2 ( x 0 , λ 0 ) , since K 2 is l.s.c., we conclude from Lemma 2.1 (iv) that there exists a sequence { y n } with y n ∈ K 2 ( x n , λ n ) such that y n → y 0 . It follows ( x n , y n , λ n , ε n ) → ( x 0 , y 0 , λ 0 , 0 ) . As x n ∗ ∈ Ψ ˜ 1 ( ε n ) , we have

In addition, by the continuity of f and e 1 , we know that there exists some n 0 such that

where cl ( B 1 ) is the closed unit ball of Z . Let M = f ( x 0 , y 0 , λ 0 ) + B Z . Then, M is a bounded closed subset of Z and

Since C 1 is c.u.c., we can further conclude that

In fact, suppose to the contrary that f ( x 0 , y 0 , λ 0 ) ∉ C 1 ( x 0 ) ∩ M . Then, by the closedness of the set C 1 ( x 0 ) ∩ M , there exsits an open neighborhood V 0 of the origin in Z such that

For the aforementioned open set V 0 , since f is continuous and C 1 is c.u.c., there exists n 1 ≥ n 0 such that, for every n ≥ n 1 ,

This, together with (3.4), yields

which contradicts (3.2). Hence, (3.3) is valid. It follows

Similarly, we can prove that h ( x 0 ∗ , y ∗ ) ∈ C 2 ( x 0 ∗ ) for all y ∗ ∈ graph S 1 − 1 .

Hence, x 0 ∗ ∈ Ψ 1 = Ψ ˜ 1 ( 0 ) ⊆ V , which is impossible as x n ∗ ∉ V for all n . Therefore, Ψ ˜ 1 is u.s.c. at 0.

Since K 1 is u.s.c. and compact-valued, we deduce from Lemma 2.1 (i) that K 1 is closed. Then, by proceeding the same arguments as used earlier, we can further prove that Ψ 1 is a closed subset of A × Λ . As both A and Λ are compact, the set Ψ 1 = Ψ ˜ 1 ( 0 ) is also compact.

(II) (BEP) is LP well-posed in the generalized sense if and only if Ψ 1 is nonempty.

In fact, if (BEP) is LP well-posed in the generalized sense, then Ψ 1 is obviously nonempty. Conversely, if Ψ 1 is nonempty, then, for each ε ∈ R + , Ψ ˜ 1 ( ε ) is nonempty as Ψ 1 = Ψ ˜ 1 ( 0 ) ⊆ Ψ ˜ 1 ( ε ) . Let { x n ∗ = ( x n , λ n ) } ⊆ A × Λ be an LP approximating solution sequence of (BEP). Then, there exists a sequence { ε n } ⊆ R + converging to 0 such that, for each n ∈ N ,

This means x n ∗ = ( x n , λ n ) ∈ Ψ ˜ 1 ( ε n ) for all n ∈ N . In addition, by (I), Ψ ˜ 1 is u.s.c. and compact-valued at 0. So, we can conclude from Lemma 2.1 (iii) that there is a subsequence { x n k ∗ } of { x n ∗ } converging to some point x 0 ∗ ∈ Ψ ˜ 1 ( 0 ) = Ψ 1 . Hence, (BEP) is LP well-posed in the generalized sense.□

Remark 3.2

From the proof of Theorem 3.1, we can see that the property of upper semi-continuity for the approximating solution mapping Ψ ˜ 1 is extremely important for proving LP well-posedness in the generalized sense for (BEP). Very recently, Hung and Hai [6] specially studied approximating solution mappings for bilevel vector equilibrium problems with fixed domination structure. They gave several sufficient conditions to ensure upper semi-continuity for their approximating solution mappings.

Remark 3.3

Theorem 3.1 extend Theorem 3.1 of Han and Gong [42] from vector equilibrium problems with fixed domination structure to bilevel vector equilibrium problem with variable domination structure.

Example 3.1

Let X = Y = R , Y = Z = R 2 , A = [ 0 , 2 ] , Λ = [ 0 , 1 ] , B = A × Λ = [ 0 , 2 ] × [ 0 , 1 ] . For each x , y ∈ A , λ ∈ Λ , let

For any x ∗ = ( x , λ 1 ) ∈ B , y ∗ = ( y , λ 2 ) ∈ B , let

Obviously, both X and Λ are nonempty and compact subsets. In addition, we can check easily that the assumptions (i)–(iv) of Theorem 3.1 are satisfied. Further, by simple calculation, we can derive that Ψ 1 = [ 0 , 1 ] × [ 0 , 1 ] , which is nonempty. Thus, all the conditions of Theorem 3.1 are satisfied. It follows from Theorem 3.1 that (BEP) is LP well-posed in the generalized sense. In fact, by computation, we have, for each λ ∈ Λ and ε ≥ 0 ,

From this, we can see that (BEP) is LP well-posed in the generalized sense.

Theorem 3.2

Let A and Λ be nonempty compact subsets of X and W, respectively. Assume that

1. C 1 and C 2 are c.u.c.;

2. K 1 is u.s.c. and compact-valued, K 2 is l.s.c.;

3. f is − C 1 -continuous;

4. for each y ∗ ∈ graph S 1 − 1 , h ( ⋅ , y ∗ ) is − C 2 -continuous;

5. for each x ∈ A and x ∗ ∈ B , C 1 ( x ) and C 2 ( x ∗ ) both have bounded closed bases.

Proof

The proof is similar to that of Theorem 3.1.

We can prove equation (3.5) by employing equation (3.1) and the assumptions given in this theorem. Then, we can complete the proof by proceeding the rest as that of Theorem 3.1.

In fact, suppose to the contrary that equation (3.5) is not true, i.e., f ( x 0 , y 0 , λ 0 ) ∉ C 1 ( x 0 ) . Then, by the closedness of C 1 ( x 0 ) , there exists a real number ε > 0 , such that

where B ε = { x ∈ Z : ‖ x ‖ < ε } is the open ball in Z centered at the origin 0 with radius ε . Observing that B ε is balanced, we have B ε = − B ε . Since e 1 is continuous at x 0 and f is − C 1 -continuous at ( x 0 , y 0 , λ 0 ) and the sequence { ( x n , y n , λ n , ε n ) } converges to ( x 0 , y 0 , λ 0 , 0 ) , there must exist N ∈ N + such that, for any n > N ,

This, together with (3.1), yields