Lower semicontinuity of approximate solution mappings for parametric generalized vector equilibrium problems

The vector equilibrium problem is a unified model of several problems, for example, the vector optimization problem, the vector variational inequality problem, the vector complementarity problem and the vector saddle point problem. In the literature, existence results for various types of vector equilibrium problems have been investigated intensively, e.g., see [1‐4] and the references therein. The stability analysis of the solution mappings for VEP is an important topic in vector equilibrium theory. Recently, the semicontinuity, especially the lower semicontinuity, of solution mappings to parametric vector equilibrium problems has been studied in the literature, see [5‐16]. In the mentioned results, the lower semicontinuity of solution mappings to parametric generalized strong vector equilibrium problems is established under the assumptions of monotonicity and compactness. Very recently, Han and Gong [17] studied the lower semicontinuity of solution mappings to parametric generalized strong vector equilibrium problems without the assumptions of monotonicity and compactness.

On the other hand, exact solutions of the problems may not exist in many practical problems because the data of the problems are not sufficiently ‘regular’. Moreover, these mathematical models are solved usually by numerical methods which produce approximations to the exact solutions. So it is impossible to obtain an exact solution of many practical problems. Naturally, investigating approximate solutions of parametric equilibrium problems is of interest in both practical applications and computations. Anh and Khanh [18] considered two kinds of approximate solution mappings to parametric generalized vector quasiequilibrium problems and established the sufficient conditions for their Hausdorff semicontinuity (or Berge semicontinuity). Among many approaches for dealing with the lower semicontinuity and continuity of solution mappings for parametric vector variational inequalities and parametric vector equilibrium problems, the scalarization method is of considerable interest. By using a scalarization method, Li and Li [19] discussed the Berge lower semicontinuity and Berge continuity of an approximate solution mapping for a parametric vector equilibrium problem.

Motivated by the work reported in [17‐19], in this paper we aim to establish efficient conditions for the lower semicontinuity of an approximate solution mapping for a parametric generalized vector equilibrium problem involving set-valued mappings. By using a scalarization method, we obtain the lower semicontinuity of an approximate solution mapping for such a problem without the assumptions of monotonicity and compactness.

Throughout this paper, let X and Y be real Hausdorff topological vector spaces, and let Z be a real topological space. We also assume that C is a pointed closed convex cone in Y with its interior int C ≠ ∅. Let Y ∗ be the topological dual space of Y. Let C ∗ : = { ξ ∈ Y ∗ : 〈 ξ , y 〉 ≥ 0 , ∀ y ∈ C } be the dual cone of C, where 〈 ξ , y 〉 denotes the value of ξ at y. Since int C ≠ ∅, the dual cone C ∗ of C has a weak^∗ compact base. Let e ∈ int C. Then B e ∗ : = { ξ ∈ C ∗ : 〈 ξ , e 〉 = 1 } is a weak^∗ compact base of C ∗.

Suppose that K is a nonempty subset of X and F : K × K → 2 Y ∖ { ∅ } is a set-valued mapping. We consider the following generalized vector equilibrium problem (GVEP) of finding x 0 ∈ K such that

When the set K and the mapping F are perturbed by a parameter μ which varies over a set M of Z, we consider the following parametric generalized vector equilibrium problem (PGVEP) of finding x 0 ∈ K ( μ ) such that

where K : M → 2 X ∖ { ∅ } is a set-valued mapping, F : B × B × M ⊂ X × X × Z → 2 Y ∖ { ∅ } is a set-valued mapping with K ( M ) = ⋃ μ ∈ M K ( μ ) ⊂ B. For each ε > 0 and μ ∈ M, the approximate solution set of (PGVEP) is defined by

where e ∈ int C. For each ξ ∈ B e ∗ and ( ε , μ ) ∈ R + × M, by S ˜ ξ ( ε , μ ) we denote the ξ-approximate solution set of (PGVEP), i.e.,

Definition 2.1 Let D be a nonempty convex subset of X. A set-valued mapping G : X → 2 Y is said to be:

Definition 2.2 [17]

Let M and M 1 be topological vector spaces. Let D be a nonempty subset of M. A set-valued mapping G : M → 2 M 1 is said to be uniformly continuous on D if, for any neighborhood V of 0 ∈ M 1, there exists a neighborhood U 0 of 0 ∈ M such that G ( x 1 ) ⊆ G ( x 2 ) + V for any x 1 , x 2 ∈ D with x 1 − x 2 ∈ U 0.

Definition 2.3 [20]

Let M and M 1 be topological vector spaces. A set-valued mapping G : M → 2 M 1 is said to be:

The following lemma plays an important role in the proof of the lower semicontinuity of the solution mapping S ˜ ( ⋅ , ⋅ ).

Lemma 2.4 [[21], Theorem 2]

The union Γ = ⋃ i ∈ I Γ i of a family of l.s.c. set-valued mappings Γ i from a topological space X into a topological space Y is also an l.s.c. set-valued mapping from X into Y, where I is an index set.

In this section, we establish the lower semicontinuity of the approximate solution mapping for (PGVEP) at the considered point ( ε 0 , μ 0 ) ∈ R + × M with ε 0 > 0.

Firstly, using the same argument as in the proof given in [[22], Lemma 3.1], we can prove the following useful result.

Lemma 3.1 For each ε > 0, μ ∈ M, if for each x ∈ K ( μ ), F ( x , K ( μ ) , μ ) + C is a convex set, then

Proof For any x ∈ ⋃ ξ ∈ C ∗ ∖ { 0 } S ˜ ξ ( ε , μ ), there exists ξ ′ ∈ C ∗ ∖ { 0 } such that x ∈ S ˜ ξ ′ ( ε , μ ). Thus, we can obtain that x ∈ K ( μ ) and inf z ∈ F ( x , y , μ ) ξ ′ ( z ) + ε ≥ 0, ∀ y ∈ K ( μ ). Then, for each y ∈ K ( μ ) and z ∈ F ( x , y , μ ), ξ ′ ( z ) + ε ≥ 0, which arrives at z ∉ − int C. It then follows that, for each z ∈ F ( x , y , μ ),

which gives that x ∈ S ˜ ( ε , μ ). Hence, ⋃ ξ ∈ C ∗ ∖ { 0 } S ˜ ξ ( ε , μ ) ⊆ S ˜ ( ε , μ ). Conversely, let x ∈ S ˜ ( ε , μ ) be arbitrary. Then x ∈ K ( μ ) and F ( x , y , μ ) + ε e ⊆ Y ∖ − int C, ∀ y ∈ K ( μ ). Thus, we have

and hence

Because F ( x , K ( μ ) , μ ) + C is a convex set, by the well-known Edidelheit separation theorem (see [23], Theorem 3.16), there exist a continuous linear functional ξ ∈ Y ∗ ∖ { 0 } and a real number γ such that

for all z ∈ F ( x , K ( μ ) , μ ), c ∈ C and c ˆ ∈ − int C. Since C is a cone, we have ξ ( c ˆ ) ≤ 0 for all c ˆ ∈ − int C. Thus, ξ ( c ˆ ) ≥ 0 for all c ˆ ∈ C, that is, ξ ∈ C ∗. Moreover, it follows from c ∈ C, c ˆ ∈ − int C and the continuity of ξ that ξ ( z ) + ε ≥ 0 for all z ∈ F ( x , K ( μ ) , μ ). Thus, for all y ∈ K ( μ ), we have inf z ∈ F ( x , y , μ ) ξ ( z ) + ε ≥ 0, i.e., x ∈ S ˜ ξ ( ε , μ ) ⊆ ⋃ ξ ∈ C ∗ ∖ { 0 } S ˜ ξ ( ε , μ ). □

Theorem 3.2 We assume that for any given ξ ∈ B e ∗, there exists δ > 0 such that the ξ-approximate solution set S ˜ ξ ( ⋅ , ⋅ ) exists in [ ε 0 , δ ) × N ( μ 0 ), where N ( μ 0 ) is a neighborhood of μ 0. Assume further that the following conditions are satisfied:

Then the ξ-approximate solution mapping S ˜ ξ : [ ε 0 , δ ) × N ( μ 0 ) → 2 X is l.s.c. at ( ε 0 , μ 0 ).

Proof Suppose to the contrary that S ˜ ξ ( ⋅ , ⋅ ) is not l.s.c. at ( ε 0 , μ 0 ), then there exist x 0 ∈ S ˜ ξ ( ε 0 , μ 0 ) and a neighborhood W 0 of 0 X ∈ X. For any neighborhoods J ( ε 0 ) and U ( μ 0 ) of ε 0 and μ 0, respectively, there exist ε ′ ∈ J ( ε 0 ) ∩ [ ε 0 , δ ) and μ ′ ∈ U ( μ 0 ) such that ( x 0 + W 0 ) ∩ S ˜ ξ ( ε ′ , μ ′ ) = ∅. In particular, there exist sequences { ε n } ↓ ε 0 and { μ n } → μ 0 such that

For the above W 0, there exists a neighborhood W 1 of 0 X ∈ X such that

We define a ξ-set-valued mapping H ξ : [ 0 , δ ) → 2 X by

Notice that H ξ ( 0 ) = S ˜ ξ ( ε 0 , μ 0 ) ≠ ∅. Next, we claim that H ξ is l.s.c. at 0. Suppose to the contrary that H ξ is not l.s.c. at 0, then there exist x ¯ ∈ H ξ ( 0 ) and a neighborhood O 0 of 0 X ∈ X. For any neighborhood U of 0, there exists ε ∈ U such that ( x ¯ + O 0 ) ∩ H ξ ( ε ) = ∅. In particular, there exists a nonnegative sequence { ε n ′ } ↓ 0 such that

Since H ξ ( 0 ) ≠ ∅, we choose x ∗ ∈ H ξ ( 0 ). Since ε n ′ → 0, there exists ε n 0 ′ such that

We claim that ε 0 ε 0 + ε n 0 ′ x ¯ + ε n 0 ′ ε 0 + ε n 0 ′ x ∗ ∈ H ξ ( ε n 0 ′ ). In fact, since x ¯ ∈ H ξ ( 0 ) and x ∗ ∈ H ξ ( 0 ), for any y ∈ K ( μ 0 ), we have inf t ∈ F ( x ¯ , y , μ 0 ) ξ ( t ) + ε 0 ≥ 0 and inf k ∈ F ( x ∗ , y , μ 0 ) ξ ( k ) + ε 0 ≥ 0. Then, for any u ∈ F ( x ¯ , y , μ 0 ),

and for any v ∈ F ( x ∗ , y , μ 0 ),

By the C-concavity of F ( ⋅ , y , μ 0 ), we have that

It follows that, for any w ∈ F ( ε 0 ε 0 + ε n 0 ′ x ¯ + ε n 0 ′ ε 0 + ε n 0 ′ x ∗ , y , μ 0 ), there exist z ¯ ∈ F ( x ¯ , y , μ 0 ), z ∗ ∈ F ( x ∗ , y , μ 0 ) and c ′ ∈ C such that w = ε 0 ε 0 + ε n 0 ′ z ¯ + ε n 0 ′ ε 0 + ε n 0 ′ z ∗ + c ′. It follows from the linearity of ξ that ξ ( w ) − ε 0 ε 0 + ε n 0 ′ ξ ( z ¯ ) − ε n 0 ′ ε 0 + ε n 0 ′ ξ ( z ∗ ) = ξ ( c ′ ) ≥ 0, which gives that ξ ( w ) ≥ ε 0 ε 0 + ε n 0 ′ ξ ( z ¯ ) + ε n 0 ′ ε 0 + ε n 0 ′ ξ ( z ∗ ). For all w ∈ F ( ε 0 ε 0 + ε n 0 ′ x ¯ + ε n 0 ′ ε 0 + ε n 0 ′ x ∗ , y , μ 0 ), by (3.5) and (3.6), we have

This implies that inf z ∈ F ( ε 0 ε 0 + ε n 0 ′ x ¯ + ε n 0 ′ ε 0 + ε n 0 ′ x ∗ , y , μ 0 ) ξ ( z ) + ε n 0 ′ + ε 0 ≥ 0, that is, ε 0 ε 0 + ε n 0 ′ x ¯ + ε n 0 ′ ε 0 + ε n 0 ′ x ∗ ∈ H ξ ( ε n 0 ′ ). By (3.4), we get that ε 0 ε 0 + ε n 0 ′ x ¯ + ε n 0 ′ ε 0 + ε n 0 ′ x ∗ ∈ ( x ¯ + O 0 ) ∩ H ξ ( ε n 0 ′ ), which contradicts (3.3). Therefore, H ξ is l.s.c. at 0. Since H ξ is l.s.c. at 0, for above x 0 ∈ S ˜ ξ ( ε 0 , μ 0 ) = H ξ ( 0 ) and for above W 1, there exists a balanced neighborhood V 0 of 0 such that ( x 0 + W 1 ) ∩ H ξ ( ε ) ≠ ∅, ∀ ε ∈ V 0. In particular, from { ε n } ↓ ε 0, there exits N 0 ∈ N such that ( x 0 + W 1 ) ∩ H ξ ( ε N 0 − ε 0 ) ≠ ∅. Let x ′ ∈ ( x 0 + W 1 ) ∩ H ξ ( ε N 0 − ε 0 ).

For any ε ¯ > 0, since e ∈ int C, there exists δ 0 > 0 such that

Since F ( ⋅ , ⋅ , ⋅ ) is uniformly continuous on K ( M ) × K ( M ) × N ( μ 0 ), for above δ 0 B Y, there exists a neighborhood V 1 of 0 ∈ B, a neighborhood U 1 of 0 ∈ B and a neighborhood N 1 of 0 ∈ M, for any ( x 1 , y 1 , μ 1 ) , ( x 2 , y 2 , μ 2 ) ∈ K ( M ) × K ( M ) × N ( μ 0 ) with x 1 − x 2 ∈ V 1, y 1 − y 2 ∈ U 1 and μ 1 − μ 2 ∈ N 1, we have

Since K is H-u.s.c. at μ 0, for above U 1, there exists a neighborhood U 1 ( μ 0 ) of μ 0 such that

We see that x ′ ∈ K ( μ 0 ). Since K is l.s.c. at μ 0, for V 1 ∩ W 1, there exists a neighborhood U 2 ( μ 0 ) of μ 0 such that

It follows from μ n → μ 0 that there exists a positive integer N 0 ′ ≥ N 0 such that μ N 0 ′ ∈ U 1 ( μ 0 ) ∩ U 2 ( μ 0 ) ∩ U ( μ 0 ) ∩ ( μ 0 + N 1 ). Noting that (3.9) and (3.10), we obtain

and

By (3.12), we choose

Next, we prove that x ″ ∈ S ˜ ξ ( ε N 0 ′ , μ N 0 ′ ). For any y ′ ∈ K ( μ N 0 ′ ), by (3.11), there exists y 0 ∈ K ( μ 0 ) such that y ′ − y 0 ∈ U 1. It follows from (3.13) that x ″ − x ′ ∈ V 1. Noting that μ N 0 ′ ∈ U ( μ 0 ) ∩ ( μ 0 + N 1 ) and (3.8), we have

By (3.7), we have

Hence, for any y ∈ K ( μ N 0 ′ ) and z ″ ∈ F ( x ″ , y ′ , μ N 0 ′ ), there exist c ″ ∈ C and z ′ ∈ F ( x ′ , y , μ 0 ) such that

It follows from the linearity of ξ that ξ ( z ″ ) + ε ¯ ≥ ξ ( z ′ ) for all ε ¯ > 0. This leads to ξ ( z ″ ) ≥ ξ ( z ′ ). Thus

Hence x ″ ∈ S ˜ ξ ( ε N 0 ′ , μ N 0 ′ ). Also, since x ′ ∈ ( x 0 + W 1 ) and by (3.2) and (3.13), we have

This means that ( x 0 + W 0 ) ∩ S ˜ ξ ( ε N 0 ′ , μ N 0 ′ ) ≠ ∅, which contradicts (3.1). This completes the proof. □

Theorem 3.3 We assume that for any given ξ ∈ B e ∗, there exists δ > 0 such that the approximate solution set S ˜ ξ ( ⋅ , ⋅ ) exists in [ ε 0 , δ ) × N ( μ 0 ). Suppose that conditions (i)-(iv) as in Theorem  3.2 are satisfied. Assume further that for each x ∈ K ( μ 0 ), F ( x , K ( μ 0 ) , μ 0 ) + C is a convex set. Then the approximate solution mapping S ˜ : [ ε 0 , δ ) × N ( μ 0 ) → 2 X is l.s.c. at ( ε 0 , μ 0 ).

Proof Since F ( x , K ( μ 0 ) , μ 0 ) + C is a convex set for each x ∈ K ( μ 0 ), by virtue of Lemma 3.1, it holds that S ˜ ( ε 0 , μ 0 ) = ⋃ ξ ∈ B e ∗ S ˜ ξ ( ε 0 , μ 0 ). It follows from Theorem 3.2 that for each ξ ∈ B e ∗, S ˜ ξ ( ⋅ , ⋅ ) is l.s.c. at ( ε 0 , μ 0 ). Thus, in view of Lemma 2.4, we obtain that S ˜ ( ⋅ , ⋅ ) is l.s.c. at ( ε 0 , μ 0 ). □

The following example illustrates all of the assumptions in Theorem 3.3.

Example 3.4 Let Y = R 2, C = R + 2 : = { ( x 1 , x 2 ) ∈ R 2 : x 1 ≥ 0 , x 2 ≥ 0 } and Z = X = R. Let B ( 0 , 1 2 ) be the closed ball of radius 1 / 2 in R 2. Let B = [ − 2 , 2 ], M = [ − 1 , 1 ] and the set-valued mapping F : B × B × M → 2 Y be defined by

where w ( x , y , μ ) : = y 2 ( 2 μ − 1 ) + x ( y − x + 1 ) − 3 y + 2 and v ( x , y , μ ) : = y 2 ( 2 μ − 1 ) − x 2 + 2 x y + 3. Define a set-valued mapping K : M → 2 X for all μ ∈ M, by K ( μ ) : = [ − 2 + μ , 2 + μ ] ∩ [ − 2 , 2 ]. We choose e = ( 1 , 1 ) ∈ int C, ε 0 = 2.5, μ 0 = 0 and ξ = ( 1 , 0 ). We can see that B ( 1 , 1 ) ∗ = { ( x 1 , x 2 ) : x 1 + x 2 = 1 , x 1 , x 2 ≥ 0 } and 1 ∈ S ˜ ( 1 , 0 ) ( ε 0 , 0 ). Further, for any μ ∈ ( − 1 , 1 ), there exists ε ∈ [ 2.5 , 4.5 ) such that 1 ∈ S ˜ ( 1 , 0 ) ( ε , μ ). Hence, S ˜ ( 1 , 0 ) ( ⋅ , ⋅ ) exists in [ 2.5 , 4.5 ) × [ − 1 , 1 ]. It is easy to observe that for any y ∈ K ( 0 ), F ( ⋅ , y , 0 ) is C-concave on K ( 0 ). Clearly, condition (ii) is true. It is obvious that K ( M ) = [ − 2 , 2 ]. Let N ( μ 0 ) = [ − 1 , 1 ], we can see that F ( ⋅ , ⋅ , ⋅ ) is uniformly continuous on K ( M ) × K ( M ) × N ( μ 0 ). Finally, we can check that for each x ∈ [ − 2 , 2 ], F ( x , [ − 2 , 2 ] , 0 ) + C is a convex set. Applying Theorem 3.3, we obtain that S ˜ is l.s.c. at ( 2.5 , 0 ).

Example 3.5 Let Y = R 2, C = R + 2 and Z = X = R. Let B = [ − 2 , 2 ], M = [ − 1 , 1 ] and the set-valued mapping F : B × B × M → 2 Y be defined by

Define a set-valued mapping K : M → 2 X for all μ ∈ M, by K ( μ ) : = [ 0 , 1 ]. We choose e = ( 1 , 1 ) ∈ int C, ε 0 = 0.5, μ 0 = 0. Then, all the assumptions of Theorem 3.3 are satisfied except (iii). Indeed, taking y = 1, x 1 = 0, x 2 = 1 and t = 0.5, we have

but ( − 2.5 , − 0.25 ) ∉ C. The direct computation shows that

Clearly, we see that S ˜ ( ⋅ , ⋅ ) is even not l.s.c. at ( ε 0 , μ 0 ) since F ( ⋅ , y , μ 0 ) is not C-concave on K ( μ 0 ).