In this section, we discuss the existence and closedness of the solution sets of symmetric generalized quasi-variational inclusion problems by using the Kakutani-Fan-Glicksberg fixed point theorem.
Theorem 8 For each , assume for the problem (SQVIP
α
) that
-
(i)
is usc in with nonempty convex closed values and is lsc in with nonempty closed values;
-
(ii)
is usc in with nonempty convex compact values if (or ) and is lsc in with nonempty convex values if ;
-
(iii)
for all , ;
-
(iv)
for all , the set is convex;
-
(v)
the set is closed.
Then the (SQVIP
α
) has a solution, i.e., there exist such that , and
Moreover, the solution set of the (SQVIP
α
) is closed.
Proof Similar arguments can be applied to three cases. We present only the proof for the case where .
Indeed, for all , define mappings: by
and
-
(a)
Show that and are nonempty convex sets.
Indeed, for all and , for each , , are nonempty. Thus, by assumptions (i), (ii) and (iii), we have and are nonempty. On the other hand, by the condition (iv), we also have , are convex.
-
(b)
We will prove and are upper semicontinuous in with nonempty closed values.
First, we show that is upper semicontinuous in with nonempty closed values. Since A is a compact set, by Lemma 6(ii), we need only to show that is a closed mapping. Indeed, let a net such that , and let such that . Now we need to show that . Since and is upper semicontinuous with nonempty closed values by Lemma 6(i), hence is closed, thus we have . Suppose the contrary . Then such that
(1)
By the lower semicontinuity of , there is a net such that , . Since , we have
(2)
By the condition (v) and (2), we have
(3)
This is a contradiction between (3) and (1). Thus, . Hence, is upper semicontinuous in with nonempty closed values. Similarly, we also have is upper semicontinuous in with nonempty closed values.
-
(c)
Now we need to prove the solution set .
Define the set-valued mappings by
and
Then , are upper semicontinuous and , , and are nonempty closed convex subsets of .
Define the set-valued mapping by
Then H is also upper semicontinuous and , is a nonempty closed convex subset of .
By Lemma 7, there exists a point such that , that is,
which implies that , , and . Hence, there exists , , such that , , satisfying
and
i.e., (SQVIP
α
) has a solution.
-
(d)
Now we prove that is closed. Indeed, let a net : . We need to prove that . Indeed, by the lower semicontinuity of , for any , there exists such that . Since , there exists , , , such that
and
Since , are upper semicontinuous in with nonempty closed values, by Lemma 6(i), we have , are closed. Thus, , . Since , are upper semicontinuous in with nonempty compact values, there exists and such that , (taking subnets if necessary). By the condition (v) and , we have
and
This means that . Thus, is a closed set.
□
If , , , and , , with , are set-valued mappings, and is a nonempty closed convex cone. Then (SQVIP
α
) becomes (SSVQEP) studied in [6].
In this special case, we have the following corollary.
Corollary 9 For each , assume for the problem (SSVQEP) that
-
(i)
is continuous in with nonempty convex closed values;
-
(ii)
is usc in with nonempty convex compact values;
-
(iii)
for all , ;
-
(iv)
for all , the set is convex;
-
(v)
the set is closed.
Then the (SSVQEP) has a solution, i.e., there exist and , such that , and
and
Moreover, the solution set of the (SSVQEP) is closed.
Remark 10 Chen et al. [6] obtained an existence result of (SSVQEP). However, the assumptions in Theorem 3.1 in [6] are different from the assumptions in Corollary 9. The following example shows that all assumptions of Corollary 9 are satisfied. But Theorem 3.1 in [6] is not fulfilled.
Example 11 Let , , and let , and
and
We show that assumptions of Corollary 9 are easily seen to be fulfilled. Hence, by Corollary 9, (SSVQEP) has a solution. But F is neither type II C-lower semicontinuous nor C-concave at . Thus, Theorem 3.1 in [6] does not work.
If , , , , and , , with , are set-valued mappings, and is a nonempty closed convex cone. Then (SQVIP
α
) becomes (SGSVQEP) studied in [5].
In this special case, we have the following corollary.
Corollary 12 For each , assume for the problem (SGSVQEP) that
-
(i)
is continuous in A with nonempty convex closed values;
-
(ii)
is usc in A with nonempty convex compact values;
-
(iii)
for all , ;
-
(iv)
for all , the set is convex;
-
(v)
the set is closed.
Then the (SGSVQEP) has a solution, i.e., there exist and , such that , and
and
Moreover, the solution set of the (SGSVQEP) is closed.
Remark 13 In [5], Plubtieng-Sitthithakerngkiet also obtained an existence result of (SGSVQEP). However, the assumptions in Theorem 3.1 in [5] are different from the assumptions in Corollary 12. The following example shows that in this special case, all assumptions of Corollary 12 are satisfied. But Theorem 3.1 in [5] is not fulfilled.
Example 14 Let , , and let , and
and
We show that all assumptions of Corollary 12 are satisfied. So, by this corollary, the considered problem has solutions. However, F is not lower -continuous at . Also, Theorem 3.1 in [5] does not work.
If , and , for each and , are set-valued mappings, and is a nonempty closed convex cone. Then (SQVIP
α
) becomes (SVQEP) studied in [21].
In this special case, we also have the following corollary.
Corollary 15 Assume for the problem (SVQEP) that
-
(i)
S is continuous in A with nonempty convex closed values;
-
(ii)
for all , ;
-
(iii)
for all , the set is convex;
-
(iv)
the set is closed.
Then the (SVQEP) has a solution, i.e., there exists such that
Moreover, the solution set of the (SVQEP) is closed.
The following example shows that in this special case, all assumptions of Corollary 15 are satisfied. But Theorem 3.3 in [21] is not fulfilled.
Example 16 Let X, Y, Z, A, B, C as in Example 14, and let , and
We show that all assumptions of Corollary 15 are satisfied. So, (SVQEP) has a solution. However, G is not upper C-continuous at . Also, Theorem 3.3 in [21] does not work.
The following example shows that all assumptions of Corollary 9, Corollary 12 and Corollary 15 are satisfied. However, Theorem 3.1 in [6], Theorem 3.1 in [5] and Theorem 3.3 in [21] are not fulfilled. The reason is that G is not properly C-quasiconvex.
Example 17 Let A, B, X, Y, Z, C as in Example 14, and let , , , and
We show that all assumptions of Corollary 9, Corollary 12 and Corollary 15 are satisfied. However, G is not properly C-quasiconvex at . Thus, it gives the case where Corollary 9, Corollary 12 and Corollary 15 can be applied but Theorem 3.1 in [6], Theorem 3.1 in [5] and Theorem 3.3 in [21] do not work.