1. Introduction
It is known to all that the Ky Fan minimax theorem acts a significant role in many fields ( [1]). There are massive articles to study Ky Fan minimax inequality problems for vector-valued mappings and set-valued mappings. Chen [2] proved some Ky Fan minimax inequalities under some different assumptions. Zhang and Li [3] obtained two types of vector set-valued various minimax theorems by applying a fixed point theorem. Zhang and Li [4] investigated three types of Ky Fan minimax inequalities by using Ky Fan section theorem and KFG fixed point theorem. However, the domain assumptions of objection function of these results obtained were convex. Gao [5] investigated some matrix inequalities for the Fan product and the Hadamard Product of Matrices.
The number of papers about Ky Fan minimax inequalities for vector (set)-valued mappings with nonconvex domain assumption is very small. Motivated by these works, we establish some new vector various Ky Fan minimax inequalities with nonconvex domain structure. At the same time, we obtain some existence results.
2. Preliminaries
Let V be a topological sapces with Hausdorff structure, and P be a cone with pointed closed convex structure. We give the signs:
1) if
,
,
, then b is weakly minimal element in Λ;
2) if
,
,
, then b is weakly maximal element in Λ. The marginal set-valued functions
and
are u.s.c. and closed-valued in the setting of continuity of K and compactness of
.
Definition 2.1 Ref. [6] Let
.
K is said to P-u.s.c. if
,
,
of v s.t.
K isP-l.s.c. if −K is P-u.s.c.
Clearly, if K is P-u.s.c., then
is u.s.c.,
.
Lemma 2.1 Let
be compact and
,
.
(i) If K is P-l.s.c., then the weakly minimal element of
is nonempty.
(ii) If K is P-u.s.c., then the weakly maximal element of
is nonempty.
Proof. (i) Let
.
There exists
such that
Thus, by the assumption of
, we have
(ii) Similar way of (i).
3. Vector Various Ky Fan Minimax Inequalities
Theorem 3.1 Let
be compact.
(i) If
,
is P-l.s.c.;
,
is P-l.s.c.; K is P-convexlike in f.v. andK is P-concavelike in its s.v., then
s.t.
(ii) If
,
is P-u.s.c.;
,
is P-u.s.c.; K is P-concavelike in f.v. and K is P-convexlike in s.v., then
s.t.
Proof. (i) Let
. Define the multifunction G by the formula
Since
is P-l.s.c. and Lemma 2.1, G is closed-valued, for each
. We claim that
(1)
Indeed, if not, then there exists
such that
. Namely,
,
. Particularly, taking
, we have that
Hence (1) holds. Thus,
,
Namely,
. Since X0 is compact and G is closed-valued, there is a finite subset
such that
By virtue of G,
,
s.t.
Then, we let
Clearly, M is a convex set in
. In fact, let
and
. Thus,
s.t.
By assumptions,
s.t.
Namely,
By the assumption of α, we have
. Next, by using separation theorem of convex sets, there exists
such that
(2)
Letting
and
, by (2), we have
and
,
. By the assumption of α and the definition of M,
and
Thus,
. Since
by (2),
Namely,
By the arbitrariness of t, we have that
. Because K is P-concave like
in its second variable,
s.t.
Then, we have that
such that
(3)
Since
is P-l.s.c., the weakly minimal element of
is non-empty.
Suppose that
and
. Namely,
Then, by the strong separation theorem of convex sets, there exists a linearcontinuous function
such that
(4)
By (4), letting
,
and
By assumptions, there is
s.t.
Then,
By (3),
such that
Thus,
Then,
By the assumption of v, we have that
Remark 3.2 In Theorm 3.1,
can be nonconvex set. Hence, the result is differents from ones in [2] [3] [4].
4. Applications
In the following, the vector equilibrium problem and lexicographic vector equilibrium problem are considered: Let
.
(VEP) find
such that
Let V be
;
. The lexicographic cone of Rn is defined:
(LVEP) find
such that
Theorem 4.1 Assume that
is compact and:
(i)
,
is P-l.s.c.;
(ii) K is P-convexlike in f.v. and K is P-concavelike in s.v.;
(iii)
.
Then,
which is a solution of VEP.
Proof. By applying vector various minimax inequality,
s.t.
Then,
Thus,
By assumption (iii) and
,
By virtue of the above vector various Ky Fan minimax theorem, we can obtain the existence result in the general conditions, which is to verify easily than ones in the literatures.
Theorem 4.2 Assume that X0 is compact and:
(i)
,
is P-l.s.c.;
(ii)K is P-convexlike in f.v. and K is P-concavelike in s.v.;
(iii)
.
Then, there is
which is a solution of LVEP.
Proof. Similar to the proof of Theorem 4.1 and
, one can show that the result holds as well.
5. Concluding Remark
We obtain some new vector various Ky Fan minimax inequalities in the setting of nonconvex domain. As applications, we obtained some existence results for VEP and LVEP with nonconvex domain assumptions, respectively. These results improve and generalize the relevant ones in the papers.
Acknowledgements
The author is grateful to the referees for suggestions. This paper was funded by the project of Yunnan education department (No: 2022J0601).