Generalized Knaster-Kuratowski-Mazurkiewicz type theorems and applications to minimax inequalities

Knaster-Kuratowski-Mazurkiewicz type theorems play an important role in nonlinear analysis, optimization, and applied mathematics. Since the first well-known result, many international efforts have been made to develop sufficient conditions for the existence of points intersection (and their applications) in increasingly general settings: Gconvex spaces [21, 23], L-convex spaces [12], and FCspaces [8, 9]. Applications of Knaster-Kuratowski-Mazurkiewicz type theorems, especially in existence studies for variational inequalities, equilibrium problems and more general settings have been obtained by many authors, see e.g. recent papers [1, 2, 3, 8, 18, 24, 26] and the references therein. In this paper we propose a definition of generalized KnasterKuratowski-Mazurkiewicz mappings to encompass R-KKM mappings [5], L-KKM mappings [11], T-KKM mappings [18, 19], and many recent existing mappings. Knaster-KuratowskiMazurkiewicz type theorems are established in general topological spaces to generalize known results. As applications, we develop in detail general types of minimax theorems. Our results are shown to improve or include as special cases several recent ones in the literature..


INTRODUCTION
xistence of solutions takes a central place in the optimization theory.Studies of the existence of solutions of a problem are based on existence results for important points in nonlinear analysis like fixed points, maximal points, intersection points, etc.
One of the most famous existence theorems in nonlinear analysis is the classical KKM theorem, which has been generalized by many authors.For example see [1,2,3,4,6,10,22,23,27].In early forms of this fundamental result, convexity assumptions played a crucial role and restricted the ranges of applicable areas.After, various generalized linear/convex structures have been proposed and corresponding types of KKM mappings have been defined together with these spaces, such as [3,6,21] investigated G-convex spaces, Ding [7][8][9] introduced the concept of a FCspace and then Khanh and Quan [18,19], Khanh, Lin and Long [14], Khanh and Long [15,16] and, Khanh, Long and Quan [17] generalized and unified the previous spaces into a notion called a GFC-space.
Applications of KKM-type theorems, especially in existence studies for variational inequalities, equilibrium problems and more general settings have been obtained by many authors, see e.g.recent papers [1,2,3,8,18,24,26] and the references therein.
To avoid in a stronger sense convexity structures in investigating KKM-type theorems, in this paper we propose a definition of a generalized type of KKM mappings in terms of a FLS-space and use it to establish generalized KKM type theorems.As applications we focus only on minimax and saddlepoint problems, which also generalize or improve recent results in the literature [3, 5, 6, 10,...].
The outline of the paper is as follows.Section 2 contains definitions and preliminary facts for our later use.In Section 3, we give our main results.This section contains generalized KKM-type theorems, a Ky Fan type matching theorem and discuss their consequences in some particular cases.In section 4, we obtain the sufficient conditions for the solutions existence of minimax and saddlepoint problems.
Generalized Knaster-Kuratowski-Mazurkiewicz type theorems and applications to minimax inequalities We recall now some definitions for our later use.For a set X , by X  2 and X we denote the family of all nonempty subsets, and the family of nonempty finite subsets, respectively.Let Z , X be topological spaces and ) is compactly open (compactly closed, resp.) in Z and for each nonempty compact subset is compact subset of Z .N , Q , and R denote the set of the natural numbers, the set of rational numbers, and that of the real numbers, respectively, and , n stands for the n -simplex with the vertices being the unit vectors 1 e , 2 e , ..., becomes an GFC -space as defitioned in [18][19][20].If in addition and becomes an FC -space in [7,8].The Example 1 below shows that in general the inverse is not true.Definition 2 (See [18][19][20]).Let ( X , Y , ) be a GFC -space and Z be a topological space.Let We will need the following well-known result.

Lemma 1 ([7]
).Let Y be a set, X be a topological space and X Y F 2 : . The following statements are equivalent.
2. for each compact subset and each

2.
We say that a set-valued mapping , the family which have the generalized L -KKM property is denoted by L -KKM(X,Y,Z).

Remark 2 Note that the Definition 5 (i) is a generalization of the Definition 2.1 of [11]. We also see that every L -T -KKM mapping is a T -KKM mapping when N is a continuous singlevalued mapping. If in addition
and T is the identity map then L -T -KKM mapping becomes an R -KKM mapping of [5] and thw Definition 2.2 of [7].
The following example shows that the Definition 5 (i) contains the Definition 2. be upper semicontinuous with compact valued from a compact space X to Y. Then T(X) is compact.

Lemma 3 Let
) , , ( Y X be a GFC-space and Z be a topological space.Then be a continuous partition of unit associated with this covering and Where However, in view of the definitions of ) ( 0 z J and of the partition be set-valued mappings.Assume that the following conditions hold 3. there are .By (iii), there is By (iv) and Lemma 1, we have Thus we arrive at the conclusion Remark 3 Theorem 1 unifies and generalizes Theorem 3.2 of [5], Theorem 3.2 of [11] and Theorem 3.2 of [21] under much weaker assumptions.By Lemma 3, Theorem 1 improves the assertion (iii 1 ) of Theorem 2.2 of [19].
The following example shows that we cannot use of known results in FC -spaces of [7] or GFCconvex spaces of [18][19][20], but is easily investigated by FLS -spaces.are defined as follows

Example 2 Let
We can see that F is not T -KKM.Indeed, we choose Hence the results in [18][19][20] are out of use for this case.
To apply our Theorem we have Proof.We define a set-valued mapping , and any and by Lemma Remark 4 Theorem 2 contains Theorem 1 of [21], Theorem 3.1, 3.2 and 3.3 of [7] and Theorem 3.1 of [18].Theorem 3 Let (X,Y, ) be a FLS -space and Z be topological spaces.Let Then there exists   [19] and Theorem 3.1 of [8].
As a consequence of the generalized L -T -KKM theorems, we prove a generalization of the Ky fan type matching theorem.
Theorem 4 Let (X,Y, ) be a FLS . This contradicts the fact that F is L -T -KKM.Thus the conclusion of Theorem 2 follows Theorem 4.

KY FAN TYPE MINIMAX INEQUALITIES
In this section, by applying L -T -KKM theorems, we shall establish some new Ky Fan type minimax inequalities and saddle point problems.
intersection property, i.e. all finite intersections of sets of this family are nonempty.The class of all mappings X be a FLS -space and Z be topological spaces.Let

Theorem 2
property,i.e., (ii)of Theorem 1 is fulfilled.If we choose ) of Theorem 1 are satisfied.Moreover it is easy to see that F is transfer compactly closed-valued.By Theorem 1Let (X,Y, ) be a FLS -space and Z be topological spaces.beset-valued mappings.Assume that Y is an L-S-subset of itself.Let the following conditions hold1.F is L-T-KKM and transfer compactly closed-valued;2.
. It follows that H is L -T -KKM.By (i), H is transfer compactly closed-valued.Clearly, all conditions of Theorem 2 are satisfied.