Remarks on Weakly KKM Maps in Abstract Convex Spaces

Abstract

A KKM space is an abstract convex space satisfying the KKM principle. We obtain variants of the KKM principle for KKM spaces related to weakly KKM maps and indicate some applications of them. These results properly generalize the corresponding ones in 𝐺-convex spaces and 𝜙_𝐴-spaces (𝑋,𝐷;{𝜙_𝐴}_{𝐴∈⟨𝐷⟩}). Consequently, results by Balaj 2004, Liu 1991, and Tang et al. 2007 can be properly generalized and unified.

1. Introduction

Since the appearance of generalized convex (simply, 𝐺-convex) spaces in 1993 [1], the concept has been challenged by several authors who aimed to obtain more general concepts. In fact, a number of modifications or imitations of the concept have followed, for example, 𝐿-spaces due to Ben-El-Mechaiekh et al. [2], spaces having property (𝐻) due to Huang [3], FC-spaces due to Ding [4, 5], and others. It is known that all of such examples belong to the class of 𝜙_𝐴-spaces and are particular forms of 𝐺-convex spaces; see [6]. Some authors also tried to generalize the Knaster-Kuratowski-Mazurkiewicz theorem (simply, the KKM principle) [7] for their own setting. They introduced various types of generalized KKM maps and obtained modifications of known results. Recently, we proposed new concepts of abstract convex spaces and KKM spaces [8–11] which are proper generalizations of 𝐺-convex spaces.

In 1991, Liu [12] obtained a form of the KKM principle and applied it to SupInfSup inequalities of von Neumann type and of Ky Fan type. Motivated by this work, Balaj [13] introduced the concept of weakly 𝐺-KKM mappings for 𝐺-convex spaces and obtained related results on intersections and the Fan type or the Sion type minimax inequalities. Moreover, based on the misconception that FC-spaces generalize 𝐺-convex spaces, Tang et al. [14] introduced the so-called W-G-F-KKM mapping and claimed to obtain similar results for the so-called FC-spaces.

In the present paper, our aim is to show that such basic results for weakly KKM maps on 𝐺-convex spaces can be extended on the more general KKM spaces [6, 10, 11], a particular type of abstract convex spaces satisfying the general KKM principle. These results properly generalize the corresponding ones in 𝐺-convex spaces and 𝜙_𝐴-spaces (𝑋,𝐷;{𝜙_𝐴}_{𝐴∈⟨𝐷⟩}). We note that some applications of them in [12–14] can also be generalized to KKM spaces. Consequently, most results in [12–14] can be properly generalized and unified.

2. Abstract convex spaces

In this section, we follow mainly [6, 8, 9, 15].

Let ⟨𝐷⟩ denote the set of all nonempty finite subsets of a set 𝐷.

Definition 2.1. An abstract convex space(𝐸,𝐷;Γ) consists of a set 𝐸, a nonempty set 𝐷, and a multimap Γ∶⟨𝐷⟩⊸𝐸 with nonempty values. One may denote Γ_𝐴∶=Γ(𝐴) for 𝐴∈⟨𝐷⟩.
For any 𝐷^′⊂𝐷, the Γ-convex hull of 𝐷^′ is denoted and defined byco_Γ𝐷^′∶={Γ_𝐴∣𝐴∈⟨𝐷^′⟩} (2.1)(co is reserved for the convex hull in vector spaces).
A subset 𝑋 of 𝐸 is called a Γ-convex subset of (𝐸,𝐷;Γ) relative to 𝐷^′ if for any 𝑁∈⟨𝐷^′⟩, one has Γ_𝑁⊂𝑋, that is, co_Γ𝐷^′⊂𝑋. This means that (𝑋,𝐷^′;Γ|_{⟨𝐷′⟩}) itself is an abstract convex space called a subspace of (𝐸,𝐷;Γ).
When 𝐷⊂𝐸, the space is denoted by (𝐸⊃𝐷;Γ). In such case, a subset 𝑋 of 𝐸 is said to be Γ-convex if, for any 𝐴∈⟨𝑋∩𝐷⟩, one has Γ_𝐴⊂𝑋. In case 𝐸=𝐷, let (𝐸;Γ)∶=(𝐸,𝐸;Γ).

Example 2.2. In [6, 16], we gave plenty of examples of abstract convex spaces. Here we give only two of them as follows.
(1) Usually, a convexity space(𝐸,𝒞)in the classical sense consists of a nonempty set 𝐸 and a family 𝒞 of subsets of 𝐸 such that 𝐸 itself is an element of 𝒞 and 𝒞 is closed under arbitrary intersection. For details, see [17], where the bibliography lists 283 papers. For any subset 𝑋⊂𝐸, its 𝒞-convex hull is defined and denoted by Co_𝒞𝑋∶=⋂{𝑌∈𝒞∣𝑋⊂𝑌}. We say that 𝑋 is 𝒞-convex if 𝑋=Co_𝒞𝑋. Now we can consider the map Γ∶⟨𝐸⟩⊸𝐸 given by Γ_𝐴∶=𝐶𝑜_𝒞𝐴. Then (𝐸,𝒞) becomes our abstract convex space (𝐸;Γ).
(2) A generalized convex space or a 𝐺-convex space(𝐸,𝐷;Γ) consists of a topological space 𝐸, a nonempty set 𝐷, and a multimap Γ∶⟨𝐷⟩⊸𝐸 such that for each 𝐴∈⟨𝐷⟩ with the cardinality |𝐴|=𝑛+1, there exists a continuous function 𝜙_𝐴∶Δ_𝑛→Γ(𝐴) such that 𝐽∈⟨𝐴⟩ implies 𝜙_𝐴(Δ_𝐽)⊂Γ(𝐽).

Here, Δ_𝑛 is a standard 𝑛-simplex with vertices {𝑒_𝑖}^𝑛_𝑖=0, and Δ_𝐽 the face of Δ_𝑛 corresponding to 𝐽∈⟨𝐴⟩; that is, if 𝐴={𝑎₀,𝑎₁,…,𝑎_𝑛} and 𝐽={𝑎_𝑖0,𝑎_𝑖1,…,𝑎_𝑖𝑘}⊂𝐴, then Δ_𝐽=co{𝑒_𝑖0,𝑒_𝑖1,…,𝑒_𝑖𝑘}.

We have established a large amount of literature on 𝐺-convex spaces; see [1, 16, 18–21] and references therein.

Recently, we are concerned with another variant of 𝐺-convex spaces as follows [6].

Definition 2.3. A 𝜙_𝐴-space(𝑋,𝐷;{𝜙_𝐴}_{𝐴∈⟨𝐷⟩}) (2.2)consists of a topological space 𝑋, a nonempty set 𝐷, and a family of continuous functions 𝜙_𝐴∶Δ_𝑛→𝑋 (i.e., singular 𝑛-simplexes) for 𝐴∈⟨𝐷⟩ with the cardinality |𝐴|=𝑛+1.

Example 2.4. The following are typical examples of 𝐺-convex spaces and 𝜙_𝐴-spaces: (1) any nonempty convex subset of a topological vector space (t.v.s.);(2) [22] a convex space due to Komiya;(3) [23] a convex space due to Lassonde; (4)[24, 25] a 𝐶-space (or an 𝐻-space) due to Horvath;(5)[2] an 𝐿-space due to Ben-El-Mechaiekh et al.;(6)[3] a topological space 𝑌 is said to have property (𝐻) if, for each 𝑁={𝑦₀,…,𝑦_𝑛}∈⟨𝑌⟩, there exists a continuous mapping 𝜑_𝑁∶Δ_𝑛→𝑌;(7) [4, 5, 26, 27] (𝑌,{𝜑_𝑁}) is said to be an FC-space if 𝑌 is a topological space and for each 𝑁={𝑦₀,…,𝑦_𝑛}∈⟨𝑌⟩, where some elements in 𝑁 may be the same, there exists a continuous mapping 𝜑_𝑁∶Δ_𝑛→𝑌;(8)any 𝐺-convex space is clearly a 𝜙_𝐴-space. The converse also holds.

Proposition 2.5 (see [6]). A𝜙_𝐴-space(𝑋,𝐷;{𝜙_𝐴}_{𝐴∈⟨𝐷⟩})can be made into a𝐺-convex space(𝑋,𝐷;Γ).

Proof. This can be done in two ways. (1)For each 𝐴∈⟨𝐷⟩, by putting Γ_𝐴∶=𝑋, we obtain a trivial 𝐺-convex space (𝑋,𝐷;Γ).(2) Let {Γ^𝛼}_𝛼 be the family of maps Γ^𝛼∶⟨𝐷⟩⊸𝑋 giving a 𝐺-convex space (𝑋,𝐷;Γ^𝛼) such that 𝜙_𝐴(Δ_𝑛)⊂Γ^𝛼_𝐴 for each 𝐴∈⟨𝐷⟩ with |𝐴|=𝑛+1. Note that, by (1), this family is not empty. Then, for each 𝛼 and each 𝐴∈⟨𝐷⟩ with |𝐴|=𝑛+1, we have𝜙_𝐴(Δ_𝑛)⊂Γ^𝛼_𝐴,𝜙_𝐴(Δ_𝐽)⊂Γ^𝛼_𝐽for𝐽⊂𝐴. (2.3) Let Γ∶=⋂_𝛼Γ^𝛼, that is, Γ_𝐴=⋂_𝛼Γ^𝛼_𝐴. Then𝜙_𝐴(Δ_𝑛)⊂Γ_𝐴,𝜙_𝐴(Δ_𝐽)⊂Γ_𝐽for𝐽⊂𝐴. (2.4)Therefore, (𝑋,𝐷;Γ) is a 𝐺-convex space.

Therefore, 𝐺-convex spaces and 𝜙_𝐴-spaces are essentially the same.

For a 𝐺-convex space (𝑋,𝐷;Γ), a multimap 𝐺∶𝐷⊸𝑋 is called a KKM map if Γ_𝐴⊂𝐺(𝐴) for each 𝐴∈⟨𝐷⟩.

Proposition 2.6 (see [6]). For a𝜙_𝐴-space(𝑋,𝐷;{𝜙_𝐴}_{𝐴∈⟨𝐷⟩}), any map𝑇∶𝐷⊸𝑋satisfying𝜙_𝐴(Δ_𝐽)⊂𝑇(𝐽)foreach𝐴∈⟨𝐷⟩,𝐽∈⟨𝐴⟩, (2.5)is a KKM map on a 𝐺-convex space (𝑋,𝐷;Γ).

Proof. Define Γ∶⟨𝐷⟩⊸𝑋 by Γ_𝐴∶=𝑇(𝐴) for each 𝐴∈⟨𝐷⟩. Then (𝑋,𝐷;Γ) becomes a 𝐺-convex space. In fact, for each 𝐴 with |𝐴|=𝑛+1, we have a continuous function 𝜙_𝐴∶Δ_𝑛→𝑇(𝐴)=∶Γ(𝐴) such that 𝐽∈⟨𝐴⟩ implies 𝜙_𝐴(Δ_𝐽)⊂𝑇(𝐽)=∶Γ(𝐽). Moreover, note that Γ_𝐴⊂𝑇(𝐴) for each 𝐴∈⟨𝐷⟩ and hence 𝑇∶𝐷⊸𝑋 is a KKM map on a 𝐺-convex space (𝑋,𝐷;Γ).

Remark 2.7. In [14], its authors repeated Ding's false claim in a large number of his own papers as follows. “Recently, Ding [4] introduced FC-space which extended 𝐺-convex space further and proved the corresponding KKM theorem. From this, many new KKM-type theorems and applications were founded in FC-spaces." For Ding's claim, see [5, 26, 27] and references of [6]. One wonders how a pair (𝑋,{𝜑_𝐴}) could extend the triple (𝑋,𝐷;Γ).

The concept of KKM maps for 𝐺-convex spaces is refined as follows.

Definition 2.8. Let (𝐸,𝐷;Γ) be an abstract convex space and 𝑍 a set. For a multimap 𝐹∶𝐸⊸𝑍 with nonempty values, if a multimap 𝐺∶𝐷⊸𝑍 satisfies𝐹(Γ_𝐴)⊂𝐺(𝐴)∶=_𝑦∈𝐴𝐺(𝑦)∀𝐴∈⟨𝐷⟩, (2.6)then 𝐺 is called a KKM map with respect to 𝐹. A KKM map𝐺∶𝐷⊸𝐸 is a KKM map with respect to the identity map 1_𝐸.

A multimap 𝐹∶𝐸⊸𝑍 is said to have the KKM property and called a𝔎-map if, for any KKM map 𝐺∶𝐷⊸𝑍 with respect to 𝐹, the family {𝐺(𝑦)}_𝑦∈𝐷 has the finite intersection property. We denote𝔎(𝐸,𝑍)∶={𝐹∶𝐸⊸𝑍∣𝐹isa𝔎-map}. (2.7)

Similarly, when 𝑍 is a topological space, a ℜℭ-map is defined for closed-valued maps 𝐺, and a ℜ𝔒-map for open-valued maps 𝐺. Note that if 𝑍 is discrete, then three classes 𝔎, ℜℭ, and ℜ𝔒 are identical. Some authors use the notation KKM(𝐸,𝑍) instead of ℜℭ(𝐸,𝑍).

Example 2.9. The above terminology unifies various concepts in other author's usage as follows.
(1) Every abstract convex space in our sense has a map 𝐹∈𝔎(𝐸,𝑍) for any nonempty set 𝑍. In fact, for each 𝑥∈𝐸, choose 𝐹(𝑥)∶=𝑍 or 𝐹(𝑥)∶={𝑧₀} for some 𝑧₀∈𝑍.
If 1_𝐸∈𝔎(𝐸,𝐸), then 𝑓∈𝔎(𝐸,𝑍) for any function 𝑓∶𝐸→𝑍. If 𝐸 and 𝑍 have any topology, this holds for ℜℭ or ℜ𝔒 for any continuous 𝑓.
(2) For a 𝐺-convex space (𝑋,𝐷;Γ) and a topological space 𝑍, we defined the classes 𝔎,ℜℭ,ℜ𝔒 of multimaps 𝐹∶𝑋⊸𝑍 [16]. It is known that for a 𝐺-convex space (𝑋,𝐷;Γ), we have the identity map 1_𝑋∈ℜℭ(𝑋,𝑋)∩ℜ𝔒(𝑋,𝑋); see [19–21]. Moreover, for any topological space 𝑌, if 𝐹∶𝑋→𝑌 is a continuous single-valued map or if 𝐹∶𝑋⊸𝑌 has a continuous selection, then 𝐹∈ℜℭ(𝑋,𝑌)∩ℜ𝔒(𝑋,𝑌).
(3) Let (𝑋,𝐷;Γ) be a 𝐺-convex space, 𝑌 a nonempty set, and 𝑇∶𝑋→2^𝑌, 𝑆∶𝐷→2^𝑌 two mappings. We say that 𝑆 is a generalized G-KKM mapping [13] with respect to 𝑇 if for each 𝐴∈⟨𝐷⟩, 𝑇(Γ(𝐴))⊂𝑆(𝐴). If 𝑌 is a topological space, 𝑇∶𝑋→2^𝑌 is said to have the G-KKM property if for any map 𝑆∶𝐷→2^𝑌 generalized 𝐺-KKM with respect to 𝑇, the family {𝑆(𝑧)∣𝑧∈𝐷} has the finite intersection property.
This simply tells that 𝑆 is a KKM map with respect to 𝑇 and 𝑇∈ℜℭ(𝑋,𝑌).
(4) Let (𝑋,𝜑_𝐴) be an FC-space, 𝑌 a nonempty set, and 𝑇,𝑆∶𝑋→2^𝑌 two mappings. We say that 𝑆 is a generalized KKM mapping [14] with respect to 𝑇 if for each 𝐴∈⟨𝑋⟩, each 𝐵∈⟨𝐴⟩, 𝑇(𝜑_𝐴(Δ_𝐵))⊂𝑆(𝐵). If 𝑌 is a topological space, 𝑇∶𝑋→2^𝑌 is said to have the F-KKM property if for any map 𝑆∶𝑋→2^𝑌 generalized KKM with respect to 𝑇, the family {𝑆(𝑧)∣𝑧∈𝑋} has the finite intersection property.
Note that (𝑋,𝜑_𝐴) becomes a 𝐺-convex space (𝑋,𝐴;Γ) for each 𝐴∈⟨𝑋⟩ with Γ(𝐵)∶=𝜑_𝐴(Δ_𝐵) for each 𝐵∈⟨𝐴⟩. Then 𝑆|_𝐴∶𝐴→2^𝑌 is a KKM map with respect to 𝑇∶𝑋→2^𝑌 and 𝑇∈ℜℭ(𝑋,𝑌).
(5) Let (𝑋,𝜑_𝐴) be an FC-space, 𝑌 a nonempty subset, and 𝑆∶𝑌→2^𝑋. Then 𝑆 is a generalized F-KKM mapping [14] if for each finite subset ^0𝑥0𝐴^02𝑑𝑐={𝑦₀,…,𝑦_𝑛} of 𝑌, there exists a finite subset 𝐴={𝑥₀,…,𝑥_𝑛} of 𝑋 such that for any subset 𝐵={𝑥_𝑖0,…,𝑥_𝑖𝑘} of 𝐴,𝜑_𝐴(co{𝑒_𝑖0,…,𝑒_𝑖𝑘})⊂^𝑘_𝑗=0𝑆(𝑦_𝑖𝑗). (2.8)

A generalized F-KKM map in the above sense can be made into a KKM map on a 𝐺-convex space (𝑋⊃𝑌;Γ) where Γ(𝐴^{0𝑥002𝑑𝑐})∶=𝑆(𝐴^{0𝑥002𝑑𝑐}) and 𝜙_{0𝑥002𝑑𝑐𝐴}∶=𝜑_𝐴 as above.

3. The KKM spaces

We introduced the following in [6].

Definition 3.1. For an abstract convex topological space (𝐸,𝐷;Γ), the KKM principle is the statement 1_𝐸∈ℜℭ(𝐸,𝐸)∩ℜ𝔒(𝐸,𝐸).
A KKM space is an abstract convex topological space satisfying the KKM principle.

In our recent work [9], we studied elements or foundations of the KKM theory on abstract convex spaces and noticed that many important results therein are related to KKM spaces. Moreover, in [10, 11], a fundamental theory and its applications on KKM spaces are extensively investigated.

Example 3.2. We give examples of KKM spaces. (1)Every 𝐺-convex space is a KKM space; see [19–21].(2)A connected ordered space (𝑋,≤) can be made into an abstract convex topological space (𝑋⊃𝐷;Γ) for any nonempty 𝐷⊂𝑋 by defining Γ_𝐴∶=[min𝐴,max𝐴]={𝑥∈𝑋∣min𝐴≤𝑥≤max𝐴} for each 𝐴∈⟨𝐷⟩. Further, it is a KKM space; see [15, Theorem 5(i)].(3)The extended long line 𝐿^∗ can be made into a KKM space (𝐿^∗⊃𝐷;Γ); see [15]. In fact, 𝐿^∗ is constructed from the ordinal space 𝐷∶=[0,Ω] consisting of all ordinal numbers less than or equal to the first uncountable ordinal Ω, together with the order topology. Recall that 𝐿^∗ is a generalized arc obtained from [0,Ω] by placing a copy of the interval (0,1) between each ordinal 𝛼 and its successor 𝛼+1 and we give 𝐿^∗ the order topology. Now let Γ∶⟨𝐷⟩⊸𝐿^∗ be the one as in (2).
But 𝐿^∗ is not a 𝐺-convex space. In fact, since Γ{0,Ω}=𝐿^∗ is not path connected, for 𝐴∶={0,Ω}∈⟨𝐿^∗⟩ and Δ₁∶=[0,1], there does not exist a continuous function 𝜙_𝐴∶[0,1]→Γ_𝐴 such that 𝜙_𝐴{0}⊂Γ{0}={0} and 𝜙_𝐴{1}⊂Γ{Ω}={Ω}. Therefore, (𝐿^∗⊃𝐷;Γ) is not 𝐺-convex.

Therefore, the concepts of KKM spaces properly generalize those of 𝐺-convex spaces and 𝜙_𝐴-spaces.

From the definition of the KKM map, we have the following form of Fan's matching theorem.

Theorem 3.3. Let (𝐸,𝐷;Γ) be a KKM space and 𝑆∶𝐷⊸𝐸 a map satisfying what follows: (1)𝑆(𝑧) is open (resp., closed) for each 𝑧∈𝐷;(2)𝑋=⋃_𝑧∈𝑀𝑆(𝑧) for some 𝑀∈⟨𝐷⟩. Then there exists an 𝑁∈⟨𝐷⟩ such thatΓ_𝑁∩_𝑧∈𝑁𝑆(𝑧)≠∅. (3.1)

Proof. Let 𝐺∶𝐷⊸𝐸 be a map given by 𝐺(𝑧)∶=𝐸⧵𝑆(𝑧) for 𝑧∈𝐷. Then 𝐺 has closed (resp., open) values. Suppose, on the contrary to the conclusion, that for any 𝑁∈⟨𝐷⟩, we have Γ_𝑁∩⋂_𝑧∈𝑁𝑆(𝑧)=∅, that is, Γ_𝑁⊂𝐸⧵⋂_𝑧∈𝑁𝑆(𝑧)=⋃_𝑧∈𝑁(𝐸⧵𝑆(𝑧))=𝐺(𝑁). Therefore, 𝐺 is a KKM map. Since (𝐸,𝐷;Γ) is a KKM space, there exists a 𝑦∈⋂_𝑧∈𝑀𝐺(𝑧)=⋂_𝑧∈𝑀(𝑋⧵𝑆(𝑧)). Hence, 𝑦∉𝑆(𝑧) for all 𝑧∈𝑀. This violates condition (2).

Corollary 3.4. Let(𝐸,𝐷;Γ)be a KKM space,𝐴∈⟨𝐷⟩,{𝑀_𝑧|𝑧∈𝐴}an open or closed cover of𝐸. Then there exists a nonempty subset𝐵of𝐴such thatΓ(𝐵)∩⋂{𝑀_𝑧∣𝑧∈𝐵}≠∅.

Corollary 3.5 (see [13, Lemma 1]). Let(𝑋,𝐷;Γ)be a𝐺-convex space, 𝐴∈⟨𝐷⟩,{𝑀_𝑧∣𝑧∈𝐴}an open or closed cover of𝑋. Then there exists a nonempty subset𝐵of𝐴such thatΓ(𝐵)∩⋂{𝑀_𝑧∣𝑧∈𝐵}≠∅.

Balaj [13] deduced Corollary 3.5 from a previous result of the present author.

Corollary 3.6 (see [14, Theorem 3.2]). Let(𝑋,𝜑_𝐴)be anFC-space, 𝐴∈⟨𝑋⟩,{𝑀_𝑥∣𝑥∈𝐴}an open or closed cover of𝑋. Then there exists a nonempty subset𝐵of𝐴such that𝜑_𝐴(Δ_𝐵)∩⋂{𝑀_𝑥∣𝑥∈𝐵}≠∅.

This is a very particular form of Corollary 3.5 with Γ(𝐵)∶=𝜑_𝐴(Δ_𝐵). In fact, (𝑋,𝜑_𝐴) becomes a 𝐺-convex space (𝑋,𝐴;Γ) with Γ(𝐵)∶=𝜑_𝐴(Δ_𝐵) for each 𝐴∈⟨𝑋⟩ and each 𝐵∈⟨𝐴⟩.

Note also that our proof of Theorem 3.3 is much more simple than that of [14, Theorem 3.2].

4. Weakly KKM maps

Definition 4.1. Let (𝐸,𝐷;Γ) be an abstract convex space and 𝑍 a set. For a multimap 𝐹∶𝐸⊸𝑍 with nonempty values, if a multimap 𝐺∶𝐷⊸𝑍 satisfies𝐹(𝑥)∩𝐺(𝐴)≠∅∀𝐴∈⟨𝐷⟩andall𝑥∈Γ(𝐴), (4.1)then 𝐺 is called a weakly KKM map with respect to 𝐹.

Clearly, each KKM map with respect to 𝐹 is weakly KKM, and a weakly KKM map 𝐺∶𝐷⊸𝐸 with respect to the identity map 1_𝐸 is simply a KKM map.

Example 4.2. (1) When 𝑋∶=𝐸 is a nonempty subset of a vector space, 𝐹 is said to be trappable by 𝐺^𝑐 iff 𝐺 is not weakly KKM with respect to 𝐹, where 𝐺^𝑐(𝑎)∶=𝑍⧵𝐺(𝑎) for each 𝑎∈𝐷 [12].
(2) Let (𝑋,𝐷;Γ) be a 𝐺-convex space, 𝑌 a nonempty set, and 𝑇∶𝑋→2^𝑌,𝑆∶𝐷→2^𝑌 two mappings [13]. We say that 𝑆 is weakly G-KKM mapping with respect to 𝑇 if for each 𝐴∈⟨𝐷⟩ and any 𝑥∈Γ(𝐴), 𝑇(𝑥)∩𝑆(𝐴)≠∅.
(3) Let (𝑋,𝜑_𝐴) be an FC-space, 𝑌 a nonempty set, and 𝑇,𝑆∶𝑋→2^𝑌 two mappings [14]. We say that 𝑆 is weakly generalized F-KKM mapping with respect to 𝑇 (for short, W-G-F-KKM mapping with respect to 𝑇) if for each 𝐴∈⟨𝑋⟩, each 𝐵⊂𝐴 and any 𝑥∈𝜑_𝐴(Δ_𝐵), 𝑇(𝑥)∩𝑆(𝐵)≠∅.

In 1991, Liu [12] obtained a form of the KKM principle. Motivated by the form, we deduce the following generalization.

Theorem 4.3. Let(𝑋,𝐷;Γ)be a compact KKM space,𝑌a nonempty set, and𝐹∶𝑋⊸𝑌and𝐺∶𝐷⊸𝑌maps such that(1)𝐺 is weakly KKM map with respect to 𝐹(2)for each 𝑧∈𝐷, the set {𝑥∈𝑋∣𝐹(𝑥)∩𝐺(𝑧)≠∅} is closed. Then there exists an 𝑥₀∈𝑋 such that 𝐹(𝑥₀)∩𝐺(𝑧)≠∅ for each 𝑧∈𝐷.

Proof. Suppose that the conclusion does not hold. Then for each 𝑥∈𝑋, there exists a 𝑧∈𝐷 such that 𝐹(𝑥)∩𝐺(𝑧)=∅. Define an open set𝑀_𝑧∶={𝑥∈𝑋∣𝐹(𝑥)∩𝐺(𝑧)=∅}for𝑧∈𝐷. (4.2)Since 𝑋 is compact, there is an 𝐴∈⟨𝐷⟩ such that {𝑀_𝑧∣𝑧∈𝐴} is an open cover of 𝑋. Then, by Corollary 3.4, there exist a subset 𝐵 of 𝐴 and a point𝑥₀∈Γ(𝐵)∩{𝑀_𝑧∣𝑧∈𝐵}≠∅. (4.3)Since 𝐺 is weakly KKM with respect to 𝐹 and 𝑥₀∈Γ(𝐵), we have 𝐹(𝑥₀)∩𝐺(𝐵)≠∅.
On the other hand, 𝑥₀∈⋂{𝑀_𝑧∣𝑧∈𝐵} implies 𝐹(𝑥₀)∩𝐺(𝑧)=∅ for all 𝑧∈𝐵 and hence, 𝐹(𝑥₀)∩𝐺(𝐵)=∅. This is a contradiction.

Corollary 4.4 (see [13, Theorem 2]). Let(𝑋,𝐷;Γ)be a compact𝐺-convex space, 𝑌a nonempty set, and𝑇∶𝑋⊸𝑌and𝑆∶𝐷⊸𝑌two maps such that(i)𝑆 is weakly 𝐺-KKM map with respect to 𝑇;(ii)for each 𝑧∈𝐷, the set {𝑥∈𝑋∣𝑇(𝑥)∩𝑆(𝑧)≠∅} is closed. Then there exists an 𝑥₀∈𝑋 such that 𝑇(𝑥₀)∩𝑆(𝑧)≠∅ for each 𝑧∈𝐷.

The following is an immediate consequence of Corollary 4.4.

Corollary 4.5 (see [14, Theorem 3.3]). Let(𝑋,𝜑_𝐴)be a compactFC-space, 𝑌a nonempty set, and𝑇,𝑆∶𝑋⊸𝑌two maps such that(i)𝑆 is 𝑊-𝐺-𝐹-KKM map with respect to 𝑇;(ii) for each 𝑧∈𝐷, the set {𝑥∈𝑋∣𝑇(𝑥)∩𝑆(𝑧)≠∅} is closed. Then there exists an 𝑥₀∈𝑋 such that 𝑇(𝑥₀)∩𝑆(𝑧)≠∅ for each 𝑧∈𝐷.

Example 4.6. In [13, Remark 1], it is noted that condition (2) in Theorem 4.3 is satisfied if 𝑌 is a topological space, 𝐹 is upper semicontinuous, and 𝐺 has closed values. In this case, [12, Theorem 2.1] is for a compact convex subset 𝑋=𝐷 of a topological vector space, and [14, Theorem 3.4] for a compact FC-space (𝑋,𝜑_𝐴).

We have the following variant of Theorem 4.3.

Theorem 4.7. Let(𝑋,𝐷;Γ)be a KKM space,𝑌a nonempty set, and𝐹∶𝑋⊸𝑌and𝐺∶𝐷⊸𝑌maps such that(1)𝐺 is weakly KKM with respect to 𝐹;(2)the set {𝑥∈𝑋∣𝐹(𝑥)∩𝐺(𝑧)≠∅} is either all closed or all open for all 𝑧∈𝐷. Then for each 𝐴∈⟨𝐷⟩ there exists an 𝑥₀∈Γ(𝐴) such that 𝐹(𝑥₀)∩𝐺(𝑧)≠∅ for all 𝑧∈𝐴.

Proof. Let 𝐴∈⟨𝐷⟩. Then it is easily checked that the subspace (𝑋,𝐴;Γ|_⟨𝐴⟩) is also a KKM space. Now the conclusion follows from the same argument in the proof of Theorem 4.3.

Corollary 4.8 (see [13, Theorem 3]). Let(𝑋,𝐷;Γ)be a𝐺-convex space, 𝑌a nonempty set, and𝑇∶𝑋⊸𝑌and𝑆∶𝐷⊸𝑌maps such that(i)𝑆 is weakly 𝐺-KKM map with respect to 𝑇;(ii) the set {𝑥∈𝑋∣𝑇(𝑥)∩𝑆(𝑧)≠∅} is either all closed or all open for all 𝑧∈𝐷. Then for each 𝐴∈⟨𝐷⟩ there exists an 𝑥₀∈Γ(𝐴) such that 𝑇(𝑥₀)∩𝑆(𝑧)≠∅ for all 𝑧∈𝐴.

From Corollary 4.8, we have the following.

Corollary 4.9 (see [14, Theorem 3.5]). Let(𝑋,𝜑_𝐴)be anFC-space, 𝜑_𝐴(Δ_𝑛)anFC-subspace for each 𝐴∈⟨𝑋⟩,𝑌a nonempty set, and𝑇,𝑆∶𝑋⊸𝑌two maps such that(i)𝑆 is 𝑊-𝐺-𝐹-KKM map with respect to 𝑇;(ii)the set {𝑥∈𝑋∣𝑇(𝑥)∩𝑆(𝑧)≠∅} is either all closed or all open for all 𝑧∈𝐷. Then for each 𝐴∈⟨𝑋⟩ there exists an 𝑥₀∈𝜑_𝐴(Δ_𝑛) such that 𝑇(𝑥₀)∩𝑆(𝑧)≠∅ for each 𝑧∈𝜑_𝐴(Δ_𝑛).

Remark 4.10. In [13, Remark 2], it is noted that condition (3.2) in Theorem 4.7 is satisfied if 𝑌 is a topological space and either 𝐹 is upper semicontinuous and 𝐺 has closed values or 𝐹 is lower semicontinuous and 𝐺 has open values. This is exploited in [14, Theorem 3.6].

As an example of applications of Theorem 4.3, we give the following.

Theorem 4.11. Let(𝑋,𝐷;Γ)be a compact KKM space,𝑌a topological space. Let𝑇∶𝑋⊸𝑌be an upper semicontinuous map,𝜓∶𝐷×𝑌→ℝ, 𝜑∶𝑋×𝑌→ℝ two functions and𝛽=inf_𝑥∈𝑋sup_{𝑦∈𝑇(𝑥)}𝜑(𝑥,𝑦).Suppose that(1) for each 𝑧∈𝐷, 𝜓(𝑧,⋅) is upper semicontinuous on 𝑌;(2) for any 𝜆<𝛽 and 𝑦∈𝑇(𝑥), co_Γ{𝑧∈𝐷∣𝜓(𝑧,𝑦)<𝜆}⊂{𝑥∈𝑋∣𝜑(𝑥,𝑦)<𝜆}.

Proof. Just follow that of [13, Theorem 4].

Corollary 4.12. In Theorem 4.11,(𝑋,𝐷;Γ)can be replaced by a compact 𝐺-convex space without affecting its conclusion.

Note that Corollary 4.12 contains some known forms of the Fan type minimax inequalities; see [13].

Corollary 4.13 (see [14, Theorem 4.1]). Let(𝑋,𝜑_𝐴)be a compactFC-space and 𝑌a topological space. Let𝑇∶𝑋⊸𝑌be a u.s.c. map,𝑓,𝑔∶𝑋×𝑌→ℝtwo functions, and𝛽=inf_𝑥∈𝑋sup_{𝑦∈𝑇(𝑥)}𝑓(𝑥,𝑦).Suppose that(i) for each 𝑧∈𝑋, 𝑔(𝑧,⋅) is u.s.c. on 𝑌;(ii)for any 𝜆<𝛽 and 𝑦∈𝑇(𝑥), if for each 𝐴∈⟨𝑋⟩ and 𝐵∈⟨𝐴∩{𝑥∈𝑋∣𝑔(𝑥,𝑦)<𝜆}⟩ one has 𝜑_𝐴(Δ_𝐵)⊂{𝑥∈𝑋∣𝑓(𝑥,𝑦)<𝜆}.

5. Further remarks

Until now, in this paper, we showed that basic results in [12] for topological vector spaces, in [13] for 𝐺-convex spaces, and in [14] for FC-spaces, are all extended to KKM spaces. Therefore, most of their applications in each paper can be also generalized to KKM spaces. The readers can show this in case they are urgently needed. Finally, note that results in [14] are all particular to corresponding ones for 𝐺-convex spaces.