CUBO A Mathematical Journal Vol.10, N o ¯ 04, (1–13). December 2008 Remarks on KKM Maps and Fixed Point Theorems in Generalized Convex Spaces Sehie Park The National Academy of Sciences, Republic of Korea, Department of Mathematical Sciences, Seoul National University, Seoul 151–747, KOREA email: shpark@math.snu.ac.kr ABSTRACT Various types of φA-spaces (X, D; {φA}A∈〈D〉) are simply G-convex spaces. Various types of generalized KKM maps on φA-spaces are simply KKM maps on G-convex spaces. Therefore, our G-convex space theory can be applied to various types of φA- spaces. As such examples, we obtain KKM type theorems and a very general fixed point theorem on φA-spaces. RESUMEN Varios tipos de φA-espacios (X, D; {φA}A∈〈D〉) son simplemente espacios G-convexos. Varios tipos de aplicaciones KKM generalizadas sobre φA-espacios son aplicaciones simplemente KKM sobre espacios G-convexos. Por lo tanto, nuestra teoria de espacios G-Convexos puede ser aplicada a varios tipos de φA-espacios. Como ejemplo obtenemos teoremas do tipo KKM y un teorema general de punto fijo sobre φA-espacios. Key words and phrases: Abstract convex space, generalized (G-) convex space, φA-space, L-spaces, F C-spaces, property (H); H-condition. Math. Subj. Class.: 47H04, 47H10, 49J27, 49J35, 54H25, 91B50. 2 Sehie Park CUBO 10, 4 (2008) 1 Introduction The KKM theory, first called by the author, is the study on applications of equivalent formulations of the KKM principle due to Knaster, Kuratowski, and Mazurkiewicz. The KKM principle provides the foundations for many of the modern essential results in diverse areas of mathematical sciences. Since 1993, the author has initiated the study of the KKM theory on generalized convex spaces (or G-convex spaces) (X, D; Γ) as a common generalization of various general convexities without linear structures due to other authors. We have established within such a frame the foundations of the KKM theory, as well as fixed point theorems and many other equilibrium results for multimaps. This direction of study has been followed by a number of other authors. In the last decade, some authors who introduced spaces of the form (X, {ϕA}) having a family {ϕA} of continuous functions defined on simplices claimed that such spaces generalize G-convex spaces without giving any justifications or proper examples. In fact, a number of modifications or imitations of the G-convex spaces have followed; for example, L-spaces due to Ben-El-Mechaiekh et al. [1], spaces having property (H) due to Huang [10], F C-spaces due to Ding [6,7], convexity structures satisfying the H-condition [22], and others. Some authors also tried to generalize the KKM principle for their own settings. They introduced various types of generalized KKM maps; for example, generalized KKM maps on L-spaces [5,20], generalized R-KKM maps [2,8,9], and many others. In order to destroy such inadequate concepts and to upgrade the KKM theory, recently, we proposed new concepts of abstract convex spaces and the KKM spaces which are proper general- izations of G-convex spaces and adequate to establish the KKM theory; see [15-18]. Moreover, we noticed that all spaces of the form (X, {ϕA}) can be unified to φA-spaces (X, D; {φA}A∈〈D〉) or spaces having a family {φA}A∈〈D〉 of singular simplices. In the present note, we show that various types of φA-spaces (X, D; {φA}A∈〈D〉) are simply G-convex spaces, and various types of generalized KKM maps on φA-spaces are simply KKM maps on G-convex spaces. Therefore, our G-convex space theory can be applied to various types of φA- spaces. As such examples, we obtain KKM type theorems and a very general fixed point theorem on φA-spaces. 2 Abstract convex spaces In this section, we follow mainly [15,16]. Let 〈D〉 denote the set of all nonempty finite subsets of a set D. Definition. An abstract convex space (E, D; Γ) consists of nonempty sets E, D, and a multimap Γ : 〈D〉 ⊸ E with nonempty values. We may denote ΓA := Γ(A) for A ∈ 〈D〉. CUBO 10, 4 (2008) Remarks on KKM maps and Fixed Point Theorems ... 3 Examples. In [15-17], we gave plenty of examples of abstract convex spaces. Here we give only two classes of them as follows: 1. 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 [21], where the bibliography lists 283 papers. 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 φA : ∆n → Γ(A) such that J ∈ 〈A〉 implies φA(∆J ) ⊂ Γ(J). Here, ∆n is a standard n-simplex with vertices {ei} n i=0, and ∆J the face of ∆n correspond- ing to J ∈ 〈A〉; that is, if A = {a0, a1, . . . , an} and J = {ai0 , ai1 , . . . , aik } ⊂ A, then ∆J = co{ei0 , ei1 , . . . , eik }. For G-convex spaces; see [11-14,19] and references therein. From now on, in an abstract convex space (E, D; Γ), E is assumed to be a topological space. Definition. Let (E, D; Γ) be an abstract convex space and Z a topological space. For a multimap F : E ⊸ Z with nonempty values, if a multimap G : D ⊸ Z satisfies F (ΓA) ⊂ G(A) := ⋃ y∈A G(y) for all A ∈ 〈D〉, then G is called a KKM map with respect to F . A KKM map G : D ⊸ E is a KKM map with respect to the identity map 1E . A multimap F : E ⊸ Z is called a KC-map [resp., a KO-map] if, for any closed-valued [resp., open-valued] KKM map G : D ⊸ Z with respect to F , the family {G(y)}y∈D has the finite intersection property. The following is the origin of the KKM theory; see [11,12]. The KKM Principle. Let D be the set of vertices of an n-simplex ∆n and G : D ⊸ ∆n be a KKM map (that is, co A ⊂ G(A) for each A ⊂ D) with closed [resp., open] values. Then ⋂ z∈D G(z) 6= ∅. 3 φA-spaces Recently, there have appeared authors in [2,3,6-10,20,22] and others who introduced spaces of the form (X, {ϕA}). Some of them tried to rewrite some results on G-convex spaces by simply replacing Γ(A) by ϕA(∆n) everywhere and claimed to obtain generalizations without giving any justifications or proper examples. 4 Sehie Park CUBO 10, 4 (2008) Motivated by this fact, we are concerned with a reformulation of the class of G-convex spaces as follows [17]: Definition. A φA-space (X, D; {φA}A∈〈D〉) consists of a topological space X, a nonempty set D, and a family of continuous functions φA : ∆n → X (that is, singular n-simplices) for A ∈ 〈D〉 with the cardinality |A| = n + 1. Any G-convex space is a φA-space. The converse also holds: Theorem 1. A φA-space (X, D; {φA}A∈〈D〉) can be made into a G-convex space (X, D; Γ). Proof. This can be done in two ways. (1) For each A ∈ 〈D〉, by putting ΓA := X, we obtain a trivial G-convex space (X, D; Γ). (2) Let {Γα}α be the family of maps Γ α : 〈D〉 ⊸ X giving a G-convex space (X, D; Γα). Note that, by (1), this family is not empty. Then, for each α and each A ∈ 〈D〉 with |A| = n + 1, we have φA(∆n) ⊂ Γ α A and φA(∆J ) ⊂ Γ α J for J ⊂ A. Let Γ := ⋂ α Γ α , that is, ΓA := ⋂ α Γ α A for each A ∈ 〈D〉. Then φA(∆n) ⊂ ΓA and φA(∆J ) ⊂ ΓJ for J ⊂ A. Therefore, (X, D; Γ) is a G-convex space. Consequently, G-convex spaces and φA-spaces are essentially the same. Definition. For a φA-space (X, D; {φA}A∈〈D〉), any map T : D ⊸ X satisfying φA(∆J ) ⊂ T (J) for each A ∈ 〈D〉 and J ∈ 〈A〉 is called a KKM map. Theorem 2. (1) A KKM map G : D ⊸ X on a G-convex space (X, D; Γ) is a KKM map on the corresponding φA-space (X, D; {φA}A∈〈D〉). (2) A KKM map T : D ⊸ X on a φA-space (X, D; {φA}) is a KKM map on a new G-convex space (X, D; Γ). Proof. (1) This is clear from the definition of a KKM map on a G-convex space. (2) Define Γ : 〈D〉 ⊸ X by ΓA := 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 φA : ∆n → T (A) =: Γ(A) CUBO 10, 4 (2008) Remarks on KKM maps and Fixed Point Theorems ... 5 such that J ∈ 〈A〉 implies φA(∆J ) ⊂ T (J) =: Γ(J). Moreover, note that ΓA ⊂ T (A) for each A ∈ 〈D〉 and hence T : D ⊸ X is a KKM map on a G-convex space (X, D; Γ). The following is a KKM theorem for φA-spaces. The proof is just a simple modification of the corresponding one in [12,13,19]: Theorem 3. For a φA-space (X, D; {φA}A∈〈D〉), let G : D ⊸ X be a KKM map with closed [resp., open] values. Then {G(z)}z∈D has the finite intersection property. (More precisely, for each N ∈ 〈D〉 with |N| = n + 1, we have φN (∆n) ∩ ⋂ z∈N G(z) 6= ∅). Further, if (3) ⋂ z∈M G(z) is compact for some M ∈ 〈D〉, then we have ⋂ z∈D G(z) 6= ∅. Proof. Let N = {z0, z1, . . . , zn}. Since G is a KKM map, for each vertex ei of ∆n, we have φN (ei) ∈ G(zi) for 0 ≤ i ≤ n. Then ei 7→ φ −1 N G(zi) is a closed [resp., open] valued map such that ∆k = co{ei0, ei1 , . . . , eik } ⊂ ⋃k j=0 φ −1 N G(zij ) for each face ∆k of ∆n. Therefore, by the original KKM principle, ∆n ⊃ ⋂n i=0 φ −1 N G(zi) 6= ∅ and hence φN (∆n) ∩ ( ⋂ z∈N G(z) ) 6= ∅. The second conclusion is clear. Remarks. (1) We may assume that, for each a ∈ D and N ∈ 〈D〉, G(a) ∩ φN (∆n) is closed [resp., open] in φN (∆n). This is said by some authors that G has finitely closed [resp., open] values. However, by replacing the topology of X by its finitely generated extension, we can eliminate “finitely”; see [13]. (2) For X = ∆n, if D is the set of vertices of ∆n and Γ = co, the convex hull, Theorem 3 reduces to the original KKM principle and its open version; see [11,12]. (3) If D is a nonempty subset of a topological vector space X (not necessarily Hausdorff), Theorem 3 extends Fan’s KKM lemma; see [11,12]. (4) Note that any KKM theorem on spaces of the form (X, {ϕA}) can not generalize the original KKM principle or Fan’s KKM lemma. 4 Examples of φA-spaces In this section, we give some examples of spaces of the form (X, {ϕA}) given by other authors: (I) In 1998, Ben-El-Mechaiekh et al. [1] defined an L-space (E, Γ), which is a particular form of our G-convex space (X, D; Γ) for the case E = X = D. Some authors incorrectly claimed that the class of L-spaces contains our class of G-convex spaces; for example, [4,5], which contain a 6 Sehie Park CUBO 10, 4 (2008) number of particular results (with certain defects) of known ones. (II) In 2003, the authors of [20] considered the L-space. (III) [10] A topological space Y is said to have property (H) if, for each N = {y0, . . . , yn} ∈ 〈Y 〉, there exists a continuous mapping ϕN : ∆n → Y . (IV) [6,7] (Y, {ϕN }) is said to be a F C-space if Y is a topological space and for each N = {y0, . . . , yn} ∈ 〈Y 〉 where some elements in N may be same, there exists a continuous mapping ϕN : ∆n → Y . This definition appears in a large number of papers of the same author and his followers. Note that for each N , there should be infinitely many ϕN ’s. The author of [6,7] wrote in more than one dozen papers that: “It is easy to see that the class of F C-spaces includes the classes of convex sets in topological vector spaces, C-spaces (or H-spaces) [20], G-convex spaces, L-convex spaces [1], and many topological spaces with abstract convexity structure as true subclasses. Hence, it is quite reasonable and valuable to study various nonlinear problems in F C-spaces.” There he failed to give any justification or any proper example of his space which is not G-convex. One wonders how could a pair (Y, {ϕN }) generalize a triple (X, D; Γ). (V) In [22], a pair (Y, C) is introduced, where Y is a topological space and C is a family of subsets of Y such that (Y, C) is similar to the convexity space in the classical sense. A pair (X, C) is said to have the selection property with respect to a topological space S if every multimap F : S ⊸ X admits a single-valued continuous selection whenever F is lower semicontinuous and nonempty closed convex valued. A pair (Y, C) is said to satisfy H-condition if C has the following property: (H) For each finite subset {y0, . . . , yn} ⊂ Y , there exists a continuous mapping f : ∆N → conv{y0, . . . , yn}, where ∆N is the standard n-simplex, such that f (∆J ) ⊂ conv{yj : j ∈ J} for each nonempty subset J ⊂ N = {0, 1, . . . , n}, where conv denotes the closed convex hull. For these definitions, we note the following remarks: (i) A pair (Y, C) is a particular form of our abstract convex space (E, D; Γ) with Y = E = D and ΓA:=conv(A) = ⋂ {B ∈ C | A ⊂ B} for A ∈ 〈Y 〉. Then (Y, C) becomes our abstract convex space (Y ; Γ). (ii) The selection property would be better to call the Michael selection property. (iii) A pair (Y, C) satisfying the H-condition is a particular form of our G-convex space (X, D; Γ) with Y = X = D such that Γ is closed-valued. The following new result gives an example of φA-spaces: CUBO 10, 4 (2008) Remarks on KKM maps and Fixed Point Theorems ... 7 Theorem 4. If an abstract convex space (E, D; Γ) has the Michael selection property with respect to a simplex and if Γ is closed-valued, then we have a φA-space (E, D; {φA}A∈〈D〉). Corollary 4.1. [22, Theorem 1] If a pair (Y, C) has the selection property with respect to any simplex, then the pair satisfies the H-condition. In fact, just following the proof of [22, Theorem 1], we can easily deduce the more general Theorem 4. Moreover, in [22], several results on the pairs (Y, C) satisfying the H-condition are obtained. Some of such results are particular forms of known results on G-convex spaces. 5 Various KKM maps A number of authors tried to generalize the concept of KKM maps on particular forms of φA-spaces. In this section, we show that all of them are particular forms of our KKM maps. (I) In 2003 [20, Definition 2], for an L-space (X, Γ) and a topological space Y , a correspondence G : Y ⊸ X is called a generalized KKM-correspondence, if for all A = {y0, y1, . . . , yn} ∈ 〈Y 〉, there exists a subset B = {x0, x1, . . . , xn} ∈ 〈X〉, such that for all J ⊆ {0, 1, . . . , n}, it is satisfied that φB (∆J ) ⊆ ⋃ j∈J G(yj ). Note that a generalized KKM-correspondence becomes simply our KKM map on a φA-space (X, D; {φA}A∈〈D〉) by putting D := Y and, for any A ∈ 〈D〉, by defining φA(∆|A|−1) := φB(∆|B|−1) for B ∈ 〈X〉 corresponding to A. (II) In 2003 [2, Definition 2.1], for a nonempty set X and a topological space Y , T : X → 2Y is said to be generalized relatively KKM (R-KKM) mapping if for any N = {x0, x1, . . . , xn} ∈ 〈X〉, there exists a continuous mapping φN : ∆n → Y such that, for each ei0 , ei1 , · · · , eik , φN (∆k) ⊂ k ⋃ j=0 T xij , where ∆k is a standard k-simplex of ∆n with vertices ei0 , ei1 , · · · , eik . For a φA-space (Y, X; {φN }N∈〈X〉), T : X → 2 Y is simply a KKM map. (III) Let X be a nonempty set and Y be a topological space with property (H). In 2005 [10], T : X → 2Y is said to be a generalized R-KKM mapping if for each {x0, . . . , xn} ∈ 〈X〉, there 8 Sehie Park CUBO 10, 4 (2008) exists N = {y0, . . . , yn} ∈ 〈Y 〉 such that ϕN (∆k) ⊂ k ⋃ j=0 T xij , for all {i0, . . . , ik} ⊂ {0, . . . , n}. Similarly to (II), a generalized R-KKM map T : X → 2Y is simply a KKM map for the φA-space (Y, X; {φA}A∈〈X〉). The author of [8] claimed as follows: “The above class of generalized R-KKM mappings in- cludes those classes of KKM mappings, H-KKM mappings, G-KKM mappings, generalized G-KKM mappings, generalized S-KKM mappings, GLKKM mappings and GMKKM mappings defined in topological vector spaces, H-spaces, G-convex spaces, G-H-spaces, L-convex spaces and hypercon- vex metric spaces, respectively, as true subclasses.” This is partially incorrect. In view of this claim and Theorem 2, so many variants of KKM type theorems in [2-10,20,22] and a large number of other papers can be reduced to the ones in our G-convex space theory. We should recognize that, in the KKM theory on G-convex spaces, every argument is related to the finite intersection property of functional values of KKM maps having closed [resp., open] values, in other words, related to some N ∈ 〈D〉 in (X, D; Γ). (IV) Motivated by a large number of recent works on generalized KKM maps, we introduced the following definition in [19]: Let (X, D, Γ) be a G-convex space and I a nonempty set. A map F : I ⊸ X is called a generalized KKM map provided that for each N ∈ 〈I〉, there exists a function σ : N → D such that Γσ(M) ⊂ F (M ) for each M ∈ 〈N〉. In [19], a unified account on results for such maps was given; for example, the KKM type theorem, characterizations of such maps, an equilibrium theorem implying minimax inequalities, variational inequalities, and so on. A little later than [19], similar results appeared in [4,5], which has trivial defects in certain aspects. 6 Various KKM type theorems For particular forms of G-convex spaces, some authors obtained KKM type theorems or equivalents which can not be applicable even to the KKM principle for (∆n, V ; co) or to the Ky Fan lemma for (X ⊃ D; co), where X is a topological vector space. In this section, we give two KKM type theorems which improve corresponding ones in [2,20]: Theorem 5. Let X be a topological space, D a nonempty set, and G : D ⊸ X a map such that CUBO 10, 4 (2008) Remarks on KKM maps and Fixed Point Theorems ... 9 (1) G is transfer closed-valued [that is, ⋂ z∈D G(z) = ⋂ z∈D G(z)]; (2) there exists a∗ ∈ Y with G(a∗) compact. Then, there exists a G-convex space (X, D; Γ) such that G is a KKM map if and only if ⋂ z∈D G(z) 6= ∅. Proof. (Necessity) Follows from Theorem 3. (Sufficiency) Choose an x∗ ∈ ⋂ z∈D G(z) 6= ∅. Define a map Γ : 〈D〉 ⊸ X given by the constant function Γ(A) = {x∗} for all A ∈ 〈D〉 with |A| = n + 1, and a function φA : ∆n → Γ(A) by φA(λ) = x ∗ for all λ ∈ ∆n. Then it is easy to verify that, with this G-convex space (X, D; Γ), G is a KKM map. Corollary 5.1. [20, Theorem 1] Let X and Y be topological spaces and Γ : Y → X a transfer closed-valued correspondence on Y such that there exists y∗ ∈ Y with cl[Γ(y∗)] compact. Then, there exists an L-structure on X such that Γ is a generalized KKM-correspondence if and only if ⋂ y∈Y Γ(y) 6= ∅. Recall that several generalizations of [20, Theorem 1] already appeared in [19]. Theorem 6. For a φA-space (Y, D; {φN }N∈〈D〉), let T : D → 2 Y be a map such that T (z) is nonempty and closed for each z ∈ D. (i) If T is a KKM map, then for each N ∈ 〈D〉 with |N| = n + 1, φN (∆n) ∩ ⋂ x∈N T (x) 6= ∅. (ii) If the family {T (z) | z ∈ Z} has finite intersection property, then T is a KKM map. Proof. (i) Apply Theorem 3. (ii) Just follow the sufficiency part of Theorem 5. The following is the key result in [2] with almost a page proof: Corollary 6.1. [2, Theorem 3.1] Let X be a nonempty set and Y be a topological space. Let T : X → 2Y be a set-valued mapping such that T (x) is nonempty and compactly closed in Y for each x ∈ X. (i) If T is a generalized R-KKM mapping, then for each N = {x0, x1, . . . , xn} ∈ 〈X〉, φN (∆n) ∩ ( ⋂ x∈N T x ) 6= ∅, where φN is the continuous mapping in touch with N in definition of a generalized R-KKM map- 10 Sehie Park CUBO 10, 4 (2008) ping. (ii) If the family {T (x) | x ∈ X} has finite intersection property, then T is a generalized R-KKM mapping. Proof. Switch the topology of Y to its compactly generated extension [13]. Then we can eliminate ‘compactly’ and apply Theorem 6. Remark. In [2], its authors used the partition of unity subordinated to a cover of φN0 (∆n) which should be assumed Hausdorff. They claim that, applying their Theorem 3.1, they obtained new theorems which unify and extend many known results in recent literature. However, theirs are all disguised forms of known results and their practical applicability is doubtful. 7 Fixed points of B-maps In this section, the well-known better admissible class B on G-convex spaces [14] can be introduced on φA-spaces. A φA-space (E, D; {φA}A∈〈D〉) is an abstract convex space (E, D; Γ), where ΓN := φN (∆n) for each N ∈ 〈D〉 with |N| = n + 1, and hence there is a continuous function φN : ∆n → ΓN . This (E, D; Γ) is not necessarily a G-convex space. Definition. Let (E, D; Γ) be an abstract convex space, X a nonempty subset of E and Y a topological space. We define the better admissible class B of maps from X into Y as follows: F ∈ B(X, Y ) ⇐⇒ F : X ⊸ Y is a map such that, for any ΓN ⊂ X, where N ∈ 〈D〉 with the cardinality |N| = n+1, and for any continuous function p : F (ΓN ) → ∆n, there exists a continuous function φN : ∆n → ΓN such that the composition ∆n φN −→ ΓN F |Γ N ⊸ F (ΓN ) p −→ ∆n has a fixed point. Note that φN (∆n) is a compact subset of X. Recall that for a φA-space (E, D; {φA}A∈〈D〉), by letting ΓN := φN (∆n), the above definition works. There are a large number of examples of B-maps; see [14] and references therein. We introduce particular types of subsets of abstract convex uniform spaces adequate to estab- lish our fixed point theory. In fact, as in [14], we introduce the Klee approximability of ranges of maps: Definition. Let (E, D; {φA}A∈〈D〉; U) be a uniform φA-space. A subset K of E is said to be Klee approximable if, for each entourage U ∈ U, there exists a continuous function h : K → E satisfying CUBO 10, 4 (2008) Remarks on KKM maps and Fixed Point Theorems ... 11 (1) (x, h(x)) ∈ U for all x ∈ K; (2) h(K) ⊂ φN (∆n) for some N ∈ 〈D〉 with |N| = n + 1; and (3) there exist a continuous function p : K → ∆n such that h = φN ◦ p. Especially, for a subset X of E, K is said to be Klee approximable into X whenever the range h(K) ⊂ φN (∆n) ⊂ X for some N ∈ 〈D〉 in condition (2). We have given a lot of examples of Klee approximable subsets in [14]. Now we have the following generalizations of the main result of [14]: Theorem 7. Let (E, D; {φA}A∈〈D〉; U) be a uniform φA-space, X ⊂ Y subsets of E, and F : Y ⊸ Y a map such that F |X ∈ B(X, Y ) and F (X) is Klee approximable into X. Then F has the almost fixed point property (that is, for any U ∈ U, there exist xU ∈ X such that F (xU ) ∩ U [xU ] 6= ∅). Further if (E, U) is Hausdorff, F is closed, and F (X) is compact in Y , then F has a fixed point x0 ∈ Y (that is, x0 ∈ F (x0)). Proof. Since K := F (X) is a Klee approximable into X, for each symmetric entourage U ∈ U, there exists a continuous function h : K → X satisfying conditions (1) - (3) of the definition of Klee approximable subsets, and we have ∆n φN −→ ΓN F |Γ N ⊸ K p −→ ∆n for some N ∈ 〈D〉 with |N| = n + 1 and ΓN := φN (∆n) ⊂ X. Let p ′ := p|F (ΓN ). Since F |X ∈ B(X, Y ), the composition p ′ ◦ (F |ΓN ) ◦ φN : ∆n ⊸ ∆n has a fixed point aU ∈ ∆n. Let xU := φN (aU ). Then aU ∈ (p ′ ◦ F ◦ φN )(aU ) = (p ′ ◦ F )(xU ) and hence xU = φN (aU ) ∈ (φN ◦ p ′ ◦ F )(xU ). Since h = φN ◦ p by definition, we have xU = h(yU ) for some yU ∈ (F |ΓN )(xU ). Therefore, for each entourage U ∈ U, there exist points xU ∈ X and yU ∈ F (xU ) such that (xU , yU ) = (h(yU ), yU ) ∈ U . So, for each U , there exist xU , yU ∈ X such that yU ∈ F (xU ) and yU ∈ U [xU ]. Now suppose that F is closed and F (X) is compact. Since F (X) is relatively compact, we may assume that the net yU in F (X) converges to some x0 ∈ F (X). Since (xU , yU ) ∈ U for each U ∈ U, by the Hausdorffness of E, the net xU also converges to x0. Since the graph of F is closed in Y × Y and (xU , yU ) ∈ Gr(F ), we have (x0, y0) ∈ Gr(F ) and hence we have x0 ∈ F (x0). This completes our proof. 12 Sehie Park CUBO 10, 4 (2008) Note that, by choosing particular subclass of multimaps or particular types of φA-spaces, we can deduce a large number of known or new fixed point theorems from Theorem 7. Received: December 2007. Revised: December 2007. References [1] H. Ben-El-Mechaiekh, S. Chebbi, M. Florenzano and J.-V. Llinares, Abstract con- vexity and fixed points, J. Math. Anal. Appl. 222 (1998), 138–150. [2] L. Deng and X. Xia, Generalized R-KKM theorems in topological space and their applica- tions, J. Math. Anal. Appl. 285 (2003), 679–690. [3] L. Deng and M.G. Yang, Coincidence theorems with applications to minimax inequalities, section theorem, best approximation and multiobjective games in topological spaces, Acta Math. Sinica, English Ser. 22 (2006), 1809–1818. [4] X.P. Ding, Abstract convexity and generalizations of Himmelberg type fixed-point theorems, Comp. Math. Appl. 41 (2001), 497–504. [5] X.P. Ding, Generalized L-KKM type theorems in L-convex spaces with applications, Comp. Math. Appl. 43 (2002), 1240–1256. [6] X.P. Ding, Maximal element theorems in product FC-spaces and generalized games, J. Math. Anal. Appl. 305 (2005), 29–42. [7] X.P. Ding, Generalized KKM type theorems in F C-spaces with applications (I), J. Glob. Optim. 36 (2006), 581–596. [8] X.P. Ding, New generalized R-KKM type theorems in general topological spaces and appli- cations, Acta Math. Sinica, English Ser. (2006) DOI: 10.1007/s10114-005-0876-y. [9] X.P. Ding, Y.C. Liou and J.C. Yao, Generalized R-KKM type theorems in topological spaces with applications, Appl. Math. Letters 18 (2005), 1345–1350. [10] J. Huang, The matching theorems and coincidence theorems for generalized R-KKM mapping in topological spaces, J. Math. Anal. Appl. 312 (2005), 374–382. [11] S. Park, Ninety years of the Brouwer fixed point theorem, Vietnam J. Math. 27 (1999), 193–232. [12] S. Park, Elements of the KKM theory for generalized convex spaces, Korean J. Comput. & Appl. Math. 7 (2000), 1–28. [13] S. Park, Remarks on topologies of generalized convex spaces, Nonlinear Func. Anal. Appl. 5 (2000), 67–79. CUBO 10, 4 (2008) Remarks on KKM maps and Fixed Point Theorems ... 13 [14] S. Park, A unified fixed point theory in generalized convex spaces, Acta Math. Sinica, English Ser. 23(8) (2007), 1509–1526. [15] S. Park, On generalizations of the KKM principle on abstract convex spaces, Nonlinear Anal. Forum 11 (2006), 67–77. [16] S. Park, Elements of the KKM theory on abstract convex spaces, J. Korean Math. Soc. 45(1) (2007), 1–27. [17] S. Park, Various subclasses of abstract convex spaces for the KKM theory, Proc. Nat. Inst. Math. Sci. 2(4) (2007), 35–47. [18] S. Park, Equilibrium existence theorems in the KKM spaces, Nonlinear Analysis (2007), doi:10.1016/j.na.2007.10.058. [19] S. Park and W. Lee, A unified approach to generalized KKM maps in generalized convex spaces, J. Nonlinear Convex Anal. 2 (2001), 157–166. [20] M.C. Sánchez, J.-V. Llinares and B. Subiza, A KKM-result and an application for binary and non-binary choice function, Econom. Th. 21 (2003), 185–193. [21] B.P. Sortan, Introduction to Axiomatic Theory of Convexity, Kishyneff, 1984. [Russian with English summary] [22] S.-w. Xiang and H. Yang, Some properties of abstract convexity structures on topological spaces, Nonlinear Analysis 67 (2007), 803–808. N1-ParkCUBO