Improving KKM Theory on General Metric Type Spaces


 A generalized metric type space is a generic name for various spaces similar to hyperconvex metric spaces or extensions of them. The purpose of this article is to introduce some KKM theoretic works on generalized metric type spaces and to show that they can be improved according to our abstract convex space theory. Most of these works are chosen on the basis that they can be improved by following our theory. Actually, we introduce abstracts of each work or some contents, and add some comments showing how to improve them.


Introduction
The KKM theory, rst called by the author in 1992 [17], is the study on applications of equivalent formulations or generalizations of the KKM theorem due to Knaster, Kuratowski, and Mazurkiewicz in 1929. The KKM theorem is one of the most well-known and important existence principles and provides the foundations for many of the modern essential results in diverse areas of mathematical sciences. Since the theorem and its many equivalent formulations or extensions are powerful tools in showing the existence of solutions of a lot of problems in pure and applied mathematics, many scholars have been studying its further extensions and applications.
The KKM theory was rst devoted to convex subsets of topological vector spaces mainly by Ky Fan and Granas, and later to the so-called convex spaces by Lassonde, to c-spaces by Horvath and others, to G-convex spaces mainly by the present author. In 2006-09, we proposed new concepts of abstract convex spaces and partial KKM spaces which are proper generalizations of G-convex spaces and adequate to establish the KKM theory. Now the KKM theory becomes the study of abstract convex spaces due to ourselves in 2006 [28] and we obtained a large number of new results in such frame. For the history of the KKM theory, see our recent article [29].
Almost independently to the above progress, Khamsi in 1996 [11] found that hyperconvex metric spaces are partial KKM spaces, and obtained some KKM theoretic results. Motivated by his work, in the last decade, many authors obtained some KKM theoretic results on hyperconvex metric spaces as well as newly de ned spaces like NR-metric spaces, metric type spaces, cone metric spaces, cone b-metric spaces, tvs-cone metric spaces, circular metric spaces, modular metric spaces, convex metric spaces, etc. These type of spaces may be called generalized metric type spaces as a generic name.
The purpose of this article is to introduce some KKM theoretic works on generalized metric type spaces and to show that they can be improved according to our abstract convex space theory.
In Section 2, we introduce some basic things on our abstract convex spaces as a preliminary. Section 3 deals with various properties of (partial) KKM spaces that can be applied various particular types of spaces.
In Section 4, we recall previous studies on generalized metric type spaces and basic facts on them. Section 5 deals with several works on the generalized metric type spaces. Most of these works are chosen on the basis that they can be improved by following our KKM theory on abstract convex spaces. Actually, we introduce abstracts of each work or some contents of them, and add some comments showing how to improve them. Finally in Section 6 with the conclusion, we propose some suggestions to improve the study of general metric type Spaces. This paper is parallel to our previous work [32] in 2020 in the sense that we repeat some introductory part there for reader's convenience. Note that there are scores of new metric type spaces appeared in the metric xed point theory related to contractive type maps or nonexpansive maps. But we will not concerned with them in this paper.
In order to introduce a most general form of the KKM type theorem, consider the following related four conditions for a map G : D Z with a topological space Z: Recall the following one of the most general KKM type theorems in [25] and others: Furthermore, The following fact was given in [21].

The KKM theory on abstract convex spaces
In the rst half of this section, we introduce some typical KKM theoretic results on abstract convex spaces appeared mainly in [23,24,29]. Of course we can not repeat whole works on that subject. So in the second half of this section, we state some subjects of our study on the KKM theory without mentioning the literature. This is equivalent to each one of the following and others: The Fan matching property Another nite intersection property The geometric property or the section property The Fan-Browder xed point property Existence of maximal elements Analytic formulation Minimax inequality Analytic alternative An abstract convex space (E, D; Γ) is called a partial KKM space whenever the following partial condition of (0) holds: This is equivalent to each partial one (that is, closed-valued case) of the above list.
Theorem. For a compact partial KKM space (X; Γ), the following statements hold.

Several types of minimax inequalities Several types of variational inequalities
Let (X; Γ ) and (Y; Γ ) be abstract convex spaces. For their product, we can de ne Γ X×Y (A) : Theorem. For a compact product partial KKM space (E; Γ) := (X × Y; Γ X×Y ), the following statements hold.
The basic minimax theorem Generalized von Neumann-Sion minimax theorem Collective xed point theorem The von Neumann-Fan intersection theorem The Fan type analytic alternative Generalized Nash-Fan type equilibrium theorem Since we rst named the KKM theory in 1992 [17], we have tried to improve the theory by applying such basic theory of abstract convex spaces. In fact, we have studied the KKM theory with respect to the following list of subjects: Let A be a nonempty bounded subset of a metric type space (M, δ). Then we de ne as follows by adopting Khamsi's way [11]: For an x ∈ M and ε > , let It is amazing that, in metric type spaces, when we do not know whether open balls are open and closed balls are closed; see [16].
Here, we need the following extra requirement: This condition holds for any metric space. We introduce new de nitions: A Γ-convex subset of (M ⊃ D; Γ) is said to be subadmissible.
Remark. For a metric space M, (M ⊃ D; Γ) is given in [3], where Γ A := ad(A). This is a metric type space.

Some articles on generalized metric type spaces
In this section we introduce some articles which can be improved by following our methods as in the previous sections. Most of these works are chosen on the basis that they can be improved by following our theory. Actually, we introduce abstracts or some part of each paper, and add certain comments like how to improve them.

C
: In a metric space (M, d), we denote B(x, r) for the closed ball about x ∈ M and of radius r ≥ . For a nonempty bounded subset A ⊂ M, set: With such preparation, Khamsi obtained the following: for any x , . . . , xn ∈ X. Note that, according to our current terminology, (H, X; co) is an abstract convex space by De nition 3, and it is a KKM space where H has the nitely generated topology. Therefore, it has a large number of properties as shown in Section 3.
In fact, Khamsi showed only a few KKM theoretic properties of (H, X; co).
Park [19] -NA 37 (1999) F I : In this paper, we obtain a Ky Fan type matching theorem for open covers, a coincidence theorem, a Fan-Browder type xed point theorem, a Brouwer-Schauder-Rothe type xed point theorem, and other results for hyperconvex spaces. Those are usually obtained for a convex space setting or a generalized convex space setting. Our arguments in this paper are based on the KKM theorem due to Khamsi [11,Theorem 4].

C
: This paper seems to be a supplement to Khamsi's previous paper in the sense that this adds up few more KKM theoretic properties of hyperconvex metric spaces. , and, hence, is a generalized convex space (or a G-convex space) due to Park. Therefore, most results in this paper are simple consequences of the corresponding known ones for c-spaces or G-convex spaces. Moreover, the author sometimes assumes super uous restrictions and some of his proofs are unnecessarily lengthy and complicated. [14] -NA 39 (2000) F I : In this paper, we rst establish a characterization of the Knaster-Kuratowski and Mazurkiewicz principle in hyperconvex metric spaces which in turn leads to a characterization theorem for a family of subsets with the nite intersection property in such a setting. As applications we give hyperconvex versions of Fan's celebrated minimax principle and Fanąŕs best approximation theorem for set-valued mappings. These in turn are applied to obtain formulations of the Browder-Fan xed point theorem and the Schauder-Tychono xed point theorem in hyperconvex metric spaces for set-valued mappings. Finally, existence theorems for saddle points, intersection theorems and Nash equilibria are also obtained. Our results unify and extend several of the results cited above.

C
: As we have already seen, Horvath showed in 1993 that hypoconvex metric spaces are his c-spaces and hence G-convex spaces. However, the KKM theory on G-convex spaces was already wellestablished around 1997, and hence, the preceding KKM theoretic studies on hyperconvex metric spaces were out-of-date from the beginning.

Zafarani [28] -Liége 73(2-3) (2004)
A : In this paper, we give a brief survey of some recent generalization of the Fan-KKM theorem. We introduce a new convex structure on a nonempty set M which contains all di erent concepts of convexity. Some approximate xed point theorems will be established for the multivalued mappings with S-KKM property on the Φ spaces. We also obtain a generalized Fan matching theorem, a generalized Fan-Browder type theorem, and a new version of Sadovskii's xed point theorem.

C
: Many results on the KKM theory on various types of spaces are introduced. Zafarani's Γconvex spaces are motivated by our G-convex spaces and the same to our original abstract convex spaces in 2006. But he did not establish any theory on his spaces. He adopted a KKM theorem, a very particular form of our Theorem C. Many terms in this paper are obsolete; for example, the KKM class, the S-KKM class, and the generalized S-KKM class belong to our KC class in our abstract convex space theory.
Zafarani introduced NR-metric spaces, which generalize hyperconvex metric spaces and are G-convex spaces, and hence satisfy all results in Section 3.

Amini -Fakhar -Zafarani [1] -NA 60 (2005)
A : We introduce the class of KKM-type mappings on metric spaces and establish some xed point theorems for this class. We also obtain a generalized Fan's matching theorem, a generalized Fan-Browder's type theorem, and a new version of Fan's best approximation theorem.

C
: On the surface, this is a very nice paper. However, the authors adopted inadequate terminology of Chang-Yen. For example, the class of KKM type mapping is KC in our works. See the next article. : In this paper, we did not assume any topology for an abstract convex spaces, but we changed our mind later. We also clari ed some inadequate usage of terminology; for example, S-KKM class.

Amini -Fakhar -Zafarani [2] -NA 66 (2007)
A : We de ne KKM mappings and S-KKM mappings similarly to in the case of convex spaces for abstract convex spaces. Some approximate xed point theorems will be established for the multifunction with the S-KKM property on Φ-spaces. We also obtain a new version of Sadovskii's xed point theorem in topological spaces.

C
: In this paper, we notice the following: (1) Here abstract convex spaces mean spaces having the routine convexity structure, (2) Chang -Yen's KKM class [5] should be replaced by the KC class.

Shahzad -Markin [33] -JMAA 337 (2008)
A : In this paper, for a commuting pair consisting of a point-valued nonexpansive self-mapping t and a set-valued nonexpansive self-mapping T of a hyperconvex metric space (or a CAT(0) space) X, we look for a solution to the problem of existence of z ∈ E ⊂ X such that d(z, y) = d(y, E) and z = t(z) ∈ T(z).

C
: Hyperconvex metric spaces and complete CAT(0) spaces are Horvath spaces and hence KKM spaces. Therefore, they have many properties as shown in Section 3. d(a, µ(a, b) µ(a, b) Corollary 17. Suppose that the identity mapping X ∈ KC(X, X), then any continuous mapping f : X → X such that cl f (X) is compact, has a xed point. Some of the above results extend corresponding ones of Amini et al. [1] for metric spaces. Moreover, Lemma 16 has already a far-reaching extended form in Park [21], see Theorem D in Section 2.

Khamsi -Hussain [12] -NA 73 (2010)
A : In this work we discuss some recent results about KKM mappings in cone metric spaces. We also discuss the xed point existence results of multivalued mappings de ned on such metric spaces. In particular we show that most of the new results are merely copies of the classical ones and do not necessitate the underlying Banach space nor the associated cone. F I : It is worth mentioning the pioneering work of Quilliot (in 1983) who introduced the concept of generalized metric spaces. His approach was very successful and is used by many (see the references in Jawhari et al. in 1986). It is our belief that cone metric spaces are a special case of generalized metric spaces. In this work, we introduce a metric type structure in cone metric spaces and show that classical proofs related to KKM mappings proved in [35] do apply almost identically in these metric type spaces. This approach suggests that any extension of known xed point results to cone metric spaces is redundant. Moreover the underlying Banach space and the associated cone subset are not necessary.

Hussain -Shah [8] -CMA 62 (2011)
A : In this paper we establish some topological properties of the cone b-metric spaces and then improve some recent results about KKM mappings in the setting of a cone b-metric space. We also prove some xed point existence results for multivalued mappings de ned on such spaces.
F I : In this work, with the structure of a cone b-metric space, we shall establish some topological properties of the cone b-metric spaces. We also prove and extend some results of Khamsi and Hussain [12] and illustrate our work in this setting with examples.

C
: The authors also adopted the obsolete Chang -Yen KKM class. : The author showed that any cone normed space (X, ||| · |||) is a partial KKM space and, hence, satis es a large number of the KKM theoretic results in Section 3. : The main purpose of this paper is to study some topological nature of circular metric spaces and deduce some xed point theorems for maps satisfying the KKM property. We also investigate the solvability of a variant of a quasi-equilibrium problem as an application.

C
: In this paper, let M be a circular metric space and X be a subadmissible subset of M. A multimap G : X M is said to be a KKM map if for each A ∈ X we have ad(A) ⊂ G(A).
Note that (M, X; ad) is an abstract convex space.
In this paper, as in the many metric type spaces, inadequate KKM class instead of KC is used. Certain results can be improved by adopting current language in the KKM theory.

Park [27] -JNAS-ROK 54(2) (2015)
A : For a long period, the study of hyperconvex metric spaces introduced by Aronszajn and Panitchpackdi in 1956 was concentrated to the xed point property of nonexpansive maps. However, since Khamsi in 1996 found a KKM theorem for such spaces, there have appeared a large number of works on them related to the KKM theory. In the present review, we follow the various stages of developments of the KKM theory of hyperconvex metric spaces. In fact, we introduce abstracts of articles on such theory and give comments or generalizations of the results there if necessary. We show that many results in those articles are consequences of (partial) KKM space theory developed by ourselves from 2006. C : While preparing this paper, we noticed some papers on useless studies of the KKM theory and they were criticized in several papers; for example [26].

Jain et al. [10] -MDPI (2016)
A : The purpose of this paper is to present a new approach to study the existence of xed points for multivalued F-contraction in the setting of modular metric spaces. In establishing this connection, we introduce the notion of multivalued F-contraction and prove corresponding xed point theorems in complete modular metric space with some speci c assumption on the modular. Then we apply our results to establish the existence of solutions for a certain type of non-linear integral equations.

C
: No KKM theoretic result was given in this paper.  [12] introduced a metric type structure in cone metric spaces and showed that classical proofs related to KKM maps proved in [36] do carry almost identically in these metric type spaces. This approach suggests that any extension of known xed point results to cone metric spaces is redundant.
We found that there are some similar results in [12,13,29]. Our principal aim in this article is to obtain generalized results unifying the corresponding ones in [12] and [22]. : In this paper, we present a modular version of KKM and generalized KKM mappings and then we establish a characterization of generalized KKM mappings in modular spaces. Also we prove an analogue to KKM principle in modular spaces. Moreover, as an application, we give some su cient conditions which guarantee existence of solutions of minimax problems in which we get Fan's minimax inequality in modular spaces.

C
: Generalized KKM maps de ned by Chang -Zhang [4] in 1991 were extended by several authors. Finally, Lee [15] in 2016 showed that they are simply KKM maps in abstract convex spaces.
In This means that (Yρ , X; co) is a partial KKM space and, hence, satis es a large number of KKM theoretic facts in Section 3.

Jafari et al. [9] -Optimization 66(3) (2017)
A : This paper deals with equilibrium problems in the setting of metric spaces with a continuous convex structure. We extend Fan's 1984 KKM theorem to convex metric spaces in order to employ some weak coercivity conditions to establish existence results for suitable local Minty equilibrium problems, where the involved bifunctions are φ-quasimonotone. By an approach which is based on the concept of the strong φsign property for bifunctions, we obtain existence results for equilibrium problems which generalize some results in the literature.

C
: Fan's 1984 KKM theorem can be extended to Theorem C in Section 2 and a convex metric space is a partial KKM space. Therefore, a convex metric space satis es many properties in Section 3.

Park [30] -NAF 22(2)
A : In 1996, Khamsi established the KKM theorem for hyperconvex metric spaces and applied it to obtain a Schauder type xed point theorem. This line of study has been followed by a large number of authors. In this article, we show that the KKM theorem, best approximation theorem, and the Schauder type xed point theorem for hyperconvex metric spaces due to Khamsi can be extended to partial KKM metric spaces.
Park [31] -RNA 2(2) (2019) A : Since Khamsi found a KKM theorem for hyperconvex metric spaces in 1996, there have appeared a large number of works on them related to the KKM theory. In our previous review [42], we followed the various stages of developments of the KKM theory of hyperconvex metric spaces. In fact, we introduced abstracts of articles on such theory and gave comments or generalizations of the results there if necessary. We noted that many results in those articles are consequences of (partial) KKM space theory developed by ourselves from 2006. The present survey is a continuation of [42] and aims to collect further generalizations of hyperconvex metric spaces related to the KKM theory.

A
: In this paper we obtain a best approximations theorem for set-valued mappings in G-convex spaces. As applications, we derive results on the best approximations in hyperconvex and normed spaces. The obtain results generalize many known results in the literature.

C
: G-convex spaces are obsolete now.

Conclusion
The study of generalized metric type spaces seems to be quite active. From Khamsi's pioneering work, most of the results are closely related to the KKM theory initiated by the present author. However, we note that the following will be helpful to peoples working this eld: 1. Most of works on generalized metric type spaces do not re ect current states of the KKM theory.
2. Most of authors does not know the mapping classes KC and KO, and adopted the obsolete class KKM due to Chang and Yen.
3. Most of authors does not know the concepts of KKM spaces or partial KKM spaces having a large number of properties as shown in Section 3. 4. Generalized metric type spaces should have some special characters other than the KKM theoretic results in Section 3.

5.
Most of results on generalized metric type spaces seem to be similar ones imitating some previous ones.