From Simplices to Abstract Convex Spaces: A brief history of the KKM theory

Abstract

We review briefly the history of the KKM theory from the original KKM theorem on simplices in 1929 to the birth of the new partial KKM spaces by the following steps.

(1) We recall some early equivalent formulations of the Brouwer fixed point theorem and the KKM theorem.

(2) We summarize Fan’s foundational works on the KKM theory from 1960s to 1980s.

(3) We note that, in 1983-2005, basic results in the theory were extended to convex spaces by Lassonde, to H-spaces by Horvath, and to G-convex spaces due to Park.

(4) In 2006, we introduced the concept of abstract convex spaces (E,D;Γ) on which we can construct the KKM theory. Moreover, abstract convex spaces satisfying an abstract form of the KKM theorem were called partial KKM spaces. Now the KKM theory becomes the study of such spaces.

(5) Various properties hold for partial KKM spaces and many new types of such spaces are introduced. We state a metatheorem for common properties or applications of such spaces.

(6) Finally, we introduce the partial KKM space versions of the von Neumann minimax theorem, the von Neumann intersection lemma, the Nash equilibrium theorem, and the Himmelberg fixed point theorem.

Keywords

• KM space,partial KKM principle,minimax theorem,equilibrium theorem,fixed point theorem

References

1. [1] S. Park, Some coincidence theorems on acyclic multifunctions and applications to KKM theory, Fixed Point Theory and Applications (K.-K. Tan, ed.), pp.248–277, World Sci. Publ., River Edge, NJ, 1992.
2. [2] S. Park, Ninety years of the Brouwer fixed point theorem, Vietnam J. Math. 27 (1999), 193–232.
3. [3] S. Park, On generalizations of the KKM principle on abstract convex spaces, Nonlinear Anal. Forum 11(1) (2006), 67–77.
4. [4] S. Park, Fixed point theorems on KC-maps in abstract convex spaces, Nonlinear Anal. Forum 11(2) (2006), 117–127.
5. [5] S. Park, Remarks on KC-maps and KO-maps in abstract convex spaces, Nonlinear Anal. Forum 12(1) (2007), 29–40.
6. [6] S. Park, Examples of KC-maps and KO-maps on abstract convex spaces, Soochow J. Math. 33(3) (2007), 477–486.
7. [7] S. Park, Various subclasses of abstract convex spaces for the KKM theory, Proc. National Inst. Math. Sci. 2(4) (2007), 35–47.
8. [8] S. Park, Generalized convex spaces, L-spaces, and FC-spaces, J. Global Optim. 45(2) (2009), 203–210.

9. [9] S. Park, Elements of the KKM theory on abstract convex spaces, J. Korean Math. Soc. 45(1) (2008), 1–27.
10. [10] S. Park, Equilibrium existence theorems in KKM spaces, Nonlinear Anal. 69(2008), 4352–4364.
11. [11] S. Park, New foundations of the KKM theory, J. Nonlinear Convex Anal. 9(3) (2008), 331–350.
12. [12] S. Park, Remarks on weakly KKM maps in abstract convex spaces, Inter. J. Math. Math. Sci. vol.2008, Article ID 423596, 10pp., doi:10.1155/2008/423596.
13. [13] S. Park, Remarks on fixed points, maximal elements, and equilibria of economies in abstract convex spaces, Taiwan. J. Math. 12(6) (2008), 1365–1383.
14. [14] S. Park, Fixed point theory of multimaps in abstract convex uniform spaces, Nonlinear Anal. 71 (2009), 2468–2480.
15. [15] S. Park, Remarks on the partial KKM principle, Nonlinear Anal. Forum 14 (2009), 51–62.
16. [16] S. Park, From the KKM principle to the Nash equilibria, Inter. J. Math. & Stat. 6(S10) (2010), 77–88.
17. [17] S. Park, The KKM principle in abstract convex spaces : Equivalent formulations and applications, Nonlinear Anal. 73 (2010), 1028–1042.
18. [18] S. Park, Applications of the KKM theory to fixed point theory, J. Nat. Acad. Sci., ROK, Nat. Sci. Ser. 50(1) (2011), 21–69..
19. [19] S. Park, A history of the KKM Theory, J. Nat. Acad. Sci., ROK, Nat. Sci. Ser. 56(2) (2017), 1–51.
20. [20] S. Park, Various examples of the KKM spaces, Nonlinear Anal. Forum 23(1) (2018), 1–19.