The rise and fall of MC-spaces

Abstract

In 1994, Llinares introduced mc-spaces and began to study KKM theoretic results on them. Since 1998, he became an L-space theorist and repeated to claim that his mc-spaces generalize G-convex spaces without any justifications. Later he insisted that his mc-spaces are the same as L-spaces. Hence his study on mc-spaces is useless now as the L-space case shown by our previous works. The present article is a continuation of our previous works on L-spaces and concerns with the rise and fall of mc-spaces. This paper will be an important record for the history of the KKM theory.

Keywords

• KKM theorem, Fan's 1961
• G-convex space
• KKM lemma
• (partial) KKM space
• mc-space
• abstract convex space
• L-space

References

1. [1] H. Ben-El-Mechaiekh, S. Chebbi, M. Florenzano, and J.V. Llinares, Abstract convexity and fixed points, J. Math. Anal. Appl. 222 (1998) 138–150.
2. [2] H. Ben-El-Mechaiekh, S. Chebbi, and M. Florenzano, A generalized KKMF principle, J. Math. Anal. Appl. 309 (2005) 583–590.
3. [3] G.L. Cain Jr. and L. Gonzalez, The Knaster-Kuratowski-Mazurkiewitz theorem and abstract convex spaces, J. Math. Anal. Appl. 338 (2008) 563–571.
4. [4] L. González, S. Kilmer, and J. Rebaza, From a KKM theorem to Nash equilibria in L-spaes, Top. Appl. 155 (2007) 165–170.
5. [5] C. Horvath, Contractibility and generalized convexity, J. Math. Anal. Appl. 156 (1991) 341-357.
6. [6] C.D. Horvath and J.V. Llinares Ciscar, Maximal elements and fixed points for binary relations on topological ordered spaces, J. Math. Econom. 25 (1996) 291–306.
7. [7] W. Kulpa and A. Szymanski, Applications of general infimum principles to fixed-point theory and game theory, Set-valued Anal. 16 (2008) 375–398.
8. [8] J.V. LLinares, Abstract convexity. Fixed points and applications, Ph.D. Thesis. Univ. de Alicante, Spain. 1994.

9. [9] J.V. LLinares, Existence of maximal elements in a binary relation relaxing the convexity condition, WP-AD 95-10, Jan. 1995. Appeared ResearchGate on 15 May 2014.
10. [10] J.V. Llinares, Unified treatment of the problem of existence of maximal elements in bindary relations. A characterization, J. Math. Econom. 29 (1998) 285–302.
11. [11] J.V. LLinares, Existence of equilibrium in generalized games with non-convex strategy spaces, Working Paper CEPREMAP Couverture Orange no.98-01, Janvier 1998.
12. [12] J.V. LLinares, Abstract convexity, some relations and applications, Working Paper CEPREMAP Couverture Orange no.98- 03, Mars 1998.
13. [13] J.V. LLinares, Existence of equilibrium in generalized games with abstract convexity structure, J. Optim. Theory Appl. bf 105(1) (2000) 149–160.
14. [14] J.V. LLinares, Abstract convexity. Some relations and applications, Optimization 51(6) (2002) 797–818.
15. [15] H. Lu, A section theorem in topological ordered spaces and its applications to the existence of Pareto equilibria for multi- objective games, 2009 Inter. Joint Conf. Artificial Intelligence, IEEE, DOI 10.1109/JCAI.2009.14 3
16. [16] D.T. Luc, E. Sarabi, and A. Soubeyran, Existence of solutions in variational relation problems without convexity, J. Math. Anal. Appl. 364 (2010) 544–555.
17. [17] Q. Luo, KKM and Nash equilibria type theorems in topological ordered spaces, J. Math. Anal. Appl. 264 (2001) 262–269. doi:10.1006/jmaa.2001.7624
18. [18] S. Park, Another five episodes related to generalized convex spaces, Nonlinear Funct. Anal. Appl. 3 (1998), 1–12.
19. [19] S. Park, On generalizations of the KKM principle on abstract convex spaces, Nonlinear Anal. Forum 11 (2006), 67–77.
20. [20] S. Park, Remarks on KC-maps and KO-maps in abstract convex spaces, Nonlinear Anal. Forum 12(1) (2007) 29–40.
21. [21] S. Park, Examples of KC-maps and KO-maps on abstract convex spaces, Soochow J. Math. 33(3) (2007) 477–486.
22. [22] S. Park, Elements of the KKM theory on abstract convex spaces, J. Korean Math. Soc. 45(1) (2008) 1–27.
23. [23] S. Park, New foundations of the KKM theory, J. Nonlinear Convex Anal. 9(3) (2008a), 331–350.
24. [24] S. Park, Comments on the KKM theory on ? A -spaces, PanAmerican Math. J. 18(2) (2008b), 61–71.
25. [25] S. Park, Generalized convex spaces, L-spaces, and FC-spaces, J. Global Optim. 45(2) (2009) 203–210.
26. [26] S. Park, The KKM principle in abstract convex spaces: Equivalent formulations and applications, Nonlinear Anal. 73 (2010) 1028–1042.
27. [27] S. Park, The rise and decline of generalized convex spaces, Nonlinear Anal. Forum 15 (2010) 1–12.
28. [28] S. Park, Comments on abstract convexity structures on topological spaces, Nonlinear Anal. 72 (2010) 549–554.
29. [29] S. Park, Several episodes in recent studies on the KKM theory, Nonlinear Anal. Forum 15 (2010) 13–26.
30. [30] S. Park, New generalizations of basic theorems in the KKM theory, Nonlinear Anal. 74 (2011) 3000–3010.
31. [31] S. Park, Remarks on simplicial spaces and L* -spaces of Kulpa and Szymanski, Comm. Appl. Nonlinear Anal. 19(1) (2012) 59–69.
32. [32] S. Park, A review of the KKM theory on pi A -spaces or GFC-spaces, Advances in Fixed Point Theory 3(2) (2013) 353–382.
33. [33] S. Park, A unified approach to generalized KKM maps, J. Nat. Acad. Sci., ROK, Nat. Sci. Ser. 55(1) (2016) 1–20.
34. [34] S. Park, The rise and fall of L-spaces, Adv. Th. Nonlinear Anal. Appl. 4(3) (2020) 152–166.
35. [35] S. Park, The rise and fall of L-spaces, II, Adv. Th. Nonlinear Anal. Appl. 5(1) (2021) 1–15.
36. [36] S. Park, Revisit to Generalized KKM maps, to appear.
37. [37] S. Park and H. Kim, Admissible classes of multifunctions on generalized convex spaces, Proc. Coll. Natur. Sci. SNU 18 (1993) 1–21.
38. [38] M.C. Sanchez, J.-V. Llinares, and B. Subiza, A KKM-result and an application for binary and non-binary choice functions, Economic Theory 21 (2003) 185–193.