The rise and fall of L-spaces, II

Abstract

In 2005, Ben-El-Mechaiekh, Chebbi, and Florenzano obtained a generalization of Ky Fan's 1984 KKM theorem on the intersection of a family of closed sets on non-compact convex sets in a topological vector space. They also extended the Fan-Browder fixed point theorem to multimaps on non-compact convex sets. Since then several groups of the L-space theorists introduced coercivity families and applied them to L-spaces, H-spaces, etc. In this article, we show that better forms of such works can be deduced from a general KKM theorem on abstract convex spaces in our previous works. Consequently, all of the known KKM theoretic results on L-spaces related coercivity families are extended to corresponding better forms on abstract convex spaces. This article is a continuation of our \cite{38} and a revised and extended version of \cite{34}.

Keywords

• KKM theorem
• Fan's 1961
• 1984 KKM theorem
• Fan-Browder fixed point theorem
• minimax inequality
• abstract convex space
• (partial) KKM space

References

1. [1] N. Altwaijry, S. Ounaies, and S. Chebbi, Generalized convexity and applications to fixed points and equilibria, J. Fixed Point Theory Appl. (2018):3 https://doi.org/10.1007/s11784-018-0517-6
2. [2] H. Ben-El-Mechaiekh, Approximations and selections methods for set-valued maps and fixed point theory, Book Chapter, Research Gate, 05 Dec. 2016.
3. [3] H. Ben-El-Mechaiekh, S. Chebbi, M. Florenzano, and J.V. Llinares, Fixed point theorem without convexity, Working Paper 97-22 Economics Series, 11 April 1997, Departamento de Economia Universidad Carlos ill de Madrid CaIle Madrid.
4. [4] 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.
5. [5] H. Ben-El-Mechaiekh, S. Chebbi, and M. Florenzano, A generalized KKMF principle, J. Math. Anal. Appl. 309 (2005) 583–590.
6. [6] S.Y. Chang, A generalization of KKM principle and its applications, Soochow J. Math. 15 (1989), 7-17.
7. [7] S. Chebbi, Minimax inequality and equilibria with a generalized coercivity, J. Appl. Anal. 12 (2006), 117–125.
8. [8] S. Chebbi, Intersection, fixed points and minimax inequalities with a generalized convexity in H-spaces, Arab J. Math. Sc. 12(1) (2006) 7–15.

9. [9] S. Chebbi, Some non-compact quasi-variational inequalities, Nonlinear Anal. 71(12) (2009) e1684–e1687.
10. [10] S. Chebbi, Intersection and minimax inequality with a generalized convexity in H-spaces, ResearchGate, 21 May 2015.
11. [11] S. Chebbi, P. Gourdel, and H. Hammami, A generalization of Fan’s matching theorem, J. Fixed Point Theory Appl. 9 (2011) 117–124.
12. [12] S. Chebbi and B. Samet, Noncompact equilibrium points for set-valued maps, Abstract Appl. Anal. 2014, Article ID 959612, 4pp. http://dx.doi.org/10.1155/2014/959612
13. [13] K. Fan, A generalization of Tychonoff’s fixed point theorem, Math. Ann. 142 (1961) 305–310.
14. [14] K. Fan, Fixed-point and related theorems for noncompact convex sets, [in: Game Theory and Related Topics (O. Moeshlin and D. Pallaschke, Eds.), pp.151–156, North-Holland, Amsterdam, 1979.
15. [15] K. Fan, Some properties of convex sets related to fixed point theorems, Math. Ann. 266 (1984) 519–537.
16. [16] P. Gourdel and H. Hammami, Applications of generalized Ky Fan’s matching theorem in minimax and variational inequality, 2007. ffhalshs-00204627f
17. [17] H. Hammami. A generalized FKKM theorem and variational inequality, 2007. ffhalshs-00204601ff
18. [18] C. Horvath, Contractibility and generalized convexity, J. Math. Anal. Appl. 156 (1991) 341-357.
19. [19] B. Knaster, K. Kuratowski, und S. Mazurkiewicz, Ein Beweis des Fixpunktsatzes für n-Dimensionale Simplexe, Fund. Math. 14 (1929) 132–137.
20. [20] M. Lassonde, On the use of KKM multifunctions in fixed point theory and related topics, J. Math. Anal. Appl. 97 (1983) 151-20l.
21. [21] S. Park, Ninety years of the Brouwer fixed point theorem, Vietnam J. Math. 27 (1999) 187–222.
22. [22] S. Park, Remarks on topologies of generalized convex spaces, Nonlinear Funct. Anal. Appl. 5 (2000) 67–79.
23. [23] S. Park, Remarks on KC-maps and KO-maps in abstract convex spaces, Nonlinear Anal. Forum 12(1) (2007) 29–40.
24. [24] S. Park, Examples of KC-maps and KO-maps on abstract convex spaces, Soochow J. Math. 33(3) (2007) 477–486.
25. [25] S. Park, A brief history of the KKM theory, RIMS Kôkyûroku, Kyoto Univ. 1643 (2009) 1–16.
26. [26] S. Park, Generalizations of the Nash equilibrium theorem in the KKM theory, Takahashi Legacy, Fixed Point Theory Appl., vol. 2010, Article ID 234706, 23pp. doi:10.1155 /2010/234706.
27. [27] S. Park, The KKM principle in abstract convex spaces: Equivalent formulations and applications, Nonlinear Anal. 73 (2010) 1028–1042.
28. [28] S. Park, A genesis of general KKM theorems for abstract convex spaces, J. Nonlinear Anal. Optim. 2 (2011) 133–146.
29. [29] S. Park, Remarks on certain coercivity in general KKM theorems, Nonlinear Anal. Forum 16 (2011) 1–10.
30. [30] S. Park, Applications of some basic theorems in the KKM theory [in: The series of papers on S. Park’s Contribution to the Development of Fixed Point Theory and KKM Theory], Fixed Point Theory Appl. vol.2011, 2011:98. DOI:10.1186/1687- 1812-2011-98.
31. [31] S. Park, Evolution of the 1984 KKM theorem of Ky Fan, Fixed Point Theory Appl. vol.2012, 2012:146. DOI:10.1186/1687- 1812-2012-146.
32. [32] S. Park, On some new Ky Fan type minimax inequalities in abstract convex spaces, Nonlinear Analysis and Convex Analysis (NACA 2011, Busan), II, pp. 141–161, Yokohama Publ., Yokohama, 2012.
33. [33] S. Park, A genesis of general KKM theorems for abstract convex spaces: Revisited, J. Nonlinear Anal. Optim. 4(1) (2013) 127–132.
34. [34] S. Park, Generalizations of the KKMF principle having coercing families, J. Nonlinear Anal. Optim. 4(2) (2013) 30–40.
35. [35] S. Park, Comments on “Some remarks on Park’s abstract convex spaces”, Nonlinear Anal. Forum 20 (2015) 161–165.
36. [36] S. Park, Some use of weak topologies in the KKM theory, RIMS Kôkyûroku, Kyoto Univ. 2065 (Aug. 31 - Sep. 2, 2016), Apr. 2018, 51–62.
37. [37] S. Park, On further examples of partial KKM spaces, J. Fixed Point Theory, 2019, 2019:10 (18 June, 2019) 1–18.
38. [38] S. Park, The rise and fall of L-spaces, Adv. Th. Nonlinear Anal. Appl. 4(3) (2020) 152–166.
39. [39] S. Park and H. Kim, Admissible classes of multifunctions on generalized convex spaces, Proc. Coll. Natur. Sci. SNU 18 (1993) 1–21.