Some new equivalents of the Brouwer fixed point theorem

Abstract

This is to recollect the equivalent formulations of the Brouwer fixed point theorem. We collect a large number of recently known sources of such equivalents. More recently, Jinlu Li obtained two fixed point theorems on newly defined quasi-point-separable topological vector spaces. His theorems extend the Tychonoff fixed point theorem on locally convex t.v.s. However, we note that his new theorems are logically equivalent to the Brouwer fixed point theorem. Consequently, we add up our large list of such equivalents.

Keywords

• Abstract convex space
• KKM theorem
• KKM class of multimaps
• $\mathfrak{A}_c^\kappa$
• $\mathfrak{B}$
• $\mathfrak{K}$
• $\mathfrak{KC}$
• $\mathfrak{KO}$

Kaynakça

1. [1] M. Balaj, A common fixed point theorem with applications to vector equilibrium problems, Appl. Math. Lett. 23 (2010) 241-245.
2. [2] H. Ben-El-Mechaiekh and R. Dimand, The von Neumann minimax theorem Revisited, Fixed Point Theory Applications, Banach Center Publications 77 (2007) 27-33.
3. [3] O. Bueno and J. Cortina, Existence of projected solutions for generalized Nash equilibrium problems, J. Optim. Theory Appl. 191 (2021) 344?362. https://doi.org/10.1007/s10957-021-01941-9
4. [4] S. Caraman and L. Caraman, Brouwer's fixed point theorem and the madeleine moment, J. Math. Arts (2018) 1-18.
5. [5] S. Cobzas, Fixed point theorems in locally convex spaces - The Schauder mapping method, Fixed Point Theory Appl. vol.2006, Article ID 57950, Pages 1-13. DOI 10.1155/FPTA/2006/5795
6. [6] J. Cortina and R. Fierro, Direct and inverse maximum theorems, and some applications, February 1, 2022. arXiv:2201.13136v1 [math.OC]
7. [7] J.A. De Loera, X. Goaoc, F.E. Meunier, and N.H. Mustafa, The discrete yet ubiquitous theorems of Caratheodory, Helly, Sperner, Tucker, and Tverberg, Bull. Amer. Math. Soc. 56(3) (2019) 415?511. https://doi.org/10.1090/bull/1653.
8. [8] K. Fan, A generalization of Tychonoff's fixed point theorem, Math. Ann. 142 (1961) 305-310.

9. [9] R. Fierro, An intersection theorem for topological vector spaces and applications, J. Optim. Theory Appl. 191 (2021) 118-133. https://doi.org/10.1007/s10957-021-01927-7
10. [10] A. Granas, KKM-maps and their applications to nonlinear problems, The Scottish Book (R. D. Mauldin, ed.), pp.45-61. Birkhäuser, Boston, 1981.
11. [11] G.H. Greco and M.P. Moschen, A minimax inequality for marginally semi-continuous functions, Minimax Theory and Applications (B. Ricceri and S. Simons, eds.), pp.41-51. Kluwer Acad. Publ., Netherlands, 1998.
12. [12] C.D. Horvath and M. Lassonde, Intersection of sets with n-connected unions, Proc. Amer. Math. Soc. 125 (1997) 1209- 1214.
13. [13] A. Idzik, W. Kulpa, and P. Mackowiak, Equivalent forms of the Brouwer fixed point theorem, I. Topol. Meth. Nonlinear Anal. 44(1) (2014) 263-276.
14. [14] A. Idzik, W. Kulpa, and P. Mackowiak, Equivalent forms of the Brouwer fixed point theorem, II. Topol. Meth. Nonlinear Anal. 57(1) (2021) 57-71.
15. [15] W. Kulpa, On Shapley KKMS theorem, Acta Univ. Carolinae. Math. Phys. 46(2) (2005) 51-54.
16. [16] T. Le, C.L. Van, N.-S. Pham, and C. Saglam, Sperner lemma, fixed point theorems, and the existence of equilibrium, mpra.ub.uni-muenchen.de 2020.
17. [17] J. Li, Split Nash equilibria for related noncooperative strategic games and applications to economics, Preprint. 22 Nov 2016.
18. [18] J. Li, An extension of Tychonoff's fixed point theorem to pseudonorm adjoint topological vector spaces, Optimizations, Published online: 07 Jul 2020, https://doi.org/10.1080/ 02331934.2020.1789639.
19. [19] J. Li, The fixed point property of quasi-point-separable topological vector spaces, to appear.
20. [20] J. Mawhin, Simple proofs of various fixed point and existence theorems based on exterior calculus, Math. Nach. 278 (2005) 1607-1614.
21. [21] O.R. Musin, KKM type theorems with boundary conditions, J. Fixed Point Theory Appl. 19 (2017) 2037-2049. DOI 10.1007/s11784-016-0388-7
22. [22] L.D. Muu and X.T. Le, On fixed point approach to equilibrium problem, arXiv:2104.13284v1 [math.OC] 27 Apr 2021.
23. [23] S. Park, A generalization of the Brouwer fixed point theorem, Bull. Korean Math. Soc. 28 (1991) 33-37.
24. [24] S. Park, Ninety years of the Brouwer fixed point theorem, Vietnam J. Math. 27 (1999) 187-222.
25. [25] S. Park, The KKM principle in abstract convex spaces: Equivalent formulations and applications, Nonlinear Anal. 73 (2010) 1028-1042.
26. [26] 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.2011:98. doi:10.1186/1687-1812- 2011-98.
27. [27] S. Park, The Fan minimax inequality implies the Nash equilibrium theorem, Appl. Math. Lett. 24 (2011) 2206-2210.
28. [28] S. Park, Remarks on marginally closed-valued KKM maps and related matters, Nonlinear Anal. Forum 17 (2012) 23-30.
29. [29] S. Park, Recent applications of the Fan-KKM theorem, Nonlinear Analysis and Convex Analysis, RIMS Kôkyûroku, Kyoto Univ. 1841 (2013) 58-68.
30. [30] S. Park, A unified generalization of the Brouwer fixed point theorem, J. Fixed Point Theory 2020, 2020:5
31. [31] S. Park, One hundred years of the Brouwer fixed point theorem, J. Nat. Acad. Sci., ROK, Nat. Sci. Ser. 60(1) (2021) 1-77.
32. [32] S. Park, Extending KKM theory to a large scale logical system, [H.-C. Lai Memorial Issue] J. Nonlinear Convex Anal. 22(6) (2021) 1045-1055.
33. [33] S. Park, and K.S. Jeong, Fixed point and non-retract theorems - Classical circular tours, Taiwan. J. Math. 5 (2001) 97-108.
34. [34] H. Petri, and M. Voorneveld, No bullying! A playful proof of Brouwer's fixed-point theorem, J. Math. Econ. 78 (2018) 1-5.
35. [35] Y. Shimalo, Combinatorial proof of Kakutani's fixed point theorem, 1811.08454v1[math.DS] 20 Nov 2018.
36. [36] Y. Tanaka, Kakutani's fixed point theorem for multifunctions with sequentially at most one fixed point and the minimax theorem for two-person zero-sum games: A constructive analysis, Advances in Fixed Point Theory 2(2) (2012) 120-134.
37. [37] P. Tkacz and M. Turza'nski, An n-dimensional version of Steinhaus' chessboard theorem, Top. Appl. 155 (2008) 354-361.
38. [38] M. Turza'nski, Equilibrium theorems as the consequence of the Steinhaus chessboard theorem, Topology Proceedings 25 (2000) 645-653.
39. [39] H. Uzawa, Walras's existence theorem and Brouwer's fixed point theorem, Economic Stud. Quart. 8 (1962) 59-62.
40. [40] M. Vrahatis, A short proof and generalization of Miranda's existence theorem, Proc. Amer. Math. Soc. 107(3) (1989) 701-703.
41. [41] Z.F. Yang, Multipermutation-based intersection theorem and its applications, J. Optim. Theory Appl. 104(2) (2000) 477-487.
42. [42] J. Yu, N.-F. Wang, and Z. Yang, New proofs of equivalence results between equilibrium theorems and fixed-point theorems, Fixed Point Theory Appl. 2016: 69.