Fixed point property and degree of coincidence

W. Kulpa

https://orcid.org/0000-0002-6913-1282

Poland

Cardinal Stefan Wyszyński University in Warsaw image/svg+xml

M. Turzanski

https://orcid.org/0000-0002-3700-2558

Poland

Cardinal Stefan Wyszyński University in Warsaw image/svg+xml

|

Accepted: 2024-09-05

|

Published: 2025-04-01

DOI: https://doi.org/10.4995/agt.2025.21711
Funding Data

Downloads

Keywords:

KKM function family, universal function, degree of coincidence, frame

Supporting agencies:

This research was not funded

Abstract:

We introduce the concept of degree of coincidence to measure the divergence from the fixed point property in case a space does not have the FPP.

Show more Show less

References:

R. Cauty, Solution du problème de point fixe de Schauder, Fundamenta. Mathematicae 170 (2001), 231-246. https://doi.org/10.4064/fm170-3-2

M. Dodson, A Brouwer type coincidence theorem and the fundamental theorem of algebra, Canadian Mathematical Bulletin 27 (1984), 478-480. https://doi.org/10.4153/CMB-1984-075-9

T. Fujimoto, N. G. A. Karunathilake, and R. Ranade, A proof of the Fundamental Theorem of Algebra via Reich's coincidence theorem and a reason why there exists no proof based on a simple application of Brouwer's fixed point theorem, Kagawa University Economic Review 89, no. 1 (2016), 1-13.

A. Granas, KKM-Maps, The Scottish Book; Mathematics from the Scottish Café with Selected Problems from the New Scottish Book, R. Mauldin ed., 2nd Edition, Birkhäuser, 2015; Chapter 5, 34-48. https://doi.org/10.1007/978-3-319-22897-6_5

A. Granas and J. Dugunji, Fixed point theory, Springer Monographs in Mathematics, 2003. https://doi.org/10.1007/978-0-387-21593-8

H. Hahn, Mengentheoretische Charakterisierung der stetigen Kurve, Sitzungsber. Akad. Wiss Wien 123 (1914), 2433-2489.

W. Holsztynski, Une generalization du theoreme de Brouwer sur les points invariants, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. et Physics 12 (1964), 602-607.

W. Holsztynski, Universal mappings and fixed point theorems,Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. et Physics 15 (1967), 421-426.

W. Holsztynski, A remark on the universal mappings of 1-dimensional continua, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. et Physics 15 (1967), 525-527.

A. Idzik, W. Kulpa, and P. Mackowiak, Equivalent forms of the Brouwer fixed point theorem II, Topological Methods in Nonlinear Analysis 57, no. 1 (2021), 57-71. https://doi.org/10.12775/TMNA.2020.036

B. Knaster, K. Kuratowski, and S. Mazurkiewicz, Ein beweis des fixpunktsatzes für n-dimensionale simplexe, Fundamenta Mathematicae (in German) 14 (1929), 132-137. https://doi.org/10.4064/fm-14-1-132-137

W. Kulpa, An integral criterion for coincidence property, Radovi Matematicki 6 (1990), 313-321.

W. Kulpa, Sandwich type theorems, Acta Universitatis Carolinae, Mathematica et Physica 35, no. 2 (1994), 45-52.

W. Kulpa, A. Szymanski, and M. Turzanski, Function and colorful extensions of the KKM theorem, Topological Methods in Nonlinear Analysis 56, no. 1 (2020), 313-324. https://doi.org/10.12775/TMNA.2020.015

K. Kuratowski, Introduction to set theory and topology, revised second English ed. Number 101 in International Series of Monographs in Pure and Applied Mathematics. Warsaw: Pergamon Press, 1972.

K. Kuratowski and H. Steinhaus, Une application géométrique du théorème de Brouwer sur les points invariant, Bulletin de L'Académie Polonaise des Sciences CLIII, no. 3-4 (1953), 83-86.

S. Mazurkiewicz, Sur les lignesde Jordan, Fundamenta Mathematicae 1 (1920), 166-209. https://doi.org/10.4064/fm-1-1-166-209

R. L. Moore, Concerning the cut-points of continuous curves and of other closed and connected point-sets, Proc. Natl. Acad. Sci. USA 161, no. 4 (1923), 101-106. https://doi.org/10.1073/pnas.9.4.101

S. Park, One hundred years of Brouwer's fixed point theorem, J. Nat. Acad. Sci. ROK 60, no. 1 (2021), 1-77.

S. Park, A history of the KKM theory, https://www.researchgate.net/publication/324388798.

H. Schimer, A Brouwer type coincidence theorem, Canadian Mathematical Bulletin 9 (1966), 443-446. https://doi.org/10.4153/CMB-1966-053-3

L.-G. Svensson, Large indivisibles: An analysis with respect to price equilibrium and fairness, Econometrica 51, no. 4 (1983), 939-954. https://doi.org/10.2307/1912044

M. Szymik, Homotopies and the universal fixed point property, Order 32 (2015), 301-311. https://doi.org/10.1007/s11083-014-9332-x

Show more Show less