Hostname: page-component-7bb8b95d7b-5mhkq Total loading time: 0 Render date: 2024-09-18T03:37:39.787Z Has data issue: false hasContentIssue false
  • English
  • Français

On a homotopy invariant for simplexes and its application

Published online by Cambridge University Press:  17 April 2009

Chung-Wei Ha
Chung-Wei Ha
Affiliation:
Department of Mathematics, National Tsing Hua University, Hsin Chu, Taiwan
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Based on a congruence relation of K. Fan in the integer labelling of pseudomanifolds, we follow an idea of H. Sies and define a mod 2 homotopy invariant d for a class of continuous functions defined on an n-simplex into ℝn+1 ＼ {0} satisfying a certain boundary condition. Using the homotopy invariance of d, generalised coincidence theorems are proved which unify and extend some existence theorems of K. Fan. Our results also contain Kakutani's fixed point theorem and a new covering property of simplexes of Shapley's type.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 55 , Issue 1 , February 1997 , pp. 73 - 80
Copyright
Copyright © Australian Mathematical Society 1997

References

[1]Aubin, J.-P., Optima and equilibria: an introduction to nonlinear analysis (Springer-Verlag, Berlin, Heidelberg, New York, 1993).CrossRefGoogle Scholar
[2]Fan, K., ‘On the equilibrium value of a system of convex and concave functions’, Math. Z. 70 (1958), 271–280.CrossRefGoogle Scholar
[3]Fan, K., ‘A covering property of simplexes’, Math. Scand. 22 (1968), 17–20.CrossRefGoogle Scholar
[4]Fan, K., ‘A combinatorial property of pseudomanifolds and covering properties of simplexes’, J. Math. Anal. Appl. 31 (1970), 68–80.CrossRefGoogle Scholar
[5]Ha, C.W., ‘Minimax and fixed point theorems’, Math. Ann. 248 (1980), 73–77.CrossRefGoogle Scholar
[6]Ha, C.W., ‘On Ky Fan's covering theorems for simplexes’, Bull. Austral. Math. Soc. 52 (1995), 247–252.CrossRefGoogle Scholar
[7]Ichiishi, T., ‘Alternative version of Shapley's theorem on closed coverings of a simplex’, Proc. Amer. Math. Soc. 104 (1988), 759–763.Google Scholar
[8]Kakutani, S., ‘A generalization of Brouwer's fixed point theorem’, Duke Math. J. 8 (1941), 457–459.CrossRefGoogle Scholar
[9]Krasnoselskii, M.A., Topological methods in the theory of nonlinear integral equations (Pergamon Press, New York, 1964).Google Scholar
[10]Sies, H., ‘Topological degree and Sperner's lemma’, Fund. Math. 118 (1983), 135–149.CrossRefGoogle Scholar