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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On the existence of fixed points in a totally ordered set
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by F. T. Metcalf and T. H. Payne PDF
Proc. Amer. Math. Soc. 31 (1972), 441-444 Request permission

Abstract:

The Tarski fixed point theorem, concerning an isotone mapping of a partially ordered set into itself, is extended to mappings which are not necessarily isotone, but which must map a totally ordered set into itself. The key requirement is that the function be continuous (in a certain sense) whenever it is “decreasing". Then $a \leqq f(a),f(b) \leqq b$, and $a \leqq b \Rightarrow$ the existence of a fixed point of $f$.
References
  • Garrett Birkhoff, Lattice theory, 3rd ed., American Mathematical Society Colloquium Publications, Vol. XXV, American Mathematical Society, Providence, R.I., 1967. MR 0227053
  • Alexander Abian, A fixed point theorem for nonincreasing mappings, Boll. Un. Mat. Ital. (4) 2 (1969), 200–201. MR 0244110
Additional Information
  • © Copyright 1972 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 31 (1972), 441-444
  • DOI: https://doi.org/10.1090/S0002-9939-1972-0286722-7
  • MathSciNet review: 0286722