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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality 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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Fixed points of order preserving multifunctions
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by R. E. Smithson PDF
Proc. Amer. Math. Soc. 28 (1971), 304-310 Request permission

Abstract:

Let $F:X \to X$ be a multifunction on a partially ordered set $(X, \leqq )$. Suppose for each pair ${x_1} \leqq {x_2}$ and for each ${y_1} \in F({x_1})$ there is a ${y_2} \in F({y_2})$ such that ${y_1} \leqq {y_2}$. Then sufficient conditions are given such that multifunctions $F$ satisfying the above condition will have a fixed point. These results generalize the Tarski Theorem on complete lattices, and they also generalize some results of S. Abian and A. B. Brown, Canad. J. Math 13 (1961), 78-82. By similar techniques two selection theorems are obtained. Further, some related results on quasi-ordered and partially ordered topological spaces are proved. In particular, a fixed point theorem for order preserving multifunctions on a class of partially ordered topological spaces is obtained.
References
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Additional Information
  • © Copyright 1971 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 28 (1971), 304-310
  • MSC: Primary 06.20; Secondary 54.00
  • DOI: https://doi.org/10.1090/S0002-9939-1971-0274349-1
  • MathSciNet review: 0274349