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Proceedings of the American Mathematical Society

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



Fixed points of order preserving multifunctions

Author: R. E. Smithson
Journal: Proc. Amer. Math. Soc. 28 (1971), 304-310
MSC: Primary 06.20; Secondary 54.00
MathSciNet review: 0274349
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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.

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Keywords: Partially ordered sets, fixed point theorems on partially ordered sets, multivalued functions on partially ordered sets, order preserving functions, fixed points for order preserving multivalued functions, partially ordered topological spaces, fixed points of order preserving multifunctions on partially ordered topological spaces
Article copyright: © Copyright 1971 American Mathematical Society