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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Fixed point structures

Authors: T. B. Muenzenberger and R. E. Smithson
Journal: Trans. Amer. Math. Soc. 184 (1973), 153-173
MSC: Primary 54H25; Secondary 54F05, 54F20
MathSciNet review: 0328900
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Abstract: A fixed point structure is a triple $ (X,\mathcal{P},\mathcal{F})$ where X is a set, $ \mathcal{P}$ a collection of subsets of X, and $ \mathcal{F}$ a family of multifunctions on X into itself together with a set of axioms which insure that each member of $ \mathcal{F}$ has a fixed point. A fixed point structure for noncontinuous multifunctions on semitrees is established that encompasses fixed point theorems of Wallace-Ward and Young-Smithson as well as new fixed point theorems for partially ordered sets and closed stars in real vector spaces. Also two other fixed point structures are presented that subsume fixed point theorems of Tarski-Ward-Smithson on semilattices and, more generally, partially ordered sets. Also the Davis-Ward converse to this last fixed point theorem is obtained.

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Keywords: Fixed points, partial order, chain topology, fixed points for a class of multifunctions, arcwise connected spaces, fixed point structures, semitrees
Article copyright: © Copyright 1973 American Mathematical Society