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 | References | Similar Articles | Additional Information

Abstract: A fixed point structure is a triple where *X* is a set, a collection of subsets of *X*, and a family of multifunctions on *X* into itself together with a set of axioms which insure that each member of 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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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1973-0328900-X

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