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 be a multifunction on a partially ordered set . Suppose for each pair and for each there is a such that . Then sufficient conditions are given such that multifunctions 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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DOI:
https://doi.org/10.1090/S0002-9939-1971-0274349-1

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