Fixed point structures

Muenzenberger, T. B.; Smithson, R. E.

doi:10.1090/S0002-9947-1973-0328900-X

Fixed point structures
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by T. B. Muenzenberger and R. E. Smithson PDF

Trans. Amer. Math. Soc. 184 (1973), 153-173 Request permission

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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