Fixed Points of Order Preserving Multifunctions

Let F:X-*X be a multifunction on a partially ordered set (X, ¿). Suppose for each pair «iá*2 and for each yi<EF(xi) there is a yi&F{y¿) such that yièyiThen 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, Cañad. 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. 1. Multifunctions on partially ordered sets. Tarski's result on the existence of a fixed point for an isotone function on a complete lattice is well known (see Birkhoff [3, p. 115, Theorem ll]), and a number of related results have also been published. For example, S. Abian and A. B. Brown [2] published results on nondecreasing maps on a partially ordered set, and A. Abian [l] obtained a result for nonincreasing functions on a totally ordered set. Then in 1954, L. E. Ward, Jr. [ó] published several results for continuous order preserving functions on quasi-ordered and partially ordered topological spaces. The purpose of the present paper is to present results analogous to these for multivalued functions. In this paper a multifunction F:X—*Y is a correspondence such that 0¥"F(x)E Y for each xEX where 0 is the empty set. Multifunctions will be denoted by F, G, etc. Let F be a property of sets. Then a multifunction F is said to be point F in case F(x) has property P for each x in the domain. Finally a fixed point of F is a point x such that xEF(x). Let F:X—+Yand let ^ denote a relation on X and a relation on Y. Then we shall use the following two conditions throughout this paper. Received by the editors March 13, 1970. AMS 1969 subject classifications. Primary 9620, 5485, 5465; Secondary 0630, 5456.