Fixed points of order preserving multifunctions

Abstract:

Let $F:X \to X$ be a multifunction on a partially ordered set $(X, \leqq )$. Suppose for each pair ${x_1} \leqq {x_2}$ and for each ${y_1} \in F({x_1})$ there is a ${y_2} \in F({y_2})$ such that ${y_1} \leqq {y_2}$. Then 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, 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.