Fixed points of order preserving multifunctions

Abstract

Let F : X → X F:X \to X be a multifunction on a partially ordered set ( X , ≦ ) (X, \leqq ) . Suppose for each pair x 1 ≦ x 2 {x_1} \leqq {x_2} and for each y 1 ∈ F ( x 1 ) {y_1} \in F({x_1}) there is a y 2 ∈ F ( y 2 ) {y_2} \in F({y_2}) such that y 1 ≦ y 2 {y_1} \leqq {y_2} . Then sufficient conditions are given such that multifunctions F 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.