Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: COMMON FIXED POINT THEOREMS FOR COMPATIBLE SELF-MAPS OF HAUSDORFF TOPOLOGICAL SPACES | COMMON FIXED POINT THEOREMS FOR COMPATIBLE SELF-MAPS OF HAUSDORFF TOPOLOGICAL SPACES GERALD F. JUNGCK Received 13 July 2004 and in revised form 3 February 2005 The concept of proper orbits of a map g is introduced and results of the following type are obtained. If a continuous self-map g of a Hausdorff topological space X has relatively compact proper orbits then g has a fixed point. In fact g has a common fixed point with every continuous self-map f of X which is nontrivially compatible with g. A collection of metric and semimetric space fixed point theorems follows as a consequence. Specifically a theorem by Kirk regarding diminishing orbital diameters is generalized and a fixed point theorem for maps with no recurrent points is proved. 1. Introduction Let g be a mapping of a topological space X into itself. Let N denote the set of positive integers and w N u 0 . For x e X 0 x is called the orbit of g at x and defined by G x gk x k e w where go x x. Thus if n e w the orbit of g at gn x is the set G gn x gk x k e w and k n . Clearly G gn x c 0 x for n e N. And if X has a metric or semimetric d we will designate the diameter of a set M c X by 8 M which of course is defined 8 M sup d x y x y e M . The purpose of this paper is to introduce the concept of proper orbits and to demonstrate its role in obtaining fixed points. We use cl A to denote the closure of the set A. Definition . Let g be aself-map of a topological space X andlet x e X. The orbit G x of g at x is proper if and only if G x x or there exists n nx e N such that cl G gn x is a proper subset of cl G x . If G x is proper for each x e M c X we will say that g has proper orbits on M. If M X we say g has proper orbits. The concept of proper orbits generalizes the concept of diminishing orbital diameters which was introduced by Belluce and Kirk 1 in 1969. They introduced the concept of mappings with diminishing orbital diameters to obtain fixed point theorems for non-expansive self-maps of metric spaces.
