Đang chuẩn bị nút TẢI XUỐNG, xin hãy chờ
Tải xuống
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 1.1. 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.
