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: EPSILON NIELSEN FIXED POINT THEORY | EPSILON NIELSEN FIXED POINT THEORY ROBERT F. BROWN Received 11 October 2004 Revised 17 May 2005 Accepted 21 July 2005 Let f X X be a map of a compact connected Riemannian manifold with or without boundary. For e 0 sufficiently small we introduce an e-Nielsen number Ne f that is a lower bound for the number of fixed points of all self-maps of X that are e-homotopic to f. We prove that there is always a map g X X that is e-homotopic to f such that g has exactly Ne f fixed points. We describe procedures for calculating Ne f for maps of 1-manifolds. Copyright 2006 Robert F. Brown. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction Forster has applied Nielsen fixed point theory to the study of the calculation by computer of multiple solutions of systems of polynomial equations using a Nielsen number to obtain a lower bound for the number of distinct solutions 4 . Because machine accuracy is finite the solution procedure requires approximations but Forster s information is still applicable to the original problem. The reason is that sufficiently close functions on well-behaved spaces are homotopic and the Nielsen number is a homotopy invariant. The point of view of numerical analysis concerning accuracy is described by Hildebrand in his classic text 5 in the following way. Generally the numerical analyst does not strive for exactness. Instead he attempts to devise a method which will yield an approximation differing from exactness by less than a specified tolerance. The work of Forster does not involve an initially specified tolerance. In particular although the homotopy between two sufficiently close maps is through maps that are close to both Forster puts no limitation on the homotopies he employs. The purpose of this paper is to introduce a type of Nielsen fixed point theory that does assume .
