tailieunhanh - Báo cáo hóa học: " MERGING OF DEGREE AND INDEX THEORY"

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: MERGING OF DEGREE AND INDEX THEORY | MERGING OF DEGREE AND INDEX THEORY MARTIN VATH Received 14 January 2006 Revised 19 April 2006 Accepted 24 April 2006 The topological approaches to find solutions of a coincidence equation f. x f2 x can roughly be divided into degree and index theories. We describe how these methods can be combined. We are led to a concept of an extended degree theory for function triples which turns out to be natural in many respects. In particular this approach is useful to find solutions of inclusion problems F x e O x . As a side result we obtain a necessary condition for a compact AR to be a topological group. Copyright 2006 Martin Vath. 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 There are many situations where one would like to apply topological methods like degree theory for maps which act between different Banach spaces. Many such approaches have been studied in literature and they roughly divide into two classes as we explain now. All these approaches have in common that they actually deal in a sense either with coincidence points or with fixed points of two functions given two functions f1 f2 X Y the coincidence points on A Q X are the elements of the set coinA fl 2 x e A I fi x f2 x x e A x e f1-1 f2 x we do not mention A if A X . The fixed points on B Q Y are the elements of the image of coin f1 f2 in B that is they form the set fixB f1 fi y e B I 3x y f1 x f2 x y e B y e f2 f- y we do not mention B if B Y . There is a strong relation of this definition with the usual definition of fixed points of a single or multivalued map the coincidence and Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2006 Article ID 36361 Pages 1-30 DOI FPTA 2006 36361 2 Merging of degree and index theory fixed points of a pair 1 f2 of functions corresponds to the usual .