tailieunhanh - Báo cáo hóa học: " A FIXED POINT THEOREM FOR ANALYTIC FUNCTIONS VALENTIN MATACHE"

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: A FIXED POINT THEOREM FOR ANALYTIC FUNCTIONS VALENTIN MATACHE | A FIXED POINT THEOREM FOR ANALYTIC FUNCTIONS VALENTIN MATACHE Received 4 August 2004 We prove that each analytic self-map of the open unit disk which interpolates between certain M-tuples must have a fixed point. 1. Introduction Let U denote the open unit disk centered at the origin and T its boundary. For any pair of distinct complex numbers z and w and any positive constant k we consider the locus of all points z in the complex plane C having the ratio of the distances to w and z equal to k that is we consider the solution set of the equation K - wl k. K - z k We denote that set by A z w k and following 1 call it the Apollonius circle of constant k associated to the points z and w. The set A z w k is a circle for all values of k other than 1 when it is a line. In this paper we consider z w e U show that if z w then necessarily A z w V 1 - w 2 1 - z 2 meets the unit circle twice consider the arc on the unit circle with those endpoints situated in the same connected component of C A z w V 1 - w 2 1 - z 2 as z and denote it by rz w. We prove that if Z z1 . zN and w w1 . wN are N-tuples with entries in U such that zj wj for all j 1 . N and N T u U w j 1 then each analytic self-map of U interpolating between Z and w must have a fixed point. The next section contains the announced fixed point theorem Theorem . Copyright 2005 Hindawi Publishing Corporation Fixed Point Theory and Applications 2005 1 2005 87-91 DOI 88 A fixed point theorem for analytic functions 2. The fixed point theorem For each eie e T and k 0 the set HD ei0 k z e U eie - z 2 k 1 - z 2 called the horodisk with constant k tangent at eie is an open disk internally tangent to T at eie whose boundary HC e 0 k z e U Ịeie - z 2 k 1 - z 2 is called the horocycle with constant k tangent at eie. The center and radius of HC eie k are given by eie _ k . C 1 k R 1 k respectively. One should note that HD eie k extends to exhaust U as k - TO. Let p be a self-map of U. For