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Báo cáo hóa học: " A DISCRETE FIXED POINT THEOREM OF EILENBERG AS A PARTICULAR CASE OF THE CONTRACTION PRINCIPLE"

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 DISCRETE FIXED POINT THEOREM OF EILENBERG AS A PARTICULAR CASE OF THE CONTRACTION PRINCIPLE | A DISCRETE FIXED POINT THEOREM OF EILENBERG AS A PARTICULAR CASE OF THE CONTRACTION PRINCIPLE JACEK JACHYMSKI Received 6 November 2003 We show that a discrete fixed point theorem of Eilenberg is equivalent to the restriction of the contraction principle to the class of non-Archimedean bounded metric spaces. We also give a simple extension of Eilenberg s theorem which yields the contraction principle. 1. Introduction The following theorem see e.g. Dugundji and Granas 2 Exercise 6.7 pages 17-18 was presented by Samuel Eilenberg on his lecture at the University of Southern California Los Angeles in 1978. I owe this information to Professor Andrzej Granas. This result is a discrete analog of the Banach contraction principle BCP and it has applications in automata theory. Theorem 1.1 Eilenberg . Let X be an abstract set and let Rn f 0 be a sequence of equivalence relations in X such that i X X X R0 3 R1 3 ii n n 0 Rn A the diagonal in X X X iii given a sequence xn f 0 such that xn xn 1 e Rn for all n e N0 there is an x e X such that xn x e Rn for all n e N0. If F is a self-map ofX such that given n e N0 and x y e X x y e Rn Fx Fy e Rn 1 1.1 then F has a unique fixed point x and Fnx x e Rn for each x e X and n e N0. The letter N0 denotes the set of all nonnegative integers. A direct proof of Theorem 1.1 will be given in Section 2. However our main purpose is to show that Eilenberg s theorem ET is equivalent to the restriction of BCP to the class of non-Archimedean bounded metric spaces. This will be done in Section 3. Recall that a metric d on a set X is called non-Archimedean or an ultrametric see de Groot 1 or Engelking 3 page 504 if d x y maxi d x z d z y V x y z e X. 1.2 Copyright 2004 Hindawi Publishing Corporation Fixed Point Theory and Applications 2004 1 2004 31-36 2000 Mathematics Subject Classification 46S10 47H10 54H25 URL http dx.doi.org 10.1155 S1687182004311010 32 A discrete theorem of Eilenberg Then in fact d x y max d x z d z y if d x z d z y and .