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 . Dugundji and Granas 2 Exercise 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 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 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 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. Copyright 2004 Hindawi Publishing Corporation Fixed Point Theory and Applications 2004 1 2004 31-36 2000 Mathematics Subject Classification 46S10 47H10 54H25 URL http 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 .
