A discrete fixed point theorem of Eilenberg as a particular case of the contraction principle
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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 )∞ n=0 be a sequence of equivalence relations in X such that × X = R0 ⊇ R1 ⊇ · · · ; (i) X (ii) ∞ n=0 Rn = ∆, the diagonal in X × X; (iii) given a sequence (xn )∞ n=0 such that (xn ,xn+1 ) ∈ Rn for all n ∈ N0 , there is an x ∈ X such that (xn ,x) ∈ Rn for all n ∈ N0 . If F is a self-map of X such that given n ∈ N0 and x, y ∈ X, (x, y) ∈ Rn =⇒ (Fx,F y) ∈ Rn+1 ,
(1.1)
then F has a unique fixed point x∗ and (F n x,x∗ ) ∈ Rn for each x ∈ X and n ∈ 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) ≤ max d(x,z),d(z, y) 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
∀x, y,z ∈ X.
(1.2)
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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 therefore, each nonArchimedean metric space has the extraordinary geometric property that each three points of it are vertices of an isosceles triangle. We notice that non-Archimedean metrics are useful tools in many problems of fixed point theory (see, e.g., [6, proofs of Theorems 3, 4, and 5] on connections between some nonlinear contractive conditions, [7, 8] on converses to contraction theorems, [5, Example 1] concerning a comparison of two fixed point theorems of the Meir-Keeler type). Moreover, there is also a variant of the BCP for self-maps of a non-Archimedean metric space proved by Prieß-Crampe [10] (also see Petalas and Vidalis [9]). It turns out that, in general, the contraction principle cannot be derived from ET since each mapping satisfying assumptions of the latter theorem need not be surjective unless its domain is a singleton (cf. Corollary 3.3). Therefore, in Section 4 we establish a slight extension of ET (cf. Theorem 4.1) which is strong enough to yield the contraction princi´ [4] goes in a similar direction; however, he was able to obtai
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