A contraction theorem in fuzzy metric spaces
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A fixed point theorem is proved. Moreover, fuzzy Edelstein’s contraction theorem is described. Finally, the existence of at least one periodic point is shown. 1. Introduction After Zadeh pioneering’s paper [15], where the Theory of Fuzzy Sets was introduced, hundreds of examples have been supplied where the nature of uncertainty in the behavior of a given system possesses fuzzy rather than stochastic nature. Non-stationary fuzzy systems described by fuzzy processes look as their natural extension into the time domina. From different viewpoints they were carefully studied. Fixed-point theory for contraction type mappings in fuzzy metric space is closely related to the fixed-point theory for the same type of mappings in probabilistic metric space of Menger type (see [10, 13]). The concept of fuzzy metric spaces recently have been introduced in different ways by many authors [1, 2, 8]. George and Veeramani [3, 4] modified the concept of fuzzy metric space which has been introduced by Kramosil and Mich´alek [9] and obtained a Hausdorff topology for this kind of fuzzy metric space. Here, we claim that if (X,M, ∗) is a fuzzy metric space, and A a contractive mapping of X into itself such that there exists a point x ∈ X whose sequence of iterates (An (x)) contains a convergent subsequence (Ani (x)); then ξ = limi→∞ Ani (x) ∈ X is a unique fixed point. In addition, we can prove fuzzy Edelstein’s contraction theorem. Note that this happen when we consider the fuzzy metric space in the George and Veeramani’s sense. In addition, it is claimed that fuzzy Edelstein’s contraction theorem is true whenever we consider the fuzzy metric space in the Kramosil and Mich´alek’s sense. Finally, the existence of at least one periodic point will be proved and two question would arise. In order to do this, we recall some concepts and results that will be required in the sequel. Definition 1.1 [12]. A binary operation ∗ : [0,1 × [0,1] → [0,1] is a continuous t-norm if ([0,1], ∗) is a topological monoid with unit 1 such that a ∗ b ≤ c ∗ d whenever a ≤ c and b ≤ d, and a,b,c,d ∈ [0,1]. Copyright © 2005 Hindawi Publishing Corporation Fixed Point Theory and Applications 2005:3 (2005) 257–265 DOI: 10.1155/FPTA.2005.257
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A contraction theorem in fuzzy metric spaces
Definition 1.2 [3]. The 3-tuple (X,M, ∗) is said to be a fuzzy metric space if X is an arbitrary set, ∗ is a continuous t-norm and M is a fuzzy set on X 2 ×]0, ∞[ satisfying the following conditions: for all x, y,z ∈ X and t,s > 0, (i) M(x, y,t) > 0, (ii) M(x, y,t) = 1 if and only if x = y, (iii) M(x, y,t) = M(y,x,t), (iv) M(x, y,t) ∗ M(y,z,s) ≤ M(x,z,t + s), (v) M(x, y, ·) :]0, ∞[→ [0,1] is continuous. Lemma 1.3 [5]. M(x, y, ·) is nondecreasing for all x, y ∈ X. In order to introduce a Hausdorff topology on the fuzzy metric space, the following definitions are needed. Definition 1.4 [3]. Let (X,M, ∗) be a fuzzy metric space. The open ball B(x,r,t) for t > 0 with center x ∈ X and radius r, 0 < r < 1, is defined as B(x,r,t) = { y ∈ X : M(x, y,t) > 1 − r }. The family {B(x,r,t) : x ∈ X
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