On Fuzzy -Contractive Mappings in Fuzzy Metric Spaces
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Research Article On Fuzzy ε-Contractive Mappings in Fuzzy Metric Spaces Dorel Mihet¸ Received 24 December 2006; Accepted 1 March 2007 Recommended by Donal O’Regan
We answer into affirmative an open question raised by A. Razani in 2005. An essential role in our proofs is played by the separation axiom in the definition of a fuzzy metric space in the sense of George and Veeramani. Copyright © 2007 Dorel Mihet¸. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Preliminaries In this section, we recall some definitions and results that will be used in the sequel. Definition 1.1 (see [1]). A triple (X,M, ∗), where X is an arbitrary set, ∗ is a continuous t-norm, and M is a fuzzy set on X 2 × (0, ∞), is said to be a fuzzy metric space (in the sense of George and Veeramani) if the following conditions are satisfied for all x, y ∈ X and s,t > 0: (GV-1) M(x, y,t) > 0; (GV-2) M(x, y,t) = 1 if and only if x = y; (GV-3) M(x, y,t) = M(y,x,t); (GV-4) M(x, y, ·) is continuous; (GV-5) M(x,z,t + s) ≥ M(x, y,t) ∗ M(y,z,s). Note (see [2]) that the “separation” condition (GV-2) means that M(x,x,t) = 1 ∀x ∈ X, ∀t > 0, x = y =⇒ M(x, y,t) < 1 ∀t > 0.
(1.1)
Definition 1.2 (see [1]). Let (X,M, ∗) be a fuzzy metric space. A sequence (xn )n∈N in X is said to be convergent if there is x ∈ X such that limn→∞ M(xn ,x,t) = 1 for each t > 0
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Fixed Point Theory and Applications
(the notation limn→∞ xn = x or xn → x will be used). A mapping f : X → X is said to be continuous if f (xn ) → f (x) whenever (xn ) is a sequence in X convergent to x. Definition 1.3 (see [3]). Let (X,M, ∗) be a fuzzy metric space and 0 < ε < 1. A mapping f : X → X is called fuzzy ε-contractive if M( f (x), f (y),t) > M(x, y,t) whenever 1 − ε < M(x, y,t) < 1. The next continuity lemma can be found in [4] (also see [5, Theorem 12.2.3]). Lemma 1.4. Let (X,M, ∗) be a fuzzy metric space. If limn→∞ xn = x and limn→∞ yn = y, then limn→∞ M(xn , yn ,t) = M(x, y,t) for all t > 0. 2. Main results The following theorem has been proved by Razani in [3]. Theorem 2.1 (see [3, Theorem 3.3]). Let (X,M, ∗) be a fuzzy metric space, where the continuous t-norm is defined as a ∗ b = min{a,b}. Suppose f is a fuzzy ε-contractive selfmapping of X such that there exists a point x ∈ X whose sequence of iterates ( f n (x)) contains a convergent subsequence ( f ni (x)). Then ξ = limi→∞ f ni (x) is a periodic point, that is, there is a positive integer k such that f k (ξ) = ξ. In [3, Question 3.7], it has been asked whether Theorem 2.1 would remain true if ∗ is replaced by an arbitrary t-norm. With Theorem 2.3, we answer into affirmative this question. In the proofs of our theorems, we need the following. Lemma 2.2. Every fuzzy ε-contractive mapping in a fuzzy metric space is continuous. Proof. The continuity of the fuzzy ε-contractive mapping f is an immediate consequence of the implication
M(x, y,t) > 1 − ε =⇒ M f (x), f (y
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