Fuzzy cone metric spaces

Abstract

In this study, we define the fuzzy cone metric space, the topology induced by this space and some related results of them. Also we state and prove the fuzzy cone Banach contraction theorem. c ⃝2015 All rights reserved.

Some topological properties of fuzzy cone metric spaces

“…In [10], we generalized the concept of fuzzy metric space of George and Veeramani by replacing the (0, ∞) interval by int(P ) where P is a cone as follows: A fuzzy cone metric space is a 3-tuple (X, M, * ) such that P is a cone of E, X is nonempty set, * is a continuous t-norm and M is a fuzzy set on X 2 ×int (P ) satisfying the following conditions, for all x, y, z ∈ X and t, s ∈ int (P ) (that is t θ, s θ)…”

“…In [10], it was proven that every fuzzy cone metric space (X, M, * ) induces a Hausdorff first-countable topology τ f c on X which has as a base the family of sets of the form {B(x, r, t) : x ∈ X, 0 < r < 1, t θ}, where B(x, r, t) = {y ∈ X : M (x, y, t) > 1 − r} for every r with 0 < r < 1 and t θ. A fuzzy cone metric space (X, M, * ) is called complete if every Cauchy sequence in it is convergent, where a sequence {x n } is said to be a Cauchy sequence if for any ε ∈ (0, 1) and any t θ there exists a natural number n 0 such that M (x n , x m , t) > 1 − ε for all n, m ≥ n 0 , and a sequence {x n } is said to converge to x if for any t θ and any r ∈ (0, 1) there exists a natural number n 0 such that M (x n , x, t) > 1 − r for all n ≥ n 0 [10].…”

“…After Zadeh [13] introduced the theory of fuzzy sets, many authors have introduced and studied several notions of metric fuzziness ( [2], [3], [4], [8], [9]) and metric cone fuzziness ( [1], [10] from different points of view).…”

“…By modifying the concept of metric fuzziness introduced by George and Veeramani [4],Öner et al [10] studied the notion of fuzzy cone metric spaces. In particular, they proved that every fuzzy cone metric space generates a Hausdorff first-countable topology.…”

Fixed point theorems in fuzzy cone metric spaces

“…In 2015, Oner et al [17] introduced the notation of fuzzy cone metric space which generalized the notation of fuzzy metric space by George and Veeramani. They also presented some structural properties of fuzzy cone metric spaces and proved a fixed point theorem under a fuzzy cone contraction condition.…”