Abstract
We generalize the Hausdorff fuzzy metric in the sense of Rodríguez‐López and Romaguera, and we introduce a new M∞‐fuzzy metric, where M∞‐fuzzy metric can be thought of as the degree of nearness between two fuzzy sets with respect to any positive real number. Moreover, under ϕ‐contraction condition, in the fuzzy metric space, we give some common fixed point theorems for fuzzy mappings.
Highlights
The concept of fuzzy sets was introduced initially by Zadeh 1 in 1965
Let A be a nonempty subset of a fuzzy metric space X, M, ∗
Let C X be the set of all nonempty compact subsets of a fuzzy metric space X, M, ∗, A, B ∈ C X, t > 0, according to 26, the Hausdorff fuzzy metric HM on C X × C X × 0, ∞ is defined as HM A, B, t min inf M a, B, t, inf M A, b, t a∈A
Summary
The concept of fuzzy sets was introduced initially by Zadeh 1 in 1965. After that, to use this concept in topology and analysis, many authors have expansively developed the theory of fuzzy sets and application 2, 3. In the theory of fuzzy topological spaces, one of the main problems is to obtain an appropriate and consistent notion of fuzzy metric space This problem was investigated by many authors 4–13 from different points of view. Journal of Applied Mathematics a Hausdorff fuzzy metric, where Hausdorff fuzzy metric can be thought of as the degree of nearness between two crisp nonempty compact sets with respect to any positive real number In this present investigation, considering the Hausdorff-Pompeiu metric and theories on fuzzy metric spaces in the sense of George and Veeramani together, we study the degree of nearness between two fuzzy sets as a natural generalization of the degree of nearness between two crisp sets, in turn, it helps in studying new problems in fuzzy topology. Under φ-contraction condition, we give some common fixed point theorems in the fuzzy metric space on fuzzy sets
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