Some Results on Fuzzy metric space and some examples of fuzzy b-metric space

Abstract

The problem of constructing a satisfactory theory of fuzzy metric spaces has been investigated by several researchers from different point of view. The concept of fuzzy sets was introduced by Zadeh. Following fuzzy metric space and fuzzy b−metric space modified by Kramosil, Mickalek-George and Veeramani using continuous triangular norm. A binary operation ∗ : [0, 1] × [0, 1] → [0, 1] is a continuous triangular norm t-norm, if ∗ is associative, commutative, continuity, monotonicity and 1 acts as identity element. Some typical examples of t−norm are product t−norm, minimum t−norm, lukasiewitz t− norm and hamacher t−norm. In our work we used minimum triangular t−norm and Banach fixed point theorem to prove fixed point theorem in Fuzzy metric space and discuss some examples of Fuzzy b−metric space. Letting (X, M, ∗) be a complete fuzzy metric space and T : X → X is a continuous function satisfying the condition M(T x, T y, t) ≥ min {M(x, T x, t), M(y, T y, t), M(x, y, t)} and limt→∞ M(x, y, t) = 1 , where x, y ∈ X, x 6= y and M is a Fuzzy set. We proved T has a fixed point in X. Moreover we proved some examples of fuzzy b− metric space under minimum t−norm condition.