Fixed point theorem on strong fuzzy metric space

Abstract

The problem of constructing a satisfactory theory of fuzzy metric spaces has been investigated by several authors from different point of view. The concept of fuzzy sets was introduced by L. A. Zadeh. Following fuzzy metric space modified by I. Kramosil, J. Mickalek -George and Veeramani using continuous triangular norm. Modern fuzzy technique is used to generalized some conventional and latest results. A fuzzy metric space is an ordered triple (X, M,∗ ) such that X is a set, ∗ is continuous triangular norm and M is a function defined on X × X × [0, ∞) with value in [0,1] satisfying certain axioms and M is called fuzzy metric on X. The objective of this paper is to prove fixed point theorem of strong fuzzy metric space by using control function under minimum continuous triangular norm condition. That is (X, M,∗ ) be a complete strong fuzzy metric space with minimum continuous triangular norm ∗ and T is self-mapping in X. If there exists control function ψ and λi = λi(t), i = 1, 2, 3, 4, 5 with λi =≥ 0 and λ1 + λ2 + λ3 + 2λ4 + λ1 < 1 such that ψ[M(Tu, Tv, t)] ≤ λ1ψ[M(u, Tu, t)] + λ2ψ[M(v, Tv, t)] + λ3ψ[M(Tu, v, t)] + λ4ψ[M(u, Tv, t)] + λ5ψ[M(u, v, t)]. Then T has a unique fixed point in X. In addition, we illustrated some examples of strong fuzzy metric space.