Paper_Ahsar (Proceedings of IICMA 2009, UGM – Yogyakarta)
Dhage in 1992 generalized the concept of metric space to be D-metric space. The concept of D-metric space has been modified to be D*-metric space by Sedghi and Shobe [4]. According to
the concept of fuzzy sets [7], gives a new concept in metric space. The combination between the
concepts of metric space and fuzzy set bring fuzzy metric spaces. There are many researchers in
fuzzy metric spaces. They give some alternatives definition of fuzzy metric space. One of the
alternatives was given by George and Veeramani [2]. Based on the concepts of George and
Veeramani, some concepts on metric spaces were extended by many author’s, one of them is the
Banach fixed point theorem by Gregori and Sapena [1]. In similar way, Sedghi and Shobe [5] have
generalized the concept of D*-metric space to be M-fuzzy metric space. Based on the concepts of
Sedghi and Shobe, some concepts on D*-metric spaces were extended by many author’s, one of
them is the fixed point theorem ([3], [5], [6]).
In this paper we extend the Banach fixed point theorem in M-fuzzy metric space. First, we
define the concepts of convergence and Cauchy sequences in M-fuzzy metric space. After that, we
need to define the suitable concepts of continuous, uniform continuous, and contraction mappings
in M-fuzzy metric space. By all of those concepts, we give the characterization of continuous
mapping in convergence sequences. Finally, we give the Banach fixed point theorem in M-fuzzy
metric space.
