This paper introduces the concept of Fg-metric Space. We generalize G-metric space. Some supporting examples are given.

Metric spaces are generalized by many scholars. Recently, Khatami and Mirzavaziri use a mapping called t-definer to popularize the triangle inequality give generalization of notion metric, which is *-metric. In this paper, we prove that every *-metric space metrizable. Also, study total boundedness completeness spaces.

in this paper, the notion of $psi -$weak contraction cite{rhoades} isextended to fuzzy metric spaces. the existence of common fixed points fortwo mappings is established where one mapping is $psi -$weak contractionwith respect to another mapping on a fuzzy metric space. our resultgeneralizes a result of gregori and sapena cite{gregori}.

we prove a related fixed point theorem for n mappings which arenot necessarily continuous in n fuzzy metric spaces using an implicit relationone of them is a sequentially compact fuzzy metric space which generalizeresults of aliouche, et al. [2], rao et al. [14] and [15].

In this paper we obtain a unique common xed point theorem for sixweakly compatible mappings in G-metric spaces.