FIXED POINT THEOREMS IN FUZZY METRIC SPACES AND MODULAR METRIC SPACES

The notions of fuzzy metric spaces and of modular metric space are from different sources. Grabiec proved two fixed point theorems in fuzzy metric spaces in the sense of Kramosil and Michalek. In this paper, we introduced a generalized fuzzy metric space which does not require the left continuity as was the case in the sense of Kramosil and Michalek. Two corresponding fixed point results of Grabiec are extended. We also discuss the theorem of Vasuki and show that this result is equivalent to the first fixed point theorem of Grabiec. We also use our new extensions to recover three fixed point theorems recently proved by Martinez-Moreno et al. in modular metric spaces. Several gaps and misleading concepts appeared in the paper of Martinez-Moreno et al. are corrected.