Minimization theorems for fixed point theorems in fuzzy metric spaces and applications

Abstract In this paper, we establish a minimization theorem in fuzzy metric spaces and make use of the result to obtain a fixed point theorem in fuzzy metric spaces which is similar to the Downing-Kirk fixed point theorem in metric spaces. As the consequences, we obtain Caristi's fixed point theorem and Ekeland's variational principle in fuzzy metric spaces and also give a direct simple proof of the equivalence between these two theorems. Some applications of these results to probabilistic metric spaces are presented.