Summary: We establish a coincidence theorem for set-valued mappings in fuzzy metric spaces with a view to generalizing Downing-Kirk’s fixed point theorem [D. Downing
and W. A. Kirk
, Pac. J. Math. 69, 339-346 (1977; Zbl 0357.47036
)] in metric spaces. As consequences, we obtain J. Caristi
’s coincidence theorem [Trans. Am. Math. Soc. 215, 241-251 (1976; Zbl 0305.47029
)] for set-valued mappings and a more general type of I. Ekeland
’s variational principle [J. Math. Anal. Appl. 47, 324-353 (1974; Zbl 0286.49015
)] in fuzzy metric spaces. Further, we also give a direct simple proof of the equivalence between these two theorems in fuzzy metric spaces. Some applications of these results to probabilistic metric spaces are presented.