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.