Abstract
In this paper, we establish a coincidence theorem for set-valued mappings in fuzzy metric spaces with a view to generalizing Downing-Kirk's fixed point theorem in metric spaces. As consequences, we obtain Caristi's coincidence theorem for set-valued mappings and a more general type of Ekeland's variational principle 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.
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