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Coincidence theorems for set-valued mappings and Ekeland’s variational principle in fuzzy metric spaces. (English) Zbl 0867.54018
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 [{\it D. Downing} and {\it W. A. Kirk}, Pac. J. Math. 69, 339-346 (1977; Zbl 0357.47036)] in metric spaces. As consequences, we obtain {\it 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 {\it 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.

##### MSC:
 54A40 Fuzzy topology 54H25 Fixed-point and coincidence theorems in topological spaces
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##### References:
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