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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.

54A40Fuzzy topology
54H25Fixed-point and coincidence theorems in topological spaces
Full Text: DOI
[1] Caristi, J.: Fixed point theorems for mappings satisfying inwardness conditions. Trans. amer. Math. soc. 215, 241-251 (1976) · Zbl 0305.47029
[2] Chang, S. S.; Luo, Q.: Set-valued caristi’s fixed point theorem and Ekeland’s variational principle. Appl. math. Mech. 10, 119-121 (1989) · Zbl 0738.49009
[3] Downing, D.; Kirk, W. A.: A generalization of caristi’s theorem with applications to nonlinear mapping theory. Pacific J. Math. 69, 339-346 (1977) · Zbl 0357.47036
[4] Ekeland, I.: On the variational principle. J. math. Anal. appl. 47, 324-353 (1974) · Zbl 0286.49015
[5] Ekeland, I.: Nonconvex minimization problems. Bull. amer. Math. soc. (New series) 1, 443-474 (1979) · Zbl 0441.49011
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[7] Hadžíc, O.: Fixed point theorems for multi-valued mappings in some classes of fuzzy metric spaces. Fuzzy sets and systems 29, 115-125 (1989) · Zbl 0681.54023
[8] He, P. J.: The variational principle in fuzzy metric spaces and its applications. Fuzzy sets and systems 45, 389-394 (1992) · Zbl 0754.54005
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[13] Schweizer, B.; Sklar, A.: Statistical metric spaces. Pacific J. Math. 10, 313-334 (1960) · Zbl 0091.29801