Abstract
The purpose of this paper is to generalize Caristi's fixed point theorem to fuzzy mappings and to obtain two formally stronger Caristi type fixed point and common fixed point theorems for set-valued mappings and sequences of set-valued mappings. The results presented in this paper generalize the corresponding results of Caristi [ Trans Amer. Math. Soc. 215 (1976) 241–251] and Kasahara [ Math. Seminar Notes 35 (2) (1975) 229–232]. In addition, by a simple method the equivalence between Caristi's fixed point theorem for fuzzy mappings and Ekeland's variational principle is shown. This result improves and extends some important results of Dancs et al., Kirk, Shi, and Sullivan.
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