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Caristi’s fixed point theorem for fuzzy mappings and Ekeland’s variational principle. (English) Zbl 0842.54041

Summary: 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 J. Caristi [Trans. Am. Math. Soc. 215, 241–251 (1976; Zbl 0305.47029)] and S. Kasahara [Math. Semin. Notes, Kobe Univ. 3, No. 2, Paper No. XXXV (1975; Zbl 0341.54056)]. 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 S. Dancs, M. Hegedüs and P. Medvegyev [Acta Sci. Math. 46, 381–388 (1983; Zbl 0532.54030)], W. A. Kirk [Colloq. Math. 36, 81–86 (1976; Zbl 0353.53041)], S. Z. Shi [Adv. Math., Beijing 16, 203–206 (1987; Zbl 0621.54030)] and F. Sullivan [Proc. Am. Math. Soc. 83, 345–346 (1981; Zbl 0468.54021)].

MSC:

54H25 Fixed-point and coincidence theorems (topological aspects)
54A40 Fuzzy topology
47H10 Fixed-point theorems
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References:

[1] Caristi, J., Fixed point theorems for mappings satisfying inwardness conditions, Trans. Amer. Math. Soc., 215, 241-251 (1976) · Zbl 0305.47029
[2] Chang, Shi-sen, Fixed degree for fuzzy mappings and generalization of Fan’s theorem, Fuzzy Sets and Systems, 24, 1, 103-112 (1987) · Zbl 0645.54014
[3] Dancs, M. H., A general ordering and fixed point principle in complete metric spaces, Acta Sci. Math., 46, 381-388 (1983) · Zbl 0532.54030
[4] Ekland, I., On the variational principle, J. Math. Anal. Appl., 47, 324-357 (1974)
[5] Kashara, S., On fixed points in partially ordered sets and Kirk-Caristi theorem, Math. Seminar Notes, Kobe Univ., 35, 2, 229-232 (1975)
[6] Kirk, W. A., Caristi’s fixed point theorem and metric convexity, Colloquim Math., 36, 81-86 (1976) · Zbl 0353.53041
[7] Shi, S. Z., On the equivalence between Ekeland’s variational principle and Caristi’s fixed point theorem, Advan. Math., 16, 2, 203-206 (1987) · Zbl 0621.54030
[8] Sullivan, F., A characterization of complete metric spaces, (Proc. Amer. Math. Soc., 83 (1981)), 345-346 · Zbl 0468.54021
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