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Minimization theorems and fixed point theorems in generating spaces of quasi-metric family. (English) Zbl 0986.54015

Summary: Following the approaches of O. Kada, T. Suzuki and W. Takahashi [Math. Jap. 44, No. 2, 381-391 (1996; Zbl 0897.54029)], we define a family of weak quasi-metrics in a generating space of quasi-metric family. By using a family of weak quasi-metrics, we prove a Takahashi-type minimization theorem, a generalized Ekeland variational principle and a general Caristi-type fixed point theorem for set-valued maps in complete generating spaces of quasi-metric family. Also, following the approach of J.-P. Aubin [Optima and equilibria, Grad. Texts Math. 140 (1998; Zbl 0930.91001)], we prove another fixed point theorem for set-valued maps in complete generating spaces of quasi-metric family without the assumption of lower semicontinuity. From our results in complete generating spaces of quasi-metric family, we obtain the corresponding theorems for set-valued maps in complete fuzzy metric spaces.

MSC:

54A40 Fuzzy topology
54H25 Fixed-point and coincidence theorems (topological aspects)
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