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Some minimization theorems in generating spaces of quasi-metric family and applications. (English) Zbl 0893.47036

The authors establish new nonconvex minimization theorems in complete generating spaces of a quasi-metric family. W. Takahashi [Pitman Res. Notes Math. 252, 397-406 (1991; Zbl 0760.47029)] proved a nonconvex minimization theorem in complete metric spaces which was generalized by J. S. Ume [Math. Jap. 40, 109-114 (1994; Zbl 0813.47074)]. J. S. Jung, Y. J. Cho, and J. K. Kim [Fuzzy Sets Syst. 61, No. 2, 199-207 (1994; Zbl 0845.54004)] established a Takahashi-type minimization theorem in complete fuzzy metric spaces introduced by O. Kaleva and S. Seikkala [ibid. 12, 215-229 (1984; Zbl 0558.54003)]. Also they obtained the analogue of the Downing-Kirk fixed point theorem wit the help of the minimization theorem. Recently, Chang et al [“Coincidence point theorems and minimization theorems in fuzzy metric spaces”, to appear in Fuzzy Sets Syst.] introduced the concept of generating spaces of quasi-metric family, which generalizes the fuzzy metric spaces of Kaleva and Seikkala and Menger probabilistic metric spaces. They also presented several fixed point theorems and Takahashi-type minimization theorems in complete generating spaces of quasi-metric family.
The present work generalizes the results of Takahashi and Ume and Downing-Kirk’s fixed point theorem in the same spaces. The corresponding results in fuzzy metric spaces and probabilistic metric spaces setting are presented also. The results obtained by the authors extend and improve upon the corresponding results of the various authors mentioned above.

MSC:

47H10 Fixed-point theorems
47S40 Fuzzy operator theory
54C60 Set-valued maps in general topology
58E30 Variational principles in infinite-dimensional spaces
54A40 Fuzzy topology
54E70 Probabilistic metric spaces
47S50 Operator theory in probabilistic metric linear spaces
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