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Common fixed point theorems and variational principle in generating spaces of quasi-metric family. (English) Zbl 0986.54016
Summary: We introduce the concept of order relation in generating spaces of quasi-metric family and establish common-fixed-point theorems for set-valued mappings in this spaces. As consequences, we also give a variational principle, fixed point theorems for single-valued and set-valued mappings and equivalence between them in complete generating spaces of quasi-metric family. Some applications of these results to fuzzy metric spaces in the sense of O. Kaleva and S. Seikkala [ibid. 12, 215-229 (1984; Zbl 0558.54003)] and to probabilistic metric spaces are presented.
MSC:
54A40Fuzzy topology
54H25Fixed-point and coincidence theorems in topological spaces
54C60Set-valued maps (general topology)
58E35Variational inequalities (global problems)
49J40Variational methods including variational inequalities