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Ekeland’s variational principle and Caristi’s coincidence theorem for set-valued mappings in probabilistic metric spaces. (English) Zbl 0880.47030
The authors prove a common generalization of I. Ekeland’s variational principle [Bull. Am. Math. Soc., New Ser. 1, 443-474 (1979; Zbl 0441.49011)] and J. Caristi’s coincidence theorem [Trans. Am. Math. Soc. 215, 241-251 (1976; Zbl 0305.47029)] for set-valued mappings in probabilistic metric spaces. They also give a direct proof of the equivalence of these two theorems in probabilistic metric spaces, generalizing a previous result of S.Z. Shi [Advan. Math., Beijing, 16, 203-206 (1987; Zbl 0621.54030)].
47H04Set-valued operators
47H10Fixed point theorems for nonlinear operators on topological linear spaces
60B99Probability theory on general structures
49R50Variational methods for eigenvalues of operators (MSC2000)
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