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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)].
##### MSC:
 47H04 Set-valued operators 47H10 Fixed point theorems for nonlinear operators on topological linear spaces 60B99 Probability theory on general structures 49R50 Variational methods for eigenvalues of operators (MSC2000)
##### References:
 [1] A.Bharucha-Reid, Fixed point theorems in probabilistic analysis,Bull. Amer. Math. Soc. 82 (1976), 641–657. · Zbl 0339.60061 · doi:10.1090/S0002-9904-1976-14091-8 [2] G.Bocsan, On some fixed point theorems in probabilistic metric spaces,Math. Balkanica 4, (1974), 67–70. [3] G. L.Cain, Jr. and R. H.Kasriel, Fixed and periodic points of local contraction mappings on probabilistic metric spaces,Math. Systems Theory 9 (1976), 289–297. [4] J.Caristi, Fixed point theorems for mappings satisfying inwardness conditions,Trans. Amer. Math. Soc. 215 (1976), 241–251. · doi:10.1090/S0002-9947-1976-0394329-4 [5] S. S.Chang, A common Fixed point theorem for commuting mappings,Proc. Amer. Math. Soc. 83, (1981), 645–652. · doi:10.1090/S0002-9939-1981-0627712-7 [6] S. S.Chang, On some fixed point theorems in PM-spaces,Z. Wahrsch. Verw. Gebiete 63 (1983), 463–477. · doi:10.1007/BF00533720 [7] S. S.Chang and Q.Luo, Set-valued Caristi’s fixed point theorem and Ekeland’s variational principle,Appl. Math. and Mech. 10 (1989), 119–121. · Zbl 0738.49009 · doi:10.1007/BF02014818 [8] S. S.Chang, Y. Q.Chen and J. L.Guo, Ekeland’s variational principle and Caristi’s fixed point theorem in probabilistic metric spaces,Acta Math. Appl. Sinica 3 (1991), 217–230. [9] S. S.Chang, Y. J.Cho and S. M.Kang,Probabilistic Metric Spaces and Nonlinear Operator Theory, Sichuan University Press, Chengdu, P. R. China, 1994. [10] Gh. Constantin, On some classes of contraction mappings in Menger spaces,Sem. Teoria Prob. Apl., Timisoara 76 (1985). [11] M. H.Dancs and P.Medvegyev, A general ordering and fixed point principle in complete metric spaces,Acta Sci. Math. 46 (1983), 381–388. [12] I.Ekeland, Nonconvex minimization problems,Bull. Amer. Math. Soc. (New Series)1 (1979), 431–474. · Zbl 0441.49011 · doi:10.1090/S0273-0979-1979-14595-6 [13] O.Hadžić, Some theorems on the fixed points in probabilistic metric and random normed spaces,Bull. Un. Mat. Ital. 13 (6)19 (1982), 381–391. [14] O.Hadžić, Fixed points theorems for multi-valued mappings in probabilistic metric spaces with a convex structure,Review of Research, Faculty of Science, Math. Series, Univ. of Novi Sad 17 (1) (1987), 39–51. [15] S.Park, On extensions of the Caristi-Kirk fixed point theorem,J. Korean Math. Soc. 19 (1983), 143–151. [16] B.Schweizer and A.Sklar, Statistical metric spaces,Pacific J. Math. 10 (1960), 313–334. [17] B.Schweizer, A.Sklar and E.Thorp, The metrization of statistical metric spaces,Pacific J. Math. 10 (1960), 673–675. [18] B.Schweizer and A.Sklar,Probabilistic metric spaces, North-Holland, 1983. [19] H.Sherwood, On E-spaces and their relation to other classes of probabilistic metric spaces,J. London Math. Soc. 44 (1969), 441–449. · Zbl 0167.46202 · doi:10.1112/jlms/s1-44.1.441 [20] H.Sherwood, Complete probabilistic metric spaces,Z. Wahrsch. Verw., Gebiete 20 (1971), 117–128. · doi:10.1007/BF00536289 [21] S. Z.Shi, The equivalence between Ekeland’s variational principle and Caristi’s fixed point theorem,Advan. Math. 16 (1987), 203–206. [22] M.Stojakovic, Common fixed point theorems in complete metric and probabilistic metric spaces,Bull. Austral. Math. Soc. 36 (1987), 73–88. · Zbl 0601.54056 · doi:10.1017/S0004972700026319 [23] N. X.Tan, Generalized probabilistic metric spaces and fixed point theorems,Math. Nachr. 129 (1986), 205–218. · Zbl 0603.54049 · doi:10.1002/mana.19861290119