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The probabilistic stability for a functional equation in a single variable. (English) Zbl 1212.39036
By using the fixed point method, the author deals with the probabilistic Hyers-Ulam stability and the generalized Hyers-Ulam-Rassias stability of the functional equation \[ \mu\circ f \circ\eta=f \] where \(\eta:X\to X, \mu:Y\to Y\) are given functions and \(f\) is the unknown mapping from \(X\) to a probabilistic metric space \((Y,F,T_{M})\) with \(T_{M}(a,b)=\min(a,b)\) and probabilistic distance \(F\).

MSC:
39B52 Functional equations for functions with more general domains and/or ranges
39B82 Stability, separation, extension, and related topics for functional equations
47H10 Fixed-point theorems
54E70 Probabilistic metric spaces
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[1] T. Aoki, On the stability of the linear transformation in Banach spaces, J. M. Soc. Japan, 2 (1950), 64–66. · Zbl 0040.35501
[2] J. A. Baker, The stability of certain functional equations, Proc. Amer. Math. Soc., 112 (1991), 729–732. · Zbl 0735.39004
[3] L. Cădariu and V. Radu, Fixed point method for the generalized stability of functional equations in single variable, Fixed Point Theory and Applications, vol. 2008, Article ID 749392 (2008), 15 pages. · Zbl 1146.39040
[4] S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific (River Edge, NJ, 2002). · Zbl 1011.39019
[5] G. L. Forti, Hyers-Ulam stability of functional equations in several variables, Aequationes Math., 50 (1995), 143–190. · Zbl 0836.39007
[6] P. Găvruţa, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184 (1994), 431–436. · Zbl 0818.46043
[7] A. Gianyi, Z. Kaiser and Z. Páles, Estimates to the stability of functional equations, Aequationes Math., 73 (2007), 125–143. · Zbl 1121.39029
[8] O. Hadžić, A generalization of the contraction principle in PM-spaces, Zb. Rad. Prirod. Mat. Fak. Univ. u Novom Sadu, 10 (1980), 13–21.
[9] O. Hadžić and E. Pap, Fixed Point Theory in PM Spaces, Kluwer Academic Publ. (2001).
[10] O. Hadzic, E. Pap and V. Radu, Generalized contraction mapping principles in probabilistic metric spaces, Acta Math. Hungar., 101 (2003), 131–148. · Zbl 1050.47052
[11] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. USA, 27 (1941), 222–224. · Zbl 0061.26403
[12] D. H. Hyers, G. Isac and Th. M. Rassias, Stability of Functional Equations in Several Variables (Basel, 1998). · Zbl 0907.39025
[13] I. Istrăţescu, On generalized Menger spaces, Boll. UMI (5) 13-A (1976), 95–100.
[14] C. F. K. Jung, On generalized complete metric spaces, Bull. Amer. Math. Soc., 75 (1969), 113–116. · Zbl 0194.23801
[15] Z. Kominek, On a local stability of the Jensen functional equation, Demonstratio Math., 22 (1989), 499–507. · Zbl 0702.39007
[16] W. A. J. Luxemburg, On the convergence of successive approximations in the theory of ordinary differential equations, II, Nederl. Akad. Wetensch. Proc. Ser. A 61 = Indag. Math., 20 (1958), 540–546. · Zbl 0084.07703
[17] B. Margolis and J. B. Diaz, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc., 74 (1968), 305–309. · Zbl 0157.29903
[18] D. Miheţ and V. Radu, On the stability of the additive Cauchy functional equation in random normed spaces, J. Math. Anal. Appl., 343 (2008), 567–572. · Zbl 1139.39040
[19] M. Mirmostafaee, M. Mirzavaziri and M. S. Moslehian, Fuzzy stability of the Jensen functional equation, Fuzzy Sets and Systems, 159 (2008), 730–738. · Zbl 1179.46060
[20] A. K. Mirmostafee and M. S. Moslehian, Fuzzy versions of Hyers-Ulam-Rassias theorem, Fuzzy Sets and Systems, 159 (2008), 720–729. · Zbl 1178.46075
[21] V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory, Cluj-Napoca, IV(1) (2003), 91–96. · Zbl 1051.39031
[22] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297–300. · Zbl 0398.47040
[23] B. Schweizer and A. Sklar, Probabilistic Metric Spaces, North Holland (1983). · Zbl 0546.60010
[24] V. M. Sehgal and A. T. Bharucha-Reid, Fixed points of contraction mappings on PM-Spaces, Math. Syst. Theory, 6 (1972), 97–100. · Zbl 0244.60004
[25] N. X. Tan, Generalized probabilistic metric spaces and fixed point theorems, Math. Nachr., 129 (1986), 205–218. · Zbl 0603.54049
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