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Modeling uncertainty and nonlinearity by probabilistic metric spaces. (English) Zbl 1393.54014

Fodor, János (ed.) et al., Advances in soft computing, intelligent robotics and control. Cham: Springer (ISBN 978-3-319-05944-0/hbk; 978-3-319-05945-7/ebook). Topics in Intelligent Engineering and Informatics 8, 259-272 (2014).
Summary: Many problems occurring in engineering, e.g., robotics and control, require mathematical models which cover uncertainties and nonlinearity. We present here one such model: the theory of probabilistic metric spaces. This theory is based on the idea that, since the value of the distance in measurement is always unprecise and uncertain, the value of the distance have to be a probability distribution function. The theory of fuzzy metric spaces, as another model for uncertainty, is closely related to the theory of probabilistic metric spaces. We consider nonlinear random equations using fixed point methods in probabilistic metric spaces. Probabilistic metric spaces, some constructions methods of triangle functions and some important classes of probabilistic metric spaces as those of Menger, Wald, transformation-generated, are recalled. Based on some additional properties of t-norms the corresponding generalizations of fixed point theorems in probabilistic metric spaces are obtained.
For the entire collection see [Zbl 1294.68019].

MSC:

54E70 Probabilistic metric spaces
54H25 Fixed-point and coincidence theorems (topological aspects)
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[1] 1.Arbeiter, M.: Random recursive constructions of self-similar fractal measures. The noncompact case. Prob. Theory Related Fields 88, 497-520 (1991) · Zbl 0723.60040 · doi:10.1007/BF01192554
[2] 2.Beg, I.: Approximation of random fixed points in normed spaces. Nonlinear Analysis 51, 1303-1372 (2002) · Zbl 1034.47027 · doi:10.1016/S0362-546X(01)00902-6
[3] 3.Chang, S.S.: On the theory of probabilistic metric spaces with applications. Acta Math. Sinica, New Series 1, 366-377 (1985) · Zbl 0614.60010 · doi:10.1007/BF02564846
[4] 4.Drossos, C.A.: Stochastic Menger spaces and convergence in probability. Rev. Roum. Math. Pures Appl. 22, 1069-1076 (1977) · Zbl 0372.60014
[5] 5.Erber, T., Harmon, B.N., Latal, H.G.: The origin of hysteresis in simple magnetic systems. Advances in Chemical Physics 20, 71-134 (1971)
[6] 6.Hadžić, O.: Some theorems on the fixed point in probabilistic metric and random normed spaces. Boll. Unione Mat. Ital. 1-B, 381-391 (1982) · Zbl 0488.47028
[7] 7.Hadžić, O., Pap, E.: Fixed Point Theory in Probabilistic Metric Spaces. Kluwer Academic Publishers, Dordrecht (2001)
[8] 8.Hadžić, O., Pap, E.: A fixed point theorem for multivalued mappings in probabilistic metric spaces and an application in fuzzy metric spaces. Fuzzy Sets and Systems 127, 333-344 (2002) · Zbl 1002.54025 · doi:10.1016/S0165-0114(01)00144-0
[9] 9.Hadžić, O., Pap, E.: Probabilistic multi-valued contractions and decomposable measures. Internat. J. Uncertain. Fuzziness Knowledge-Based Systems 10, 59-74 (2002) · Zbl 1060.54017 · doi:10.1142/S0218488502001831
[10] 10.Hadžić, O., Pap, E.: New classes of probabilistic contractions and applications to random operators. In: Cho, Y.J., Kim, J.K., Kang, S.M. (eds.) Fixed Point Theory and Applications, pp. 97-119. Nova Science Publishers, Hauppauge (2003) · Zbl 1069.54026
[11] 11.Hadžić, O., Pap, E., Budinčević, M.: Countable extension of triangular norms and their applications to fixed point theory in probabilistic metric spaces. Kybernetika 38, 363-381 (2002) · Zbl 1265.54127
[12] 12.Hadžić, O., Pap, E., Budinčević, M.: A generalization of Tardiff’s fixed point theorem and applications to random equations. Fuzzy Sets and Systems 156, 124-135 (2005) · Zbl 1086.54018 · doi:10.1016/j.fss.2005.04.007
[13] 13.Hadžić, O., Pap, E., Radu, V.: Some generalized contraction mapping principles in probabilistic metric spaces. Acta Math. Hungarica 101, 111-128 (2003)
[14] 14.Hicks, T.L.: Fixed point theory in probabilistic metric spaces. Univ. u Novom Sadu, Zb. Rad. Prirod.-Mat. Fak. Ser. Mat. 13, 63-72 (1983) · Zbl 0574.54044
[15] 15.Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer Academic Publishers, Dordrecht (2000) · Zbl 0972.03002 · doi:10.1007/978-94-015-9540-7
[16] 16.Klement, E.P., Mesiar, R., Pap, E.: Triangular norms: Basic notions and properties. In: Klement, E.P., Mesiar, R. (eds.) Logical, Algebraic, Analytic, and Probabilistic Aspects of Triangular Norms, pp. 17-60. Elsevier (2005) · Zbl 1078.03022
[17] 17.Klement, E.P., Mesiar, R., Pap, E.: Semigroups and triangular norms. In: Klement, E.P., Mesiar, R. (eds.) Logical, Algebraic, Analytic, and Probabilistic Aspects of Triangular Norms, pp. 63-93. Elsevier (2005) · Zbl 1082.06013
[18] 18.Klement, E.P., Mesiar, R., Pap, E.: Archimedean components of triangular norms. J. Aust. Math. Soc. 78, 1-17 (2005) · Zbl 1087.20041 · doi:10.1017/S1446788700008065
[19] 19.Menger, K.: Statistical metric. Proc. Nat. Acad. USA 28, 535-537 (1942) · Zbl 0063.03886 · doi:10.1073/pnas.28.12.535
[20] 20.Nelsen, R.B.: An Introduction to Copulas. Springer, New York (1999) · Zbl 0909.62052 · doi:10.1007/978-1-4757-3076-0
[21] 21.Olsen, L.: Random geometrically graph directed self-similary multifractals. Longman, Harlow (1994) · Zbl 0801.28002
[22] 22.Pap, E.: Null-Additive Set Functions. Kluwer Academic Publishers, Dordrecht (1995) · Zbl 0856.28001
[23] 23.Pap, E.: Pseudo-additive measures and their applications. In: Pap, E. (ed.) Handbook of Measure Theory, vol. II, pp. 1403-1465. Elsevier, Amsterdam (2002) · Zbl 1018.28010 · doi:10.1016/B978-044450263-6/50036-1
[24] 24.Pap, E., Hadžić, O., Mesiar, R.: A fixed point theorem in probabilistic metric spaces and applications in fuzzy set theory. J. Math. Anal. Appl. 202, 433-449 (1996) · Zbl 0855.54043 · doi:10.1006/jmaa.1996.0325
[25] 25.Radu, V.: Lectures on probabilistic analysis. Surveys. Lectures Notes and Monographs Series on Probability, Statistics & Applied Mathematics 2. Universitatea de Vest din Timişoara (1994) · Zbl 0927.60003
[26] 26.Riedel, T.: On sup-continuous triangle functions. J. Math. Anal. Appl. 184, 382-388 (1994) · Zbl 0802.60022 · doi:10.1006/jmaa.1994.1207
[27] 27.Schweizer, B., Sklar, A.: Espaces métriques aléatoires. Comptes Rendus Acad. Sci. Paris 247, 2092-2094 (1958) · Zbl 0085.12503
[28] 28.Schweizer, B., Sklar, A.: Statistical metric spaces. Pacific J. Math. 10, 313-334 (1960) · Zbl 0091.29801 · doi:10.2140/pjm.1960.10.313
[29] 29.Schweizer, B., Sklar, A.: Probabilistic Metric Spaces. North-Holland, New York (1983) · Zbl 0546.60010
[30] 30.Schweizer, B., Smítal, J.: Measures of chaos and a spectral decomposition of dynamical systems on the interval. Trans. Amer. Math. Soc. 344, 737-754 (1994) · Zbl 0812.58062 · doi:10.1090/S0002-9947-1994-1227094-X
[31] 31.Sehgal, V.M.: Some fixed point theorems in functional analysis and probability. Ph.D. Thesis, Wayne State University, Detroit (1966)
[32] 32.Sehgal, V.M., Bharucha-Reid, A.T.: Fixed points of contraction mappings on probabilistic metric spaces. Math. Syst. Theory 6, 97-102 (1972) · Zbl 0244.60004 · doi:10.1007/BF01706080
[33] 33.Sherstnev, A.N.: Random normed spaces: problems of completeness. Kazan. Gos. Univ. Učen. Zap. 122, 3-20 (1962) · Zbl 0178.52404
[34] 34.Sherwood, H.: Complete probabilistic metric spaces and random variables. Ph.D. Thesis, University of Arizona (1965)
[35] 35.Soós, A.: Random fractals using contraction methods in probabilistic metric spaces. Ph.D. Thesis, University of Cluj-Napoca (2002)
[36] 36.Tardiff, R.M.: Contraction maps on probabilistic metric spaces. J. Math. Anal. Appl. 165, 517-523 (1992) · Zbl 0773.54033 · doi:10.1016/0022-247X(92)90055-I
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