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On random topological structures. (English) Zbl 1254.46073

The authors present a review of random normed spaces, their topological structure and of random functional analysis. The article contains a survey of such topics as triangular norms, topological structure of random normed spaces, random functional analysis, fuzzy normed spaces. The paper contains a lot of definitions, examples, theorems with proofs. In particular, Theorem 6.15 states that if \(W\) is a closed subspace of a random normed space \((V,\mu ,T)\) and if two of the spaces \(V\), \(W\) and \(V/W\) are complete, then so is the third one. Theorem 6.16 is an analog of the open mapping theorem: If \(T\) is a random bounded linear operator from a random normed space \((V,\mu ,T)\) onto a random normed space \((V',\nu ,T)\), then \(T\) is an open mapping. Moreover, Theorem 6.18 is an analog of the closed graph theorem.

MSC:

46S50 Functional analysis in probabilistic metric linear spaces
60B99 Probability theory on algebraic and topological structures
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[1] A. N. \vSerstnev, “On the concept of a stochastic normalized space,” Doklady Akademii Nauk SSSR, vol. 149, pp. 280-283, 1963.
[2] O. Had and E. Pap, Fixed Point Theory in Probabilistic Metric Spaces, vol. 536 of Mathematics and Its Applications, Kluwer Academic, Dordrecht, The Netherlands, 2001.
[3] O. Had and E. Pap, “New classes of probabilistic contractions and applications to random operators,” in Fixed Point Theory and Applications, pp. 97-119, Nova Science Publishers, Hauppauge, NY, USA, 2003. · Zbl 1069.54026
[4] B. Schweizer and A. Sklar, Probabilistic Metric Spaces, North-Holland Series in Probability and Applied Mathematics, North-Holland, New York, NY, USA, 1983. · Zbl 0546.60010
[5] K. Menger, “Statistical metrics,” Proceedings of the National Academy of Sciences of the United States of America, vol. 28, pp. 535-537, 1942. · Zbl 0063.03886 · doi:10.1073/pnas.28.12.535
[6] C. Alsina, B. Schweizer, and A. Sklar, “On the definition of a probabilistic normed space,” Aequationes Mathematicae, vol. 46, no. 1-2, pp. 91-98, 1993. · Zbl 0792.46062 · doi:10.1007/BF01834000
[7] C. Alsina, B. Schweizer, and A. Sklar, “Continuity properties of probabilistic norms,” Journal of Mathematical Analysis and Applications, vol. 208, no. 2, pp. 446-452, 1997. · Zbl 0903.46075 · doi:10.1006/jmaa.1997.5333
[8] B. Lafuerza Guillén, J. A. Rodríguez Lallena, and C. Sempi, “A study of boundedness in probabilistic normed spaces,” Journal of Mathematical Analysis and Applications, vol. 232, no. 1, pp. 183-196, 1999. · Zbl 0945.46056 · doi:10.1006/jmaa.1998.6261
[9] B. Lafuerza-Guillén, “D-bounded sets in probabilistic normed spaces and in their products,” Rendiconti di Matematica e delle sue Applicazioni. Serie VII, vol. 21, no. 1-4, pp. 17-28, 2001. · Zbl 1053.46059
[10] B. Lafuerza-Guillén, “Finite products of probabilistic normed spaces,” Radovi Matemati\vcki, vol. 13, no. 1, pp. 111-117, 2004. · Zbl 1167.54310
[11] R. Saadati and M. Amini, “D-boundedness and D-compactness in finite dimensional probabilistic normed spaces,” Indian Academy of Sciences. Proceedings. Mathematical Sciences, vol. 115, no. 4, pp. 483-492, 2005. · Zbl 1096.46048 · doi:10.1007/BF02829810
[12] A. George and P. Veeramani, “On some results in fuzzy metric spaces,” Fuzzy Sets and Systems, vol. 64, no. 3, pp. 395-399, 1994. · Zbl 0843.54014 · doi:10.1016/0165-0114(94)90162-7
[13] A. George and P. Veeramani, “On some results of analysis for fuzzy metric spaces,” Fuzzy Sets and Systems, vol. 90, no. 3, pp. 365-368, 1997. · Zbl 0917.54010 · doi:10.1016/S0165-0114(96)00207-2
[14] V. Gregori and S. Romaguera, “Some properties of fuzzy metric spaces,” Fuzzy Sets and Systems, vol. 115, no. 3, pp. 485-489, 2000. · Zbl 0985.54007 · doi:10.1016/S0165-0114(98)00281-4
[15] V. Gregori and S. Romaguera, “On completion of fuzzy metric spaces,” Fuzzy Sets and Systems, vol. 130, no. 3, pp. 399-404, 2002. · Zbl 1010.54002 · doi:10.1016/S0165-0114(02)00115-X
[16] V. Gregori and S. Romaguera, “Characterizing completable fuzzy metric spaces,” Fuzzy Sets and Systems, vol. 144, no. 3, pp. 411-420, 2004. · Zbl 1057.54010 · doi:10.1016/S0165-0114(03)00161-1
[17] R. Saadati, A. Razani, and H. Adibi, “A common fixed point theorem in L-fuzzy metric spaces,” Chaos, Solitons and Fractals, vol. 33, no. 2, pp. 358-363, 2007. · Zbl 05231135 · doi:10.1016/j.chaos.2006.01.023
[18] R. Saadati and J. H. Park, “On the intuitionistic fuzzy topological spaces,” Chaos, Solitons and Fractals, vol. 27, no. 2, pp. 331-344, 2006. · Zbl 1083.54514 · doi:10.1016/j.chaos.2005.03.019
[19] R. Saadati, “On the L-fuzzy topological spaces,” Chaos, Solitons and Fractals, vol. 37, no. 5, pp. 1419-1426, 2008. · Zbl 1142.54318 · doi:10.1016/j.chaos.2006.10.033
[20] R. Saadati, “Notes to the paper Fixed points in intuitionistic fuzzy metric spaces and its generalization to L-fuzzy metric spaces,,” Chaos, Solitons and Fractals, vol. 35, no. 1, pp. 176-180, 2008. · Zbl 1151.54034 · doi:10.1016/j.chaos.2006.01.023</u>
[21] R. Saadati and S. M. Vaezpour, “Some results on fuzzy Banach spaces,” Journal of Applied Mathematics & Computing, vol. 17, no. 1-2, pp. 475-484, 2005. · Zbl 1077.46060 · doi:10.1007/BF02936069
[22] L. C\uadariu and V. Radu, “Fixed points and stability for functional equations in probabilistic metric and random normed spaces,” Fixed Point Theory and Applications, vol. 2009, Article ID 589143, 18 pages, 2009. · Zbl 1187.39033 · doi:10.1155/2009/589143
[23] S.-S. Chang, Y. J. Cho, and S. M. Kang, Nonlinear Operator Theory in Probabilistic Metric Spaces, Nova Science Publishers, Huntington, NY, USA, 2001.
[24] O. Had and D. Nikolić-Despotović, “Some fixed point theorems in random normed spaces,” Analele Universitatii de Vest Timisoara, Seria. Matematica, vol. 17, no. 1, pp. 39-47, 1979. · Zbl 0448.47049
[25] B. Singh, M. S. Chauhan, and R. Gujetiya, “Common fixed point theorems in fuzzy normed space,” Indian Journal of Mathematics and Mathematical Sciences, vol. 3, no. 2, pp. 181-186, 2007. · Zbl 1171.54339
[26] J.-Z. Xiao and X.-H. Zhu, “Topological degree theory and fixed point theorems in fuzzy normed space,” Fuzzy Sets and Systems, vol. 147, no. 3, pp. 437-452, 2004. · Zbl 1108.54010 · doi:10.1016/j.fss.2004.01.003
[27] I. Beg, “Approximation in random normed spaces,” Indian Journal of Pure and Applied Mathematics, vol. 28, no. 10, pp. 1435-1442, 1997. · Zbl 0901.46066
[28] S. Cobza\cs, “Best approximation in random normed spaces,” in Advances in Mathematics Research. Vol. 5, pp. 1-18, Nova Science Publishers, Hauppauge, NY, USA, 2003. · Zbl 1073.46515
[29] I. Gole\ct, “Approximation theorems in probabilistic normed spaces,” Novi Sad Journal of Mathematics, vol. 38, no. 3, pp. 73-79, 2008. · Zbl 1224.54071
[30] I. Gole\ct, “Approximation of random signals in probabilistic normed spaces,” Journal of Approximation Theory and Applications, vol. 2, no. 1, pp. 57-67, 2006.
[31] S. M. Vaezpour and F. Karimi, “t-best approximation in fuzzy normed spaces,” Iranian Journal of Fuzzy Systems, vol. 5, no. 2, pp. 93-99, 2008. · Zbl 1171.46051
[32] C. Alsina, “On the stability of a functional equation arising in probabilistic normed spaces,” in General Inequalities, Vol. 5 (Oberwolfach, 1986), vol. 80, pp. 263-271, Birkhäuser, Basel, Switzerland, 1987. · Zbl 0633.60029
[33] A. Ghaffari, A. Alinejad, and M. E. Gordji, “On the stability of general cubic-quartic functional equations in Menger probabilistic normed spaces,” Journal of Mathematical Physics, vol. 50, no. 12, Article ID 123301, 7 pages, 2009. · Zbl 1241.39010 · doi:10.1063/1.3269920
[34] M. Eshaghi Gordji and M. B. Savadkouhi, “Stability of mixed type cubic and quartic functional equations in random normed spaces,” Journal of Inequalities and Applications, vol. 2009, Article ID 527462, 9 pages, 2009. · Zbl 1176.39022 · doi:10.1155/2009/527462
[35] D. Mihe\ct and V. Radu, “On the stability of the additive Cauchy functional equation in random normed spaces,” Journal of Mathematical Analysis and Applications, vol. 343, no. 1, pp. 567-572, 2008. · Zbl 1139.39040 · doi:10.1016/j.jmaa.2008.01.100
[36] D. Mihe\ct, R. Saadati, and S. M. Vaezpour, “The stability of the quartic functional equation in random normed spaces,” Acta Applicandae Mathematicae, vol. 110, no. 2, pp. 797-803, 2010. · Zbl 1195.46081 · doi:10.1007/s10440-009-9476-7
[37] D. Mihet, R. Saadati, and S. M. Vaezpour, “The stability of an additive functional equation in Menger probabilistic \varphi -normed spaces,” Mathematica Slovaca. In press.
[38] R. Saadati, S. M. Vaezpour, and Y. J. Cho, “A note to paper “On the stability of cubic mappings and quartic mappings in random normed spaces”,” Journal of Inequalities and Applications, vol. 2009, Article ID 214530, 6 pages, 2009. · Zbl 1176.39024 · doi:10.1155/2009/214530
[39] P. Hájek, Metamathematics of Fuzzy Logic, vol. 4 of Trends in Logic-Studia Logica Library, Kluwer Academic, Dordrecht, The Netherlands, 1998. · Zbl 0937.03030
[40] L. A. Zadeh, “Fuzzy sets,” Information and Computation, vol. 8, pp. 338-353, 1965. · Zbl 0139.24606 · doi:10.1016/S0019-9958(65)90241-X
[41] E. P. Klement, R. Mesiar, and E. Pap, Triangular Norms, vol. 8 of Trends in Logic-Studia Logica Library, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000. · Zbl 0972.03002
[42] E. P. Klement, R. Mesiar, and E. Pap, “Triangular norms. Position paper. I. Basic analytical and algebraic properties,” Fuzzy Sets and Systems, vol. 143, no. 1, pp. 5-26, 2004. · Zbl 1038.03027 · doi:10.1016/j.fss.2003.06.007
[43] E. P. Klement, R. Mesiar, and E. Pap, “Triangular norms. Position paper. II. General constructions and parameterized families,” Fuzzy Sets and Systems, vol. 145, no. 3, pp. 411-438, 2004. · Zbl 1059.03012 · doi:10.1016/S0165-0114(03)00327-0
[44] E. P. Klement, R. Mesiar, and E. Pap, “Triangular norms. Position paper. III. Continuous t-norms,” Fuzzy Sets and Systems, vol. 145, no. 3, pp. 439-454, 2004. · Zbl 1059.03013 · doi:10.1016/S0165-0114(03)00304-X
[45] O. Had, E. Pap, and M. Budin, “Countable extension of triangular norms and their applications to the fixed point theory in probabilistic metric spaces,” Kybernetika, vol. 38, no. 3, pp. 363-382, 2002.
[46] G. Deschrijver and E. E. Kerre, “On the relationship between some extensions of fuzzy set theory,” Fuzzy Sets and Systems, vol. 133, no. 2, pp. 227-235, 2003. · Zbl 1013.03065 · doi:10.1016/S0165-0114(02)00127-6
[47] J. A. Goguen, “L-fuzzy sets,” Journal of Mathematical Analysis and Applications, vol. 18, pp. 145-174, 1967. · Zbl 0145.24404 · doi:10.1016/0022-247X(67)90189-8
[48] K. T. Atanassov, “Intuitionistic fuzzy sets,” Fuzzy Sets and Systems, vol. 20, no. 1, pp. 87-96, 1986. · Zbl 0631.03040 · doi:10.1016/S0165-0114(86)80034-3
[49] D. H. Mu\vstari, “The linearity of isometric mappings of random normed spaces,” Kazan. Gos. Univ. Uchen. Zap., vol. 128, no. 2, pp. 86-90, 1968.
[50] V. Radu, “Some remarks on quasi-normed and random normed structures,” Seminar on Probability Theory and Applications, vol. 159, 2003.
[51] I. Gole\ct, “Some remarks on functions with values in probabilistic normed spaces,” Mathematica Slovaca, vol. 57, no. 3, pp. 259-270, 2007. · Zbl 1150.54030 · doi:10.2478/s12175-007-0021-8
[52] K. Hensel, “Uber eine neue Begrundung der Theorie der algebraischen Zahlen,” Jahrestagung der Deutsche Mathematiker-Vereinigung, vol. 6, pp. 83-88, 1897. · JFM 30.0096.03
[53] A. K. Mirmostafaee and M. S. Moslehian, “Stability of additive mappings in non-Archimedean fuzzy normed spaces,” Fuzzy Sets and Systems, vol. 160, no. 11, pp. 1643-1652, 2009. · Zbl 1187.46068 · doi:10.1016/j.fss.2008.10.011
[54] T. M. Apostol, Mathematical Analysis, Addison-Wesley, Reading, Mass, USA, 2nd edition, 1974. · Zbl 0309.26002
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