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Stability of mixed type cubic and quartic functional equations in random normed spaces. (English) Zbl 1176.39022

Summary: We obtain the stability result for the following functional equation in random normed spaces (in the sense of Sherstnev) under arbitrary \(t\)-norms
\[ f(x+2y)+f(x - 2y)=4[f(x+y)+f(x - y)] - 24f(y) - 6f(x)+3f(2y). \]

MSC:

39B82 Stability, separation, extension, and related topics for functional equations
39B52 Functional equations for functions with more general domains and/or ranges
46B09 Probabilistic methods in Banach space theory
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References:

[1] Ulam SM: Problems in Modern Mathematics. John Wiley & Sons, New York, NY, USA; 1940. · Zbl 0137.24201
[2] Hyers DH: On the stability of the linear functional equation.Proceedings of the National Academy of Sciences of the United States of America 1941, 27: 222-224. 10.1073/pnas.27.4.222 · Zbl 0061.26403 · doi:10.1073/pnas.27.4.222
[3] Rassias ThM: On the stability of the linear mapping in Banach spaces.Proceedings of the American Mathematical Society 1978,72(2):297-300. 10.1090/S0002-9939-1978-0507327-1 · Zbl 0398.47040 · doi:10.1090/S0002-9939-1978-0507327-1
[4] Gajda Z: On stability of additive mappings.International Journal of Mathematics and Mathematical Sciences 1991,14(3):431-434. 10.1155/S016117129100056X · Zbl 0739.39013 · doi:10.1155/S016117129100056X
[5] Aczél J, Dhombres J: Functional Equations in Several Variables, Encyclopedia of Mathematics and Its Applications. Volume 31. Cambridge University Press, Cambridge, UK; 1989:xiv+462. · Zbl 0685.39006 · doi:10.1017/CBO9781139086578
[6] Aoki T: On the stability of the linear transformation in Banach spaces.Journal of the Mathematical Society of Japan 1950, 2: 64-66. 10.2969/jmsj/00210064 · Zbl 0040.35501 · doi:10.2969/jmsj/00210064
[7] Bourgin DG: Classes of transformations and bordering transformations.Bulletin of the American Mathematical Society 1951, 57: 223-237. 10.1090/S0002-9904-1951-09511-7 · Zbl 0043.32902 · doi:10.1090/S0002-9904-1951-09511-7
[8] Găvruţa P: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings.Journal of Mathematical Analysis and Applications 1994,184(3):431-436. 10.1006/jmaa.1994.1211 · Zbl 0818.46043 · doi:10.1006/jmaa.1994.1211
[9] Hyers DH, Isac G, Rassias ThM: Stability of Functional Equations in Several Variables, Progress in Nonlinear Differential Equations and Their Applications. Volume 34. Birkhäuser, Boston, Mass, USA; 1998:vi+313. · Zbl 0907.39025
[10] Isac G, Rassias ThM: On the Hyers-Ulam stability of[InlineEquation not available: see fulltext.]-additive mappings.Journal of Approximation Theory 1993,72(2):131-137. 10.1006/jath.1993.1010 · Zbl 0770.41018 · doi:10.1006/jath.1993.1010
[11] Rassias ThM: On the stability of functional equations and a problem of Ulam.Acta Applicandae Mathematicae 2000,62(1):23-130. 10.1023/A:1006499223572 · Zbl 0981.39014 · doi:10.1023/A:1006499223572
[12] Rassias ThM: On the stability of functional equations in Banach spaces.Journal of Mathematical Analysis and Applications 2000,251(1):264-284. 10.1006/jmaa.2000.7046 · Zbl 0964.39026 · doi:10.1006/jmaa.2000.7046
[13] Jun K-W, Kim H-M: The generalized Hyers-Ulam-Rassias stability of a cubic functional equation.Journal of Mathematical Analysis and Applications 2002,274(2):267-278. · Zbl 1021.39014 · doi:10.1016/S0022-247X(02)00415-8
[14] Park W-G, Bae J-H: On a bi-quadratic functional equation and its stability.Nonlinear Analysis: Theory, Methods & Applications 2005,62(4):643-654. 10.1016/j.na.2005.03.075 · Zbl 1076.39027 · doi:10.1016/j.na.2005.03.075
[15] Chung JK, Sahoo PK: On the general solution of a quartic functional equation.Bulletin of the Korean Mathematical Society 2003,40(4):565-576. · Zbl 1048.39017 · doi:10.4134/BKMS.2003.40.4.565
[16] Lee SH, Im SM, Hwang IS: Quartic functional equations.Journal of Mathematical Analysis and Applications 2005,307(2):387-394. 10.1016/j.jmaa.2004.12.062 · Zbl 1072.39024 · doi:10.1016/j.jmaa.2004.12.062
[17] Najati A: On the stability of a quartic functional equation.Journal of Mathematical Analysis and Applications 2008,340(1):569-574. 10.1016/j.jmaa.2007.08.048 · Zbl 1133.39030 · doi:10.1016/j.jmaa.2007.08.048
[18] Park C-G: On the stability of the orthogonally quartic functional equation.Bulletin of the Iranian Mathematical Society 2005,31(1):63-70. · Zbl 1117.39020
[19] Chang S, Cho YJ, Kang SM: Nonlinear Operator Theory in Probabilistic Metric Spaces. Nova Science, Huntington, NY, USA; 2001:x+338. · Zbl 1080.47054
[20] Schweizer B, Sklar A: Probabilistic Metric Spaces, North-Holland Series in Probability and Applied Mathematics. North-Holland, New York, NY, USA; 1983:xvi+275. · Zbl 0546.60010
[21] Sherstnev AN: On the notion of a random normed space.Doklady Akademii Nauk SSSR 1963, 149: 280-283. · Zbl 0127.34902
[22] Hadžić O, Pap E: Fixed Point Theory in Probabilistic Metric Spaces, Mathematics and Its Applications. Volume 536. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2001:x+273.
[23] Hadžić O, Pap E, Budinčević M: Countable extension of triangular norms and their applications to the fixed point theory in probabilistic metric spaces.Kybernetika 2002,38(3):363-382. · Zbl 1265.54127
[24] Baktash, E.; Cho, YJ; Jalili, M.; Saadati, R.; Vaezpour, SM, On the stability of cubic mappings and quadratic mappings in random normed spaces, No. 2008, 11 (2008) · Zbl 1165.39022
[25] Miheţ D, Radu V: On the stability of the additive Cauchy functional equation in random normed spaces.Journal of Mathematical Analysis and Applications 2008,343(1):567-572. 10.1016/j.jmaa.2008.01.100 · Zbl 1139.39040 · doi:10.1016/j.jmaa.2008.01.100
[26] Miheţ D: The probabilistic stability for a functional equation in a single variable.Acta Mathematica Hungarica 2009,123(3):249-256. 10.1007/s10474-008-8101-y · Zbl 1212.39036 · doi:10.1007/s10474-008-8101-y
[27] Miheţ D: The fixed point method for fuzzy stability of the Jensen functional equation.Fuzzy Sets and Systems 2009,160(11):1663-1667. 10.1016/j.fss.2008.06.014 · Zbl 1179.39039 · doi:10.1016/j.fss.2008.06.014
[28] Miheţ, D.; Saadati, R.; Vaezpour, SM, The stability of the quartic functional equation in random normed spaces (2009) · Zbl 1195.46081
[29] Saadati, R.; Vaezpour, SM; Cho, YJ, A note to paper “On the stability of cubic mappings and quartic mappings in random normed spaces”, No. 2009, 6 (2009) · Zbl 1176.39024
[30] Eshaghi Gordji, M.; Ebadian, A.; Zolfaghari, S., Stability of a functional equation deriving from cubic and quartic functions, No. 2008, 17 (2008) · Zbl 1160.39334
[31] Eshaghi Gordji M, Khodaei H: Solution and stability of generalized mixed type cubic, quadratic and additive functional equation in quasi-Banach spaces.Nonlinear Analysis: Theory, Methods & Applications 2009,71(11):5629-5643. 10.1016/j.na.2009.04.052 · Zbl 1179.39034 · doi:10.1016/j.na.2009.04.052
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