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On the stability of a \(k\)-cubic functional equation in intuitionistic fuzzy \(n\)-normed spaces. (English) Zbl 1360.39022
Let \(k\) be any real number. The functional equation \[ f(kx+y)+f(kx-y) = kf(x+y)+kf(x-y) + 2k(k^2 - 1) f(x) \] for all \(x, y \in \mathbb{R}\) is called the \(k\)-cubic functional equation. It is known that \(f(x) = x^3\) satisfies this equation. The authors prove some stability results concerning this functional equation in intuitionistic fuzzy \(n\)-normed spaces.

MSC:
39B82 Stability, separation, extension, and related topics for functional equations
46S40 Fuzzy functional analysis
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[1] Aoki, T., On the stability of the linear transformation in Banach spaces, J. Math. Soc. Jpn., 2, 64-66, (1950) · Zbl 0040.35501
[2] Bag, T.; Samanta, S.K., Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math., 11, 687-705, (2003) · Zbl 1045.46048
[3] Bag, T.; Samanta, S.K., Fuzzy bounded linear operators, Fuzzy Sets Syst., 151, 513-547, (2005) · Zbl 1077.46059
[4] Chang, S.C.; Mordesen, J.N., Fuzzy linear operators and fuzzy normed linear spaces, Bull. Cal. Math. Soc., 86, 429-436, (1994) · Zbl 0829.47063
[5] Felbin, C., The completion of fuzzy normed linear space, J. Math. Anal. Appl., 174, 428-440, (1993) · Zbl 0806.46083
[6] Felbin, C., Finite dimensional fuzzy normed linear spaces II, J. Anal., 7, 117-131, (1999) · Zbl 0951.46043
[7] Gahler, S., Linear-2-normierte raume, Math. Nachr., 28, 1-43, (1965) · Zbl 0142.39803
[8] Gunawan, H.; Mashadi, M., On n-normed spaces, Int. J. Math. Math. Sci., 27, 631-639, (2001) · Zbl 1006.46006
[9] Hyers, D.H., On the stability of the linear functional equation, Proc. Natl. Acad. Sci., USA, 27, 222-224, (1941) · Zbl 0061.26403
[10] Jun, K.W.; Kim, H.M., The generalized Hyers-Ulam-Rassias stability of a cubic functional equation, J. Math. Anal. Appl., 274, 867-878, (2002) · Zbl 1021.39014
[11] Koh, H., Kang, D.: On the Mazur-Ulam problem in non-Archimedean fuzzy 2-normed spaces. J. Ineq. Appl. 2013, 507 (2013) · Zbl 1302.46064
[12] Malceski, R., Strong n-convex n-normed spaces, Mat. Bilten, 21, 81-102, (1997) · Zbl 1010.46024
[13] Mirmostafaee, A.K.; Mirzavaziri, M.; Moslehian, M.S., Fuzzy stability of the Jensen functional equation, Fuzzy Sets Syst., 159, 730-738, (2008) · Zbl 1179.46060
[14] Mirmostafaee, A.K.; Moslehian, M.S., Fuzzy versions of Hyers-Ulam-Rassias theorem, Fuzzy Sets Syst., 159, 720-729, (2008) · Zbl 1178.46075
[15] Mirmostafaee, A.K.; Moslehian, M.S., Fuzzy approximately cubic mappings, Inf. Sci., 178, 3791-3798, (2008) · Zbl 1160.46336
[16] Narayanan, Al.; Vijayabalaji, S., Fuzzy n-normed linear space, Int. J. Math. Sci., 24, 3963-3977, (2005) · Zbl 1095.46512
[17] Narayanan, Al.; Vijayabalaji, S.; Thillaigovindan, N., Intuitionisitic fuzzy bounded linear operators, Iran. J. Fuzzy Syst., 4, 89-101, (2007) · Zbl 1139.47060
[18] Park, C., Alaca, C.: An introduction to 2-fuzzy n-normed linear spaces and a new perspective to the Mazur-Ulam problem. J. Ineq. Appl. 2012, 14 (2012) · Zbl 1280.46049
[19] Park, C., Eshaghi Gordji, M., Gaemi, M.B., Majani, M.: Fixed point and approximately octic mappings in non-Archimedean 2-normed spaces. J. Ineq. Appl. 2012, 289 (2012) · Zbl 1280.39019
[20] Rassias, J.M., On approximation of approximately linear mappings by linear mappings, J. Funct. Anal., 46, 126-1307, (1989) · Zbl 0482.47033
[21] Rassias, J.M., On approximation of approximately linear mappings by linear mappings, Bull. Sci. Math., 108, 445-446, (1984) · Zbl 0599.47106
[22] Rassias, J.M., Solution of a problem of Ulam, J. Approx. Th., 57, 268-273, (1989) · Zbl 0672.41027
[23] Rassias, T.M., On the stability of the linear mapping in Banach spaces, Proc. Am. Math. Soc., 72, 297-300, (1978) · Zbl 0398.47040
[24] Rassias, T.M., On the stability of functional equations and a problem of Ulam, Acta Appl. Math., 62, 23-130, (2000) · Zbl 0981.39014
[25] Ravi, K.; Arunkumar, M.; Rassias, J.M., Ulam stability for the orthogonality general Euler-Lagrange type functional equation, Int. J. Math. Stat., 3, 36-46, (2008) · Zbl 1144.39029
[26] Ulam S.M.: Problems in Modern Mathematics. Chap. VI, Science (eds.). Wiley, New York (1960)
[27] Vijayabalaji, S.; Thillaigovindan, N., Best approximation sets in α-n-normed space corresponding to intuitionistic fuzzy n-normed linear space, Iran. J. Fuzzy Syst., 5, 57-69, (2008) · Zbl 1177.46055
[28] Xu, T.Z., Rassias, J.M.: Stability of general multi Euler-Lagrange quadratic functional equations in non-Archimedean fuzzy n-normed linear spaces. Adv. Differ. Equ. 2012, 119 (2012) · Zbl 1346.39041
[29] Xu, T.Z., Rassias, J.M., Xu, W.X.: Stability of a general mixed additive-cubic functional equations in non-Archimedean fuzzy normed spaces. J. Math. Phys. 51, 093508 (2010) · Zbl 1309.30029
[30] Xu, T.Z., Rassias, J.M., Xu, W.X.: Intuitionistic fuzzy stability of a general mixed additive-cubic equation. J. Math. Phys. 51, 063519 (2010) · Zbl 1311.46066
[31] Xu, T.Z.; Rassias, J.M.; Xu, W.X., A fixed point approach to the stability of a general mixed additive-cubic functional equation in quasi fuzzy normed spaces, Int. J. Phys. Sci., 6, 313-324, (2011)
[32] Xu, T.Z.; Rassias, J.M.; Xu, W.X., Generalized Hyers-Ulam stability of a general mixed additive-cubic functional equation in quasi-Banach spaces, Acta Math. Sinica, 28, 529-560, (2012) · Zbl 1258.39016
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