## Fuzzy versions of Hyers-Ulam-Rassias theorem.(English)Zbl 1178.46075

Summary: We introduce three reasonable versions of fuzzy approximately additive functions in fuzzy normed spaces. More precisely, we show under some suitable conditions that an approximately additive function can be approximated by an additive mapping in a fuzzy sense.

### MSC:

 46S40 Fuzzy functional analysis 39B52 Functional equations for functions with more general domains and/or ranges 39B82 Stability, separation, extension, and related topics for functional equations 26E50 Fuzzy real analysis 46S50 Functional analysis in probabilistic metric linear spaces
Full Text:

### References:

 [1] Amyari, M.; Moslehian, M.S., Approximately ternary semigroup homomorphisms, Lett. math. phys., 77, 1-9, (2006) · Zbl 1112.39021 [2] Aoki, T., On the stability of the linear transformation in Banach spaces, J. math. soc. Japan, 2, 64-66, (1950) · Zbl 0040.35501 [3] Baak, C.; Moslehian, M.S., Stability of $$J^*$$-homomorphisms, Nonlinear anal.—TMA, 63, 42-48, (2005) · Zbl 1085.39026 [4] Bag, T.; Samanta, S.K., Finite dimensional fuzzy normed linear spaces, J. fuzzy math., 11, 3, 687-705, (2003) · Zbl 1045.46048 [5] Bag, T.; Samanta, S.K., Fuzzy bounded linear operators, Fuzzy sets and systems, 151, 513-547, (2005) · Zbl 1077.46059 [6] Cheng, S.C.; Mordeson, J.N., Fuzzy linear operator and fuzzy normed linear spaces, Bull. Calcutta math. soc., 86, 429-436, (1994) · Zbl 0829.47063 [7] Czerwik, S., Functional equations and inequalities in several variables, (2002), World Scientific River Edge, NJ · Zbl 1011.39019 [8] Felbin, C., Finite dimensional fuzzy normed linear space, Fuzzy sets and systems, 48, 239-248, (1992) · Zbl 0770.46038 [9] Găvruta, P., A generalization of the hyers – ulam – rassias stability of approximately additive mappings, J. math. anal. appl., 184, 431-436, (1994) · Zbl 0818.46043 [10] Hyers, D.H., On the stability of the linear functional equation, Proc. nat. acad. sci. U.S.A., 27, 222-224, (1941) · Zbl 0061.26403 [11] Hyers, D.H.; Isac, G.; Rassias, Th.M., Stability of functional equations in several variables, (1998), Birkhäuser Basel · Zbl 0894.39012 [12] Jung, S.-M., Hyers – ulam – rassias stability of functional equations in mathematical analysis, (2001), Hadronic Press Palm Harbor · Zbl 0980.39024 [13] Katsaras, A.K., Fuzzy topological vector spaces II, Fuzzy sets and systems, 12, 143-154, (1984) · Zbl 0555.46006 [14] Kramosil, I.; Michalek, J., Fuzzy metric and statistical metric spaces, Kybernetica, 11, 326-334, (1975) [15] Krishna, S.V.; Sarma, K.K.M., Separation of fuzzy normed linear spaces, Fuzzy sets and systems, 63, 207-217, (1994) · Zbl 0849.46058 [16] Mirzavaziri, M.; Moslehian, M.S., A fixed point approach to stability of a quadratic equation, Bull. braz. math. soc., 37, 3, 361-376, (2006) · Zbl 1118.39015 [17] Moslehian, M.S., Approximately vanishing of topological cohomology groups, J. math. anal. appl., 318, 2, 758-771, (2006) · Zbl 1098.39020 [18] Rassias, Th.M., On the stability of the linear mapping in Banach spaces, Proc. amer. math. soc., 72, 297-300, (1978) · Zbl 0398.47040 [19] Ulam, S.M., Problems in modern mathematics, (1964), Science Editions Wiley, New York, (Chapter VI, Some Questions in Analysis: §1, Stability) · Zbl 0137.24201 [20] Xiao, J.-Z.; Zhu, X.-H., Fuzzy normed spaces of operators and its completeness, Fuzzy sets and systems, 133, 389-399, (2003) · Zbl 1032.46096 [21] A.K. Mirmostafaee, M. Mirzavaziri, M.S. Moslehian, Fuzzy stability of the Jensen functional equation, Fuzzy Sets and Systems (2007), doi:10.1016/j.fss.2007.07.011. · Zbl 1179.46060
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.