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Kim, Chang Il; Han, Giljun; Shim, Seong-A

Fuzzy stability of the generalized version of drygas functional equation. (English) Zbl 1442.39031

J. Appl. Math. 2014, Article ID 514192, 11 p. (2014).
Summary: We consider the functional equation \(f(a x + y) + f(a x - y) - a(a + 1) f(x) - a(a - 1) f(- x) - [f(y) + f(- y)]+ k [f(x + y) + f(x - y) - 2 f(x) - f(y) - f(- y)] = 0\) for a fixed rational number \(a\) with \(a \neq 1, - 1,0\) and a fixed real number \(k\). We study the solution of the equation between linear spaces and prove the generalized Hyers-Ulam stability for it when the target space is a fuzzy normed space.

MSC:

39B52 Functional equations for functions with more general domains and/or ranges
39B82 Stability, separation, extension, and related topics for functional equations
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\textit{C. I. Kim} et al., J. Appl. Math. 2014, Article ID 514192, 11 p. (2014; Zbl 1442.39031)
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[1] Ulam, S. M., A Collection of Mathematical Problems. A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8, xiii+150 (1960), New York, NY, USA: Interscience Publishers, New York, NY, USA · Zbl 0086.24101
[2] Hyers, D. H., On the stability of the linear functional equation, Proceedings of the National Academy of Sciences of the United States of America, 27, 222-224 (1941) · Zbl 0061.26403
[3] Aoki, T., On the stability of the linear transformation in Banach spaces, Journal of the Mathematical Society of Japan, 2, 64-66 (1950) · Zbl 0040.35501
[4] Moslehian, M. S.; Rassias, T. M., Stability of functional equations in non-Archimedean spaces, Applicable Analysis and Discrete Mathematics, 1, 2, 325-334 (2007) · Zbl 1257.39019
[5] Rassias, Th. M., On the stability of the linear mapping in Banach spaces, Proceedings of the American Mathematical Society, 72, 2, 297-300 (1978) · Zbl 0398.47040
[6] Cholewa, P. W., Remarks on the stability of functional equations, Aequationes Mathematicae, 27, 1-2, 76-86 (1984) · Zbl 0549.39006
[7] Cieplinski, K., Applications of fixed point theorems to the Hyers-Ulam stability of functional equations—a survey, Annals of Functional Analysis, 3, 1, 151-164 (2012) · Zbl 1252.39032
[8] Czerwik, St., On the stability of the quadratic mapping in normed spaces, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 62, 59-64 (1992) · Zbl 0779.39003
[9] Găvruta, P., A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, Journal of Mathematical Analysis and Applications, 184, 3, 431-436 (1994) · Zbl 0818.46043
[10] Mirzavaziri, M.; Moslehian, M. S., A fixed point approach to stability of a quadratic equation, Bulletin of the Brazilian Mathematical Society, 37, 3, 361-376 (2006) · Zbl 1118.39015
[11] Kim, H.-M.; Rassias, J. M.; Lee, J., Fuzzy approximation of Euler-Lagrange quadratic mappings, Journal of Inequalities and Applications, 2013, article 358, 1-15 (2013) · Zbl 1286.39014
[12] Mirmostafaee, A. K.; Moslehian, M. S., Fuzzy almost quadratic functions, Results in Mathematics, 52, 1-2, 161-177 (2008) · Zbl 1157.46048
[13] Mirmostafaee, A. K.; Moslehian, M. S., Fuzzy versions of Hyers-Ulam-Rassias theorem, Fuzzy Sets and Systems, 159, 6, 720-729 (2008) · Zbl 1178.46075
[14] Katsaras, A. K., Fuzzy topological vector spaces. II, Fuzzy Sets and Systems, 12, 2, 143-154 (1984) · Zbl 0555.46006
[15] Cheng, S. C.; Mordeson, J. N., Fuzzy linear operators and fuzzy normed linear spaces, Bulletin of the Calcutta Mathematical Society, 86, 5, 429-436 (1994) · Zbl 0829.47063
[16] Kramosil, I.; Michálek, J., Fuzzy metrics and statistical metric spaces, Kybernetika, 11, 5, 336-344 (1975) · Zbl 0319.54002
[17] Bag, T.; Samanta, S. K., Finite dimensional fuzzy normed linear spaces, Journal of Fuzzy Mathematics, 11, 3, 687-705 (2003) · Zbl 1045.46048
[18] Drygas, H.; Gupta, A. K., Quasi-inner products and their applications, Advances in Multivariate Statistical Analysis, 13-30 (1987), Dordrecht, The Netherlands: Reidel, Dordrecht, The Netherlands
[19] Faiziev, V. A.; Sahoo, P. K., On the stability of Drygas functional equation on groups, Banach Journal of Mathematical Analysis, 1, 1, 43-55 (2007) · Zbl 1130.39023
[20] Najati, A.; Moghimi, M. B., Stability of a functional equation deriving from quadratic and additive functions in quasi-Banach spaces, Journal of Mathematical Analysis and Applications, 337, 1, 399-415 (2008) · Zbl 1127.39055
[21] Kim, C. I.; Han, G.; Shim, S.-A., Hyers-Ulam stability for a class of quadratic functional equations via a typical form, Abstract and Applied Analysis, 2013 (2013) · Zbl 1291.39053
[22] Mirmostafaee, A. K.; Mirzavaziri, M.; Moslehian, M. S., Fuzzy stability of the Jensen functional equation, Fuzzy Sets and Systems, 159, 6, 730-738 (2008) · 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.