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On the generalized Hyers-Ulam stability of an \(n\)-dimensional quadratic and additive type functional equation. (English) Zbl 1449.39028

Summary: We investigate the generalized Hyers-Ulam stability of a functional equation \(f \left(\sum_{j=1}^n x_j\right) +(n - 2) \sum_{j = 1}^nf(x_j) - \sum_{1 \leq i < j \leq n} f(x_i + x_j) = 0\).

MSC:

39B82 Stability, separation, extension, and related topics for functional equations
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[1] Ulam, S. M., A Collection of Mathematical Problems (1960), New York, NY, USA: Interscience, New York, NY, USA
[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] Rassias, T. 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
[5] Găvruţa, 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
[6] Kim, G. H., On the stability of functional equations with square-symmetric operation, Mathematical Inequalities & Applications, 4, 2, 257-266 (2001) · Zbl 0990.39028
[7] Lee, Y.-H., On the stability of the monomial functional equation, Bulletin of the Korean Mathematical Society, 45, 2, 397-403 (2008) · Zbl 1152.39023
[8] Lee, Y.-H.; Jun, K.-W., A generalization of the Hyers-Ulam-Rassias stability of Jensen’s equation, Journal of Mathematical Analysis and Applications, 238, 1, 305-315 (1999) · Zbl 0933.39053
[9] Lee, Y.-H.; Jun, K.-W., On the stability of approximately additive mappings, Proceedings of the American Mathematical Society, 128, 5, 1361-1369 (2000) · Zbl 0961.47039
[10] Jun, K.-W.; Kim, H.-M., On the stability of an n-dimensional quadratic and additive functional equation, Mathematical Inequalities & Applications, 9, 1, 153-165 (2006) · Zbl 1093.39026
[11] Janfada, M.; Shourvazi, R., Solutions and the generalized Hyers-Ulam-Rassias stability of a generalized quadratic-additive functional equation, Abstract and Applied Analysis, 2011 (2011) · Zbl 1217.39035
[12] Jin, S. S.; Lee, Y.-H., Fuzzy stability of an n-dimensional quadratic and additive type functional equation, International Journal of Mathematical Analysis, 7, 29-32, 1513-1530 (2013)
[13] Jin, S. S.; Lee, Y.-H., A fixed point approach to the stability of the n-dimensional quadratic and additive functional equation, International Journal of Mathematical Analysis, 7, 29-32, 1557-1573 (2013)
[14] Jin, S. S.; Lee, Y.-H., On the stability of the n-dimensional quadratic and additive functional equation in random normed spaces via fixed point method, International Journal of Mathematical Analysis, 7, 49-52, 2413-2428 (2013)
[15] Nakmahachalasint, P., On the generalized Ulam-Gavruta-Rassias stability of mixed-type linear and Euler-Lagrange-Rassias functional equations, International Journal of Mathematics and Mathematical Sciences, 2007 (2007) · Zbl 1148.39026
[16] Lee, Y.-H., On the quadratic additive type functional equations, International Journal of Mathematical Analysis, 7, 37-40, 1935-1948 (2013)
[17] Jung, S.-M., On the Hyers-Ulam stability of the functional equations that have the quadratic property, Journal of Mathematical Analysis and Applications, 222, 1, 126-137 (1998) · Zbl 0928.39013
[18] Jin, S. S.; Lee, Y.-H., Fuzzy stability of a mixed type functional equation, Journal of Inequalities and Applications, 2011, 70 (2011) · Zbl 1272.39017
[19] Jin, S. S.; Lee, Y.-H., A fixed point approach to the stability of the mixed type functional equation, Honam Mathematical Journal, 34, 1, 19-34 (2012) · Zbl 1253.39029
[20] Jin, S. S.; Lee, Y.-H., On the stability of the mixed type functional equation in random normed spaces via fixed point method, Journal of the Korean Society of Mathematical Education. Series B. The Pure and Applied Mathematics, 19, 1, 59-71 (2012) · Zbl 1271.39027
[21] Chang, I.-S.; Lee, E. H.; Kim, H.-M., On Hyers-Ulam-Rassias stability of a quadratic functional equation, Mathematical Inequalities & Applications, 6, 1, 87-95 (2003) · Zbl 1024.39008
[22] Jin, S. S.; Lee, Y.-H., A fixed point approach to the stability of the quadratic-additive functional equation, Journal of the Korea Society of Mathematical Education. Series B. The Pure and Applied Mathematics, 18, 4, 313-328 (2011)
[23] Jin, S. S.; Lee, Y. H., Fuzzy stability of a quadratic-additive functional equation, International Journal of Mathematics and Mathematical Sciences, 2011 (2011) · Zbl 1227.39025
[24] Jin, S. S.; Lee, Y. H., On the stability of the quadratic-additive functional equation in random normed spaces via fixed point method, Journal of the Chungcheong Mathematical Society, 25, 201-215 (2012)
[25] Lee, Y. W., Stability of a generalized quadratic functional equation with Jensen type, Bulletin of the Korean Mathematical Society, 42, 1, 57-73 (2005) · Zbl 1074.39029
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