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Demonstratio Mathematica

Editor-in-Chief: Vetro, Calogero

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Volume 51, Issue 1


Stability and hyperstability of a quadratic functional equation and a characterization of inner product spaces

Gwang Hui Kim / Iz-iddine El-Fassi / Choonkil Park
Published Online: 2018-12-05 | DOI: https://doi.org/10.1515/dema-2018-0027


We have proved theHyers-Ulam stability and the hyperstability of the quadratic functional equation f (x + y + z) + f (x + y − z) + f (x − y + z) + f (−x + y + z) = 4[f (x) + f (y) + f (z)] in the class of functions from an abelian group G into a Banach space.

Keywords: Hyers-Ulam stability; hyperstability; quadratic functional equation; fixed point theorem

MS: 39B82; 39B52; 47H14; 47H10


  • [1] Ulam S. M., Problems in Modern Mathematics, Chapter IV, Science Editions, Wiley, New York, 1960Google Scholar

  • [2] Hyers D. H., On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A., 1941, 27(4), 222-224CrossrefGoogle Scholar

  • [3] Bourgin D. G., Approximately isometric andmultiplicative transformations on continuous function rings, DukeMath. J., 1949, 16(2), 385-397Google Scholar

  • [4] Aoki T., On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 1950, 2(1-2), 64-66Google Scholar

  • [5] Rassias T. M., On the stability of the linear mapping in Banach spaces, Proc. Am. Math. Soc., 1978, 72, 297-300Google Scholar

  • [6] Gajda Z., On stability of additive mappings, Int. J. Math. Math. Sci., 1991, 14(3), 431-434CrossrefGoogle Scholar

  • [7] Rassias T. M., On a modified Hyers-Ulam sequence, J. Math. Anal. Appl., 1991, 158(1), 106-113Google Scholar

  • [8] Rassias T. M., Semrl P., On the behavior of mappings which do not satisfy Hyers-Ulam stability, Proc. Am. Math. Soc., 1992, 114(4), 989-993Google Scholar

  • [9] Găvruta P., A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal, Appl., 1994, 184(3), 431-436Google Scholar

  • [10] Czerwik S., On the stability of the quadratic mapping in normed spaces, Bull. Abh. Math. Sem. Univ. Hamburg, 1992, 62(1), 59-64Google Scholar

  • [11] Gordji M. E., Rahimi A., Park C., Shin D., Ternary Jordan bi-homomorphisms in Banach Lie triple systems, J. Comput. Anal. Appl., 2016, 21(6), 1040-1045Google Scholar

  • [12] Jung S., On the Hyers-Ulam-Rassias stability of the quadratic functional equations, J. Math. Anal. Appl., 1998, 232(2), 384-339Google Scholar

  • [13] Kannappan P., Quadratic functional equation and inner product spaces, Results Math., 1995, 27(3-4), 368-372CrossrefGoogle Scholar

  • [14] Jung S., Quadratic functional equations of Pexider type, Int. J. Math. Math. Sci., 2000, 24(5), 351-359CrossrefGoogle Scholar

  • [15] Krishnan R., Kumar M. A., On the generalized Hyers-Ulam-Rassias stability of a quadratic functional equation, Int. J. Pure Appl. Math., 2006, 28, 85-94Google Scholar

  • [16] Maksa G., Páles Z., Hyperstability of a class of linear functional equations, ActaMath. Acad. Paedagog. Nyházi, 2001, 17(2), 107-112Google Scholar

  • [17] Brzde¸k J., Remarks on hyperstability of the the Cauchy equation, Aequationes Math., 2013, 86, 255-267Web of ScienceGoogle Scholar

  • [18] Brzde¸k J., Hyperstability of the Cauchy equation on restricted domains, Acta Math. Hungar., 2013, 141(1-2), 58-67Web of ScienceGoogle Scholar

  • [19] Brzde¸k J., A hyperstability result for the Cauchy equation, Bull. Austral. Math. Soc., 2014, 89(1), 33-40Google Scholar

  • [20] Gselmann E., Hyperstability of a functional equation, Acta Math. Hungar., 2009, 124(1-2), 179-188Google Scholar

  • [21] Bahyrycz A., Piszczek M., Hyperstability of the Jensen functional equation, Acta Math. Hungar., 2014, 142(2), 353-365Google Scholar

  • [22] EL-Fassi Iz., Generalized hyerstability of a Drygas functional equation on a restricted domain using Brzde¸k’s fixed point theorem, J. Fixed Point Theory Appl., 2017, 19, 2529-2540CrossrefGoogle Scholar

  • [23] Brzde¸k J., Ciepliński K., Hyperstability and superstability, Abs. Appl. Anal., 2013(2013), Art. ID 401756Google Scholar

  • [24] Piszczek M., Remark on hyperstability of the general linear equation, Aequationes Math., 2014, 88(1-2), 163-168Web of ScienceGoogle Scholar

  • [25] Brzde¸k J., Chudziak J., Páles Z., A fixed point approach to stability of functional equations, Nonlinear Anal., 2011, 74(11), 6728-6732.Google Scholar

About the article

Received: 2018-06-14

Accepted: 2018-10-31

Published Online: 2018-12-05

Published in Print: 2018-11-01

Citation Information: Demonstratio Mathematica, Volume 51, Issue 1, Pages 295–303, ISSN (Online) 2391-4661, DOI: https://doi.org/10.1515/dema-2018-0027.

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© by Gwang Hui Kim, et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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