Document Type : Research Paper


Department of Mathematics‎, ‎Faculty of Science‎, ‎Ibn Tofail University‎, ‎BP-14000‎, ‎Kenitra‎, ‎Morocco


In this paper, we prove the generalized Hyers-Ulam-Rassias stability for the Drygas functional equation
$$f(x+y)+f(x-y)=2f(x)+f(y)+f(-y)$$ in Banach spaces by using the Brz\c{d}ek's fixed point theorem. Moreover, we give a general result on the hyperstability of this equation. Our results are improvements and generalizations of the main result of M. Piszczek and J. Szczawi'{n}ska [21].


Main Subjects

[1] M. Almahalebi, On the stability of a generalization of Jensen functional equation, Acta. Math. Hungar. 154 (1) (2018), 187198.
[2] M. Almahalebi, Stability of a generalization of Cauchy’s and the quadratic functional equations, J. Fixed Point Theory Appl. (2018), 20:12.
[3] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64-66.
[4] A. Bahyrycz, M. Piszczek, Hyperstability of the Jensen functional equation, Acta Math. Hungar. 142 (2014), 353-365.
[5] D. G. Bourgin, Approximately isometric and multiplicative transformations on continuous function rings, Duke Math. J. 16 (1949), 385-397.
[6] J. Brzdek, A hyperstability result for the Cauchy equation, Bull. Aust. Math. Soc. 89 (2014), 33-40.
[7] J. Brzdek, Hyperstability of the Cauchy equation on restricted domains. Acta Math. Hungar. 141 (2013), 58-67.
[8] J. Brzdek, On Ulam’s type stability of the Cauchy additive equation, The Scientific World J. (2014), 2014:540164.
[9] J. Brzdek, Remarks on hyperstability of the Cauchy functional equation, Aequ. Math. 86 (2013), 255-267.
[10] J. Brzdek, K. Cieplinski, A fixed point approach to the stability of functional equations in non-Archimedean metric spaces, Nonlinear Anal. 74 (2011), 6861-6867.
[11] J. Brzdek, K. Cieplinski, Hyperstability and superstability, Abs. Appl. Anal. (2013), 2013:401756.
[12] J. Brzdek, D. Popa, I. Rasa, B. Xu, Ulam Stability of Operators, Mathematical Analysis and its Applications, Academic Press, 2018.
[13] H. Drygas, Quasi-inner products and their applications, Springer-Netherlands. (1987), 13-30.
[14] B. R. Ebanks, P. L. Kannappan, P. K. Sahoo, A common generalization of functional equations characterizing normed and quasi-inner-product spaces, Canad. Math. Bull. 35 (3) (1992), 321-327.
[15] P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431-436.
[16] E. Gselmann, Hyperstability of a functional equation, Acta Math. Hungar. 124 (2009), 179-188.
[17] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U. S. A. 27 (1941), 222-224.
[18] S.-M. Jung, P. K. Sahoo, Stability of a functional equation of Drygas, Aequ. Math. 64 (3) (2002), 263-273.
[19] G. Maksa, Z. Páles, Hyperstability of a class of linear functional equations, Acta Math. Acad. Paedag. Nyıregyháziensis. 17 (2001), 107-112.
[20] M. Piszczek, Remark on hyperstability of the general linear equation, Aequ. Math. in press.
[21] M. Piszczek, J. Szczawi´ nska, Hyperstability of the Drygas functional equation, J. Funct. Spaces. Appl. (2013), 2013:912718.
[22] Th. M. Rassias, On the stability of linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300.
[23] P. K. Sahoo, P. Kannappan, Introduction to functional equations, CRC Press, Boca Raton, Florida, 2011.
[24] M. Sirouni, S. Kabbaj, A fixed point approach to the hyperstability of Drygas functional equation in metric spaces, J. Math. Comput. Sci. 4 (4) (2014), 705-715.
[25] W. Smajdor, On set-valued solutions of a functional equation of Drygas, Aequ. Math. 77 (2009), 89-97.
[26] S. M. Ulam, Problems in modern mathematics, Wiley, New York, 1960.