Korean J. Math. Vol. 29 No. 4 (2021) pp.765-774
DOI: https://doi.org/10.11568/kjm.2021.29.4.765

Superstability of the $p$-radical trigonometric functional equation

Main Article Content

Gwang Hui Kim


In this paper, we solve and investigate the superstability of the $p$-radical functional equations

$$ \begin{align*} f\left(\sqrt[p]{x^{p}+y^{p}}\right) &-f\left(\sqrt[p]{x^{p}-y^{p}}\right)=\lambda f(x) g(y),\\ f\left(\sqrt[p]{x^{p}+y^{p}}\right) &-f\left(\sqrt[p]{x^{p}-y^{p}}\right)=\lambda g(x) f(y), \end{align*} $$

which is related to the trigonometric(Kim's type) functional equations, where $p$ is an odd positive integer and $f$ is a complex valued function. Furthermore, the results are extended to Banach algebras.

Article Details


[1] J. d’Alembert, Memoire sur les Principes de Mecanique, Hist. Acad. Sci. Paris, (1769), 278–286 Google Scholar

[2] M. Almahalebi, R. El Ghali, S. Kabbaj, C. Park, Superstability of p-radical functional equations related to Wilson–Kannappan–Kim functional equations, Results Math. 76 (2021), Paper No. 97. Google Scholar

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

[4] R. Badora, On the stability of cosine functional equation, Rocznik Naukowo-Dydak. Prace Mat. 15 (1998), 1–14. Google Scholar

[5] R. Badora, R. Ger, On some trigonometric functional inequalities, in Functional Equations-Results and Advances, 2002, pp. 3–15. Google Scholar

[6] J. A. Baker, The stability of the cosine equation, Proc. Am. Math. Soc. 80 (1980), 411–416. Google Scholar

[7] J. A. Baker, J. Lawrence, F. Zorzitto, The stability of the equation f (x + y) = f (x)f (y), Proc. Am. Math. Soc. 74 (1979), 242–246. Google Scholar

[8] D. G. Bourgin, Approximately isometric and multiplicative transformations on continuous function rings, Duke Math. J. 16, (1949), 385–397. Google Scholar

[9] P. W. Cholewa, The stability of the sine equation, Proc. Am. Math. Soc. 88 (1983), 631–634. Google Scholar

[10] Iz. EL-Fassi, S. Kabbaj, G. H. Kim, Superstability of a Pexider-type trigonometric functional equation in normed algebras, Inter. J. Math. Anal. 9 (58), (2015), 2839–2848. Google Scholar

[11] M. Eshaghi Gordji, M. Parviz, On the Hyers-Ulam-Rassias stability of the functional equation f(x2 + y2) = f(x) + f(y), Nonlinear Funct. Anal. Appl. 14, (2009), 413-420. Google Scholar

[12] P. Gˇavruta, On the stability of some functional equations, Th. M. Rassias and J. Tabor (eds.), Stability of mappings of Hyers-Ulam type, Hadronic Press, New York, 1994, pp. 93–98. Google Scholar

[13] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. USA 27 (1941), 222–224. Google Scholar

[14] Pl. Kannappan, The functional equation f(xy) + f(xy−1) = 2f(x)f(y) for groups, Proc. Am. Math. Soc. 19 (1968), 69–74. Google Scholar

[15] Pl. Kannappan, Functional Equations and Inequailitis with Applications, Springer, New York, 2009. Google Scholar

[16] Pl. Kannappan, G. H. Kim, On the stability of the generalized cosine functional equations, Ann. Acad. Pedagog. Crac. Stud. Math. 1 (2001), 49–58. Google Scholar

[17] G. H. Kim, The stability of the d’Alembert and Jensen type functional equations, J. Math. Anal. Appl. 325 (2007), 237–248. Google Scholar

[18] G. H. Kim, On the stability of the Pexiderized trigonometric functional equation, Appl. Math. Comput. 203 (2008), 99–105. Google Scholar

[19] G. H. Kim, Superstability of some Pexider-type functional equation, J. Inequal. Appl. 2010 (2010), Article ID 985348. doi:10.1155/2010/985348. Google Scholar

[20] G. H. Kim, Superstability of a generalized trigonometric functional equation, Nonlinear Funct. Anal. Appl. 24 (2019), 239–251. Google Scholar

[21] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Am. Math. Soc. 72 (1978), 297–300. Google Scholar

[22] S. M. Ulam, Problems in Modern Mathematics, Wiley, New York, 1964. Google Scholar