Superstability of the $p$-radical sine functional equation

Document Type : Research Paper


1 Department of Mathematics, Kangnam University, Giheung-gu, Yongin-si, Gyeonggi-do, 16979, Repub. of Korea

2 Department of Mathematics, Semnan University, P.O. Box 35195-363, Semnan, Iran


In this paper, we investigate the transferred superstability of the $p$-radical functional equation
        f\left(\sqrt[p]{\frac{x^{p}+y^{p}}{2}}\right)^{2} -f\left(\sqrt[p]{\frac{x^{p}-y^{p}}{2}}\right)^{2} =f(x)f(y)
with respect to the sine functional quation from the Pexider type $p$-radical functional equation  $f\left(\sqrt[p]{x^{p}+y^{p}}\right) +g\left(\sqrt[p]{x^{p}-y^{p}}\right)=\lambda \cdot h(x)k(y)$,     where $p$ is an odd positive integer and $f$ is a complex valued  function. Furthermore, the results are applied to the stability of the  cosine type $p$-radical functional equations. 


[1] J. d’Alembert, Memoire sur les principes de mecanique, Hist. Acad. Sci. Paris (1769), 278–286
[2] M. Almahalebi, R. El Ghali, S. Kabbaj and C. Park, Superstability of p-radical functional equations related to Wilson–Kannappan–Kim functional equations, Results Math. 76 (2021), Paper No. 97.
[3] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66.
[4] R. Badora, On the stability of cosine functional equation, Rocznik Naukowo-Dydak., Prace Mat. 15 (1998), 1–14.
[5] R. Badora and R. Ger, On some trigonometric functional inequalities, Functional Equations- Results and Advances (2002), 3–15.
[6] J.A. Baker, The stability of the cosine equation, Proc. Amer. Math. Soc. 80 (1980), 411–416.
[7] J.A. Baker, J. Lawrence and F. Zorzitto, The stability of the equation f(x + y) = f(x)f(y), Proc. Amer. Math. Soc. 74 (1979), 242–246.
[8] B. Bouikhalene, E. Elquorachi and J. M. Rassias, The superstability of d’Alembert’s functional equation on the Heisenberg group, Appl. Math. Lett. 23 (2010), 105-109.
[9] D.G. Bourgin, Approximately isometric and multiplicative transformations on continuous function rings, Duke Math. J. 16 (1949), 385–397.
[10] P.W. Cholewa, The stability of the sine equation, Proc. Amer. Math. Soc. 88 (1983), 631–634.
[11] Iz. EL-Fassi, S. Kabbaj and G.H. Kim, Superstability of a Pexider-type trigonometric functional equation in normed algebras, Int. J. Math. Anal. 9 (2015), no. 58, 2839–2848.
[12] P. Gˇavruta, On the stability of some functional equations, Th. M. Rassias and J. Tabor, Stability of mappings of Hyers-Ulam type, Hadronic Press, 1994, pp. 93–98.
[13] M. Eshaghi Gordji and M. Parviz, On the Hyers-Ulam-Rassias stability of the functional equation f( p x 2 + y2) = f(x) + f(y), Nonlinear Funct. Anal. Appl. 14 (2009), 413–420.
[14] D.H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. 27 (1941), 222–224.
[15] Pl. Kannappan, Functional equations and inequailitis with applications, Springer, 2009.
[16] Pl. Kannappan and G.H. Kim, On the stability of the generalized cosine functional equations, Ann. Acad. Pedagog. Crac. Stud. Math. 1 (2001), 49–58.
[17] G.H. Kim, Superstability of the p-radical functional equations related to Wilson and Kim’s equation, Int. J. Nonlinear Anal. Appl. 12 (2021), 571–582.
[18] G.H. Kim, Superstability and solution of the pexiderized trigonometric functional equation, Math. Statist. 8 (2020), 363–371.
[19] G.H. Kim, On the superstability of the pexider type trigonometric functional equation, J. Inequal. Appl. 2010 (2010), Doi:10.1155/2010/897123.
[20] G.H. Kim, On the Stability of the generalized sine functional equations, Acta Math. Sin., Engl. Ser. 25 (2009), 965–972
[21] G.H. Kim, The Stability of the d’Alembert and Jensen type functional equations, J. Math. Anal Appl. 325 (2007), 237–248.
[22] G.H. Kim, A Stability of the generalized sine functional equations, J. Math. Anal. Appl. 331 (2007), 886–894.
[23] G.H. Kim and Y.H. Lee, boundedness of approximate trigonometric functional equations, Appl. Math. Lett. 331 (2009), 439–443.
[24] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300.
[25] S. M. Ulam, “Problems in Modern Mathematics” Chap. VI, Science editions, Wiley, New York, 1964
Volume 14, Issue 4
April 2023
Pages 161-170
  • Receive Date: 18 December 2021
  • Revise Date: 29 April 2022
  • Accept Date: 08 May 2022