International Journal of Nonlinear Analysis and Applications
Superstability of the $p$-radical functional equations related to Wilson's and Kim's equation
Document Type : Special issue editorial
Author
Department of Mathematics, Kangnam University, Yongin, Gyeonggi, 16979, Republic of Korea
Abstract
In this paper, we solve and investigate the superstability of the $p$-radical functional equations related to the following Wilson and Kim 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*}
where $p$ is an odd positive integer and $f$ is a complex valued function. Furthermore, the results are extended to Banach algebras.
Keywords
es
[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, 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, R. Ger, On some trigonometric functional inequalities, in Functional Equations- Results and Advances,
2002, pp. 3–15.
[6] J. A. Baker, The stability of the cosine equation, Proc. Am. Math. Soc. 80 (1980), 411–416.
[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.
[8] B. Bouikhalene, E. Elqorachi, 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. Am. Math. Soc. 88 (1983), 631–634.
[11] 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.
[12] M. Eshaghi Gordji, M. Parviz, On the Hyers-Ulam-Rassias stability of the functional equation f(
p
x
2 + y
2) =
f(x) + f(y), Nonlinear Funct. Anal. Appl. 14, (2009), 413–420.
[13] P. Gǎvruta, 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.
[14] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. USA 27 (1941), 222–[15] Pl. Kannappan, The functional equation f(xy) + f(xy−1
) = 2f(x)f(y) for groups, Proc. Am. Math. Soc. 19
(1968), 69–74.
[16] Pl. Kannappan, Functional Equations and Inequailitis with Applications, Springer, New York, 2009.
[17] Pl. Kannappan, G. H. Kim, On the stability of the generalized cosine functional equations, Ann. Acad. Pedagog.
Crac. Stud. Math. 1 (2001), 49–58.
[18] G. H. Kim, The stability of the d’Alembert and Jensen type functional equations, J. Math. Anal. Appl. 325 (2007),
237–248.
[19] G. H. Kim, The stability of pexiderized cosine functional equations, Korean J. Math. 16 (2008), 103–114.
[20] G. H. Kim, Superstability of some Pexider-type functional equation, J. Inequal. Appl. 2010 (2010), Article ID
985348. doi:10.1155/2010/985348.
[21] G. H. Kim, S. H. Lee, Stability of the d’Alembert type functional equations, Nonlinear Funct. Anal. Appl. 9 (2004),
593–604.
[22] G. H. Kim, Y. H. Lee, The superstability of the Pexider type trigonometric functional equation, Aust. J. Math.
Anal. 7 (2011), Issue 2, 1–10.
[23] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Am. Math. Soc. 72 (1978),
297–300.
[24] S. M. Ulam, Problems in Modern Mathematics, Wiley, New York, 1964.
[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, 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, R. Ger, On some trigonometric functional inequalities, in Functional Equations- Results and Advances,
2002, pp. 3–15.
[6] J. A. Baker, The stability of the cosine equation, Proc. Am. Math. Soc. 80 (1980), 411–416.
[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.
[8] B. Bouikhalene, E. Elqorachi, 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. Am. Math. Soc. 88 (1983), 631–634.
[11] 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.
[12] M. Eshaghi Gordji, M. Parviz, On the Hyers-Ulam-Rassias stability of the functional equation f(
p
x
2 + y
2) =
f(x) + f(y), Nonlinear Funct. Anal. Appl. 14, (2009), 413–420.
[13] P. Gǎvruta, 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.
[14] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. USA 27 (1941), 222–[15] Pl. Kannappan, The functional equation f(xy) + f(xy−1
) = 2f(x)f(y) for groups, Proc. Am. Math. Soc. 19
(1968), 69–74.
[16] Pl. Kannappan, Functional Equations and Inequailitis with Applications, Springer, New York, 2009.
[17] Pl. Kannappan, G. H. Kim, On the stability of the generalized cosine functional equations, Ann. Acad. Pedagog.
Crac. Stud. Math. 1 (2001), 49–58.
[18] G. H. Kim, The stability of the d’Alembert and Jensen type functional equations, J. Math. Anal. Appl. 325 (2007),
237–248.
[19] G. H. Kim, The stability of pexiderized cosine functional equations, Korean J. Math. 16 (2008), 103–114.
[20] G. H. Kim, Superstability of some Pexider-type functional equation, J. Inequal. Appl. 2010 (2010), Article ID
985348. doi:10.1155/2010/985348.
[21] G. H. Kim, S. H. Lee, Stability of the d’Alembert type functional equations, Nonlinear Funct. Anal. Appl. 9 (2004),
593–604.
[22] G. H. Kim, Y. H. Lee, The superstability of the Pexider type trigonometric functional equation, Aust. J. Math.
Anal. 7 (2011), Issue 2, 1–10.
[23] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Am. Math. Soc. 72 (1978),
297–300.
[24] S. M. Ulam, Problems in Modern Mathematics, Wiley, New York, 1964.
)
Volume 12, Special Issue
December 2021Pages 571-582
- Receive Date: 09 May 2021
- Accept Date: 14 July 2021